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TENS VEC ENCE
& V RTUALPH LOSOP
f rI
lin
Intensive Science and Virtual Philosophy cuts to the heart of the ph osophyof Gilles Deleuze and of todays science wars
o L ndexpl 11
Manuel DeLanda began his career in experimental film. became acomputer artist and programmer, and is now Adjunct Professor ofPhilosophy at Columbia University. He is author of the best-sellingbooks, War in the Age of Intelligent Machines and A Thousand Yearsof Non-Linear History.
Int n tve SCI nc and VIrtual PhilosOpll'V I wnnen for lanti-Deleuzi n f phitcso h r for an i-phil () hIt III h n 1 r . f Y V J hm
J th st r of t
PHILOSOPHY I CULTURAL & MEDIA STUD IES I SC ENCE STUD ES
continuum manuel delanda
rPV'".Q(7'-- - - ------ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ......__IltANS\, I:HSAI.SNEW I>JH LCTI ONS IN PIII LOSOPIIY
ERIE EDITOR
Keith Ansell Pearson, University of Wan i .k
CO NSULTANT EDIT O R
Eric Alliez , Richard Beardsworth, Howard Caygill, Gary Gcnosko ,
Elisabeth Grosz, Michael Hardt, Diane Morgan, John Mullarkey, Paul
Patton, Stanley Shostak, Isabelle Steng ers, James Williams, David
Wood.
Transver sals explores the mo st exciting collisions within contempo rary
thought as philo soph y encounte rs nature, materiality, tim e , technology,
science , culture , politics, art and everyday life. Th e ser ies aims to
pr esent work which is both theoreti cally innovativ e and challenging,
while retaining a commitme nt to rigour and clarity, and to the power
and precision of thought.
INTEN SIVE S(~ II: NeEAN l) V11{~r UAL
PHILOSOPHY
MANUEL DELANDA
Intensive Science &.. Virtual Philosophy
Felix Guau ori: an Aberrant Introduction
Political Physics: Deleuze, Derrida and the Body Politic
FORTHCOMING
Manuel DeLand a
Gary Genosko
John Protevi
Philosophy in the ABe if Science &.. Capital Gregory Dale Adam son
II
ContinuumThe Tow er Building, 11 York Road, London SEI 7 IX370 Lexington Avenue, New York , NY 10017- 650 3
www.continuumbooks.com
First published in 2002
© Manuel DeLanda 2002
All rights reserved . No part of this publication may be reproducedor transmitted in any form or by any means, electronic or mechanical,
including photocopying, recordin g or any informati on storage or retri evalsystem, without permi ssion in writin g from the publishers.
British Lib rary Cata loguing-in-Pu blicat ion DataA catalogue record for this book is available
from the British Library.
ISBN 0- 8264 - 5622- 7 (hardback)0-8264 - 5623-5 (paperback)
Typeset by CentraServe, Saffron Walden , EssexPrint ed and bound in Great Britain byMPG Books Ltd, Bodmin, Cornwall
C tit .nt
lntroduct ion: IJl, ll'uze's W orld
The Mathemat ics of th e Virtual: Manifolds,
Vec to r Fields and Transformation Groups
2 T he Actualizat ion of th ' Virtual in pace
3 The Actualization of th e Virtual in Time
4 Virtuality and th e Laws of Physics
Appendix: Deleuze's Words
Notes
Index
9
45
82
117
157
181
241
'1'" Jltlil"! ;t "" rl ,<I :1 l'illl "LI .
\1'11<) ra uoh t s" Illl l ('!l abo u t ti ll' worl. ]Co
lntroduction: Dclcurc 's World
Th ere arc always dan gers in writing it book with a specific a ud il' lKt' in
mind . The most obvious one is the danger o f missing the targt'l
audit-nee co m plete ly I eithe r because the subj ect matter fails to gr.lb ihattention or because the sty le of presentation docs not nu -vt its
standards or e xpectations. Th en there is th e associated danger of Iw,ing
readers wh o , had not that particular target been chose n, would IM H '
formed the real audience or the book . A book may end up this \\ .I~'
without an~' read ership at all. In the world or W estern philosophy . 1,,,'e xample, history and geog raphy have co nspired to divide th is world
into tw o almost mutually exclusive camps, the Anglo~Amcrican .111<1the Continental camps, each with its own style , research priorities and
long traditions to defend , A phi losophical book which refus es [ 0 I.,k,·
sides, attem pting, for example , to present the work of a philosopher
of on e cam p in the terms and sty le of the other, may end up heing a
ho ok witho ut an audience: too Anglo~American for the Continentals.
and too Cont inental for the Anglo~Amcricans .
Such a danger is evident in a hook like this, which attempts to
present the work or the philosopher Gilles Dclcu zc to an aud ience or
analytical philosophers of science , and of scientis ts interested ill
philosoph ical questions. W hen confronte d wit h Dcleuze' s original te xts
this audience is bo und to be puzzled , and may even be repelled by t IH.',
superficial sim ilarity or these texts with books belonging to what has
cume to he known as th e ' post -mode rn' traditio n . Although as I arglH.~
in these pages Dcieuzc has absolutely nothing in com mo n with that
tradition, his expe rimenta l sty le is bound to create that impression .
Another source of difficulty is the philosophical resources whi ch Dclcuzc
brings to his project. Despite the fact that authors like Spinoza and
Leibniz, Nietzsche and Bergson. have mu ch to offer to phil osophy
tod ay. they arc not ge nera lly perceived by scientists o r analyt ical
philosophers of science as a legitimate resource. For thi s reason wh at I
"11,, IH" h lI"t ,,111 •• , 1111"1",1.11'''11 tIt 11,1,'1/, 10\\ 1'\11 .1
r, '<llI/ llrl/,/I"" 01 III' 1'1111"'''1'1". 11'"1 .1~ ,,,111,1, ,hll",,,t tllI"I"","re source S ,11111 lines 01 .lrgullwllt. 1111' 1'''.111 lIt 1111 , •• "" IIIHU,," is
1I0t jus t to make his ideas sc,'m Il'gitim.III' 10 III} illll'lIdl'd audience.
bu t also to sho v that his co n .lusions do not depen d 011 his particul : r
cho ice of resources , or th e particul ar lines of arg ume llt he uses , but
th at they arc robust to changes in theoretical assum ptio ns and st rategi's .
Clearly , if the same conclus io ns can be reached from enti rely different
points of departure and following ent irely d ifferent paths , th e valid ity
of those conclus ions is thereby stre ngthe ne d.
I must qualify thi s state ment , however, because what I attem pt
here is far from a com pre hens ive recon struction of all of Delcu zc 's
philosophical ideas. Instead, I focus on a particular ye t fundamental
aspect of his work: his ontoloBY' A philosopher ' s o nto logy is th e set
o f entit ies he or she assumes to exist in reality, th e typ es of entit ies
he or she is comm itted to assert actually exist. Although in th e history
of philosophy th ere ar e a great vari ety of ontological com mit me nts,
we can very roughly classify th ese into three main groups. For some
phil osophers reality has no ex istence ind ependently from th e human
mind that perceives it, so th eir ontology consists mostl y of mental
entities , wh ether th ese are thought as transcendent obj ect s or , on th e
cont rary , as linguisti c representati on s o r soci al conventions. Other
philosophers grant to the objects of everyday expe rience a mind
ind ependent existe nce, but remain unconvinced that th eoretical ent it
ies, whether unobservabl e relations such as ph ysical causes, or
unobservable ent ities such as el ectrons, possess suc h an ontological
autonomy . Finall y, th ere are phil osophers who grant reality full
auton omy from the human mind, disr egarding th e differen ce between
th e obse rvable and th e un ob servable , and th e anthropocentri sm this
distinction implies. These phil osophers are said to have a realist ontol
0BY . Delcuzc is suc h a reali st ph ilosopher, a fact that by itself sho uld
distinguish him from most post -m od ern phil osophies wh ich remain
basicall y non-reali st.
Reali st philosophers, on th e other hand, need not agree about th e
co nte nts of thi s mind-independent reality . In particular, Deleuze rejects
several of th e ent it ies taken for gra nted in ordinary for ms of realism .
To tak e th e m ost obv ious exam ple, in some realist approaches th e
2
"II I I III"" ,I., I" I, •"" 'I''' ,.I " I 1,,11 1"1111 01 "It, . I I." ,•.1111111 '11." 1111".1 I, 11.,,, I'" , ,,," " I ,II I II, . " "II • I "I1""'"1111 Ih.lld,IIII' \\h,IIII .. , ,,11 '1 II l>e II III' • 11"',1"".1
,h"llI " 'lilt', 01 ,111\ othl'l Ir.JIl (( '11.1"111 ,11111 v , 0 III III plllio "1'1.\
"",..111111' 1'1 " is m'I"It 'd t" explain "h.lt I .H ohl " t IIII'll 100.,,,tll
IIId "h.11 1>!"l'SI'n '.,s lhis idcntit th rouoh tinu- . Hricllv, th is sll"wthi,,', ," lsI' is d maun cal proll: J '\. , OIl1l' of th ese pr"cl'ssI'S arc ma n-ria l . lId
" ,wr Idie, so me ar e not , but eV1'1I the latter remain immane nl 10 tlH'
wor ld of matter and enl' rgy. Thus, J) ' le uze's proc 'ss onto log, hn '"k
\ ith the essent ialism th at charac te r izes naive I' .alism and, sim ul
t.1I1 -ously, re moves on' of the main o bjec tions wh i h non -real ist s IIJ.l k,
against th e postulat ion of an auto no mous reality. T he ex te nt to whic h
he indc d deprives non -r ealist s fro m thi s casy way out dcp mds, o n lh ,
othe r hand, on the det ails o f his account of how th e mt it ies th .lt
populat e realit y are produced without the need for any thing trans e nd
cnt, For thi s r eason I will not be co nce rned in thi s recon struction with
th e textual so urce of Delcuze 's ideas, nor with his sty le o f argumenta
tion o r his usc of language . In sho r t , I will not be co nce rned with
De lcuze.'s words only with Dcleuzc 's world.
T he basic plan of th e book is as foll ows. Chapte r I introdu cs the
forma l ideas needed to think about the abstract (o r rather virtual)
struc ture of dynamical processes. I draw on th e same mathemat ical
resources as Deleu ze (different ial ge ometry , gro up th eory) but , unlike
him , I do not assume th e reader is already familiar with th ese field s ,
Deleuzes grasp of th e technical details involv ed is, I hope to show,
co m ple tely adequate (by anal yti cal philosophy standards) , but his
discu ssion of technical details is so co m pressed, and assumes so much
on th e part of th e read er , that it is bound to be misinterpreted .
Chapter 1 is written as an alt ernative to his own presentation of th '
subject, guidi ng the reader step by ste p th ou gh th e different math
emat ical ideas involved (man ifo lds , transformati on groups , vec to r
field s) and giving exam ples of th e application of these abstract ideas to
th e task of modelling concre te physical processes. Despite my efforts
at unpacking as much as possibl e th e contents of Deleuzes highl y
co m pressed description s, however, th e subject matter remains techni
cal and so me readers may st ill find it hard to foll ow. I recommend
that suc h readers skip this first chapte r and, if need be, co me back to
3
It lIIH I th" pOlll t fli tilt' 1,,1111••1 I I nll n. I.. , 1I11H I 1•.11 III II ~
.1 p p lk.HiclIl' III Ic ·s, .,h..u-ac t m.1I1t '1 II I tilt IfllIll\\ III lll.lph·1
C hdptl'rs and J deal \\ ith thlo prodlll l ion III Ih.. c11 11'T"1I1 t'n li tit'S
that populate Dclcu zes world. Tlu- h'l."iic them e i, lit" t , \\ ithin .l n·d lis l
pe rspective, one does not ge t rid of essences until one repl aces them
with so me thing else, This is a burden which affect s only till' realist
philosopher given that a non -r eali st can sim ply decl are esse nces mental
entit ies or reduce them to socia l conventions. One wa)" to think about
esse nt ialism is as a theor-y of the genesis of form, that is, as a theory
of morphogenesis, in which physical ent ities are viewed as more or less
faithful realizatio ns of idea l forms. The de tails of the process of
realization arc typica lly ne ver given. lisscnccs arc thou ght to act as
models, ete rnally maintaining their identity, while part icular ent it ies are
co nceived as mere copies of these models , resem bling them with a
higher or lowe r degree of perfect ion . Dcleuze replaces the False gen C'sis
implied by these pre-existing forms which remain th e same for all t ime ,
with a theory of morphogen esis based on the notion of the d!fferent. He
co nce ives differen ce not negathoely, as lack of resemblance. but
positiv ely or productively, as that which drives a dynamical process .
The best examples are intensi..e d!ffirences, the differen ces in tempera
ture. pressure , speed, chemical concentrat ion , which are key to the
scientific explanation of the genesis of the fonn of inorganic crysta ls,
o r of the forms of organic plants and animals. Chapter 2 is concerned
with the spatial aspects of this inte nsive genesis while Chap ter 3 deals
with its temporal aspec ts .
After reconstructing Delcu zes onto logy I move on in Chapte r 4 to
give a brief acco unt of his episiemoloqy, For an)' real ist philosopher
these two areas mu st be , in fact. int imately related , This may he most
clearly seen in the case of naive realism, where truth is conce ived as a
relation of correspondence between , on on e hand, a ser ies o f facts about
the classes of entit ies populating reality and , on th e o the r , a ser ies of
sente nces exp rl.~ssing those facts. If one assumes that a class of entit ies
is defined by the esse nce wh ich its members share in com mon , it
becomes relatively simple to conclude that these classes are basically
given, and that the)' exhaust all there is to know about the world . The
ontological assumption that the world is basically closed , that entirely
novel classes of e ntit ies cannot emerge spontaneously, may now he
4
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IItII IlItllUI ,11 11 1 I' ,lll.llt t,lIl l .. m.lll. ,lh.,11I11 It I 111111. ,11 II'
\\ 1a.11 l' It'Il1 .111~ 1, ·,,11 I 1'1.1111 oph"1 .u tu.,!1\ 1111 '11111 ' to till , tr_ lilt h
'Ioll\" \11 ·\\ . hut II IS ~I('ol l th.u " " '~Cllllllhlllin lIt Ik l" II / I" 1' ,111 III
mu ..t n 'l"tl ",Hh on I ' 0 1 Iltl·"" .'''U111ptiolt .uul 't 'pl.H"I' tl)l'lI1 \\Itll
di llt-n'n l om-s.
\Vhilt· in III.., fir..t thn'e ..hapu-rs I J.IIt'mpt 10 ~' li ll1i l1,lh' lIu· c rrom-ou
.I, ..umprion o f.1 d ost,c1 \\ o rld . in Chapter 4 I t r)' to repl 'ln · 1I0t Cl llh
tlu- idea of .1 sim ple co rrespo ndence hut . h C)'OIHI that, Il) J,ohlillt' ,"
\'(':Y iJ"ll cj.truth . In ot her words, I wi ll .1 rgUt· tha t even if otu - ,Kn'pt
th,lt there are true se-n tences ex prl.'ssing rea l facts it can stil l Ill"
mai ntai ned that most of these factual sentences arc lririal , TIlt" rolt, 01till' th inker is no t so m uch to utter truths or establish facts. hut to
di stinguish among thc large population of true fact s those that ,In·
import ant and relevant from those th at arc not. Importance anJ relevance,
not truth , arc the key co nce pts in Dcleuzc 's l' pistc molog)". the task of
realism be ing to gro und these co nce pts preventing them from hdug
red uced to subject ive e valuations or soci al co nve ntions . This point can
be made cleare r if we co ntrast Dcleuzcs position not with the
lingu ist ic version of co rre spo nde nce theory but with the mathcmaucal
one. In this case a relation of correspo nde nce is postulated 10 e xist
between the sta tes of a physical objec t and the solut ions to mathematical
models capturing the essence of that ohje ct. By contrast , Dcl cuve
st ress es thc ro le of cor rectly posed problems, rather than the ir true
solutions, a problem being well posed if it captures an objective
distribut ion of the important and the un im portant , or mo re mathemat
ically , of the sinq ular and the ordinaly .
Chapte r 4 explores th is problematic epistemolo8Y and compares it with
the more fami liar axiomatic or theorematic versions which pred ominate
in the physical scie nces. To anticipate the main conclusion of the
chapte r , while in an axiomatic episte mo logy one st re sses the roll' of
qeneral laws, in a probl ematic on e laws as such disappear but without
sacrificing the obj ectivity of physical knowledge , an objectivity now
captured by distribution s of the singular and the ordinal). If such a
co nclusion can indeed be made plausible , it fo llows that despite the
fact that l reconstruct Dcl cuze to cater to an audience of scienti sts and
analyti cal philosoph ers of scie nce , nothing is yielded to the orthodox
5
pU'IltiulI lu-ld h., tll('..r- 1\\0 g l o u p " 01 tlllllkcl . (hi til(' ( 01111 ,11\ I'ollt
p!lf ,de.ll ~d('nn ,J .lIH I ,u IoI IJ l il',11 plai lo..oph)' "II11'l'g" 11'.1I 1'ilol"lll( ·;I· from
this enc-ounte r with Ik' k ul.t" till' rOnllCr rt'l 'lini ng its o l> j(·l·th·il)' hut
losing the laws it holds so ch-ar, the lat ter maint aining its rigo ur and
clarity but losing its ex clusive focus on fact s and so lutions . And more
importantly, the world itself eme rges transformed : the vcry idea that
there can be a set of true sente nces whi ch give us the facts once and
for all, an idea presupposing a d osed and finished wor ld , gives way to
an open world full of divergent processes yielding novel and unex
pected entities , the kind of world that wo uld not sit st ill lon g eno ugh
for us to take a snapshot of it and present it as the final truth .
To conclude this introduction I must say a few words conce rning
that other audience which my reconstruction may seem to overlook:
Deleuzian philosophers , as well as thinkers and artists of different kinds
who are interested in the philosophy o f Dcleuze . First of all, there is
much more to Dcl cuze 's books than just an ontology of processes and
an epistemology of prob lems . He mad e contr ibutions to such diverse
subjec ts as the nature of cinema, painting and literatu re, and he held
very specific views on th e nat ure and genes is of subjectivity and
language. For better or for worse , these are the subjec ts that have
captured the attention of most readers of Del euze , so it will come as
a surprise that I will have nothing to say about them . Ne vertheless , if
I manage to reconst ruct Dcleuze 's world these other subjec ts should
be illuminated as well , at least indircctl y: once we understand
Dcl euze ' s world we will be in a better position to und erstand what
co uld cine ma , language or subjectivity be in that world .
On th e other hand, if th is reconstructi on is to be faithful to
Dclcu ze 's world it is clear that I mu st rel y on an adequate intcrprc ta
tion of his wo rds . Therc is a cer tain violen ce whi ch Deleu zes texts
mu st endure in order to be reconstructed for an aud ience they were
no t int ended for , so whenever I break with his own way of presenting
an idea I ex plain in detail the degr ee of rupture and the reason for it
in a footnote. A different kind of violence is invol ved in wren ching his
ideas from his collaboration with Feli x Guattari. In this reconstructi on
I use Dc!euze ' s ontology and episte mology as expose d in his ea rly
text s, and use onl y those parts of his collaborative work which ca ll he
directl y traced to those ear ly texts. For thi s reason I always ascribe the
6
m u ( , 0 1 11 10 , .dl.1 t il 11 1111 . U III' ti lt IH IIII II IIII ' Ill ' III II ,ld II I ' ti ll \ I
n t' lI \\ 11t II 'Juolln ' h OIll rlu-u 101llt 1I")d 1 11 1.111 ) . tlWlt' I th t \ lfl k i ll t
dOli" 10 1)(·I('u/l· '" lIuid 1\It " 10 tilt' \ \ ,1\ III ' II ,Ilt'> lilt' 111 1'11I.11111", ,'io lidifi, .u icu (If ,I It'nninolog)' h) ,1Iw,,) !'i kl'(')HlIg it i ll ,I ~t ,I h' 0 1 1111 ,
hxillg hi'i 1t'l'l11illolog)· will 'i( '( '1II 10 'iOllh' .lkin to pin ning .10\\ II .1 Ii\'(,buth'rlly, As an .1IItidllt" I otl~'r .111 .ippc ndix whe-re I n ,I.,tt' till' h 'l"IlI "
u'I,d in Ill)' n -ron struction to all the diffe- rent h ·rmiuo!o git·s Ill' U'it'" III
his ow n t('x ts and ill his co llahorativc work, se tt ing his words li ",'('
linn' ,lgain afte r they have served their pu rpose of givi ng LIS his wurb l.
The hope is that this wo rld will ret ain all its ope nness and din'l'gl'lIn ' ,
so that the inten se cx prcss ivity and eve n madn ess so often at t ribut.-d
to De lcuzc's wo rds rna)' be see n as int egral properties of the wo rld
itse lf.
7
CIIAI' I LR I
7h !!'Iathemat ic C!I th e Virtu al:
Manifolds, Vector Fields and Traniformation Groups
or all the oncepts which populate the wo rk of Gilles Deleu ze there
is on ' that stands out for its longevity: th e conce pt of mult iplicity . T his
oncept makes its appearance in his early books and remains one of
central im po rta nce, with almost unchanged meanin g and function, until
his linal work. I Its formal definition is highly techni cal , including
·Ieme nts fro m seve ral different branches o f mathematics: differential
ge ome try, gro up theory and dynamical syste ms theory. In this chapte r
I will discuss the techni cal backgr ound need ed to define this important
concept bu t some preliminary informal remarks will prove help ful in
sett ing the stage for the formal discussion . In the first place , one may
ask what ro le the conce pt of a multiplicit y is suppose d to play and the
answer would be a re place me nt for the mu ch older phil osophical
concept of an essence. The es ence of a thing is that wh ich ex plains its
identity, that is, those fundame ntal traits without which an object
wo uld not be w hat it is. If such an esse nce is share d by man y objects ,
then possession of a co mmo n esse nce would also explain the fact that
these objec ts resemble each other and, indeed, that they form a distinct
natura l kind of things .
Let 's take one of the most trad itional illustrations of an essence .
When one asks what makes omeone a member of the human species
the answer may be , for example, be ing a 'rational animal'. The exact
definitio n of the human essence is not what is at issue here (if
rat ionality and animal ity are not co nsidered to be essential hum an
pr operties some other set will do). T he imp ortant point is that there
be some set of defining characte rist ics, and that this set explain bo th
the identity of the human species and the fact that particul ar memb ers
of the species resemble eac h ot her. In a Dcl euzian ontology, on the
othe r hand , a species (or any other natural kind ) is not defined by its
9
t ' . 'lItl.1l u.ut hill r.HIu-1 II) til(' tII l'trh"H Itc'I" I"lk c' 111.ll ~." , tI t lo
it . Ib t}h'r th.in rl'l)n·..l·nti llg tinu-h-.... t .1t'·glJnl ·... P" t u- .11r III "'tfJllt.111)
consti tuted c lltit it's, tlw rescmhlanc« of the-ir uu-tubc-r: n.p lai,wd hyhaving undergone co mmon pron ·sst.·s of natu ral sch-ctlon . .1IId the
endur ing identity of the speck-s itself guar.m lt·t.·d hy the fad th.u it has
become reproductively isolated from other species. In short , whil e an
essentialist account of speci es is basically sta tic, a morphojjenctic
account is inherently dynam ic. And while an essentialist account may
rely on factors that transcend the realm of matter and energy (et ernal
archetypes, for instance), a morphogen eti c account ge ts rid of all
transcendent factors using exclusively form -gen erating resources wh ich
arc immanent to the material world .
Animal and plant species are not, of course, th e onl y natural kind s
traditionally defined by essences. Many other natural kinds, the
chemical clements or the set of elementary particles, for e xample , arc
also typically so defined . In eac h of these cases we would need to
rep lace timeless cate gorics by historical proccsses . Yet , even if success
ful thi s replacement would take us only half-way towards our goal.
The reason is that e ven if the details of a given process account for the
resemblance am on g its products, the similarities which make us classify
them as members of the same kind, there may be similarities c!f process
wh ich still demand an explanation . And when accounting for these
common features we may be tempted to reintroduce esse nce s through
the back door. These would not be essences of object s or kind s of
obj ects, hut essences of pro cesses, yet essences nevertheless . It is in
order to break this vicious circle that mult ipliciti es are introduced .
And it is because of the ten acity of thi s circle that the concept of
multiplicity mu st be so care fully constructed , justif)'ing each ste p in
the construction by the wa)' it avoids the pitfalls of esse ntialism . To
anti cipate the concl usion I will reach after a lon g and techn ical
definitional journey: multiplicities speci fy the structu re c!f spaces c!fpossibilities, spaces whi ch, in turn , explain the regularities exhibited by
morphogeneti c processes. I will begin by defining an appropriate
notion of 'space ' , a notion whi ch must not he purely geometr ical hut
also capable of bein g linked to qu estions of process.
The term ' mult iplicity ' is close ly rel ated to th at of ' ma nifold' , a
term which designates a geome tr ical space with cert ain characte rist ic
1 0
pillpltlil 10 'f.1 P \\Iltl 1 I"tlll ,IIUlllI Ifllllllllial (.11lt1 \ liltII ltllltl 1111 (Ollllplt .mothlll' \ lllcl, 11111.11111I)11\\11111( tllltll
to 'I\ t ., hlll 'f .Hl t tunl 01 Il III tl"h.,1 1111'11I hilt III II 1111' lit ' 01
1"(lIl1t.tllt .,1 I_nul o.lnn ... lltl" IIH~ 'HI)U1UlII 01 IIIIII,lt'm IS .HI ,lIHllll1
pr .H tin ' IIlht·rih·tI fr o m t ilt' Cn'l·k!l, tht' es tcn ..i\t· u r- of ltll \t. .1Ilt!
tr. 'jl.c to rh·... in thl' formulalioll of .1 , .,rid) of plaf'ik .ll prohl t'l11!l from
tht. ... ixh't"llth n' lI l u ry on mad e it TWC(·ssar y 10 d('velop ru-w pl'ol,h'lIl
snh·illg n· ...ourccs. \Vith thi s ill mind, Hew," l k sc.,rh·s and Ph·H.· lIl'
l-cr'mat inven te d lIll' now familiar method of t'mht,tlding curves into .1
rwn .dinu-nsional sran~ on which arbitrar y axes co uld lu- fixed . Ont (.
so embedded . the fixcd axes allowed the assigIlIlH.' nt of .1 pJ.ir of
num bers, or coordinates, to every point of the curve, so that the
gl'OIlll't riC rel ation s between points could now he expres sed as n '!at ion
be tween numbers, a task for which the newly developed algehra \\ .1"
perfl.ctl y suite d. This translation sche me, in short , allowed the comhi
natorial resources of algebra to be brought to hear on the so lution of
gt·ome trical problems.The term ' ma nifold' does not l>dong to the analyt ical geo ll1 l'l ry of
Descartes and Fermat, but to the d1Jerent ial aeometry of Friedrich Gauss
and Bernhard Riemann , but the basic idea was the same : tapping into
a new reservoir of probl em-solving resources, the reservoir in thi s case
bei ng the differential and int egral calculus. In its original application
the calculus was used to solve problems involving relations between
the changes of two Of more quantities. In particular, if the se rel ations
were expressed as a rare <if change of one quan tity relative to ano ther,
the calculus allowed findin g the instantaneous value for that rat e . For
example , if the changing quantities were spatial position and tim e , one
co uld find instantaneou s values for the rate of change of one relat ive
to the other , that is, for velocity. Using this idea as a resource in
geome try involved the realization that a geom etrical object , a curved
line or surface , for instan ce, co uld also be characterized by the rate at
which some of its properties changed , for example , the rate at wh ich
its curvcrure change d between different points. Using the tools of the
calculus mathematicians could now find ' instantaneo us' values for this
rate of change , that is, the value of the curvature at a given
infinit esimall y small point.In the ear ly nin eteenth century, when Gau ss began to tap into these
II
dll it It 1111••1 I t IIUI'I ', I tll l\ld 1\ \ 11 1111111 11 1111111 III l.u t \\.1 tudn-d1l"lIlg Ih(' old l ' .l rh·'I.lIl 1I1dhod : 1111' "'Il llI.H(· \\,1 t IIIhnld,d 111 .1 Ih n 't'
dilll t'II,,,iOIl,tl sp.tn' l'o m p lt"lt' \\ itll it" 1) \\ II fl'(t'd "(' 1 01 .lX t ·" : then , thillg
tllO~1' axvx, coo rd inates wou ld he .t~sjg ,wd to (,\(T) ' point of the
~lI r1 al'C"; finall)', . the gl'ometric links between points deh>rmining the
form of the surface would be expressed as algebraic n -latlons between
the numbers. Rut Gauss realized that the calculus , focusing as it docs
o~ infinites~mal points on the surface itself (that is, op(~rating entirely
With local mformation), allowed the stud), of the sur face without anI
rejerence to a alobal embedding space. Basically , Gauss developed a method
to implant the coordinate axes on the surface itself (that is, a method
of ' coordinatizing ' the surface) and, on ce points had been so translated
into numbers, to use dilTerential (not algeb raic ) equat ions to character
ize thei r relations. As the mathematician and historian Morris Kline
obse rves , by. g<·lting rid of the global embedding space and dea ling
With the surface through its own local properties 'Gauss advanced the
to ta lly new concept that a suiface is a space in itself ,2
The idea of st lld)'ing a surfa ce as a space in itself was further
developed by Riemann . Ga uss had tack led the two-d imensional case
so one would have ex pecte d his disci ple to treat the next case, three
dimen sional curved surfaces . Instead, Riem an n wen t on to successfully
attack a m uch more gene ra l problem: that of N -d ime nsional surfaces
or spaces . It is these N-d ime ns iona l curved structures , defined excl u
sively thro ugh the ir int rin sic features, that were originally referred to
by the term ' ma nifo ld ' , Riemann' s was a vcry bold move, one tha t
took him into a realm of abst ract spaces with a varia ble number of
dimen sion s, spaces which could be studied withou t the need to embed
them intu a high er-dimension al (N't- L) space . As Morris Kline puts it :
' The geometry of space o lTered by Riemann was no t just an extension
of Gauss 's d ifTerentia l geometry . It reconsidered the whol e approach
to thc study of space . " And we co uld add that this new way of posina
spatial problems would, a few decades later in the hands of Einstein and
othe rs , co mpletely alter the way physicists approached the qu estion ofspace (or more exactly, of spacetime) .
A Dclcu zian multiplicity takes as its first defining feature these tw o
traits of a manifold : its variable number of dimen sion s and, more
importantly . the absence of a supplementary (higher) dimension impos-
1 2
1I1~ .1Il t 11111 It 100ltllll.lll/oI llIlIl , III.! III lit I , 1m ,,,,11\,",11. ,I./IuulIUIlO , I I I II lilt \\l llt · • ' tl hl l'lI l l " 11111 1 lit II l it l'll .llt •• 1 11111111
u.umu ell 11' 1' 1II .1Il \ .lIltl t ilt 0111 , hili , .ttl ll I an 01 ' .11111.1110 11 lu1011 ' II I'
III ti ll ' IIMIl\ .t lIl h, \\ hh h la.l IIU Ill·"d \\ h,ll"lll '\(" til ullit) ill on lt'l
10 IOflll .1 ",,, It'm ,' " 1' " , 'lH t ' ''l, 0 11 the- othvr h.ultl, do p ll ..'t· .....t dd illlllg
u nitv (r.g. J tlw uuitv 01 r,l lio ll.t lil) ' and ,mil1lollily de fining till' lunu.m
t'S'it'; ln') ,lilt!. lIlon ': )\'('[ ' arc takr-n 10 vxi ..l in ,I t ranscx-nd r-nt "1'.1' ~\\ hi," ".. rvc-s i.1S J container for them or in which lhe) an' (· m b ,· c1d ,·t l.
A mult iplicity, on the other hand , 'h owever man y dinu -nsion ... il 111.1)
Ii.l n ' , . , . ne\'er has a supplementary dimension 10 that which tran
"pin's upon it. This alon e mak es it natural and immanent. ' S It 111.1)' he'o bj ('Cll'" that these an' pllrl'i )'J ormal dilTl'n'n ces between concq >t s• .and
that as such , they do not ne cessaril y point to a deeper ontological
din~'[t.·I lC'e . If we~ arc to repl ace esse nces as the explanat ion of till'idcntitv of materi al obj ects and natural kind s we need to specif)' tlu
wav in wh ich multiplicities relate to the ph ysical processes which
gL' l;e rate those mat erial obj ects and kinds,
Achieving this goal implies establishing a more int imate relation
between the geometric properties of manifolds and the propcrt h-s
which define morphogenetic processes. The resou rces in this case co me
from th e theory of dynamical systems wh ere the di mensions of a
manifold ar c used to represent pr op erties of a particul ar phys ical
process or system, while the ma nifold itsel f becomes [he space ifpOSSible
states which the ph ysical system can have. " In other words, in this
theo ry manifolds are co nnec te d to material reality by thei r USl : as
models of physica l processes. When one attem pts to mod el the dynam
ical be haviou r of a particular physica l object (say , the dynamica l
behaviour of a pendul um or a bicycle , to st ick to relat ively sim ple
cases) the first step is to determine the number of relevant wap in
which such an ob ject can change (these are known as an object 's deqtees
iffreedom), and then to relate those changes to one another using the
difTcrential calculus . A pendulum , for instance , can change on ly in its
position and momentum, so it has two degrees of freedom. (A
pendulum can, of course, be melted at high tem peratures , or be
explode d bv dvnamite . These ar e , indeed, other ways in which thi s
object can ~ha~ge, the)' sim ply are not relevant ways from the point
of view of dynamics. } A bicycle , if we consider all its moving parts
13
( .HI
dlllll II IOIMI In.Ulll nld ( \ " II, d In!JUlllllll I \ llh " 1' '' ' 1 I I" It 11111111 III I
III III. h. 1I.l\ nuu 01 till H"I' t 1111 It I ,11 11 1 li lt I II I. I.IU.I n pll 1111
oll tu.il t 'lll ' o f (,tlc " lit ,I pll\ h .11 '1111I , .1 1.11 It mlllll 'l h I 111 th ,
ht'h.l\ uurr of tllt' 1111\"lt "I s, tvrn it (·It .·, ,Sillgul.lriti t's 11M)' inlhlt'Jl(\" bt,h,l\'iour h) ,u ting ,\ S aftra(t ~T\ .Ii .,r 1111
Ir,\jc'llorit':-. . \ Vllal this IIW.lIlS is th.n ,I I.ug\· numlx-r of dlllen'nl
tr,Ijt't'lorit's, :-.t,uti ng thei r cvo lutio u at \'t'r)' d ill't' rent placx-s ill ti lt'uianifoh l, m ol)' end lip in t'xactl)' th e same final sta te (t he at tractor) . .IS
long a.s all of them hl'g in somewhere within the 'sphere of inlhn-no
of the auractor (the basin oj aurdcri on). Given that , in this SI' T1St',
din....rent trajectories rna)' be attracted to the same final state . singu lar .
itivs arc said to reprl'scnt the inherent or intrinsic long-term tendencies
of a svstcm, the sta tes which the system will spontaneously tt'nd to
adopt "'in the long run as lon g as it is no t co nstrained hy other fon~c..;s,
Some singularities arc topological points , so the final sta te they d ('(IIlt.'
.1S J dest iny for the traj ectories is a stead)' state . Beside these, Poin care
also found that ce rta in closed loop s acte d as attractors and called them
"lim it cvcles' , The fina l state which traject ories att rac ted to a limit
cyc le {or periodic at tractor) arc bound to adopt is an osci llato ry state .
But whether we are dealing with steady-state, periodic or o ther
attractors what ma tters is that they are recurrent topoJo8ical features,
which means that different sets of equations, re presenting quite
different physical systems , ma y possess a similar d istribution of aurae
tors and hen ce, sim ilar lon g-term behaviour.Let me give a sim ple example of how singularit ies (as part of wh at
defines a multiplicity) lead to an entirely different wa )" of viewing the
genesis of physical forms. There are a large number of difTerent
physical st ructures which form spontaneously as their co mponents try
to meet certain energetic requirements . T hese com ponents rna)' be
constrained , for example , to seek a point of minimal free energy, like
a soap bubble , wh ich acquires its spherical form by minim izing surface
tension, or a co mmon salt crysta l, which adopts the for m of a cube by
minimizing honding energy. W e can imagine the state space of the
process whi ch leads to th ese forms as st ruc ture d by a single point
attractor (re presenti ng a point of minimal ene rgy). O ne way of
describing the situatio n would be to say that a capo/anical fo rm (a
singu lar poi nt in a manifo ld) guides a process which results in many
.. I mhh .lIl el Ih.Ihl '1\1 P,lIt ~
(11,111111111,11 • 1I.1I1t ,,1111 I. Il ,lIIl t 1i.1I1l '1 .11 \\1111 I1\\01'('11.11,) h,l u-n cI,·gr.t 01 I"'t·dum (C·.1l1a 01
I h,Ulgt· in b01h po,i1ioll .1Ite l 1Il0uH'1I1Um) ,
Ncxt , one maps 1',1('11 d'·gn't· of frn,dolll into one 01 lht, dirrn -usiun s
of a mani fold , A pendulum ' s span' of possihilit it's will m-cd a twu .
dimen sion al plane , but the bicycle: ' vill involve a ten -dimen sion al s p,lCt' ,
~fter thi~ mapping ope rat ion , the state of the object at any gi vl' ll
Instant of time becomes a single point in the manif old , whi ch is now
called a state space. In addition, we can capture in this model an
object's changes ef state if we allow the rep resent at ive point to move in
this abstract space, one tick of the clock at a tim e , describing a curve
or traj ectory . A physicist can then study the changing behaviour of an
object by st udying the behaviour of these representativ e trajecto r ies . It
is important to noti ce that eve n thou gh my exam ple Invo lves tw o
objects , what their state space captures is not their sta tic properties
but the way these properties change, that is, i t captures a process. As
with any mod el, th ere is a trade-off here: we exchange the co mplexity
of the object's changes of state for the complexity of the modelling
space . In other words, an object's instantaneous sta te, no matter how
co mplex, becomes a single po int, a great simplification, but the space
in which the object's state is embedde d becomes more complex (e .g .
the three-dimen sional space of the bicycle becomes a te n-di mensiona lsta te space) .
Besides th e great simplification achieved by modelling co mplex
dyn amical processes as trajectories in a space of possible states, there
is the added advantage that math ematicians can bring new resources to
bear to the study and solution of the physical problems involved . In
particul ar , topoioqtcal resources rna)" be used to analyse ce rtai n features
of these spaces, features wh ich det ermine recurrent or typical behaviour
common to man)" different models, and b), ex te nsion, co mmon to
man)' physical processes. The main pion eer of this approach was
ano the r great ninet eenth-century mathemati cian , Henri Poin care . Poin
care began his study not with a differential equation modelling a real
physical syste m , but with a vcry sim ple equation, so sim ple it had no
physical applicat ion, but whi ch nevertheless allowe d him to ex plore
the recurrent traits of an)' model with two degrees ef freedom. He
discovered and class ified certain special topological features of two-
'4 I ~
dlllt"t 'l1l ph) 11.11101111. lIullllhlll ph~' II ' !'i ntl (11)1('. l.hll llllt \\1111t1 ifl ~ ' n ' 1I1 .'I( (lm rtru propt"rti~''' ' 'I Ill' i.. \\ h.n 1 )..1 " \1/ 1' nu-ans \\ [u-u Ill'sa)'s th.u singular-i lil's are like ' im plicit for ms th.n df\" Illpologil.ll r.ulu-rth an gl·OlTIl· t ric ' ,'* Thi s rna )' hi' cont ras ted to the ('sSt" lIt i.llis t dpproadlin whi ch th e ex planation for the spherical 10fl11 of soap bubbles, forinstan ce , would be framed in terms of the esse nce of sphe r icity. tha tis, o f ge ome trica lly charac te rized essences acting as ideal Forms.
I will di scuss in a moment th e mean ing and re levance of th etopological nature of singularit ies . What matters at thi s point is thatsingularities, by determining long-term tendenci es, st ructure the possibilities which make up state space , and by ex tension, str uc ture th epo ssibilities op en to th e phy sical process modell ed by a state space. Inaddition, singu lari t ies tend to be recurrent that is th ey tend to, "characte r ize processes independently of their particular physical mcch -anism s. In th e exa mple above, the mechanism whi ch lead s to th eproducti on of a soap bubble is quite dilTerent from th e one leading toa salt crys tal, ye t both arc minimizing proce sses. This mechanism~independence is wha t makes singular it ies (or rather th e multiplicitiesth ey define) perfect cand idates to replace essences ."? As I said befnre,however, we mu st be care ful at thi s stage not to make sing ularit ics th eequivalent of thc esse nce of a process, To avoid this erro r I will discusssome additional formal properties of multiplicities distingui shing themfrom essences and then, as above , I will discus s th e way in whi ch th esepurely co nceptual differences connect with qu estions of physicalprocess.
The formal difference in qu estion has to do with the way essencesand multiplicities arc specified as ent it ies, Whil e essences are traditionally regarded as po ssessing a clear and distin ct nature (a clarity anddisti nct iveness also charac te rizing th e ideas whi ch appear in the mindof a phil osopher wh o grasps one of th ese esse nces) , multiplicities arc ,by design. obscure and disrinct: th e singularit ies which defin e a m ult iplicity co me in sets . and th ese sets are not given all at on ce but arest ructure d in such a \\'ay th at th ey pro8ressiYcIy specify th e nature if amult iplicity as th ey unfold foll owing recurrent seque nces. II What thismean s may be illustrat ed first by a metaphor and then given a precisetechni cal definition . Th e metaphor is that of a fertilized egg pri or toits unfolding into a fully developed orga nism with di fferentiated tissues
16
.u II I 0 1 1.1lI ( \ l"Otl ~1l0\\11 ,1 ",I".t'l/ II \' ) \\1111. 1111 11111 .111 tlIItt'Pltl .lllon 0 1 I mill \11 'I li t I II III Illd ol .m .HI "pI H' fdl "hl"I, t '.It!\ gl\ I 'll III tlw I' I I ( t'f('/~mll(tJ, • ~ It \ \ ct ,,\1 11 1 III li t t 11,1\ III I d t 1.- .11
,lIu l di..tim I 1l.lhln') llIo 'll billlngl '" IlId ,,) h.I\t.' g l\ (' 11 Ill' plt' lOllIlI"illt,II lt I .h t t"ph"d IIH' idt.'., th.u dIlItTI'llli.tlt·d ..'rll~ um-c l' l1 ll' rg l' P~I~ "I·..i\l.h' .IS ti ll' ('g.g <I t"\\·lop,... TIH' l"gg is 1I0t, 01 nH lrSl'•.111 untllllt'n 'lIli,'h';1 mass : it P()~St' SSI · S au obscure )l'l t1istillt:l ..rruc-turc dt"lilu,d h)zones of biod ll' l11 il"al comTlllr.llioll and hy po lari t h-s l's lolhlistH't1 h)' till '.IS\, I11 111 t.. t r il..ll positi on of till' yolk (o r nucleus), But e-ven though itde:t's POSSt'SS th e necessary biochem ical materials an ti ge netic i ll f~) r~llJ.tion, th ese materials and info rmat ion do not co nta in a clea r ,mel (hst llU'thiucprint of th e final organism . 11 .
Although th e egg me taphor docs provide a vivid illu st rat ion 01 d.wdis ti nct ion I am trying to draw here, it is nevertheless just a wwlul,,m alogy. Fortunate ly, th ere are technical wa)'s of dcfinin.~ th e hit',]. 01f,oHressil'e J1JCrent ialion wh ich do not rely on metapho rs. I [u- technicalresources in th is case co me from ano ther crucial ninctl'el1lh ~lTl1ll1r)'inn ova tio n. th e theory of groups. a field of mathemat ics which. tikt'th e d ifferen tial ge ome try I discussed before, e ventua lly becam e anintegral part of th e basic mathematical technolo~)' ()f.t~\'cn t ic ~ h-c('n t u~yphysics. The term 'gro up' refers to a set of entltl~s . (WI~~ specialpr op erties) and a rul e of combinat ion for th~se cntl~lcs . .1he most,important of th e pro pe rties is th e one nam ed closure , whi ch meanstha t wh en we usc th c rul e to combine an)' two entit ies in th e set, th eres ult is an e nt ity also belonging to th e set. For exam ple . the set ofposi tive integers displays closure if we usc addi tio n as a comhin~ti.onr ule : add ing together any two positiv e int egers yield s anothe r posltl\'Cinteger. that is , another clement in th e original set. H . .
Although sets of numbers (o r many othe r m ath ematical objects)rnav he used as illustrations of g roups , for th e purpose of ddlnmgpr~grcssi\'e differentiation we need to co nside r groups ~\' h~se m embersare not objec ts but tranifo rmations (and th e co mbinat ion rul e , aco nsecut ive applicat ion of th ose transformations) , For e xam ple, th e, setco nsisting of rot ati ons h)' nin et y degrees (that is. a set conta in ingrotations hy 0, 90 , 180. 270 degrees) forms a group , since any .twoco nsecut ive rotations produce a rotation also in the gro up. provided360 degr ees is taken as zero . The importance of gro ups of transforma-
17
1'011 tI•.•1 till III 1'1 II I" 101 , I I II '1 OIlI lt 11 11 11 '111 ' II till II
/II I(.lr1</III\ : II \\, ' p," 101 Ill!''' Olll 01 till I , Ollp 10111 11'11 O il I IIh, . .in
observer who did Ilol w it m -ss till' Ir.II I, lo rJll,ll io ll \\Iurld not ", ,1" 1,' ( 0
noti ce that allY change had .1·tually O("('lI'T,'d (th.u is, tilt' iS lI.11
appca ranc of the cube wo uld rem ain invarian t rc l.uivr- to this
observe r). O n the othe r hand , the cube would not remain invariant
und er ro tatio ns by, say, 45 degrees, but a sph ire wou ld. Inde d, a
sphere rem ains visually un changed under rotations by an)' amount of
degrees. Mathematically this is ex pressed by aying that the sphere has
more ~mmet1J' than th e cube relative to the rot ation transformation,
That is, degree of symme try is measured by the number of transforma
tions in a gro up th at leave a property invariant, and relation s bet ween
figures may be established if the gro up of one is included in (o r is asubgro up of) the gro up of the other.
ClaSSifying geometrical objects by their degrees of symme try repres
ents a sharp departure from the trad itional classificati on of ge ome trical
figures by th eir essences . While in th e latter approach we look for a
set of properties co mmon to all cubes, or to all spheres, gro ups do
not classify these figures on the basis of their sta tic prop erties but in
terms of how th ese figures are affect ed (o r not affect ed ) by acti ve
transformations, that is, figur es are classified by their response to events
that occur to them. I f Another way of putting this is that even though in
this new approach we are still claSSifying entit ies by a prop erty (their
degree of sym me try), thi s property is never an intr insic prop erty of
the enti ty being classified but always a property relati ve to a specific
transformation (o r group of transformations). Additionally, the sym
metry approach allows dynamic relations to ente r into the classification
in a different way. When two or more entities ar e related as the cube
and the sphe re above, that is, when the group of transformations of
on e is a subgro up of the other, it becomes possible to envision a process
which converts one ~ the entit ies int o the ocher by losing or gaining
symmetry. For example , a sphere can 'b ecome a cube' by loosing
invariance to some transformations, or to use the technical term, by
und ergoing a ~mmet1J'-breakin8 transiti on. While in the realm of pure
geometry this transmutation may see m some what abstract , and irrelev
ant to what goes on in th e worlds of physics or biology , there arc
18
111.111\ dlll·.II.III"" II I \1111 111It \ I.. 1~1I 11 ' 1.1111111111 III til 11'"1'
1 1'11 1 I t Ie 1II Ifl1,IIII',
I II ph Il .d III 01 I "III 111111 .111 1111 Ih, OIl 'h hi o ~. 11 11l 1l1l 1/\ 11 10.
,,,. III , 1111 " .unpl,-, III till 101111 III rho,,' 1'.," '1/ ,111 I'h.1 I (, .111 111 1111
11" ,'\, '11 1 "hi. h I.lk. · pl.Il" .11 '1Ilil .11 \,1111\' ot OIlW p,II .IIIIt'I. I
II, IlIpl'loIIUI" ', lill'" .nuph-) w it lUll' ,I pll\ ital "It'1Il IIo III tJ/1l 1.11.
\., .mo t lu-r , like the, riti ca] POlllls Ill' tl'mlwr,llllrt' al "llIlh \ .1"'1
, !l .11I , , 'S [rom ice to liquid, or rro m liquid to str-am , '1Ill' "rokl II
'.\ nutu-tr aspect here can be clcar l S 'I' lI il' WI: cOIllI),\/"l' tilt' g.lS .lIId
""Iid states o r a material , and if' Ior simpli 'it , W I ' .ISSUn1l' p!'r rl'etl )
Illdlill'lll gases and perfect rystal arrange ments. In these idl',ll COli
,lit ions, the gas would d i 'play invariant prop ert i , . under all I rans la
t 1< iu s, rotations and reflections, while thc solid wo uld be invarian t [Il
" lIly a subs et of these transformations. For xamplc, while tlu- 'il
cou ld h disp laced by any amo unt and r main basi ally the same (that
is, an observer wo uld b unable to tell whether a clisplacc m -nt
lie urrcd at all) the so lid wo uld rem ain visually unchanged onl und er
displacem ents which moved it one unit crystal at a time (or multip les
of that unit). In other words, the gas has more symmctry than till'
so lid , and can become the so lid by undergoing a symmetry-breaking
phase tra nsitio n . I S The metaph ori cal example I gave above, that of a
rel'tilized egg which differentiates into a fully formed organism, can
now be made quite literal: the progressive di fferentiat ion of the
spherical egg is achieved th rough a complex cascade of symmctry
br eaking phase transiti ons. 16
Let me now incorporate the idea of progressive di fferen tiation into
the co nce pt of multiplicit y by showing how it can be translated int o
state-space terms. I said before that for th e purpose of defini ng an
entity to repl ace esse nces the aspect of state space that matter ed was
its singulari ties . O ne singular ity (o r set of singular it ies) may undergo a
sym met ry-breaking transition and be co nve rted into another one.
T hese transitions are called bifurcations and may be studied by add ing
to a particular state space one or more 'control knobs' (technically,
control param et ers) which det ermine the stre ngth of externa l shoc ks
or perturbations to which the system being mod elled may be subject.
These control param et ers tend to display critical values, threshold ' of
IIIkll:'4l1, .11 \\Iull. ,I p ,lI III 111 ,11 hllllll .llIoll LIll' pl." I ' I lll".LIll I till III luI
S}llIrndry of lilt' ")""h'I II . J\ " Li ll o "p.ll t' "t1 lh llll n l It) 0 111' 1)1 Ii lit
nt tr-ac-tor, 1(11' cxn m p lc, 111.\) ' hifi.l n -.l tt' in to ,lIIlltht'r w ith 1\\C1 ~HH h
at tractors, or a point an -ac to r may hil'urcat l' into ,\ IH 'riodi c o tu -, losing
some of its ori ginal symme try . 17 Much as at t rac to r-s cor m - in recu rrent
forms, so bifurcations may define recurrent sequen((~.'i o f' such forms.
There is a sequence , for instan ce , that begin s with a point an -acto r
which, at a critical value of a control param et er, becom es unstabl e and
bifurcates into a periodic attractor. Thi s cycl ic singular ity, in turn, can
become unstabl e at another crit ica l value and und ergo a serlue ncc of
instabilities (several period-doubling bifurcations) which transform itinto a chaotic ott-actor.
Thi s symmetry-breaking cascade of bifurcations can , in turn, be
related to actual recurring sequences in physical processes . There is,
for example , a realization of the above cascade occurring in a well
studied series of distinct hydrodynamic flow patterns (steady-sta te ,
cycl ic and turbulent flow) . Each of these recurrent flow patterns
appears one after the other at well -defined critical thresholds of
temperature or speed . The seguencc of phase transit ions may be
initiated by heating a water container from below. At low temperatures
the flow of heat from top to bottom , referred to as thermal conduction,
is simple and steady , displaying onl y a hland, featureless overall
pattern , having the degree of symmetry of a gas. At a cr it ical point of
temperature , however, thi s ste ady flow sudde nly disappears and
another on e takes its place, thermal con vection, in which coherent roll s
of wat er form, rotating either clockwise or anti -clo ckwi se . Th e water
conta iner now has struct ure and, for the same reason , has lost some
symme try. As the temperature continues to intensify another threshold
is reached, the flow loses its orderly periodic form and a new pattern
takes over: turbulence. The cascade that yields the sequence conducti on
convection-turbulence is, indeed, more complicated and may be
studied in detail through the usc of a special machine called the
Coue tte-Taylor apparatus , which speeds up (rather than heats up) the
liquid mat erial. At least seven different flow patterns are rev ealed bythi s machin e, each appe aring at a specific critical point in speed, and
thanks to the simple cylindrical shape of the apparatus , each phase
20
11 .111 111011 111.\\ I II dill I I " I I Lilt d t il '" 111 1l~ 1 II \ I lllii t 11 ' III II.. I I ti ll I'fll 11 .111 lelllll .1111l1I 111 11 11 I , llIltll l I
'\ 'Ii 1.111 h.' 1'1 '11 It 0 11I 1111 1' ox.un p h-, .1 1 .1 t,ltl., III halUll ,llilln, 111.'\ lit1.lith lu lh n ·,IIi/t·d ill .1 )'11\ !'o i l ,II "!'itt' ll l, I hi, n ·.lli/.'1 Ion , ho\\ t" r-r, ht·.I1 !'1
Ilt l rt·,,· ;nhl.ll1l t' to till' 1I~lol tlh ' IlI :\lk,, 1 ( '.l ~(,ldl' . III p,lrt h Il I,H' , un like ti ll"
1,IUI '1" " hkh is mt.·dllll1/HII -lntlqll·nJl·rH, tlal' php ic,t' n '.,lil'"tio lt ill'"ht '
"I'l'd lic nwch,mi.'ims. To hl'gin with tln-n- ,11"1 ' causa l inu-r.nuons .uul
tlu-ir t' ITl'ets . To re t u r n to our e xam ple, till: How o f 1H'"t into ti ll't'tlllt"inl'r causes ,1 graded dl'l1sity ditl l'n'IKe to form , gin' l1 th'lt wau -r
t'xlMnds wlu-n heated (that is , becom es less den se) . This tklls it)
gradient , in turn , interacts with other forces like the viscosity or ti ll'wate r, their balance of power det ermining whether a s)'SIt'Tn swi ld le"
fro m one flow patt ern to the next. For exa mple , the de nsity gradit·nt
will tcnd to amplify small differen ces in movem ent (fluctuat ions) \\ luch
cou ld add so me detail to the bland ste ady-sta te How, but whk-h art'
damp ed by the viscosity of the fluid. As the flow of heat is intvnsifi...l,
how e ver , the syste m reaches a cri t ical po int at wh ich th e dl'n sity
gradiellt is strong enough to overcome viscosity, leading to till'am plification of fluctuations and allowing th e formati on of co here-nt
ro lls . Thus , a very specific seque nce of e vents underlies the transition
to convection. On the other hand, as the biologist Brian Goodwin has
pointed out , portions o f this hydrodynamic seque nce may be obse rved
in a complete ly different process, the co mplex morphogenet ic
Sl'llue llce whi ch turns a fertilized egg into a fully developed organism.
Afte r describing another instan ce of a sequence of flow paltl'rns in
hydrod ynami cs Goodwin says:
Th e point of the description is not to sugges t that morphogen et ic
patterns originate from the hydroJynamie properties of living
orga nisrns . .. What I want to emphasize is simply that many
patt ern-generating processes share with developing organisms the
characterist ic that spatial detail unfolds progressivel y simply as a
result of th e laws of the process. In th e hydrodynami c exa mple we
see how an initi ally smoo th fluid flow past a barri er goes through a
symme try- brea king event to give a spat ially periodic pattern , fol
lowed by the elaboration of local nonlinear detail whi ch devel ops
2 I
hilt ti ll') do ~o .,1 po int ' . ti ll tilt' , 'e1 1 t · ~••md umh-r g lllllllll' l ill '
whic h IH'\('r havv llll' u n ilelr ln it ) clf .1 n.uu rol light . ()n (·.It h
Ill t .l , ion , ohscurit it·l'i .Hlel 1 111l1' S of ,Jude,,\ co rn" pofl( l 10 tlu-u
c1 i,linl"l ion . IMultiplidt il' ''il .H·t' c1islingub.lwd from on e .urorln-r , but
nu t at .111 in the same manne-r as form s and tlu- terms in "hkh thc' '''
.m - incarnated . Thcv are ohj l'ctin'h' mad e and unmade according tl l
till' co ndit ions that : lct e rl1l inl' thl'i; fluent synthes is. This is IWI". 11'1',tht;)' co mbine the greate st power o f hdng difTcrenrialt·d with an
inahility to be dttfcrcnciated .JO
Altho ugh I will not stick to thi s subtle typographical distin ction ,
I ),ol,'ule distingui shes the progressive unfolding of a multiplicit j'
Ihrough broken symmetries (diffc renrioltion), from the progn'''isht'
"i lwcification of the continuous space formed by mu ltip licities as it gin'''i
rise to our world of discontinuous spatial structures (differl.·nciatio n) .
Unlik e a transcendent heaven which exists as a separate dimension from
n 'aHty, Dcleuze asks us to imagi ne a conti nuum of mul tiplicities whic-hdlj]crenciaees itself into our familiar three-dime nsional space as wel l as
its spatially struct ured conte nts ,Let me explain in what sense a co ntinuous space may he said to
beco me progressively defined giving rise to discontinuous spaces . First
of all, a space is not just a set of points , but a set together with a way
of binding these poin ts together into neiohbouThoods th rough well
defined relati ons of proximity or continuity . In our famili ar Euclidea n
gcomet ry these rel ations are specified by fixed lengths or distance"
which det ermine how close pOinL'i arc to eac h othe r. The concept of
' length ' (as we ll as rel ated ones, like 'a rea' or 'volume' ) is what is
called a metric conce pt , so the spaces of Eucl ide an geome try are known
as meeric spaces.1. There ex ist other spaces, however. whe re fixed
distances canno t define proximities since dis tances do no t remain fixed.
A topological space, for exa mple, may be stre tched without the
neighbourhoods which defin e it changing in nature . To cope with such
exotic spaces. mathematicians have devised ways of defi ning the
property of 'be ing nearby' in a way tha t does no t presup pose any
olll 01 llit ' PI ' rlOelh It) Illlhn ollH d ,· \ (·IClI"I ll'1l1 10110 \ \ .1 111 11 1.11
(llI.l lit.l l in · COurw: illit i.,lly Mllooth p'II1"" .IX'·... . dIC·IlI' (·h ,·... tlH'res ult of Sl>.ll ial bifurcatton from .1 uniform st.ll t·, bifun:.llt· to
spa tially periodic patt ern s such as sl'gnwl1 ts [iu J.1l Insec-t hodYI.
wi th in which fine r de ta il develops . . . through .1 progn'~si\'c
expression of non linearities and successive bifurcations . . . The ro ll.'
of ge ne products in such an unfolding is to stabilize a parttcular
morphogenetic pathway by facili tating a sequence of pattern transitions, res ulti ng in a particular morphology, l q
From a Deleuzian point of view, it is this uni versality (or mechanism
independen ce) of mul tip licities whi ch is high ly significant. Unlike
essences which are always abstract and general entit ies, multiplicities
arc concreee universals. That is. concre te sets of attractors (realized as
tenden cies io physical processe s) linked together by bifurcations
(r ealized as abrup t transitions in the tendencie s of physical processes) .
Unlike the generality of essences, and the resemblance with which this
gene ra lity endows instan tiations of an essence , the universality of a
multiplicity is typically diverqem: the different rea lizations of a multi
plicity bear no resemblance \..-hatsocvcr to it and there is in prin ciple
no end to the set of po tential divergent forms it may ado pt. Thi s lack
of resemblance is amplified by the fact that multiplicities give form to
processes, not to the final product, so that the end res ults of processes
realizing the same multipl icity may be highly dissimilar from each
othe r , like the spherica l soap bubble anel the cubic salt crys tal which
not only do not resemble one anothe r. but hear no similarity to thetopological point guiding th eir production.
The co nce pt of progressive differentia tion whi ch I have just defined
was mea nt , as I said , to dist inguish the obscure yet distinct natu re of
mult ip licities from the clear and disti nct identity of essences, as we ll
as from the clari ty afforded by the light of reaso n to essences grasp ed
hy the mind . A final di stinct ion must now be made : un like essences,
wh ich as abstract genera l en tit ies coe xist side by side sharply distin
gu ished from one another, concrete universa ls must be thought as
meshed tonether into a continuum. This further blurs the identity of
multipliciti es, creating zones of ind iscernihilitv where they blend into
each othe r , fonning a continuous immanent space very different from
22
•• I I II\1ltl 0 1 t Il 111011 .lId l' " pit Ol I t
\ t1 It11'11\ 1111 , I I). I. 1111 \\ l i lt •
li ll lilt I Ollll 'pl , hili 0111\ 110111111 111 1 t IIllt t pi I.kl ' 1111111111' 1I11••1 ,Ill t '
nl "'~!". lion ,'\ vr nih' I II.II",H. h ', it t'''' It . lilt ' til tu« tmn 111'1 \\ ITII II/n t ll IlIlt '
nonmctric J/,I1US is fuu d.uucutal in .1 I)"I"llli.Hl olllol,,!:) . rJ ,\1o n'o 't' r .
and this is the cr ucial point, there arc \\TIJ ·dl' f iIWd u-clmir-al \\.I\'S of
linking metric and no nmct ric span:s in such .1 wa v t hat the fo~m"-'r
become the product of the progr essive dmcr('ntiJti (J~ of the latte r . To
ex plain how such a svm mctrv- brcaking cascade would wo rk in this, ,case , I will need to tak e a bri ef det our throu gh the history ofnineteen th-century geometry.
Althou gh in that ce ntury most physicists and mathematicians though t
the struct ure of physical space was captured by Euclidean geome try.
man y othe r geome tr ies , with very different properties, had co me int o
existe nce. Some of them (such as the non -Euclidean geome try de
ve loped b)· Lohatch evskv] shared with the geometry of Euclid the
propert), of being metric . There wer-e , however , o the r geome trics
where metri c co nce pts we re not in fact fund am ental. The differential
geometry of Gauss and Riemann wh ich gave us the co nce pt of a
manifold is one exam ple, bu t there were se veral o the rs (pro jec tive
geometry, affine geometry , topology). Moreo ver, and despite the fact
that Euclidea n geometry reigned supre me, some mathem aticians
realized that its basic conce pts could in fact be derived from the
non metric co ncepts whi ch formed the foundation of the newcomers.
In par ticul ar, ano ther influential ninet eenth-century math em ati cian
Felix Klein , realized that all the geome tr ies known J to him could be
categorized by the ir invariants under gro ups of t ransformations, and
that the different gro ups were em bedded one int o the o the r .23 In
mod ern te rmino logy this is equivalent to saying that the different
ge ome tries we re re late d to each ot her by rel at ion s of bro kensym me try.
In Euclidean geome try , fo r example , lengths, angles and shapes
remain unalt ered by a group conta ining rotation s, translat ion s and
reflecti on s. This is called the gro up of riBid traniformations. T hese
metric prop ert ies, howe ver, do not remain invariant under the groups
of transforma tions characterizing othe r geomctrit·s . T here is onc
geometry , called affine Beomet')', which adds to the gro up characte riz ing
Eucl idea n geometry new transfonnation s, called linear traniformations,
under which pro perties like the parall elism or the straightness of lines
It 111.111 1 urv .ur.mt , IHIt 11111 till II It Il·d. 111111 till II I 1"11/11 '" IIlnl/lCff.,
wlm h ,.dd In II l id .mel )1111'.11 11 .111111111I .111011 tlllI I 01 1'1 0lt ·tlloll ,
tCll"Il "pon ding III , 111I1111,111111011 ,I 111 1'11 ot hlru , .11Il1 , (·t I IOIl, I I",
n plh.l)t·nl o f intt 'rn'pti ng liuN' ligh t I .l ) ·"i 011 ,I "it I (T I1. (Mort· In hui
("111 )', thi s gl'onw lr) .ldcls t ra n- Io rm.u illllS t-,lllt·d ·pro jl·t l i\'il it's' . ) lln-sc
rr.msforrnat ions do 1101 nt·n ·ss.lr ily h-a ve l .uc-l id r-an or atfi m- propt'rti t·
undl.1ng('d. d S CJ Il h.., ('asH)' pictured if we inl.lgilH' .1 film pro jed or
(which typically inc reases the mJgnitudl' of length s) and .l projr-rtion
" ..-n-cn at an angle to it (which distorts parallel lines) .
If we picture these three gt.'ometr ies as forming the leve ls of a
hierarch)' (projcct i\"c-affine-Euclide an) it is easy to sec that tlu
tra nsfo rmat ion group of each level includes the transformations of tln
leve l below it and adds new ones . In o the r words, each level POSSt'SS('S
more sym metry than the leve l bel ow it . This suggests that, as we
1110 \"C down the hierarch )' , a sym me try- breaking cascade sho uld pro ·
dun' pr ogressively more differentiated geometric spaces, and, vice
versa, that as we move up we sho uld lose differen tiat ion . For exam ple,
as we asce nd fro m Eucl idean geometry more and more figur es become
eq uiva lent to one another, fo rm ing a Jesser number if distinct classes,
Thus , while in Euclidean geome t ry tw o triangles are equivalent on ly if
their sides have the same length, in affine geometry all tri anglcs an '
the same (regardless of lengths). In other words, as we move up the
class of equivalent triangles becomes less differentiated . Or to take a
dillerent exam ple , whil e in Euclidean geometry tw o co nic sec tions
(the family of curves co ntaining circl es, ellipses, parabolas and hyper
bolas} are equivalent if th ey are both of the same type (both circles or
both parabolas) and have the same size, in affine geometry they only
need to be of the same type (rega rd less of size) to be equivalent , whil e
in pr oject ive geometry all co nic sections , without further qu alificati on.
are the same .H In sho rt, as we move up the hierarchy figures whi ch
used to be fully differentiated fro m one another become progressively
less distinct event ually blending into a single one , and vice versa, as
we move down, what used to be one and the same shape progressivcl)'
d iffere ntiates into a vari ety of shapes .
This hierarchy can be expanded to include other geo me tries, such
as differential geometry and topo log}'. The latter , for exam ple , may be
ro ughly said to co nce rn the prop erties of geometric figures wh ich
2~
I t 111.1111111 \ .111.\111 !llIdll IUlldlll '. 1111111111 • II I .1 .1 01111111 ' II II Il lI1I1.l
tn ut x, llloll i . tr,m"iIOI"I1l.ltloll \\lu, II do lIul t 1I' .,le IW\\ pOlllt " 0 1 IU "'l'
t'xis l ing ones . (fvl(ln' "'(,I<'t l)'. lcll)ll log )' ill\I,I\( ' tl.III ,lllrnlolliIJn ... , I.IIIt,(1' homeomo rp hisms ', which conrcrr n(drb) POints lilt .. nt'utb) ['Oint! .mel
wh ich can be reversed or be continuously 1II1dOlW,) Llnck-r these
transformati ons many figun~s which arc complete ly distil1(,t in Euclidcan geome try (a tri angle, a square and a circ le. for exa mp le) become
one and the same figure . since they can be deformed into one another.
In thi s sense . topology may be said to he the least d1fe rentiateJ
geome try. the one with the least number of distin ct equivalence
classes, the one in which many discontinuous forms have blended into
one cont inuous one ." Metaphorically, th e hierarchy ' topologica l
dilTerential-projecti ve- affine-Euciidean ' may be see n as re presenting
an abstract scenario for the birth of real space . As if the metric space
wh ich we inhabit and that physicists st udy and measure ",as born from
a nonmetric, topological continuum as the latter differentiated and
acquired struct ure following a ser ies of symme try· breaking transitions,
Thi s morphoceneuc view of the relation between the difTerent geo
metries is a metaphor in the sense that to math emat icians these
relations are purely logical , useful because theorems whi ch are valid at
on e level are automatically valid at the levels below it. ' 6 But thi s
cascade of broken symme tr ies rna)' be also given an ontoloqtcol dimen
sion. O ne way in which thi s scenario for the birth of metric space can
be mad e less metaphorical and more directl y onto logical, is through a
co mparison between metric and nonmctric ge ome t rical properties , on
one hand, and extensive and intensive phy sical properties, on the other.
Extensive properti es incl ude not only such metric propert ies as length ,
area and volume, but also quantities such as amount of ene rgy or
entropy , They are defined as properties which are intrinsically divisible:
if we divide a volume of matter into two equal halves we end up with
tw o volumes, each half the exte nt of the original one . Intensive
pr operties, on the other hand, arc properties such as temperatu re or
pressure, whi ch canno t be so divid ed , If we tak e a volume of water at
90 degrees of te mperature, for instan ce, and break it up into tw o
equal parts, we do not end up with two volumes at 45 degrees each ,
bu t with two volumes at the original temperature .?"
Dcleu ze argues , however , that an inten sive propert}' is not so mu ch
26
flil l Ihlt I 111111\1 1111. 1.II1 IIl U \\ llh lll.UII'., ~ .1I . hl J "hill" ""I .I. lntl ...d"ml/( III klllJ. 1111 111111" '1 .1111I 1 III .• ' 1\1 11 \ IIIII III I III 11llll id \\.11_ 1,
III, " .lIll pll, t ,m mdl 'l ,d III ' d l \ II II' d ' 11\ 111 ,11111 t lilt' tlll1l .IIIII " [rr un
Illuk l lll,.It ll l U·.l llfl' ,I lI 'IIlP, 'I .lllllt· dlll ll l'll t l 11('1\\ t ' . '1I tilt' t llP .uHIho llum portion' of th e w an-r , )'t 'l. \\1.11" pnor to tlu' Iw,ltlllg till
, h' m is .11 cquililu-ium , 011('(' th t' h '111I H'r,lIU l'l' di lll'n'm I' i.. I fl ·.lh ,d
til(' '1)"sk m will [n - aw.1)' from equilibri um , th.u is , wv con di\ iell ' it ll
Ic- mpt.' rat lln· hut in so doing we dl.mgt' the sph'm Cl ll.l!it.lIh c1) .
Indt'('cI, .IS we just saw, if tilt' h 'mpe ratu rc- c1ifl~ ' n'I )(' I ' is m.ul t' inll'n"i.·
(' lIough the SYStl' l1\ will und ergo .1 ph ase transition, losing S)' I1II1h' II")'
.,,11 1 changillg its d ynamics, dl'vL·loping the periodi c p.lltL'rn of Iluidmotion which I referred to above as 'convec tion". T hus. in a \'t' r )" n -al
w nsc, phase transit ions do divid e the temperature scale hut in so doing
Iht,), mark sudde n changes in the spatial symml't ry of a material.
Using these ne w concepts we can define the sense in which tht'
me tri c space we inhabit emerges fro m a no nme tric continuu m through
.1 cascade of broken symme tries. Th e idea wo uld he to view this
g" m'sis not as an abstract mathemat ical pr ocess but as a co nc- re te
phpical process in which an undi fferen tiated intensive space (t hat is, a
s paCl~ defined by co nti nuous inten sive properties) progressivcl )' differcn tiatcs, eventually giving rise to extensive structures (discontinuous
structures with definite metric properties) , W e can take as an illustration of thi s po int some recent developments in quantum field
theories. Although th e conce pt of spo ntaneo us sym me try breaking,
.1I1d its connec tio n with phase transitions, devel oped in rath er humble
branches of physics, like th e fields of hydrodynam ics and conde nsed
matter physics. it was e ventually incorporated into the main strcam .!"
Today, thi s conce pt is helping unify the four basic forces or physics
(gravitational, elect romagnetic, strong and weak nuclear forces) as
physicists realize that , at extremely high temperatures (the ex tre me
conditions probably pre vailing at the birth of the universe) , these
forces lose their individuality and blend int o one, highly symme tr ic ,
force . The hypothesis is that as the universe ex pande d and cooled , a
se ries of phase transitions broke the original sym metry and allowed the
four forces to differentiate from one ano thcrv '" If we conside r that , in
relativity theory, gravity is \,shat gives space its metric properties
(mo re exac tly , a gravitational field co nstitutes the metric structure of
27
,I 111111 d llll1'l1 11111," 11I,lIl1ll1ld>. 111'( II \\1 ,11 1.1 III 1111 11. ,11 ' I I II II, Itl' lIl1' rgl' . ,IS .1 (lisli llll fi l/'l\' ,11,' P '\1111 II/I" ," pllllll III II 1111, II 1\'
prop 'rty (te m p .raturc ) , the idl',l 01 .1 11 intvusrv « p,lI " ' 1\ 111' 1IIIIh 10
exte nsive ones through progrcssivl' din~'rcnl i .1t i on h" l'olll l' · 1110r, ' than
a suggestive metaphor. I I
Let me pause for a moment to sum marize th e argument so Iar . I
began by es tablishing some purely formal d ifferen ces bet ween th e
co ncepts of 'essence' and of ' m ult iplici ty' : whil e th e former co ncept
implies a unifi ed and timeless identity, th e latter lacks unity and implies
an id entity which is not giv en all at on ce but is defin ed progressively ;
and while essences bear to th eir instantiations th e same relation whi ch
a model has to its copies, that is, a relation of greater o r lesser
resemblance, multiplicities imply divergent reali zations which bear no
sim ilar ity to th em. These formal differences, I said, are insuffici ent to
character ize th e distinction between essences and multipliciti es as
immaterial ent ities whose job is to account for th e genesis of form:
replacing etern al archetyp es involves supplying an alternative expla
nation of morphogenesis in th e world. Unlike essences whi ch assume
that matter is a passive receptacle for exte rnal forms, multiplicities ar e
immanent to material processes, defining th eir spontane o us capacity to
generate pattern without exte rn al intervention. I used cer tain features
of mathematical models (state space s) to defin e th e nature of multipli
cities: a multiplicity is defined by distributions of singular ities , defining
tenden cies in a process; and by a se r ies of crit ical transitions which can
take several such di stributions em be dded within one another and
unfold th em. Finally, I said that a population of suc h conc re te
universals forms a real dimen sion of th e world , a nonmetric co ntinuo us
space whi ch progressively specifies itself giving rise to ou r familiar
metric space as well as th e discontinuous spat ial st ruc tures that inhabit
it.
No doubt, despite m y effo rts th ese remarks remain highl y meta
phorical. First of all , I have defin ed multipliciti es in terms of attractors
and bifurcations but these ar e features of mathematical m odels . Give n
th at I want th e term 'm ult iplicity' to refer to a conc re te universal ( to
replace abst ract ge ne ral essences) th e qu esti on m ay ar ise as to th e
legitimacy o f taking features of a model and reifying th em into th e
definin g traits of a real entity . Second , th e relation between a
28
'"11111111 11 111 " I 1IIII IIIpill Ill. III I II.. ,h 1111 111111 " I 111 1 .11 I .1.1 I'"
I I I I 1" " .11 1 11 111 . 1 1. 11 11111 ' 11 1. '11111 11II 11111 • • I ' " I ,
111 .,11. '11' ,1111 " '''II III" 111111 ,11" 1111 1,IId , III "111111 1111 III I dlt 11111 I.
1,,1. li-m. 111111111 .11111 ' Ih.. 1I1 ,'llphlllll ." '"11 1' 11 1 \\ i1 1 \ll\ lIh ,' 11,,111111
,I Ihllrllu ,h <l lllo lo 111 ,,1 \II.lh i 01 I,ll' p,lI" "Ih,'l II ', '/, ,'/ (I.,/, c,'/
IIl1 c"i<1 /)/1 1..111 III S(' ll.Ir.lll'd 11'0111 its \.1 1"1.11.1 .. 111,111,, '111.1111".11111111<'111 , hilI
111 addiliol1, J d, 'l.lih,d discu ssion 0 1 ho\\ 1111'S" tllp" logil.11 111\.11 i.urt
111.1 • Ill' WO\'l'1I to gl'lhl'r to C(I/I.\/[ II I ,I continullus, '('\ Ill'll'ro " '111'0 11 ,
SIl.1 Cl' , In t lu- fo llowi ng chr pll..r I w ill show in technical (k l,lil hem 11 11
l tl l1st ruet io n can bl' carried o ut and ho w th e I' sultinv cu nt inuum . 11.1
replace th e top o r least m ct ri ' level in th e hierar ch y of gl'ollll"ll'i, ' . I
\\ ill also di s uss how th int rm ,<Iiate I rve ]s may he rl' p l.H I'd b\
inl -nsiv pro cesses of ind ividuat ion whi ch yield as the ir fina l prtllhll l
tlu- fully differ ntiat ed m tric st ruc tures th at popUlal l' ~h l' ,bo tlOIl\
lvvc l, At the e nd of chapte r tw o the metaph or of a ge nes Is 0 1 nu -tr«
spa c th rough a cascade of broken sym m ' t r ies sho uld have been most!
e lim inated , and a literal acco unt taken its place.
Meanwhile, in what rem ain s of thi s cha pter I would lik 10 mak.. ,I
1I10re detailed analysis of th e nature o f multipliciti es . T he fir st s('\ 01
issues to be d iscussed will invol ve th e technical details of Del C U/." ' .
ol1to logical interpretation o f th e co ntents of sta te space. His approach
is very un orthodox as will be sho wn by a co m pariso n with th e stau
space o ntologies proposed by anal yti cal philosophers, T he n I wi ll 1110\'1'
on to a sec ond se t of issu es co ncern ing th e modal srcrus o f multipliciti cs.
Modal logic is th e bran ch o f ph ilosophy w hich deals w ith th e rdat i or~ s
between th e possible and the actua l , Here th e qu esti on to be answered IS
if state space is a space of possibl e states what is th e status of attra ' to rs
and bifurcations in relation to th ese po ssibilities? Can multiplicities 1)('interp re ted in terms of th e traditional modal catego ries, th e possibk.
and the necessary , o r do w e need to postulate an origina l form 0 1
physical m odality to characte r ize th em ? Fina lly, a third se t of issues.
that needs to be dealt with is related to th e speculat ive dimension o f
Deleuze's proj ect. Replacin g essences with soci al co nve nt ions or
subject ive beliefs is a relativel y safe mo ve , but putting in their place a
new set o f object ive ent it ies inevitabl y invol ves philosophical sp' u
lation . What gu ides thi s speculat ion? One wa y of looking at thi s
ques tion is to see Deleuze as engaged in a co ns truc t ive project guided
, . ( . 11.1111 1" 1' "/'''\ '''''' /' 1/1''. II.. I I . ( " " 1111111 Ill e I. u ll 111111 11 ,,1
\\ h.11 10 do hili \\ h,ll 10 .1\ oltl .10111 ' (lilt lit I. "" 11.11111 I 0 1 (1111 c', ,to avoid the- I'-"f> 01 ('sSI'llli.llislII, hili Ihl rl' .11( ' o IIIC' I ,11I.1 till' (' Ill'l'dto be d iscussed.
Let me begin with Del zuz ·'s ollto logie. 1 .m.lly.sis or state 'pa 'e .
Many philosophers arc today look ing at these abstract spaces as ob]e ts
of study and reflecti on . A recent shift in the analyt ical philosophy of
scie nce, for example, moving away from logic (and s t theory) and
towards an analysis of the actual math ematics used by scient ists in the ir
eve ryday pra cti ce , has brought the importance of sta te spaces to the
foreground. 32 Yet non e of the philosophers invol ved in this new
movement has attempte d such an origina l analysis of state spac e as
Del euze has. In particular, analytical philosophers see m unaware of (or
at least uncon cerned with) Poin care 's topological studies and of th e
onto logical differen ce that may be posited bet ween the recurrent
features of state space and the traj ect ori es these features det ermine.
Given that this onto log ical differen ce is key to the idea of a Deleu zian
multiplicity, I will need to ex plain how state spaces are co nstructed.
First of all , it is important to distingui sh th e different ope rato rs
invo lved in this co nstructio n. As I aid above , given a relat ion between
the changes in two (o r more) degrees of freedo m ex pressed as a rate
of change, one ope rato r, differentiation , gives us the instantaneous
value for such a rate , such as an instantaneous velocity (also kno wn as
a velocity vector). The o the r operator, integration , performs the opposite
but compleme ntary task: fro m the instantaneou s values it recon stru ctsa full traject ory or series of states .
These two operators are used in a par ticul ar order to generate the
stru ture of sta te space. Th e modelling process begins with a choice of
manifold to use as a sta te space . The n from ex perime ntal observations
of a system's changes in time, that is, fro m actual series of states as
obse rve d in th e laboratory, we create some traject ori es to begin
populating this manifold . These trajec tories, in turn , serve as the raw
mat erial for th e next step: we rep eatedl y apply th e di ffere ntiation
operato r to the trajectories , eac h application generating o ne veloci ty
vecto r and in thi s way we gene rate a velocity vectorfield. Fina lly, using
the integration operato r, we genera te from the vector field further
trajecto ries which can function as predi ctions about future observa tio ns
3°
II III I 111 ' 11 '1" 11111. I (111,,1lit Ih. 1'111 1111 ,. t II I'll' I I '
I • I • I II I II. 11 11 • I ), I. 11/1 111.1 ' "'"1I II 1' 1,1 ( 1'''111.111 II l'
1.///" /1111,,,11 .1, "//(" 1'// " ," // /1" trill 01 ,." till ' 1'1'( ,II III li lt 1 h .1 (
I I I J /1, " 1/1" I' 1.1, 11 11 ti ll o lh( Il'0rll.1I1 II • \ 11111 . Oil o ll( 10111( , "//1 «
\\'h ill' .1 p,lll i 111 .11" II"II( 11 01 or mu - '1.11curv I' ) IlIodl'l .1 UU ' 11111 (II
,111 11.11 st.lll'S 01.1 \sll' lIl ill Ihl ph~' si .JI \\ odd , Ihl \Idor 1I1·\l1 c ,111Il1I1
•
Ih.. inherent n-ndcn cics or man ' such tr: j.·cloril's, and hellll' 01 111.11'
.1 tual svstcm s, to ln-have in C(·rt.l in \\'a~'s , As mcnt ion ..d .11 1lI\(' , th.. I
tendencies are represented hv singularit i .s in th .. vector field, ,lIld I
1kleuz' not .s, lit-spite the fact tha t the precise narure of cac h sin iul.u
point is w ·1I d ·fin d on ly in th e phase portrai t (b the [orm till
ira] c tories take in its vicinity) rhe e.t islence anJ dimihullon 0\ till ('
singularit ies is already complet ly given in the vector (or din'( l ion)
field. In one mathematician " words:
Th e geometrical interpretation of the theory of different ial · (11l·11iUl~
\car ly places in evidence two abso lutely distinct rea lities :. then' I
the field of directions and the tcpoloqica! accidents wh ich 11101 )
suddenly crop up in it , as for example the ex iste nce of . .. singlll.lr
points to which no direction has been attached; and then' an', tho
integral curves with the form they take on in the vic~nity 01 Ihl
singular ities of th e field of directi on s . , . T h ~X l st ' I~ e and
distributio n of singularities are notions rela tive to the field 01 vc ' tors
defined by the differential eq uation . T he for m of the int gra l curves
is relative to the solution of thi s equatio n. The two problem ' arc
assuredly complementary, since the nature of the singu larities of.th...
field is defined by the form of th e curves in their vicinity. But It IS
no less true that the field of vec to rs on one hand and the int 'gral
curves on th e other are two essentially distinct mathematical realiti es. 14
T here are seve ra l other features of singular it ies, or more spccif ally,
of attractors, which are cr ucia l in an on to logica l analysis of state 'pa "
and which furthe r differentiate its two 'distinct mathematical rea liti 's' .
As is we ll known , the trajectories in thi s space always approach an
attractor asymptotically, that is, they approach it indif/nitely close bur
nerer reach it. 35 This means that unlike trajecto ries, which re present the
actual sta tes of objects in the wo rld, attrac to rs are never actualized,
3 I
--
II". 1111 p.1I1l1 01 .1 II Jt I 1111 \ I \t I II ,It III' IIll .u t r .u nu II f II It I 111
l h i , M' II 'iI ' th,lt ' 1Il I u l. lI ll it ' , n 'ptt ' c u t flllh Iht ' ICII1~ It rill h ,lltkllt II " of
a ,"}Stl' III, neve-r its J.d ll.ll 'il.\h' " , I )t" IHIt' t hl' i, I." k 01 .lllll.lli l\,
at tractors are ucvcrth clcss n -al am] h.\,,' dl'fillilt' dlt'lts on .\('tll~ienti ~i~s . In part icular, the)' co nfer on trajc-ctoru-s .1 cc-rtnin dt 'grn' of
sta~lh t)', called a~)'mprolic stability. it, Sma ll shol'ks rna)' d i sl ()dg(~ a
trajectory from its att ractor but as lon g as the shock is not too large
to push it out of the basin of attracti on , the traject ory will naturally
return to the stable state defined by the at tractor (a stca dv sta te in the
case of point attractors, a stable cycle in the case of periodic attract orx,
an~ so o~) , Another important feature invo lves not the stability of the
trajecto ries but that of the distribution of attractors itself (its structural
stability). Much as the stability of trajectories is measured by their
resistance to small shocks , so the sta bility of a particular distribution
of attractors is checked by submitt ing the vector field to perturbation s,
an effec t achieved by adding a small vector field to the main on e and
checking wh ether the resulting distribution of attractors is tapaJoo;caJIy
equivalent to the original one, J7 Typ ically, distributions of attrac to rs arc
str uctura lly stable and thi s, in part , is what accounts for their
recurren ce among different physical systems. On the othe r hand, if the
perturbation is large enough a distribution of attractors may cease to
be struct urally stable and change or bifurcate into a different one, Such
a bifurcation event is defined as a continuous deformation of one
vector field into another topologically inequivalent one through astruct ural instability, 38
Using the technical terms just introduced I can give now a final
definition of a multiplicity, A multiplicity is a nested set if' vector fields
related to each other by symmetry-breakino bifurcations. [oaether with the
distribut ions of allraclors which define each if' its embedded levels, This
definiti on separates out the part o f the model wh ich carries informati on
abo ut the act ua l world (t rajectories as ser ies of possible states) from
that part wh ich is, in principle, never actualized, This definition
presupposes only the two co nce pts of 'd ifferential relation ' and
'singularity ' , J will return in the next chapter to a discussion of what
~urther philosophical traniformati on these two conce pts need to und ergo
In order to be truly detached from their mathematical realization. At
this point, granting that the definition I just gave could specify a
3 2
l oltt II II , 11111v , \\ t 111 ,1\ .1 ~ \ .. h ,lt 111I1111l) ' II .,1 1.11" li t h .111 t 11111 \
.... "lIld h,I'" ( 10 1)1,11...1 I .lIef 01 p.ltlllli 01 ll\dllllhll.lIllll 1111 .... lidII I IMth 'l'lI IIll'llIhl )lllolh'll t! e"I·lop"lt·III .1 tll\(1 11111 r('"I" IIl /ll1l Ill .,
llI1i" 'I', .,1 lIIultlpli, il I mi,lt ,.\tlll1g 'illll l' II ll~l' t th,ll ,llt "I' IM'It -lll
an - re-al, \\ hill' till' Illllitiplid') iht,lI i, 11111. So I h'lt-tl/l' '1)I'"l.. nlll 01
' t!' ,lli/ ,l ' io ll ' but of dtllllJludlWn, .lIltl inlrotlu,,''i J no"'1 oll lologil ,ll
,""h-gor)' to refer to the status of lnu lti plit-itil 'S tlll'm""I,"" : II"u"hr ,
l'hi-, te- rm does not rc-h-r, of co ursc-, to the virtual n ,.llity .....hich digl t.,1
..Imulatio ns han ' made so fnuil iar , hut to a real l" ir'"tJlit)' fo rm ing .1
\ it.ll ('o mp(Hlent of th e ol*'cti\"l' world, As he writes:
TIH~ virtual is not opposed to the real hut to the actual. The d rtlllJl
isJ ulJ.y real in soJar as it rs rirw cJI , , . Indeed , the virtual mu st [u
defined as st ric t ly a part of the real obj ect - as thou gh till' ohjt'l t
had one part of itself in the virtual int o \v·hich it plunged as thou gh
into an object ive dimen sion , , , The realit y of the virtual co nsists
of the differential elem ents and relat ions along with the singul.u
points wh ich co rrespo nd to them, The reality of the virtual is
struct ure, W e mu st avoid giving the elements and rel ations that
form a struct ure an actuality whi ch they do not have, and withdraw
ing from them a reality whi ch they have. ?"
\Vhat is the modal status of the virtual? If state space traj ectories han'
the sta tus of possibilities (possible ser ies of states) what modality do
virtual multiplicities represent? This is not an casy qu estion to answer
givcn that the ontological status of even the familiar modal categoril's
is a thorny issue. So before dealing with virtuality let me discu ss the
ques t ion of possibility. Traditionally, ontological discussion of possi
bilit ies has been very controvers ial du e to their elusive nature , and in
part icular , to the difficulty of giving a clear crite rion for individuatino
them, that is, for telling when we have one instead of another
poss ibility. As a famous cr itic of mod al logic, the philosopher Willard
Van Orman Quine , jok es:
Ta ke , for instance, the possible fat man in the doorvvay; and again ,
the possible bald man in the doorway . Are they the same possible
man, or tw o pos sible men ? How do we decide? How many possible
33
"WII lht 1( ' ••r III Ih.lt dIHlr\\.I\ / II till It 1I1t'11 ' l" I 11.1, 111111 lUll
th..1II lat OIIt'S ? 110 \\ " Mil) of du-m .Ir, ' ,. l lk,· ? (h \\ ould tlwil Ih'lIlgalike llIake th em o ne? ,\ n ' not two pos.•dhh, things .,Jill"( I ~ Ihb th l'
same as saying that it is impossihl e fc)r tw o thin g,'i to h,· .,Iikl·? Or,
finally, is th e concept of ide nt ity simply inapplicable to unactualiz,«]
possibles? But what sense can be found in talk ing o f ent it ies whichcanno t be meaningfully said to he iden tical wi th themsel ves anddistinct from one anothcr-P'"
Most approach es to modal logic concentrate on langua ge , or more
specifically, on an anal ysis of se nte nces whi ch express what could hare
been, sentences such as 'If j.F.K. had not been assassinated th en th e
Vietnam War would have ende d sooner.' Given that human beings
seem capable of routinely using and making sense of these countcr fac
tual sentences, the modal logician's task is to explain this ordinary
capability." However, th e fact that linguisticall y specified possible worlds(like th e po ssibl e world wh ere j .F.K. survived) are so devoid of
st r ucture , and allow so mu ch ambiguity as to what distinguishes one
po ssible world from another, is what has prompted cr it icisms such as
Quine's . But as some philosophers have suggested, the problem here
would seem to he ,..'ith linguistic representations and their lack of
resources to st ructure possibl e worlds, and not with possibilities as
such . The philosopher of science Ronald Giere, for instance , thinks th e
extra const raints which st ruct ure state space can overcome the limitations of other modal approaches:
As Quine delights in pointing out, it is ofte n difficult to individuate
possibilities . . . [But] many models in whi ch th e syst em laws arc
expressed as differential equations provide an unambiguous cr ite r ion
to individuate the possible histories of the model. They ar c the
trajectories in sta te space co r res ponding to all possibl e initial
condit ions. Threatened ambiguities in the set of possibl e initial
condit ions can be elim inated by explicitly restrict ing th e set in th edefinition of th e th eoret ical mod el. 42
G,iere argues that state spaces may be viewed as a way of specifying
possibl e worlds for a g iven physical syste m , or at least , possibl e
34
11I1lJlll b u 11 ,1,111111'1111 IIllhl phil pUllIlIIIl"1 I lit III I 11I11
pu Ibll [u lui H .11 111'11 '11\ I III t ,Ilt 101., .. It '" 01 p,nl I I ht
illdh idtl.llil\ 0 1 tht' d llll'lt'lIl ptl"l"l lhll ' IUIOllt' \\1111111 Lilt· P·ltl I
ddilH'd I I) ' /111\'\, t·xpn ·....,·d h ti ll' dilh-n 'llti ;d l ·clll.lll lln tll ,11 lUlu tio u.dl)
rc-l.ue tilt' sni t(,!H 'S dl'gn 'l ' s of frlTdom, .Is \\,· 11 .1"1 b)' HlltllJI IIIIIJIlWf!\,
the specific ..ta tv , or po int in till' manif old . \\ Iwn ' .I "ph' III Iwgill!l it
evolut ion. Given a specific iuu ial co ndition .md a dcn-rm ini tic 1.1\\
(such as those or classical physics] OIl(: and only tIIW tr.lj ITt or) i
individuated , a fact that may he lIsc.'d to Ch.l l1(, l1gc.~ Quilw 's sn ·p t il.,1
stance. The phase portrait of any particular stale space will l»- typic,llIy
Fil led with man)' such individual traject ori es, one for each po ssihll'
in it ial condit ion. O ne may reduce thi s number bJ adding other 1.1\\s
wh ich forbid ce rta in co mbinat ions of values for th e degrees of freedom ,
that is, which make some initial cond it ions not available for J gin'lI
svstc rn , but st ill, one enos up with many possibl e histories .o4l
The problem for the phil osopher becomes what omo loq tcal SloW' tn
assign to th ese well -defin ed possibilities. One onto logical stance , which
Giere calls 'actualism" , deni es any reality to th e pos sibl e trajcctorics ,
however well individuated they ma y he. A mathematical model , in thi s
view , is simply a tool to help us in the control of particular phy sical
syste ms (that is, th e manipulation in th e laboratory of th e beh aviour:,f
real syste ms) as well as in th e predicti on of their future beha\'l(:ur. h:r
this limited purpose of predi ction and control all we need to Judge IS
the empirical adequacy of th e model : we generate one trajectory for J
given initial condit ion , then try to reproduce that particular combi
nation of valu es for the degrees of freedom in the laboratory, and
observe wh ether th e seque nce of actual states matches that pr ed icted b)'
th e traject ory. Give n th e one trajectory we associ ate with th e actual
seque nce in an expe riment , th e rest of th e population of traject ories is
merely a useful fiction , that is, ontologically unimportan~ . ·4 As Giere
argues , however, thi s ontological stan ce misses th e lact th~t th e
population of trajectories as a whole displays certain reBularities In tI~t.~
possible histories of a system, global regularities whi ch play a rol e m
shaping any on e particul ar actual history. :" To him , understandin~ a
syste m is not knowing how it actually behaves in thi s or that specific
situat ion , but knowing how it lVould behave in conditions which may in
fact not occur . And to kn ow that we need to use th e global information
t"lllboclh d 111 lilt ' popUJ..lioll oj pC " 'lhlt III IOlw • 1II101l1l.111011 whuh I
Im.l if we COlln'ntr."t ' on tilt ' 0Ilt' t l"' ln to , ) \\hhh I tclIllp,Ht,c1 withreal Sl'(IUC IICt'S o f sta tes. 4t>
As sho uld hl' d ear from ti ll" discussion in th is d I.1JlIt"r .. Dt"It'lI/c was
not an "actualist ' . He held a realist position to ward s the mod al
st ructure of sta te space but would have disagrcl'd wilh Git'n .' in his
int erpretati on of what co nst itutes that mod al st ructure . In part icular ,
in a Dcleuzian ontology one mu st em phasize that lhe regul arities
displayed . by the different possib le traj ectories are a consequence Vf the
sIngu lantles that shape the vector field . The well -defined nature of the
poss ible histories is not to be approached by a mere mention of laws
ex pressed as dlffcrcntlal equations, but by an understanding of how
such equations in fact individuate trajectories , Each pos sible sequence
of states, each possible history, is ge nerated by following at each point
of the trajectory the directions specified by the "ect or field , and any
regularities or propensities exhibited by the trajectories sho uld indeed
be ascribed to the topological accide nts or singularities of the field of
directions , As Deleu ze puts it, ' the singular it ies preside over the
gencs is' o f the trajectories. "? In o the r words, Giere is right in thinking
that state space offers more resources than language to individuate
possibilities (thus sides tepping Quine's crit icisms) but wrong in his
assessment of how the process f!f indi viduation takes place . To leave the
vector field out of our ontological analysis (that is, to mak e it int o an
auxiliary const ruct ion or yet another useful fiction) hides the real
source of th e regularities or propen sities in the population of possiblehistorics. r"
This point tends to be ob scured in traditional philosophi cal analyses
by the use of examples involving the sim plest typ e of cquation , a linear
equation . Despite the fact that of all the types of equations availab le to
physicists the linear typ e is the least typical, it happen s to be the I)'pe
that becam e dominant in classical physics. Th e vect or fields of these
differential equatio ns are extre mely simple, "the only possible attractor
of a linear dyn am ical syste m is a fixed point. Furthermore, this fixed
point is unique - a linear dyn am ical s}"stc m cannot have mo re than one
basin of attracti on . '49 In o ther cases (in co nse rvative s}"stl' IllS which are
qua si-isolated from their surro und ings) there may be no alt rac to rs at
all , only traj ectorics.Thux, in a linear conse rvative syste m (such as the
Ia,UllloIlU n t all.\llIl II' d .1 .111 I .11 11 1'1. iI\ (,I I I. ) 1111 \n 1111 ht'lti 1 ' 0
b.u •.1\ '\11uc tUII·t! tl..11 It 111 ,1\ . 101 lUll t l" ,H tll ,ll pur pn,~ ' s, ht ' ignCln'et,. I
. l .1 'CHlit t ' of loll,tr:unt s in the IIldi\ iduation 01 Ir.lit'dorics , On t u-
o ther h.uul, tilt, mort' typic<11 ~'Cfualio lls (nonlinear equations) have J
mort' t,l.,horah' distr-ibu tio n of singuIJri til's. the sta te space bein g
normal ly par-titioned in a ce llular fashion b)' many altractors and their
basin«, and these multiple at t ractors may be of different types. In these
mo rt' co m mo n cases, the vector field has too mu ch structure to be
igno H'(1.'>0 •
This argument, however .. establishes only that there arc In state
.span : othc-r constraints for the individuation of possib le histo ries, bl~t
not that they should be given a sl'parate modal status , W e could , It
wo uld see m , take singulari t ies to belong to the realm of the possibl e
and save ourse lves the trouble of introducing novel forms of phy sical
mod ality, such as virtuality. One way of doing thi s would be to take a
basin of attract ion to be merely a subse t of points of state space. Given
that sta te space is a space of possible states , any subset of it will also
he just a co llec t ion of possib ilities, 51 Yet, as I mention ed before,
despite the fact that the nature of singularities is well defined only in
the phase portrait of a system, th eir existence and distr;bution is a~read)'
give n in the vector field .swh erc they define overall flow tendencies for
the vectors, It may see m plausible to think of point attractors, for
exa mple , as just one marc point of state space , but this sing ular point
is not an available pos sibility for the system since it is never occupied
by a traj ectory, only approached by it asymptotically. Trajectories wi ll
tend to approach it ever closer but never reach it, and even when on e
speaks of the end state of a trajectory, in reality the curve is fluctuating
aro und its at t racto r , not occupying it. Strict ly speaking, as I said above,
attrac to rs arc never actualized,Thus, it see ms, a more co mple te analysis of sta te space does see m
to de mand a form of physical modality that goes beyond mere
poss ibility , Rut could not that o the r tradi tion al modal catcgor)',
necessity, do the job? After all, in classical physics' models a gennal
law rel ates all the successive points of a traj ect ory in a necessar y or
det erministi c way, and wh ich specific trajectory is gene rate d is ncccs
sarily det ermined given a particular initial sta te .S2 This is, i.ndecd, .t~u.e ,
but the relative importance of gene ral laws and particular 111111.11
37
39
I I I ' ,,, llI'd \ \\ ' knm that the cells will, - Ithl' IltH .l \ III I , ,t\s SOOIl as , I . , .t to strict dctermlll ism .
I ' I ' 111111 I th"n,lor ' su ) I~C •appear: t liS P WIlIlIll\ , ' f tl ' cells [clock - or ant i-
I I I rot lion 0 11.: •In co ntrast, t il" ir e: /l Oll II' II blc a ni chance, in th e
10ckw iseJ is ullpredict.lhlc and uncontro a e. y , '1 d at theI ' h t ay have pre\ al e
form of till' particul: I' pertur )at~ol ln ]t ~ 1m
hether a given ce ll is(' I ) . r imcnt W I necrc e w
mom ent 0 t 11' (X I ' _ t remarkable cooperationI I' l l d W e thus arnve a a
right - or c t lane ' . . . .d ore formally, severalI I de term nllsm . , State m
b .twccn c lance anc I Chance alon e will'I I for th same parameter va ue.
solutions arc pOSSI ) e . 51
decide which of these solutions is realized .
f nt for a different inte rpre ta tion of th e mod~1This line 0 argume . _ f t Ocleuze ' s own , alth ou gh It
f t te space IS III act, nostructure 0 sa ' 'I I ' Oeleuze own argum nt s
. I fl ' ontologlca ana YSls. .follo ws direct y rom llS . f h _ ible and the necessary are ol
h d ategones 0 t e POSSI eagainst th e ort 0 ox c 54 d linked dir ectlv with the
hil I' I nature an are j ,
a more gene ral p I oso~ nca d d ' b . di cussed in the remaind 'r 01f ' I saId nee e to e IS
third set 0 Issues . , uid e Oc!euze ' s speculation aboutthi s chapter: the co nstramts ~hat dg h constraint to avoid at all
I I lady mentlOn e one sue 'virt uality. lave a re I " I' . . ete rn al esse nces . Meeting
I" ' . t al mu tIp icttrc s ascosts conceptua Izmg VII' U.. h f h t modal logiC has to say
, . . ectmg mu c 0 w a ,this const ramt reqUIres reJ , th t the postulation 01
" ., d it The reason IS aabout POSSibIlity an necessl y. I ld as Quine and oth I'
ld . " longside th e actua wor ,Possible wor s eXlstmg a . I' plies a commitment to
f ' ked almost a ways imcritics have 0 ten Icmar ' . I' 55 A d it should b emphasized,
h f m of essentla Ism. n ,one or anot cr or d I hil ophe rs but also to th os
. " . I' t only to mo a p I osthi s cn tlCISm app res no . " h . t nee of alt ernate parall Iphysicists who ser iously believe m t e exts e
univ er ses. II I . es both philosophers and. ki b t th ese para e urnvers I
When thin 'mg ah"" . e of f ully fo rmed individuals populatin g th
ph ysicists assume t e eXlstenc.. di t I raises a number of qucs-ible 'orlds ThIS irnm e ia e y I
different POSSI e \\. . I' htl alte red in othe r work s?h 'ndividual eXist, s Ig y, I
tio ns: Can t e same I . ny worlds afte r scvc rah . tai thi s identIty across rna ,
Can he or s e main am I d? C Id we identify him or her aft erslight alteratio ns have accumu ate. ou
I <Illdll 10III 11..111 'I "'10 I \, 10101 111 1111.11111' t )11 0111 "'Ild , tI" 11111
III .111\ 1' .11 tHIII .1I 111111., 1 1.111' I 111 .ld\ dlllllli l I" d II". 111.111\ IIl1tl .II
I OllditiollS (all tho s« tli.u .111' IIHlud,·d \\ltlllll.1 I),\ IIHII I.II h.1 III) will l«:
eCJ uivall'lIt .I S far as the l'lld st.lle of ti ll" tr ,ljl'l tol ) IS ('ollll'rIIed. 'l he
states a traj 'c tory adopts on its way to till' I'lld stall', what l'ngilll' C'rs
call its transient sta tes and which co nst itute the bulk of the trajc tory,
may be of interest some times, but lcarl y will not be as important as
the stabl e end state , since the syste m will spend most of its tim
fluctuating around that state . On the other hand , the role of the
gen eral law will also be diminished because the behaviour of the
traj ectory at its end state , a steady-state or a cyclic beha viour, for
example , will be determined not by its pr evious states (defined by the
general law), but by the typ e of the attractor itself.
Thi s argument, again, establishes the need to consider additional
factors in the individuation of possible histories but not the need for
additional modalities , After all, is not the end state of a traj ectory
necessary? In this case too, the complexity of the distribution of
singularit ies makes a great differen ce in our interpretation of the modal
structure of state space . A state spac e with a single attractor , and a
sing le basin encompassing the entire spac e, has a unique end state for
the evolution of th e syst em. Concentrating on this atypical case ,
therefore, can mislead us into thinking that det erminism implies a
single necessary outcome, On the other hand, a space with multiple
attractors breaks the link between necessity and determinism, giving a system
a 'choice ' between different de stinies, and making the particular end
state a syste m occupies a combination of determinism and chance . For
instance, which attractor a system happen s to be in at anyone tim e is
det ermined, in part, by its contingent history: a traj ectory may be
dislodged from an attractor by an accident, a strong-e nough exte rn al
shock pushing it out of one basin and into the sphe re of influen ce of
another attractor. Furthermore, which specific distribution of attractors
a yste m has available at anyone point in its history, may be changed
by a bifurcation . When a bifurcation lead s to two alt ernative distribu
tions, only one of which can be realized, a det erministic syste m faces
further 'c ho ices' . Which alt ernative obtains, as nonlinear scientists lIya
Prigogine and Gregoire Nicolis have been arguing for decades, will be
decided by chance fluctuation s in the enviro nme nt . Speaking of the
\ ' Ill I I I III ,
\\ riu -:
III "'ll ,, \1 1111 "II ,I I'h' II "' It11111 , till IIII h ili
..1111'1'1°,1,."", 11,,1,1, I I I"'t 1.1, 11111 . tllll' 111" ,101 p .Utltll.II,
•tn ' lIIt1 0dlll n l to c!"l ll it til' Idllllll ) 01 till I 1I1111\lIlu.ll .u rd to
g U.U-,lII l t '(· it s pn'M'I"' ''lliOI1 .HTO" \\l,dtl" J IWl t ' .11" ! I,I", iL III )' t wo
dilli.'n·nl tec'hnical wa),s or.u ili,·\' jng th is d ltO( I. ( ) II W it' 1I.lIu l, OIH' C•in
cla im that t ran sworld identity is insured h} tilt.' pmis..oss ion or d particular
essence, that is, the propert)' of bein g thi s part icula r indi vidual. O n the
o the r hand . on e can deny that there arc, in fact, such transw orld
individuals. and speak sim ply of counterparts, that is, o ther possib le
individuals which closely resemble their real co unte rpart, but arc not
identical to it (in particular, they do not share the esse nce of being
precisely th is ind ivid ual) . T hese counterparts , ho wever I would share a
general esse nce . (Such as bei ng ' rat ional animals' , in the case of humanheings. '6)
The alternative o ffered by Del euze is to amid taking as gi ven full)'
formed individuals, o r what am ounts to the same thing, to always
account for the eenesis of individuals via a specific individu ation process ,
such as the developmental process which turns an embryo into an
organism. Thi s emphasis on the objective producti on of thc spatio
temporal structure and boundaries o f individuals stands in stark
contrast with the comple te lack of process medi at ing between the
poss ible and th e real in orthodox modal th inking. T he category of the
possible assumes a set of predefined forms which retain their identity
despite their non-exi sten ce, and which alread y resemble the forms
they will adopt once they become realized. In other words, un like the
indi vidu ati on process linking virtua l multiplicit ies and actua l structures,
realizing a possib ility docs no t add anything to the pre -exi sting formbut mere reality. As Deleuze writes:
What difference can there be between the existent and the no n
existent if the non-existent is alr cady possible, already included in
the co nce pt and having all the characterist ics that the co nce pt
co nfers up on it as a possibility? . . . Th e possible and the virt ual arc
. .. distingui shed by the fact that on e refers to the form of identity
in the concept , wh ereas the other designates a pure multiplicity .. .
which radically excludes the iden tical as a prior condit ion . . . To
the exte nt that the possible is open to ' realizat ion ' it is understood
as an image of the real, whil e the real is supposed to resemble the
40
po 'ilhlt I h.u I \\ II 11 I dllh , tllt III U ll til I 1.11111 \\ 11.lt , I 1t "IU I
.Itld ttl tlu (1IIll' pt "til II 11 II dill 1 douhh' lilt' "ith lilt ' .. .,\ l'1 l1 .lli, .lt io ll III (' .lk \\ It II II·M'mhl.ull ,. ,1 '\ .1 p r on 'ss no lr-ss th .111 it
doc-s with itk nt il) .u .l prim iph- . In this sense, act ua lizat ion o r
dif1~'n'ndatioll is .1lwa)'s a gt'l1uinc creation . Actual terms never
rcsernhle the singulari tit's they incarn ate . . . For a potential o r
virtual object to he actua lized is to crea te divergent lines whi ch
co rres po nd to - without resem bling - a virtual m ultip licity.n
Besides the avo idan ce of esse nt ialist thinking, Deleuze ' s speculation
abo ut virtuality is guid ed hy the closely related const raint of avoiding
lyp oloBical thinking, that sty le of thought in which ind ividuation is
achieved throu gh the creati on oj classifications and offormal criteria for
membership in those classifJcations. Although some classificat ion s are
csse ntialist, that is, use transcendent essences as the crite rio n for
membership in a class, this is not always thc case. For exam ple , unlike
Platonic esse nces whi ch are transcendent entit ies, Aristotle ' s 'natural
states ' those sta tes towards which an individual tends, and which
would' be achieved if there ",'ere not interfering forces , are not
t ranscendent bu t immanent to those individuals. But while Aristo telian
philosophy is indeed no n-essent ialist it is st ill completely typological ,
that is, co nce rned with defining the cr ite r ia which group individuals
into species , and species into gene ra. 58
For the purpose of discussing the constraints guid ing Deleu zc ' s
co nstructive project, on e historical exam ple of typological thinking is
particularl y useful. This is the classificatory pra cti ces which were
co m mo n in Euro pe in the seventeenth and eighteenth ce nturies , such
as those that led to the botanical taxonomies of Linnaeus. Simplifying
some what , we may say that these classificat ions took as a point of
departure perceived resemblances am on g fully formed individuals, fol
lowed by precise co mparisons aimed at an exhaustive listing 01 what
differed and what stayed the same amo ng those individual s. This
amounte d to a translation of their visible features int o a lingui st ic
re presentation , a tab ulation of differences and identities which allowed
the assignment of individuals to an exact place in an orde red table .
Judgmen ts of analoBl between the classes included in the table we re
used to gene rate higher-order classes , and rel at ions of opposition were
4 1
t 1.111 11 hnll lt 1\\ 1'. II 11111. 11.1". III \11 ,1.1 dilltollllllli 0 1 111111" , '1.1bor
.llt' hit'r.ln !Iii ' 01 t )" ' :0; , I ill' n " ultll1g IHCl lllgh .1I 1.1 . ClIIOlllil 's wvrc
SlIj> j>IISt'll lo n 'con,lrucl ,1 nat ur.rl ordor- \\hi ,'h \\ .l"' j lUd l1t1d (On t m lw lH,
rcganll ,'ss o f th e I:let that historical ac'c-k lcn ts m.rv h.Wt' b ro ken that
continuity . In othe r words, given the fixity of tilt.' ':io)ogkal types, time
itself did not play a constructive role in the gene ration of types, as itwould later on in Darwin 's theory of the evolution of species, ~q
Dclcu ze takes the four elements which inform these classificat ory
practices, resemblance, identity, ano lo8)' and opposition (o r co ntradict io;)
as the four categories to be avoid ed in thinking about the virtual.
Dclcuze, of course, would not den y that there arc objects in the world
whi ch resemble one another, or that there arc entit ies wh ich manage
to maintain their identity through tim e , It is ju st that resemblances and
identities must be treated as mere results of deeper physical processes,
and not as fundamental cat egories on which to base an ontology ,60
Similarly, Dcleuze would not deny the validity of making judgments of
analogy o r of establishing relations of oppos it ion, but he demands that
we give an account of that whi ch allows making such judgments or
establishing those relations. And thi s account is not to be a story about
us , about categories inherent in our minds or conventions inherent in
our socie ties , but a story about the world, that is, about the objectiv e
individuation processes which yield analogous groupings and opposedproperties. Let me illustrate this important po int.
I said before that a plant or animal species may he viewed as defined
not hy an essence hut by the process which produced it. I characterize
the process of speciation in more detail in the next chapter where I also
discuss in what sen se a species may be said to be on individual, differing
from organisms only in spatia-temporal scale, The individuation of
species consists basically of 1\\'0 separate operations: a sort ing operation
performed by natural select ion, and a conso lida tion operation per
formed by reproductive isolation, that is, by the clos ing of the gene
pool of a spe cies to exte rnal ge ne tic influences, If selection pressures
happen to be uniform in space and constant in tim e, we will tend to
find more resemblance among the members of a population than if
those select ion forces are weak or changing . Similarl y, the degree to
whi ch a species possesses a clear-cut identity will depend on the degree
to wh ich a part icular reproductiv e community is effecti vely isolated.
M ,1I1\ 1'1.1111 I II I U , 101 I .l1 l1l'll. II LUll till II I II' ,It 11\ to In,hlld"l1I1111llgl.out 11\('11 11\ (tlln (,Ill' tl. .lIl " tclIdh 1ll.11t1 1.11 \\ltll olllt'l
p l,lll t ' 1H'('j, ·:"i) .111.1 IWII( t ' pU'I 'U" " ,I It- I h-.u ,Ill 1('lwth Ukll l lt )' th.m
1',.rI,'( II)' I"t'protlm th ,') i,ol.lh' d .iniru.•ls. III ~Il Cl r.t , 1I,It" d.~gn: , ' 01
Tt'\I'mhl.mn' atu l id" ntit)' lh'pt'ml, on ('(lntingt'n t Ill,t orll':,1 dd .lll , Clltht' pron'ss o f ind ivid ua t ion, .u HI is tlt,'n -fon ' not 10 be taken lor
gr,mh' d . For the same n ·.lSOI\ , rt.'st.·mhlann· and idt.·TlIity sho uld ~10 1 ,1)(used .1S fundamental co nce pts in an ontology, hut only as (h ' r1\'.ltl\('
not ions,In addition to sho wing , case b)' case, how similar it}' and idt.·ntil)' .1ft ' ,
("(Hl t ingcnt on the details of an individuation process, th e rejection 01
static categories and esse nces mu st he e xte nde d to all natu ral kmds , not
just bio logical ones . W e mu st sho w, also case b)' case , how terms
wh ich purport to refer to natural categories in fact refer to hisIOr;ct11lj
cOnSlilUled individuals. In a way terms like 'human' arc th e easiest to
de.essentialize given that Darwin long ago gave us the mean s to think
abo ut species as historical entit ies , But what of terms like 'gold' wl\t.'n~
the esse ntialist account seems more plausible? After all, an samples 01
go ld must have ce rtain atomic properties (such as having ~ specific
ato mic number) whi ch , it can be plau sibly argued , const u utc lIwesse nce of gold. Part of the answer is that all atoms, not onl y go ld
atoms, need to be individuated in processes occurring within stars
(nucleosynthesis) , and that we can use these processes to specify what
go ld is instead of, say, giving its atomic number."! But amon'
compelling reason to re ject essentialism here wo uld be to deny that a
given sample 'of gold large enough to be held in one's hand can be
conside re d a mere sum of its atoms, hence reducible to Its atorrnc
properties. . .,In particular, much as between individual ce lls and the individual
organisms which the)' compose there are several intermediate st ruc
tures bridging the two scales (t issues , organs, organ ~ystems) ~o
between individual atoms of go ld and an individual bulk pIece of solid
material there are intermedi ately scaled structures that bridge the
micro and macro scales : individual atoms form crystals; individual
crystals form small grains; individual small grains form larger g rains ,
and so on. Both crystals and gra ins of different sizes are individuated
following specific cau sal processes, and the properties of an individual
H
hulk ',11111'1 ( 'l1h" I' lrOIll 1111 ' ("II .r] IIJIl r t llllll Itd\\( I 11 till" (
iuu-rnu-di.m - ,,(rllt(lI n · . 'Llu-r r- .ln~ '1If111 proper lit ' II! '0ld, 'llth.H
h,ning a spt.Tilic melting point , I()r vx.unpk-, \\ hid . hy dC 'hnition do
not bel ong to individual go ld ato ms sino- single atoms do not me lt .
Alth ough individual gold crysta ls rna) be said to melt . in rca litv it
takes a population of crysta ls with a minimum critica l size (a so .cal led
'rnicroclu ster ") for the melting point of the bulk sam ple to emerge .
Moreover, the prop erties of a bulk samp le do not emerge all at once
at a given cr it ical scale but appear on e at a time at different scales.v!
In conclusion, avoiding essentialist and typologica l thinking in all
realms of rea lity arc basic requirements in the construction of a
Dc lcuaian ontology. But besides these negative constraints there must be
some positive resources which we can use in thi s construction . I will
develop these resources in the following chapter from a more detailed
analysis of the intensive processes of individuation which actualize
virtual multiplicitics. The virtual, in a sense . leaves behind traces of
itself in the int ensive processes it animates, and the phil osopher' s task
may be seen as that of a detective wh o foll ows these tracks or co nnects
these clu es and in the proccss, creates a reservoir of co nce ptual
resources to be used in com plet ing the project whi ch this chapte r has
only started . This project need s to include , besides defining multiplici
ties as [ did above, a description of how a population of multiplicities
can form a virtual continuum. that is, it needs to include a theory of
virtual space. Similarly, if the term 'virtual multiplicity' is not to be
just a new label for old timeless essences , th is project m ust include a
theory of virtual time, and specify the relations which this non-actual
temporality has with actual history . Finally, the relationship between
virtuality and the laws if'phy sics need s to be discu ssed, ideally in such a
way that ge ne ral laws are replaced by univ ersal multiplicities whil e
preserving the objec t ive conte nt of physical knowledge . Getting rid of
laws, as well as of esse nces and reificd categories, can then justif)' the
introduction of the virtual as a novel dimen sion of realit y. In othe r
words. while introducing virtuality may see m like an intlati cnarv
ontological move, apparently burd~ning a reali st phil osophy with a
co m plete new set of entit ies , wh en see n as a replacement for laws and
essences it actually becomes deflationary, leading to an ultimatelyleaner ontology . '
C I I,\ I''[' [,I{ 2
The Actualization if the Virtual in Space
T he picture of a relatively undifferentiated and co nt inuo us topological
space undergoing di scontinuous transitions and progressively acquiring
detail until it conde nse s into the measurable and di visible metric space
which we inhabit , is a powerfu l metaphor for th e cosmic genesis of
spatial structure. I attempted before to remove some of its metaphorical
co nte nt by comparing the rel ation between topological and metric
spaces to that between Int ensive and extensive properties: the latt er ar c
divisible in a simple way, like lengths or volumes are, whil e th e former,
exem plified by properties like temperature or pressure , arc continuo us
and relatively indivisible . T he cascade of sym metry-breaking events
which progressively differentiates a topological space was, in tum ,
com pared to pha se transition s occurr ing at critical values of intensit y. I
gave an e xample from co nte m po rary physics where such a sce nario is
becoming literally true but the fact is th at. as a description of the
genes is of space, thi s picture remains just that, a picture.
It is time now to givc a less metaphorical account of how the
intensive can engender the extensive , or more exactly, how processes
of individuation characterized by intensive properties can yield as their
final product individuals with specific spatial structures . In the first part
of this chapte r I will discuss two different aspects of the int en sive ,
eac h illustrated 'with a specific individuation process. First I willdescribe th e process whi ch individuates biological species and from thi s
description I will extract tw o of the main co nce pts which characterize
inten sive thinking: populations and rates if'chanae. I will also show how
these co nce pts can be used to repl ace the two main feat ures of
esse ntialist thinking: fixed cypes and ideal norms. Then I will move on to
our second task, a discu ssion of how the e xte ns ive or met ric features
of individuals emerge from processes whi ch are, at least in an
approximate sense, nonmetric or topological, using as illustrat ion the
process which yields as its final product indi vidual organi sms, A more
45
tld .u lt,c1 d" l.lJli 11111 0 1 t' II1 I"\1I11 11I I \\111111\01\1 II.. til I tlt'p.UIlIlI
from o u r gt'o!Ut' lric Illd.lpho r gl\'"!l th.t! II 1' 1011111 I .111' ddllH"c1 ntlC
on9' h)' e:Ucm lC leJ but also b) qualma. In olhl 'l \\ ol d, ..111 olg,lII i'm is
defined both hy its spatial ar('hihTture, as \\"( ·11 , IS b\' tilt' di fferent
materials {hone, muscle) which gin ' thai archih '('u;n' its spl'cific
mechanical qu alities. The intensive will then Iw revea led to ht.~ behind
both the e xte nsive and the qualitative.
Let 's begin with the process of individuation of species. First o f all,
in what sense can we speak of 'individuation' here ? For centuries
biological species were one of the main exam ples of a natural kind.
Whether on e thought of natural kind s as defined by a transecndent
essence , as Plato did, or by an immanent (natural state' as did
Ari stotle, an imal and plant species provided the exe mplar of what an
abstract aeneral entity was suppos ed to be . I Charles Darwin of course
broke with this tradition by showing that species , far from bein~
eternal archetypes, are born at a particular historical time and die
through extinction in an equally historical way, but th e idea that species
ar~ individuals, not kinds, has only recentl y (and still controversially)
gain ed ground . Much of the cre dit for the new view on speci es go es
to the biologist Michael Ghi selin who has been arguing for decad es
that a species , formed th rough the double process of natural selection
and reproductive isolation, does not represent a hiBher om oloqical
calegory than the individual organisms that compose it. 2 Unlike the
relation between a natural kind and its members, which is one of
~xe,~pli fi cation or instantiation, the relation of individual species to
individual organisms is one of whole and parts, much as the relation
between an organism and the individual cells that compose it. More
over , unlike the relation between a particular instance and a gcneral
type, the relation of parts to whol e is causal: the wh ole emerges from
the ,causal interaction s between the component parts. J A new species,
for Instance , may be said to be born when a portion of an old species
becomes unable to mate with the rest . This reproductive isolation is a
causal relation between the memb ers of two sub-populations, and
morc~"er, it is a relation which must be maintained through time .
Anything that breaches the ge ne tic, mechanical o r geographical barriers
mamtaining this isolation will co mpromise the enduring gc netic identity
of a species.
l 'l l fl r h ,t1ul t " ,HI 1I1.III\chlll lllltl "11\\1111 1" l l t ,lIldll lfJllI III ,
tilt' mo ; 0 11\ ion Wh' 1 1I' 1 1l ~ dl lli I t"" t ' III I.Ilc · ' IMlI" II)', .• "'WI. il. '"
h." ., muc h I.,rgl·r r' " '11" 0 11 lh,1Il all orgol l1i'l1I sinn' it is t)'pically
t'011lprist.d of SI'\l' I",1 rt'protitH rive l'OmmulIitil-s inh.lhilillg gt.·ographi
("III )" st'paralt.·d t't.'osplt'llls. Temporally, a species also operates at
much largt.·r scales, its a\"cragl' life span heing much greater than the
lih-cvc lcs of organisms. But the fact that species arc construc te d
t hro~lgh a historical process suggests that they are, in fact , just ano ther
individual entity, one which operates at larger spatia- temporal scales
than organisms , but an individual ent ity nevertheless. One philosoph
ical consequencc of this new conce ption of species must be emphasized:
while an onto logy based on relation s between general types and
particular instances is hierarchical, each level representing a ~li fTercnt
ontological category (o rganism , species, genera), an approach III terms
of interacting parts and cmergcnt wholes leads to a fiat ontoloBY, one
made exc lusively of unique, singular individuals, diffcring in spatio
temporal scale but not in ontological status! On the othe r hand, the
new appro ach demands that we always specify a process through whi ch
a whole emergcs, a process which in a Del euzian ontology is character
ized as intensive . The process of speciation may he said to he intensive ,
first of all, because its description involves the hasic ideas of population
and heteroBeneity, tw o fundamental co ncepts which characterize a mode
of biological explanation known as population thinking. What makes thi s
form of thinking different fro m esse ntialist and typological thought is
expressed in a famous quote by one of the creators of the mod ern
synthesis of evolution and genetics , Ernst Mayr:
[For the typologist there] are a limited number o f fixed , un change
able ' ide as' underlying the observed variability lin nature], with the
eidos (idea) bein g the only thing that is fixed and real , whil e the
observed variability has no more reality than the shadows of an
object on a cave wall . . . [In contrast}, the populationist stresses
the uniquen ess of everyth ing in the organic world .. . All organisms
and organic phenomena arc composed of unique features and can be
described co llect ively only in sta t ist ical terms. Individuals, o r any
kind of organic entities , form populations of which we can deter
mine the arithmetic mean and the statistics of variation. Averages
4 7
.111 ' 1111 H,I, t III I I I• ua .1111 IltOIl. nllh fhl 111(11\1.111 .t1 III \\huh till
PClPU l.l tl Cl II ~ .m- t OlllpO""t ·«! h. l\ I' r.·,,111 \ I I I I 'I I . . U Ulllll.ll( tll'h thlCHI'i 0 1
t )(' popu anun th inker .md Iht" t\ I .Fo I . ' I . " po ogl\l .m- pnT"t·l,. 11..· 0ppo... iu-.'II r , I ll I),",o ogl st the I)'PC (ddos) is n-al J. ut! tht" varia tion .111I usron W lilt· for th e I "
l. ', { popu auonrst, the typt.' (tlw an ' rage) is an
a sst racuon and only" th . . .c vanation IS real . No tw o W J ) 'S of lookingat nature co uld be more different. "
When one views species as natural kinds wh ose memb ers sharecom mon s t f Id ' I ' ath o l e o I entica properties, th e inevitable variat ion between
Po~n7e7 "" of a
fcdlass canno t be hut an accident of history. Fro m the
o VI C'\' 0 ctc rm ining thd f fi c co mmon set of pr op erties whichFe nes a I ~ed ar,chctype, thi s vari ation is indeed quite unim~ortantor popu ation thmkers on the o th hI ' , 'that is C' f b " . er anc , vanauon , genetic variation
I Jar rom cmg ummp rta ' h f Iadaptive d 'ffi 1 0 nt IS t e ue of evo lution: witho ut. bl I cr.cnc~s >etween organisms natural select ion ,...ould bemcapa e of yieldin 'II
g an)' Improve ments in the population let Ia ow novel for r t ' a oneh , ' : s 0 emerge. Put differen tly, for populat ion th inkerseteroa~nell)' IS t estate we should expect to exist spontaneousl under
:~yst :1T~umstahncesb' while bomopeneny is a high I)' unli kely stat: whi che ro ug t a out only under ve 'fi I .abno rm II . 'C' ' ry speCl c se ect ron pressuresa } unuorrn 111 space an l ti ~ 6 M 'thinks of th " , rr ttme , oreover, whil e the typologist
e genesIs of form 111 terms of the e xpression of sin Ie tfor the populationist the forms of ' I a ypes ,1 ' orgamsms a ways evolve withico Iecttviues (re produc tive co m m unities cor I ) I I m
I. ,.< exam p e as se 1" I
ac vantageou s traits with differc t ., . ec i vc ypopulation . n ongms propagate through the
essPOtpullat ion thh inking el iminates one of the two und esirable aspects ofen ra Ism, t c existe nce f ' ,id ' f 0 pre -exlstmg arche types defin ing the
I entity 0 specie Th h, es, e ot er aspect the role which h hI , id I ' sue arc ety'pesp a) as I ea norms which their instantiat ions a '
:~ss perfect degree, is elim ina ted by ano the r k:;:::~:;::;~:r:7~:;~e norm if reaction . To illust rate this co nce pt let 's " .
diff ducti Imagll1e tv..-oerent repro ucuve com m unit ies b I · th. h bi , li e ongmg to e same spec ies but
III a ltmg ( iflcrcn t ecosystems T h [ react ithat there j h . c norm 0 react io n refers to the factere IS eno ug flexibility' i th . 'b dil ' h " n e co nnect ion between ge nes and
o I y tra its t at differen ces in th .e environment can yield diffe rent
t 1I.1I .1t 11'11 Iii till 1I1l' 1\.. 0 '011111111111111 I I \, II ,!lIlU ,11 lilt" .111 till till
.mu- 'IH't It' . l-o t t' .lIlI pll. ell l" ndlll' 0 11 tilt I .lll· 01 .1\.111.lhihl) 0 1 .1
IMrtitul.lr n '''Ollllt' (..unh ght . 11I1' r-: .lIu ph' , or .1 lMrtinlbr nutr-ie-nt] tilt.'
rates of grow th of the org,lni..ms in tht' two n )lnl1lllnitit's may in:
<Ii1lt'n'nt , with o ne co nsis ting of sll1.l lll'r o rganisms than the o the r. In
this case, rln-n- would be no point in saying that one co mm unit)'
f('prcsl'nts the no rma l, ideal, fixed phen otype, o r that it approxima tes
it to a gn'ate r degree of perfect ion . Since the phen otypes are flexible
within ce rtain lim its, all realiza tions of the genoty pe arc normal within
those limits." The concept of norm of reaction repl aces the idea of
decrees of peifection wi th that of relations between rates of chanae (in our
ex am ple , rates o f nutrient availability coupled to rat es of growth) .
Dcleuze credits Darwinism with thi s double blow to esse ntialism .
challenging stat ic classification s and the mod e of th inking th ey im ply
with a dynamic form of thought which is at once populational ond
d1!e renl ial . As he 'wri tes:
First . .. the form s do not preex ist the populati on, they are more
like sta tist ical re sults . The more a pop ulation assumes divergen t
fo rm s, the more its multiplicity divides int o mult ipliciti es of a
di fferent nature . . . the more efficie ntly it distri butes it sel f in the
mi lieu , o r divides up the milieu , ' . Second. sim ult aneously and
under th e same condit ions . . . degrees are no lon ger measured in
terms of increasing perfecti on . . . but in terms o f differential
relation s and coefficie nt s such as selection pressure, catalyt ic act ion,
speed of propagation, rate of growth, evolution , mutation . ..
Darwinism ' s two fundam ental co ntr ibutions move in th e direction
of a scie nce of multiplicities: the substitution of populat ions J or types,
and the substi tu tion of rates OT differential relat ionsJOT dCBrees.8
I said before that between organisms and the cells that are thei r
working parts there arc int ermediatel y scaled indiv idual st ru ct ures,
such as ti ssues or organs, Simi larly, be tween these organisms and the
species th ey compose there are halfway individuals called demes:
concre te reproductive com munit ies inhabiti ng a given eco system ." The
intensive prop erties of these demes, such as how den sely thei r
com ponent organisms are packed in their habitat , arc characterized hy
4 9
1\ , II" IIld. 11 111, "I \.,. :' "' I ,I I I., " . I
11 ' ti lt. II d,'lIl\l'd III
I" I' I' II ,I. • I II ' •111111" , 1111 " ,,1," ,,1"1,1 111 111 " II' II " I I
11\1" 1111111'1\1 ' 111.\1111". ,tll"II·, tll.,1 11 1....1 I I
.11, tll,tl .11 1' tI\I"l l 1111,11 pilltllli I. '" Ill ' ,'Il l I " I
11.11111,11 n'pl.ll '\'IIIl 'lIt ~1\"11 Ih,11 • lit 1'\ . 1., •
u -rms or .'ss.'nn's,Thl'S!' would hc , in .1 nutslll'II, till Ih1"1'1 , nntolo ,jc.11 dillll'lI sion s
which onstit ute th« I i-lcuzian world: thl' vi rt ual , the intcnsi vv and
Ihl' actual. r to phrase this in te rm ' of thl' m 'taphor that 01' .ncd thi s
-haptcr (and n >glecting for a mo ment the t ' mporal dim -n 'ion) the
indivi duals populat ing the actual wo rld would be like the discontinuous
spatial or metric structures which condense out of a nonrn t ric , vir tu. I
continuum. These metric individuals would ex ist at different spatial
scales, since populat ions at one scale ma y form larger emergent
ind ividuals at another scale, but altogether (fro m the smalle r indi vidu al
particles to the largest cosmic indi viduals) they would co nsti tute tho
familiar, measurable and divisible space of th e actual world. At this
point, however , we mu st make our first departure from th e geometr ic
met apho r : actual indi viduals differ fr,om each other not only in th eir
extensity (spatial st ructure and scale) but also in th eir qualities. A
species, for ex ample, possesses both an exte nsive aspect defining its
distribution in space (its division into seve ral reproductive communities
inhabiting distinct ecosyste ms) as we ll as a qualitative aspect defined by
populati on-Ierel qualit ies, distinct from th ose of individual organisms,
such as playing a particular role in a food chain or having a particular
reproductive strategy ,13 This means that inten sive individuation pro
cesses must be described in such a way th at the or igin of both
ex te nsit ies and qualities is accowlte d for .To illustrate thi s important point I would like to move to a different
level of scale, do wn from species to organisms, and discuss two
examples of inten sive pr ocesses in embryogenesis, one related to th e
pr oduction of ex tensities, th e othe r to th e pr oducti on of qualiti es. .or
more speci fically, I would like to discuss two different embryologIcal
processes, one behind the spatial structu ratio n of organisms throu gh
cellular migration , folding and invagination , and the othe r behind the
qualitatil'e dlj]'erentiation of neu tral ce lls into fully specialized muscle,
bon e, blood , nerve and othe r ce ll tvpes. !" Met aph orically, an egg may
be compared to a topological space which und ergoes a progressive
Ill' , \ 1..' ) I ,ll. "I • h,lIlI. '" till • I ' I III I.. th, I lit " I ""I Ih 0 1
tIl,' d"IlIl ' , w hu h I I" h., .II 1111 ' U I hi'll 110111 till ' Il n Ih I,ll. ' or
individual Ol"g,lIliSlllS I just nu-m iom-d. I h., r.II., 01 IIII\\th or .111
individual ,dcmc depend s on thl' hir th, death ,lI1d llligr,lI ion rat's
prevalen t In the co mmunity, as wel l as on the rate or • vailabi lit \' or
res~urces (some times referred to as the carrying apa ity ot its
environment.) A demc so defined is, ind cd , a dynam ical system, and
as such may exhibit endogeno usly ge nerate d sta ble state (attractors)
a~ well as abrupt transiti ons between stable sta tes (bifurcations), In
s llT~ple m~dels, for instance , the system consist ing of a deme coupled
t~ Its enviro nme nt ex hibits an unstabl e steady state (o ne with popula
non at zero numbers, meaning extinc tion) as well as a sta ble steady
state where population numbers match the carry ing capacity .!" More
complex attractors, such as stable cycles , app ear the moment we add
nonli~earities to the model. This may be done , for exa mple, by making
the birth-rate term more realistic to reflect the fact that there are
always nonlinear delays between the moment of birth and the moment
of sexual maturity, When the growth dynamics of a dem e are gove rned
by a periodi c attractor, the numbers characte rizing its population will
tend not to a fixed stable value but will oscillate between values , II
This simple example is meant only as an illustration of the sense in
which a dynamical process occurring in populations and defined by
coupled rates of change may be said to be inten sive , How is such an
inte~sive process rela ted to th e vir tual multipliciti es I discussed in the
pr evIOus chapte r? As I said, multiplicities consist of a structure defined
by differential rel ation s and by the singularities whi ch characte rize its
unfolding levels , These two elem ents of the virtual find their co unter
part in th i~ten,s~ve , Th e coupled rat es of birth, death , migration and
resource availability corres pond with out resemblan ce to the differential
rcl~tions that charac ter ize a multiplicit y. The co llec tive ly stable sta tes
aV~'lable to po~ula~ions (steady-s tate or periodic, in my exam ple)
c.OI resp~nd , agam WIthout any sim ilar ity, to a distribution of singulari
~Ies , ~h,s correspo~dence, in turn , is explained by th e fact that a given
mte~sl,ve ~rocess of individuation embodies a multiplicity , and the lack
of slmtl~rtty b tween the virtual and the int ensive is explained in terms
of the dl~ergent characte r of this embo dime nt, that is, by the fact that
seve ra l different processes may embody the same multiplicity, 12 Finally,
5"0
IMII 'II .1lul'plollll ,III\1 111111'1111... 11011 III lillI/ill! ' ,h. 11111,,, p,U.'n 'pn '''c-Il11 'c1 h) .1 111 11) IUIIII.·d o lg.lI ll'1 l1 l . 11 \11 In \ \ h,11 l ' II'~ I ' 1,111 "ggs
and org.lllisms hI' ,'i.l id to Ionu ,"p.ln·,..? A'i I s" id ill t ilt' pn' \'iOliSI h.ipn-r- ,
dll' d istinction betwee- n me-t r- ic ,lII d uotu m-t rh- sp.ln-s hoi! , d o w n to till'~V.l)' in whi ch neigh bourh oods (o r the linkagl's bet ween thl' points thai
form a space) are defincd~ eithe r throu gh exact Il·ngth s o r through
non -exact topological relations o f proximity. In thi s sense , the fc rtil izt.d
egg . defined mostly hy chemical gradients and polar it ies, as well as the
early emhryo defined hy neighbo urhoods with fuzzy bord ers and ill
defined qualities, may ind eed be viewed as a topol~gical space which
acquires a rigid I)' metric anatomical SlTucture as tissues, organs and organ
s),stems become progressively better defined and relatively fixed inform .
, Let' s begin with the creation of distinct spatial st ructures, starti ng
with the aggrega tion of individual cells int o difTerent neighbourh oods
or collect ives via a variet ), of adhesion processes, These neighbourhoods
do not have a well -defined metric structure . Within an)' one neigh
bourhood, the exact location of a cell is immaterial as long as there are
sufficiently man y cells with a shared history located nearby. Simil arly.
the exact number of neighh ours is not impo rtant and , at any rate. it is
always subject to statistical fluctuations, What is important arc the
local, adhesive interacti ons between ce lls (or between cells and their
ext ra-cellular matrix during migration ) . int eract ions which are t),pically
both nonlinear (sma ll changes may lead to large consequences) and
statist ical. " As the biologist Gerald Edelman has shown . the se local
int eracti ons yie ld tw o stable sta tes for co llectives: ce lls may be tightly
linked to each other by adhesion molecul es int o sheets (called epithelia)
or be loosely associated via minimal int eract ion s int o migratory groups
(referred to as mesenchym e) . Th ese two stab le states arc related to
each other by a transformation wh ich closely resembl es a phase
transition . and which leads to two different types o f cellular motion :migrati on and folding . '.
While ce llular migrations move entire co llectives into new places,
where they can inte ract with differen t co llecti,·es. ce llular folding and
m\"agmatJOn cre ate a large \"ar iety of three -dimensional struct ures
which constitute the external and int ernal spatial boundaries of an
organism , Just where a co llective migrates and what extensive struc-
,2
IIl1t ,,01 hllltitl \\alll" !tlllJltd I d.ltlfllill 01 1111'11 II Illhlll\1
1l , 1.111t11l ~ : not o llh th l' I ,lit ' 01 \ fill•. I 11111 tit ',.ul,IIIlJll 01 Ih.ditkl'l'nt .1(111"',;&011 ' 1Il Il II ' l ll l l " (., 111.1111 1 11'1 111.111\1 1IIIIIIht ' I III ' lit II
11101,'1 uh-s, \\lakh in (urn IIl l ·el i ,l lI · tllC' pll.l. u au 111011 ht'l\\('I'n th l'
t \\ () st.lhll' sta te-s}, hut .l lso th., birth .mtl d",llll 1'.1(( " of n ,lI" \\ ithin .1
('olln't in' , I? Tlu-rc is 11 0 de-tailed gl'Tll' tic cnuu-ol of till" l'x ., d numbe-r
of n·1I d ivisions, or of the exact number uf n· 1I deat hs. but ra tlu-r .1
nonlinea r feedback rela tion between birth and death rat es and till'
processes of migration and fo lding : these pr ocesses an: affected hy till"
rate at which new cel ls are born and die and, vice versa, the rates .m
strongl)" place-dep enden t and hence affected hy migrato ry and foldi ngmo tions . III
T he intensive (populational and differential) aspl~cts of th is proCl'ss
rna)' be said to be nonmetric in the following sense. Dcl euzc ofte n
spea ks of the cnexccr yec riaorous sty le of tho ught whi ch may I,,·
neccssar )' when ever we need to think abo ut non metric ent it ics .!" A
good example would be the way Edelman approa ches his ce ll colk-c
tives, where the exact number of members or their exact position is
immate r ial. Thi s attitude to wards quant itath'e exactitude is not a sign
that biologists. unli ke physicists, are less careful nr disciplined . It
indic ates, on the contrary ~ the presen ce of a more sophisticat ed
topo logical style of thought. To quote another biologist whose work
will be discussed in the following chapte r , Arth ur Wi nfree:
T he science s of life have never been admired for quantitati ve
exactitude . . _ But it cannot be said that living things are at heart
sloppy , fuzz)', ine xact , and unscientific. How docs an oceanic salmon
find its way home to spawn on the ver)" rivulet it left in O regon
three years earlier? How is a meter-long sequence of billions of
nucl eotide base-pa irs reversibly coiled without entang lement int o a
nucleu s no more than a few thousand base -pairs in d iameter? _ , .
Such miracles bespeak of reproducible precision . But that precision
is not the kind we know how to write equations about, no t th e
kind we can measure to eight deci mal places, It is a more flexibl e
exactitude whi ch evades quant ifying. like the e xact itude of a ce ll's
plasma membrane dividing the un iverse into an inside and an outside
with not even a viru s-sized hole lost some where in all that
'1111\ I ,lUll d I IMII I Illpe do 'It .•1 t .11 Ililldl , 11I1 111/(((11I "t I/ Uelll ' II, , " I 4"
Jntlt/\ of ..lr.rp fon c , ,mel 11I111' , 'q
Thus, then- is J wcll-rh-Iincd St·nSt· in whk -h the SIJ.lli.1 1 rvl.uio us
characterizing an egg or the still devel oping pJrts of an em bryo arc,
indeed, anexact yet rigorous, As migration and fo lding hegin lo yieldfinished anatomical structures, however, these non mel ri c relations
becom e progressively replaced by a less flexib le set of metric ones .
The finished product is a spatial st ructure adapted to specific fun cti ons.
Like a building or a bridge , for example , an animal mu st he able to
act under gravit)· as a load-bearing structure. On the other hand, the
spatial architecture of an organi sm is not the only factor that deter
min es its capacity to bear load s, the qualities of the mat erials making
up that archit ecture also matter: the qualities of muscle that allow it
to bear loads in tension , for instance, or the qualities of bone that
allow it to bear them in com pression. The intensive processes that
create these materials are another example of a process of progressive
differentiation, one which starts with a population of relatively undif
ferentiated cell s and yields a st ruc ture chara cterized by qualitatively
distinct cell types.
When ce lls begin their em bryological development they arc pluripo
tent, that is, they are capable of becoming any of the dilTerent typ es of
ce lls whi ch characterize the adult individual. This number vari es from
two in bacteria, to twenty or thirty for je llyfish, to about 254 for
human beings. 21 Co ntact between different cells (or between dilTerent
cellular collect ives) leads to the important phenomenon of induction.
This term refers to a complex process in which co llectives exchange
che m ical sign als which lead to the enhance ment or suppression of
cel lular dilTerentiati on . However, as the biol ogist Stuart KaulTman has
shown, these inductive signals act as non-specific sti mula (o r perturba
tions) which switch a cell am ong a variety of int ernally available stable
states. The basic idea in KaulTman' s model is that the regulatory gen es
within a cell form a complex network in which ge nes, interacting via
their products, can turn one another on or ofT. Kauffman has found
that there are certain recurrent patterns of ge ne activit)' within these
networks, patterns which exhibit the kind of homeostatic stability
.1 tlll.ll,d \\111. .1II r .u l or III1 Itl I.d 111111 tu 1..11,\1 tllll . IIl lllt ,I ,
, '.H II .l tl l.1l III' 11I.1\ 1)( (1 til 111, I (d 'I) It 1'1 I I III .. I t I \II I ( III t 1·11 I " l"Kauflm.m' 1111111. ·1 dlh'l1Il't III I'l l .Ih I IIllt IIllh lilt IHII II Ill'I 01
dilh'rl'nl n ·1I I)rH' ill ,I gl\l'11 01 ' ,111 1 m , bUI ilion' lIl1porl.lIlt!) Irum
our poinl of \ h-w, II", numla-r of n ·1I t) p" \\ hi, II .1 particular (( ·11 , '.HI
t.!i rec'lj' «1!D~·r(flrl(J '/.· into. (;i n' n .1 n 'lI \\ uh <I Slh·dfic hbl or)', ,lilt! .1
certain inductive sig'l<1 1 which can ('It,lUg" its ratt', llw outcome of tlH' ir
int eracti on will depend Oil how man)' other att r-actors exist n/.·urh)' in
the state span~ of the cell (o r more exacl ly, in the stan - sp,ln ' of til"netw ork of ge nes within the cel l) , In other words, far from din'cd }
determining the qualities of a differentiated ce ll, inducti ve signals ad
as trioo ers causing cells to swi tch from one attractor to another Iwarh)'
one, guiding a process of qualitative differentiation which follow s
attracto rs as so many stepping-stones . This property of st imulus
independence must be added to the mechanism-ind ependence I discussed
before as part of what defines ' the 'signature ' or the virtual, or put
differ ently, as part of wh at defines the traces which the virtual leav..s
in the intensive . But relative autonomy from specific stimula can he
achie ved only if the internal dynamics of a cell (o r co llec t ivities of
cells) arc rich eno ugh in endogeno usly genera ted stable states, This
co ndition is by no means guarantee d and depends on certain inten sive
properties of a network, those defining its connectivity: the number of
ge nes directl y or indirectl y influen ced by each single gen e or the
number of steps needed for the influen ce of one gen e to be propagat ed
to other genes . At critical values of connectivity a phase transition
occurs leading to the crystallization of large circuits of genes, each
displaying multiple attractors. : "Edelm an's and Kauffman's models illustrate the sense in which the
int en sive may be said to be behind the genes is of both the exte nsive
and the qualitative. Yet, neither one is a literal rendering of a simple
cascade of broken symmetries . While the ce llular neighbourhoods in
Edelman's model do illustrate how non -rigidly metric spaces may he
transformed into fixed spatial struct ures, the connection with topology
is indirect. This is even more true in Kauffman's model where the
connection with nonm etric questions is completel y indirect, mediated
by the topological invariants (such as co nnect ivity) of abstrac t spaces
0 1 I'll.. 111I11I 1c" d l 1111111 J till" .I\M!.,"I. qll .•IIIII ' , .. 1111It 1.111 , hlllll t ' .un
pit'!'> ..!luuld I ll ' '1 '1'11 Il lli .1\ till I ', t1 ) IIi WOl rau u I hUI .. fl"l'Jlll "'.'I p.lI h IIf
t he simple sym l1l\' t r}'-brt'.lk ing C.l"' ''ldt · . It is through "Ill II pin.' 11 1\",11
rc placc nu-n ts that lite ral co ntent m .l)' 1)(· unparu-d to, .lIId nu-t.rphoru-alcontent re moved from, our guiding imagl' fo r th e ac-t ualiza tion of till'
virtual in space . There is one more aspe('t of ,'mhryog"lll'sis fromwhich we can der-ive further resources to co ntinue this process of
progressive literalization. It involves looking at a dC\'eloping cmbrvo as
a process c1 assemblj' of organisms, a process which must yield individualswith the capacity to em/reo As an illustration of this point I will co ntrast
two different assembly processes , the process behind the creation o f
industrial products , as it takes place in an assembly-line factory , for
example , and th e process taking place wi thin and amo ng living ce lls
whi ch results in th e asse mbly of t issues and orga ns .
The parts of an object put together in an assembly line are typi cally
fully Eucl idean, hav ing rigid metric properties such as sizes, shapes and
positi on s, a fact that limits the kind of procedures that may be fo llowed
for their assembly. These procedures must include a rig idly channe lled
transport system (using conv eyor belts or pipes to transport rawmaterials, and wires to transport energy and informatio n) as we ll as
sequences of rigid motions to correctly position the parts relative to
one another. By con trast, the component parts used in biologica l
assembly arc defined less by rigid metric properties than by thei r
topoloqtcal connect ivity : the specific shape of a cell's membrane is less
important than its con tinuity and closure, and the specific length of a
muscle less important than its attachment points. This allows co mpo
nent parts to be not inert but adaptiv e, so that muscle lengths can changeto fit longer bones, and skin can grow and fold adaptivcly to cove r
bo th . It also permits the tra nsport processes not to be rigidly
channe lled, using simple diffusion through a fluid medium to hring th e
different parts together. Components ma y float around and randomly
co llide, using a lock-and-key mechanism to find matching patterns
with out the need for exact positioning.
All of thi s has consequences for th e capacit}' to evo lve through
mutation and selection which each of these tw o assembly processes
may have . If putting together organi sms followed an assembly-line
pattern, random mutations would have to occ ur simultaneo usly in
1I1,lll11l1I' p.lIt . ll. .l1l11ll .lI l1l p l lI t l . 11I11 III IIl d, t III "II,ld .1 \I .lhll
('1I1It) (111 whu h n.u ur.il ..t ,ll·llltlll ttllllt! (11"1 Itt l l« r u r ut t r-tn c- 0 1
"lIt h .t I,trgt' number or .. irnu lt.uu -uu 1111111111111 I I (II l tl ll l "I' , .l highl}'
iml'roll,lhl,' (' \\'1 11 . III hio logic;\1,ls"<t 'lIlllh, 011 1111 otht 'l" 1I.IIH I, muta tions
do not h.tn· to h(~ so (:oo rdin.H,·tt .11It I till gn',ldJ e nhanc-es tlu
possi bilities for evolutionary cx pcrinu-nt.uiou. As till' scien t ist Ericl Irvx ler w rites:
Because ce lls and organisms make widespread usc of diffusive
transport for energy, information and molecular parts, the evolut ion
of new processing entities (enzymes , glands) is facilitated , A genetic
change that introduces an enzyme with a new function can have
immedi at e favorable effects becau se diffu sion automatically links th e
enzyme to all other enzymes , energy sources and signal moleculesin the sam e m embrane compartment of th e cell (and often beyond) .
No new channels need to be built . .. [and ] no special space need
be set aside for the enzyme, because device placement isn't
geometric, Changes in the number of parts . . . become easy. There
are no strong geometric or transport constraints; this often allowsthe numbe r of mo lecular parts in a ce ll to be a variable, statistical
quantit)" With many copies of a part, a mutation that changes the
instruct ions for some co pies is less likely to be fatal . . . At the
level of m ultice llu lar organisms, th e striking adaptability of t issues
and organs ensures that basic requirements for viability, such as
continuity of skin and vascularization of tissues, continue to be met
despite changes in size and structure . If skin and vascular systemswere inert parts , they would require com pensating adjustments forsuch changes .25
Thi_s example illustrates another indirect way in which the metric
may be said to emerge from th e nonm etric . Unlike . a developing
embryo , a finished organism has more specialized tubes and channels
and some of its com ponents lose adaptabilit y and rigidify. T his
"metr-iza rion' is, of course, never co mplete , even when an organism
reaches maturity . But what is very significant is that, at least in thecase of multi -cellu lar animals, if organisms were not individuated in an
intensive environment which is not rigidly metric, their capacity to
n
nol\(' \\0111.1 ht ' 11·. l tI) d 1111111 I Iud Ih.lIlk IUlth lodlliu 1\ 111 .111 pllit .
lock .unl kt') lII.thlllng .1 ('lIIbl) . IlIpldogll .11 .II H I .ld.lpll\l · IMlt • 0 11
one h.md, .1S we-ll as stimulus indt 'p"nd"IH e, 011 tht ' utiII'I , cvolutu m
has an opl'n space in which to carry out its hlind ~.;t'.Irdl for 11('\\ I ~)rllls,
Put differently, hiological e volution can lx- din'rgl'nl Jud It'dd to J
prol iferation of novel ties thanks to the fact that the elements it uses to
try out new combinat ions are neit her rigid l)' connected (to specific
stim ula, to specific channels) nor intolerant to hctl' rugen d ty and
variation .
Let me summa rize what thi s discussion of embryogenesis has taught
us about the actualization of the virtual in space . Inten sive processes
possess nonmctric properties in subt le and complex ways: som etimes
they in volve the spatial co ntinuity and indi visibility of pr operties like
temperature, pressure or den sit y; other tim es the anexact yet rigorous
way in which ce llular spatia l neighb ourhoods are defined ; so metimes
what is invol ved is nothing specifically spatia l, but rather that wh ich
remains topologically invariant in a spatial process: and other t im es
spec ifically spatial capacit ies arc conce rned, such as the capability of
adaptive components to fold , stre tc h or bend . Simila rly, the final
product of an intensive process is no t ju st metric geome trically
spea king: extensive properties include some geome tric ones (like
length or volume) but also seve ral o the rs that have nothing geome tric
about them, like entropy o r amount of ene rgy; the n there are
properties which are metric, such as channelled tran sport or rigidity of
parts, but which expand the co nce pt from structure to function; lastly,
a finished product is characterized by qualities, whi ch also result from
inten sit ies but whi ch are metrically indi visible like int en sities. T hus ,
the re lation between the metric and the non metric in a process of
indi vid uation is not as simple and stra ightforward as the metaphor of a
'topo logical egg' progressively differentiating int o a ' Euclide an organ
ism ' would suggest . But what this comparison has lost in simplicity it
has, I be lieve, gained in literal adequacy.
Ha" ing clarified the relations bet ween the int ensive and the non
metr ic , in th e ne xt part of thi s chapte r I woul d like to probe more
deepl y into the nature of inten sities. Altho ugh as I said in Chapter I ,
th e term ' inte nsive property' belongs to thermod ynamics, it may be
exte nde d to co ver other areas. Indeed, my usc o f the word ' intensive'
III 1111 tI, ( 1 11'11011 0 1 Ihl 11 111 1\ 1ll1l1l lll11 II I I' II 111.1 or .111 1 III ' .1
.1111'.1(1\ .111 " h' llI lt- d II .11(' . \tl 11'1 III t I , ~ III till 1(11011 \\111 ht 10
1)('( il; till' (fl ll lle'( 11011 ht't\\(" ,u lilt l .lUli ml til 11111 111111 ,md it '(' \(' 1.,1
c xu-u srons , Afh'r thi , l OIH t'ptu,ll ( I.'nla, ,111011' l tl l1lp l t'tt'd I ,\ ill 111o,
on to di" 'u',, 011(' o f I h,lt-u/t ·' .. 11Il....1 imporLutt Iht" t' s n~ganling till"
inu-nstve. The h,)~i(' idl',) is Ih.ll 0110' .1 prol ('SS or iudividuauon is
cornph-u-d, till' iun-nsivc factors whivh dl·fim·d this procl'ss disJppt',lr
or lx-come hidden underneath the e xtensive and qualitative prOpt'rlit's
of the final produ ct. Or as Dc lcuzc puts it, ' we know intc usitv on l), .IS
already de veloped withi n extensity. and as covered over by qualities' , 110
Thi s theme of the disguising of pr ocess under product is key to
Dc lcuzc's phil osophy since his philosophical method is, at least in p.ut ,
designed to ove rcome the objective illusion fostered hy thi s
concea lme nt .Let 's begi n this discu ssion with the textbook definit ion of til<"
distinction between the inten sive and the extensive : "The rmodyuarnic
properties can be divided into two ge ne ral classes, namely intensive
and exte nsive properti es. If a quantity of matt er in a given sta te is
div ided into two equa l parts, each part will have the same value of
intensive properti es as the original, and half the value of the exte nsive
prope rt ies . Pressure , temperature, and den sity are examples of intens
ive properti es. Mass and total volume are exa mples of ex te nsive
properties. "?" Although this definit ion does point to a basic differen ce
between inte nsit ies and ex te nsities, its emphasis on divi sibility allows
it to equally appl y to qualities, such as colour or texture . But as we
just saw, a crucial part of Dclcu zc' s argument hinges precisely on the
distin ct ion between th e inten sive, on one hand , and the ex te nsive and
qualitat ive, on the othe r. Co lours are, ind eed, not di visible in ex te n
sion: _a certain patch of material of a given co lour does not yield , whe n
hroken into equal halves, two sma ller patches with half the value of its
co lour (half the hue and half the brightness). T his lack of di visibil ity
has misled some philosophe rs into failing to distinguish qualities, or
even subjectively ex perien ced intensities, such as pleasure , from
ob jective inten sive properties ,2~ Thus, we nee d a characte ristic other
than indiv isibili ty in ex te nsion to distingui sh objective inten sities fro m
qu aliti es.T here is, indeed , another way in which physicists sta te the distinc-
11011 1u- 1\\1 " II dll mu-u 1\" , li lt I till I ' tc-u 1\( \\1111. 1\\ 11 ' It 'll'il\t'
prolH'l'til" ,Hid lip ill ,I , illl pl(· \\ ,,)' (I\\O,U( ' ,1 "tid lip til , I III 0 1' " 1 t11l11o\1I)
lol rg t' r arc-a), inu -nsive propt 'rti t" do no t oldcl up hilt r.ulu-r dlt""Hc. This
avcraging ope ratio n is an objective operation . in till' !'it'n,',' th.u pladng
into co ntact tw o bodies with ditlcrcm t.'mp,·ratu n ·s will lr igger a
spontaneous diffu sion process which will equalize the tw o tl'mpt'raturcs
at some intermedi ate valu e. "? Thi s capaci t)' to spo ntaneo usly reach an
average value explains wh y temperatures or pressures canno t he
divided in exte nsion . A particular value of temperatu re or pressure ,
hein g an average, will remain the same when the bod )' possessing th ese
properties is broken into tw o or more part s. But beyond that, it points
to a dynamical aspect of inten sive properties not shared by qu alities:
differences in thermodynamic int en sities ar c capable of drh'ing a
process of equilibration in a populati on of molecul es, a process in
which these differences will tend to average themselves out. The
int ensiv e would then be distingui shed from the qualitative by the fact
that d!fferences in intensity, though not in quality, can drive flu xes of
matter o r ene rgy .
Intensive differen ces may be sharp or gradual (in which case they
arc referred to as 'g radients') bu t in either case they are nothing like
the exte rn al differences which distinguish one fully formed individual
from another. In static typologies one confronts the diversity of obj ects
in the world by a careful tabu lation of that which stays the same and
that which differs among them, Th e exte rn al difTerences between
diverse objects ar c viewed simply as a lack of similar ity so the conce pt
of differen ce plays a purely negative rol e . Int ensive or int ernal differ
ence s, such as a temperature or pressure grad ient within one and the
same body I arc, on the co ntrary , positive o r productive, forming the
basis of simple processes of individuati on . The soap bubbles and salt
crystals I mention ed in the last chapte r , for instance . arc equilibrium
structures whi ch e me rge from a process dri ven hy intensive gradic nLo;;,
o r more exact ly, from th e spontaneous tendency of the molecula r
co mpo ne nts of bubbles or crys tals to minimize a pot ential (o r minimize
an int en sive differen ce) . Given thi s morphogen eti c rol l' , it is not
surp rising that Del eu ze makes int ensive dilll 'rl'n n 's J kt} clement in
his onto logy , As he writes:
60
llll/"' '''IH 1\ li t " dun '0 II 'f It II hili dill . I t I" I I tlMI h\
whn h tlJ(' '1\' II I 11\111 11,11 , I I II I 1101 pill 111111ll"1I01l II1Il 1111nllflllWlloll , Ill 1' ''11 III Ihl pili nOIll. 111111 I \ 1'1 \ Ihm t \\ hit h h.lp
pen' ,lIul ('\, '1' IIl1n I \\lIh" '1l'IH,I I ltllrt 'l.ltt'd wit h ordt'r~ 01dil1 ~ 'r t'l1 cl' s : di lfl ' ll 'lll t " III 1t·\( ,I, 1t'llIp,'r,llUn', pn's~un', n-nvion ,
pou-nt ial, dint'n'Il('t' of ill1t 'Il,il ) , III
The first modification which mu st he made to the standard definit ion
of intensive pro perty is, the n. that the inten sities definin g a parti cu lar
physical syste m may indeed he 'divided ' but the differen ces that result
change the syste m in kind (fro m an equilibri um syste m, where
differences are cancelled, to a non-equilibrium one) . Moreover if th l"~t'
d iffere nces are made int en se enough a crit ical thres ho ld rna)' be reached
and the physical syste m in quest ion will undergo a phase transit ion . its
ex te nsive properties suffering ~ radical change in nature . Thus, rather
than indivi sibility, the key conce pt in the definit ion of the int ensive is
productive d1Jerence, as well as the rel ated conce pts of endogeno us stable
sta te (such as a thermod ynamic equilibrium state) and of cri t ical
transitions between states . How does this rel ate to th e tw o conct'Jlts
whi ch I said defined the inten sive in biology, populations and rates?
The answe r is relatively straightforward : intensive gradients arc rncas
urcd by rat es of change , and th e fluxes of matter and energy these
differen ces drive are eithe r the migratory mov ements of a molecular
population , or mov em ents of energy through such a populat ion , H In
this sense , the thermodyna mic defin it ion is di rectly related to the one
I used in biology, bu t I also made several depart ures fro m it.
W hen I descri bed population thi nkin g in evolutionary bio logy a key
issue was the role of genetic differen ces. W hile in essentialist or
typological thinking uniformity is the natural state and diffe rence what
need s special explanation , for pop ulation thinkers it is differen ce that
is unprobl ematic. Thi s use of the conce pt of differen ce alread y
constitutes an exten sion of the orig inal noti on of intensive gradient,
hut it is nevertheless related: a hiological population where ge netic
dlffercn ces have been el iminate d is as unproductive as a thermodynam ic
syste m where differen ces in temperature or pressure have been
cance lled throu gh equilibration, " Yet, the biological examples I gave
6 .
., lt ll \ I' il l\oh, ' .1 1II 0l t' l .ldh .ll lkp.lItlll l II l1l n till' 011 '11 1.1 1 d d ll ll ll oll II I
till' inu-nslve . In p.u-tic-ul.u- , unhkr- till ' 111011'1 ul.u- pClplll.lt ioll ' ludil,d in
thermodynamics, the meml u-rs of biologic.ll popul.uion s [rave .1 I.lrgl'r
repertoire of ways 10 interact with eac-h other. I ike a thermodvuamk
system, a biological population ma)' exhibit auractors (and tims hl'
defined in part bJ the tenden cies with which these singularities endow
it) but in addition its members will typi cally display com plex capacities
for interact ion which have no counterpart in the physics of heat.
An individual may be characterized by a fixed number of definite
propert ies (e xtensive and qualitative) and }'et possess an indefinite
number of capacities to c1Jeet and be c1Jeeted by other individuals . Thedegree of openness of this set of possible interactions will vary from
individual to individual. In the realm of chemistry. for instance,
different chemica l elements have different capacities to fonn novel
combinations with other elements. the capacities of carbon, for
instance , vastly outperforming those of the inert gases. In biolog)', as
we jus t saw , the flexib le capabilities of adaptive parts or the capability
to transport and match co mponents without rigid channels or position
ing procedu res, lead to even more ope n combinatorial spaces . This
ope nness is also re lated to the virtual as can be glim psed from the fact
that it dem and s fro m us th e use of modal terms (such as 'un limited
possibilities ). Deleuze , in fact , always gives a two-fo ld definition of
the vir tual (and the inten sive), using both singularities (unactua lized
tendencies) and what he calls c1Jeccs (unactualized capacities to affectand be affecte d) ,"
Unlike sing ulari ties , which arc relativ ely well studied thanks to the
developmen t of the topological approach to state space , the for ma l
study of affects is relati vely underdeveloped . Several scientists who had
previously focused on the study of singularities, howe ver, have recen tly
switched to the study of a different type of formal system which allows
the exp loration or cons tructive capacities. Stuart Kauffman and Walter
Fontana, among others, view the capacity to form novel assemblages
when objects are put into functional relations with one another as a
problem which is complement ary to that of state space , a problem whichmay also lead to the discovery of universal features analogous to those
reveal ed by classifications of attractors. Alth ough the formal s)'Ste ms
they have designed to stud)' affects ( Kauffman's random grammars,
6 2
1 0 Ill .IIl ," "l lolltl\l1\lt dW II1 I " \) III I. \ \1 II lind. I 1111 III I!J.IIl lilttllnl tlld to Iiltlth 11 1 III .lIH 1' • tI'l\ hoi, . .,Ir' ,llh \llhltd \ .•IIl ,llt',II I I ,ht ~ 1I1 tO cl lI" tlC 'llI ~ 0 1 111111 t ll lll "l lllh 11., 110 11, 1111 ludru I tl u ' ell " 0\1 '1")
of .• I't'\\ (("lurrm' t1 \\t'nrh~ {'Imam (IIt h .l ,,"tll,.l t ., I) lit 1001''1 ) ,, !lil h
11M \' turn out to hI' univ t'r~,,1. '4
\Vhilt' th e re lation [u-twcc-n inn-n..itit' s .Hld ..ingul,Irit it·s does 110t
invo lve anv dt'p,lrtufl' from th e thermodynamic dcfimuon of ' iuk U'
in" , Jddil;g capacit ies implies ('xtl'nding that definition . Let uu- fi ....tgin' a more detailed characterization of capacities and then show ho\\
the original definition may be naturallv e xte nded to include them . Anindividual organism will 'typically exhibit a variety of capabilit ies to
form assemblaaes with other individuals, organic or inorganic . A good
example is the assemblage which a walking animal form s with a pil·n·
of solid gro und (which supplies it with a surface to walk) and with a
graVitat ional field (which endows it with a given weigh t). Although the
capac ity to form an assemblage depends in part on the emcrg('nt
properties of the interacting individuals (an imal, ground, field ) it is
nevertheless not reducib le to them. We may have exhaustive know ledge about an individual' s properties and yet, not having observed it
in interaction with other individuals, know noth ing about its
capacit ies. 35
The term 'capaci ty' is elosely related to the ter-m 'a fforda nce'
introduced by James Gibso n within the context of a theory of
ecological interact ionsv'" Gibson distinguishes betw een the intrinsic
properties of things and their a tTordances . A piece of ground docs have
its own intrinsic properties determining, for example, how horizontal
or slanted, how flat, concave or convex , and how rigid it is . But to be
capable of affording support to a walking anima l is not just ano the r
intrinsic property, it is a capacity which may not be exercised if thereare no animals around. Given that capacities are relational in this sense,
what an individual affords another may depend on facto rs like their
relative spat ial scales: the surface of a pond or lake may not afford alarge animal a walking medium, but it docs to a small insect whic h can
walk on it because it is not heavy enough to break th rough the surface
tension of the water. Affordances are also symmetric, that is, the)'involve both capacities to affect and be affected . For exam ple , a ho le
in the ground affords a fleeing animal a place to hide , but such animal
l o uld .d tI d.· II "" II hllll , Ihu 111. 1 1111 ' " ' I h,lI' ' II I' Ih. ' 1tlulld
itsl II, ~illlll .l ri •.111 .1Il II 11oI1 '",1\ III I bl'l ,llI I .1 I'" .1,1111' 11111' d II d.1Il "' I
but it its..lf aftlmls nut r it ion 1;1Ih. pn ·d.lllll , ·
W e may expand th ' rnl'anillg of the krill •intr-nsi , I" 10 ill' lud l' th '
properties of assemblages, or mol" exact ly, of the prou'ss,'s wh ich
give rise to th em. An assembly process may be said to ln- charac t .r izcd
by intensive properties when it articulates het eropcncous cI .rn cnts as
such. i" In the assemblage formed by a walking animal, a pic e of
ground and a gravitational field, three heterogeneous individuals ar e
joined together as such without the need for any homogenization ,
More generally, the interactions which organisms have with the organic
and inorgan,ic components of an ecosyst em are typically of th e int ensive
kind (in the enlarged sense), an ecosystem itself being a complex
assemblage of a large number of heterogeneous components: diverse
reproductive communities of animals, plants and micro-organisms, a
geogra phical site characterized by diverse topographical and geologica l
features, and the ever diverse and changing weather patterns. Similarly,
the meaning of 'extensive ' may be enlarged to refer to the properties
of processes, such as the assembly-line process I mentioned before,
where hotnoqeneous components ar e linked together. The enlarged
meaning of 'intensive' is related to the standard definition in the
crucial role played by d!fJerence. Much as a thermodynamic intensive
process is characterized by the productive role which differences play
in the driving of fluxes, so in the enlarged sense a process is intensive
if it re lates diffirence to d!fJerence. 39 Moreover, as the exam ple of
assembly processes based on adaptive components showed, th e flexible
links which these components afford one another allow not only the
meshing of differences, but also endow the process with th e capacity
of divergent evolution, that is, the capacity to further d!fJerentiate
diffirences.
Armed with this more adequate definition of inten sive pr oc ss we
can move on to the second set of issues I said need d to be discu ssed:
the concealment of the intensive under the extensive, as \ ell as the
concealment of the concrete universals (singularities and afTect s) which
animate intensive processes . To anticipate th con .lusion I will reach
in a moment, in the case of singularities th existe nce of th e virtual is
manifested in those situations where int ensive diftt'r ' 11(,(" are not
c in. vl! ,I IIll d ll l . III tilt ' I ' II I II h,l l II
I I Illbl .1 '. lilt lit ddt. I I III' U hili .111 " , 11111 ' Ih, III
tllI lllI .h hOlllll 'I'III/ .IIIIIII , Ih.II I 1111 .1\ til " I II • I III I'll Ihlllll<'
1,IIIill' 11II .111 I ' 1'1.111 .1111111 ill \1'1 111 III \ 11111,111 t ti ll \1 ' 1 ,,1\• ..tl ll\\ III '
di l'fl'r('JI('"s ill inll 'nsi t \ III I . 1.1111. , 111.1 III ..11I1I1ll.llin ' d illi 'n 'lI' I'
Ih rou gb uni fllrmiz,llilll; , I' n~' l"l ivI · l IlId.,s Ih., \ irt u.il ,1I11 1 Illakl's till'd isappearance of process 1I11lkr product SI'(' III k-ss pr ohl crnat ic .
Althoug h thi s co nce alme nt is partly tlu- result of human int crvvnt ion ,
o f laboratory practi ces whi ch focu s on the final ' 'l uilib r ium stale or
which 'yst ' matically homogeni ze materials, for exa m pi . , it is also. n
obj .ct ive phenomen on . Any ar ea of th e world whi ch is in thermo
dynamic equi lib riurn , for instance, is an ar ea wh ere intensi c dill er
cnccs have cancelled th emselv es out, and hence an area whi ch co nn', I.th e virtual without the need for human intervention. These ar eas of
the world, in short , would constitute an objecti ve illusion.
Deleuzc argues, for exam ple , that des pite th e fact that classical
th crmodynamics yielde d valuable insights into the importance of th e
inten sive, thi s branch of physics did not provide th e foundation need ed
for a th eory of individuation given its exclusive focu s on th e final
equilibr ium state of a s)"te m . The problem with concentrat ing on the
final state is that only during th e difference-driven process can th e
equilib r ium state be see n as a virtual attractor, a state which is not
actualized yet but which is neverthel ess real since it is actively
attract ing th e successive states of th e syste m towards itself. But while
it is true that classical thermodynamics tends in this sense to under
est imate the virtual and th e intensive, 'this tendency would lead
nowhere if intensity, for it s own part, did not present a corresponding
tendency within th e extensity in which it develops and under th e
quality which co vers it. Intensity is difference, but this differen ce tends
to deny or to can cel itself out in extensity and underneath quality' .-w
In other words , while certain scie nt ific practices tend to systematicall y
down-grade th c intensive and conceal the virtual, th ese practices only
amplify an illusion which is obj ective and which is, therefore, much
harder to overcome.One way of allowing th e virtual to manifest itself is to design
expe riments or to study phenomena in circumstances wh ere intensive
differen ces are not allowed to cancel th emselv es, This is what is done
III lIu" 1.'h'l \t ' l Inll 0 1 Iht " ti t Ilt t 0 1 h.-.u , ti ll held 0 1 IIJr IfI'mcl/ llI hhn UIII Iht ·nllOd)"ll.llJlICS, \,ht'n' .111 Inlt 'n ,,' lIem of m.u n -r .1Ilt!
t' lIl'rgy cont lnuouslv lr.l\"t'rsl'S lilt' svsn-ru undt 'l" stud v .Kling a.. J.- - ,constrain t maintaining intensive- dil1~'n'nn's JliH·...• I said ill Iht' pn-vi -
ous cha pter that the varictv of atl ractors which J s)'s ll 'rn n1.1)' h.wl'
depend s on wheth er its dynam ics arc linear or nonlinear. Whi k linear
systems possess the simplest dis tribution of sing ularities, a singll~ glohal
optimum structuring the whole of state space, nonlinear ones typicallyhave multiple att racto rs (o r put more techni cally. nonlinear equations
allow for multiple so lutio ns) . To the mathemat ical distin ct ion bet ween
the linear and the nonli near , therefore , we mus t now add a thermo
dynam ic one , that betw een s)'ste ms near and Jar from equ ilibri um . As
Prigogine and Nicolis pu t it "witho ut the maintenance of an appropriate
distance from equilibrium, nonlineari ty canno t by itself give rise to
mul tipl e solutio ns. At equilibrium detailed balance introduces a further
co ndi tio n that restricts and even un iquely fixes ' the solutio n." ? In other
words, to exhibit their full co mplexity nonlinear systems need to be
driven away fro m equilibr ium, or what amounts to the same th ing,
appropr iately large differen ces in inte nsity need to be maintained by
e xte rna l co nstra ints and not allowed to ge t cance lled or be made too
small. In this sense, as these authors say, 'no nequilibrium reveals the
potentialities hidden in the nonlinea rities, potenti aliti es that remaindormant at or near equilib rium' .H
Thi s is important in the presen t co ntext because it explains the
physical source of the objecti ve illusion Deleu ze ta lks about. Take for
example a linear syste m wi th a sing le attractor . As I just said, wh ile
the system is on its way to thi s attracto r the unactualized end state is
indeed there alr eady, actively att racting the process toward s itsel f. At
this point its virt uality is relatively easy to grasp. But once the process
is over it becomes easy to over loo k the virtual nature of the end state,
even thou gh a system will never actually reach the attracto r, only
fluctuate in its vicinity. A nonlinear syste m with multiple attractors,
on the other hand, continues to display its virtuality even on ce the
syste m has settled into one of its alternative stab le states, because the
othe r alt ernatives are there all the tim e, coexisting with the one that
happen s to be actua lized . All one has to do to reveal their virtual
66
I'l l I li t I I 10 IIU ., I II II 1 111111 111 IIl1t l I II Itt. II III 111 pll h II IItl1
ol on 1"' ''111 I II ,l lt l,H !lli ll , ll1d IlIll l .lIl l ltllll ( 1 11 11 \\t tllUld. "I ttHlI I ,
n -h-r t o lilt.. ,. It , · l l loIl i \ I ' , I" hl. " I, l tt ' ,I J",nl/"I" ..." IIn1 \ 1I111.•lIlit ·"l llll I1M\(' ,.In·.ul) ,1fglll·t1 for 1hv w,.·d to h·pl.hI ' lilt pO"lhlc \\ ith ,l mUI('
.l t k 'lU.lI t· form of physical rnod.llit ),. )
A systeTll with multiple auractors, in shor t , has J gn'ah'r t"' IMdt)·
to l'xpn'ss or reveal t111~ virtual. But this I,'xpn'ssin' cap.wit)' will
depend , in turn, on the thermodynamic "zone of inu-nsity in which
the syS1l'm orl'rJtl~s: at low inten sit ies (ncar equi librium) a nonl inea r
system will in effect he linearized, that is, its pot ential complexbehaviour will no t be revealed . This procedu re has, in fact, lx-comc
ro utine in physics whe neve r trou blesom e nonlinear effects ru-ed to he
eliminate d: one simply studies the syste m in question at very low
inte nsity values for the trouble-making variable.-4-4 How ever I by fo!lo\\,
ing pro ced ures like this and systematically neg lecting the high inu- nsit yvalues at wh ich nonlinear effects arc fully expresse d , physicists promotl'
an illusion which is originally objective but wh ich now becom es
subjectively amplified . O n the othe r hand , study ing syste ms wh ich are
bo th nonlinea r and nonequ ilibrium , systems where the objective
illusion is at it wea kest , opens up windows into the virtual.
One of the tasks of a philosopher attempting to create a theory of
virt uality is to locate those areas of the world where the virt ual is st ill
ex presse d, and use the unactualized tendencies and capacit ies one
discov ers there as sources of insight into the nature of virtual
multiplicities. More exac tly, Deleu ze recommends following a very
specific philosophical me thod in which, as he says , it is
necessary to return to the interior ef scientific states ef~ajrs or bodies in
the process of belne consti tuted, in orde r to penetrate int o co nsiste ncy ,
tha t is to say, into the sphe re of the virt ual, a sphere that is only
act ualized in them. It would be necessary to 90 bock up the porh rhor
science descends, and at the very end of which logic sets its camp.:"
In other words, unli ke the linear and equili briu m approach to science
\.... hieh conce ntrates on the final product, or at best on the process of
actualizatio n but always in the direct ion of the final product , philosophy
It I' I 11111.1 Ilk, 10 ,Ic I. II III 1111111111 I, .. 1111 nlll ' 10 till 1111 I. 1'1.111
" III(h 0P,"11 d 1111 ,I.,'pl'" I 1°1'0111 'II ,.I 1',11 I hll II dill, I' IItl,II. ",.1
di"II !.' il '01111111111,.1 II 1ll" Ollll" 111 0 11<" I'd, mil',' Il'ldl 1111'1'11J '
[ollowi n J .1 (.be.H I" 01 sv rruru t rv hi ".lkin I ,'v, '111 ., ,I: . u -nsivv struct ures wo uld co nstituu tl u- co unh 'rl' rt or thl' holl Oll 1
lc 1'1, while inte nsive pr o ·( ' SS 's wo uld he Ihl' co unterpart 01 till'
int rrn ccliatc leve ls, -ach one representing a g 'o llll' l ry which is no t
fully mctri but whi h can, in fa t , be me:tricized .n Th · top Il'v"I , .111
ideally co ntinuo us and relatively undifferentiated spa " would III till'co unte rpart of the virtual . I us ' terms like ' to p' and ' bottom' hl'r, '
informally, with no sugges tion that these spac s a tu ally form ,I
hierar hical structur . A better image here would b a nested set III
spaces, with the cascade acting to unfold spaces whi h arc -mbcddcd
into on e another. Another important qualifi cation is that each on orthe spaces that comprises thi s' nested set is classified not by its
exte nsit ies or its qualities, but by its affects, that is, by its invariants
under a transformation (or group of transformations). In othe r words,
what matters about each space is its way of being affect ed (o r not
affect ed ) by specific ope rations, themselves characte rized by their
capacity to affect (to translate, rotate, project, bend, fold, st rc t h) .
Without this caveat, we could run th e danger of circularity, since th o
exte nsive properties of the bottom level would be used to define tlu
other levels as well.Thi s metaphor supplies us with a target for a theory of th e virtual:
we need to conce ive a cont inuum which yields , through progressive
differentiation , all the discontinuous individuals that populate the a tual
world. Unlike the metaphor, however, this virtual continuum canno t
be conceived as a single , homogen eous topological space , but rather as
a heterogeneous space made out of a population of multiplicities, each
of which is a topological space on its own . The virtual continuum
would be, as it were, a space if spaces, with each of its compone nt
spaces having the capacity of progressive differ entiation. Beside this
multiplication of spaces , we need a way of me shing th ese together into
a het erogen eous whole. Delcuzc, in fact , refers to th e virtual contin
uum as a plane if consistency, using the term 'consiste ncy ' in a unique
I.ollid 1111'" III till 01'1'"11. dlll,lllIlI '111111 '11111111 III 11111111 .
I II ti ll' 1111 ,'11 1\1 1" 0" " " ,dllc i. 1'," dIlC' tl1I'11I , lid 1111111 till It 10 th,
virt u: I.Let me give a con, rl'll' cxa m pl« or w h.n il " wil d III' ,II' to re- turn
to the interior of a bod y in the pr ocess or being co nstituted . Biological
categories, particularly those above '1' cies, tend to be cr iatcd by
observing similarities (or techni cally, homologies) among the anatom
ical parts of fully formed organisms, To the exte nt that the pr ocess
which generates these organisms is ignored these static classifications
conceal the virtual. But the development of a nonlinear, non equilib
rium approach to embryology has reveal ed a different, more dynamic
way of creating classifications. A good example is provided by a new
approach to the study of the tetrapod limb, a structure which can take
many divergent forms, ranging from the bird wing, to the singl e digit
limb in the horse, to the human hand and its opposed thumb, It is
very hard to define this structure in terms of the common properties
of all the adult forms, that is, by concentrating on homologies at the
level of the final product. But focusing instead on the embryological
processes that produce this structure allows the creation of a more
satisfactory classification. As on e author puts it, this new classificatory
approach 'sees limb homology as emerging from a common process
(asymmetric branching and segmenting), rather than as a pr ecisely
rep eated archetypal pattern '. 46
Returning to the int erior of the tetrapod limb as it is being
constituted would mean to reveal how on e and the sam e 'virtual limb'
is unfolded through different int ensive sequences, some blocking the
occurrence of particular bifurcations (those leading to the branching
out of digits, for example), some enabling a full ser ies to occur,
resulting in very different final products. This step in the method,
however, can only con stitute a beginning. The reason is that it st ill
relie s on the notion of similar ity or homology, even if this now
characterizes processes as opposed to products. A second step needs to
be add ed to explain the source of these process homologies. Or to put
this differently, once we have rev ealed th e intensi ve process behind a
product we still need to continue our ascent towards the virtual
structures that can onl y be glimpse d in that process but which explain
68
II I' ," 1.11 11It III 101< I 11'1'111 ' III I 11,11111 ,I dl I II loll III till 1.1111.1
• Illhl , '.I,ll I..,tt.. I ••11I.1 I ,h. I I I I I' II
01 .1 h. I 10 "111 011 1,,"111111111"
llu III t 1.1 k I • till II , I" •• I tI, ' "11' I" ItII It I" I. 1I11 1l\lJpl
[roru 11I.l tl lI' lIl .l l il (.11111111111••1 II I 111'" , 1II I1ILlIII ) .1IId 'll lid ot .111\
trao- or .Kl lI,l li t) 11t.11 till , '0111 pI 11l.1\ 11111.... 11 di -spite tl w i! .111'1',11"
h i ' h ly a bstracl n.u urv . III Il,Ir II ul.n , 1I01H' or th ese ('olln' p l 1'.111
I" '5IIPPOSC tndividuation, T hey lH'l'd to he u-ansformccl t o IWl"OIl H' fllih
pre-individual nonons so that they 'an for m th logical and pit sical h .ISI
for th e ge nesis of indi vidu als. When physicists o r mathemati ians spI·.lk
of 'di ffere nt ial relati on s ' , for example , they have in mind a part icul ar
math ematical object whi ch embodies th ose relation s : a J unclion. Such
an objec t ma y be viewed as a device whi ch maps one domain or
numbers (o r othe r ent ities) into another , or to use a more technolo
gical metaphor, as a device whi ch rec eives some inputs and maps them
into an output .?" As suc h, functions defin e mathematical individual i n
processes . For example , when a function is used to model a physical
syste m , its inputs (or ind ep endent variables) become th e dimensions of
state space , while its output (dependent variable) individuates a parti u
lar state in that space. (A series of such states forms a trajectory.)
Although Deleuze do es defin e virtual ent it ies via differential relation s
(that is, as relations between changes or differences) it is clear that Ill'
cannot co nceive of th ese relations as possessing th e form of a functi on ,
since thi s would presuppose individuality. In othe r words , th e differ
ent ial relations defining multiplicities canno t involve th e asymmetry
between dep endent and independent variables (or input and output).
If anything, th ese relations must be like ' form less functions' , wh re
inputs and outputs are not yet distingui shed, wh ere th e relation is not
a rate of change of on e quantity relative to an other, but th e rate at
whi ch two quantities change relative to each other. As Deleuze puts
, it , virtual relations must involve a purely reciprocal determination
between th eir elements, a reciprocal synthesis between pure changes
or differen ces which should not presuppose any prior individuation .so
A philosophical transformation is also need ed to lift th e virtual conte n t
from th e mathematical concept of singularity . Mu ch as virtual differ
ent ial relations must be distinguished from individuatine Juncti ons ,
virtual singularit ies should be distinguished from indi viduated sta tes .
Attract ors, for example , ma y be defin ed as special subse ts of state
I' ll v , 111.1 III 1'.11 111 11 1.11 , III ,I I II (. 1..1\ Ill' 1I0t/1I1l' 10 .I" III. 10 111.,1
IO IlSiS ll' IH), Ih.11 is, " ill. 11.<, .lbs<, III' · 0 1,, "'111.11111011. H.t/III, 'Oil i
l:nc)" is defined as the syntlicsts 1 h CI l'r0H '1ll?/I1C\ (,/\ \/II h .'
T here arc tw o se ts of issues that mu st Ill' d iscu ssi«] b,.ro n · \\1' c. n
move beyond thi s metaph o r . Both arc issues r ·Iat ing to till" CII I it ies th . t
populate the virtual. First o f all, Chapter l ' description of multiplicit
ies left unresolv ed th e qu estion of th eir nature as co nc re te unive rsal
ent it ies . In other words, I used ce rtain features of math imatical models
(the vector fields of state spaces) as a source for th e noti ons that defin e
a multiplicity but 1 did not discuss how th e properties of an actual
entity, a mathematical model, can be made into th e properties of a
virtual on e. This is a task which will involve a specific philosoph ical
transiormation of the mathematical concepts involved, a m eans of
detaching th ese concepts from th eir mathematical actualization, so to
speak. In addition to this, th e first part of this discussion need s to add
to the last chapter's characterization a description of what makes
multiplicities capable of being meshed together. 1 will argue that by
extending eac h singularity into an iriflnite series, and defining th ese
series without the use of m etric or quantitative concepts, multipliciti es
can become capable of forming a heterogeneous continuum .
The second set of issues involves going beyond singularities and into
a discussion of affect s, I said before that there ar e two special cases of
intensive processes that cry out for explanation in terms of virtuality
(or at any rate, in terms of some kind of physical modality. ) The first
case was exem plified by physical systems with multiple attractors,
syste ms which for ce on us th e problem of accounting for th e mod e of
existence of th e available ye t una ctualized tendencies. The sec ond case
was flexible assembly processes whi ch lead to an open se t of potential
combinat ions. When a process leads to a clo sed set of assemblages,
thi s set may be given by exhaust ive enumeration (that is, it ma y be
defined extensionally) elim inat ing th e need to bring in a modal
explanat ion . But if th e set is divergent (as in th e case of biological
evo lut ion) th en no exhaustive enumerati on will do since there will
always be novel assemblages not included in th e list . The qu esti on now
is, if multiplicities and their singular ities co rrespond to multiple stable
state s, what corresponds to these unactualized capacities in th e vir tual
co ntinuum? Is there another virtual entity embody ing th e capacity to
7° 7 I
'l p .III', tll.11 I ••1 ltmn (Iu't, (01 1111111 t'1 of ~ 1. 1 1t ) But II \ III t III1'IJI
.1S s l.H I'" \\ OI lld IIl1ph rh,u tl U'\ .l ln' ,llh Pel ,,"," ,) dduillt 1I I(I I\ ulll.1I1" .
lien ee n el l·tl / e....s ide:' that tht.... pn .. iIHli, idu.l1 .lSIW I I (II III l u l.l n th'" c~nonly he grasped be fore the) .lC<lu in° a wd l·dl'fillt'd iell·nt it ), in a state
space full of t raject ori es, that is, when they arc onlv ' ".lgu l'h" defined, , ,by their existence and distributi on in a vec tor field . Llnlikc trajectories , a
vect or field is not composed of Individuated sta tes, but of instantaneous
values for rates of change . Individually, these instantaneous rates (o r
infinites ima ls) have , in fact , no reality, but collec t ively they do exhibit
topological inva riants (sing ulari t ies) , and it is these invariants that
should be given ontological significance . O ntologically, however , an
invari an t of a vector field is just a topolog ical accident , a point in th e
field which happens to be stationary (more tech nically, a point at which
the zero vector is attached). Dcl euze proposes that these topologica l
accidents should be given the ontological status of an event, but given
their universality or recurren t nature, these events sho uld be seen as
ideal, not actual. A similar point applies to the bifurcations which
unfo ld th e em bedded levels of a multiplicity: each one of these
sym metry-breaking transitions sho uld be see n as an ideal event , and
not, of course , as an actual pha se transiti on . As Delcuze writes:
W hat is an idea l event ? lt is a singulari ty - or rather a set of
singularit ies or of singular points charact erizing a mathematical
cu rve , a physical sta te of affairs, a psycho logical and moral person.
Singul arit ies arc turning points and poi nts of inflect ion; bo tt lenecks ,
knots, foyers, and centers; points of fusion, condensation and
boiling; points of tears and joy, sickne ss and health, hope and
anxi cty, ' sen sitiv e points' , , , [Yet , a singularity] is cssentially pre
individual, non -persona l, and a-con ceptual. It is quite indifferent to
the individual and the co llec t ive, the person al and the imperson al,
the particular and the gen eral - and to their opposition s. Singularityis neut ral ,Sl
To com plete the characte rizat ion of multipliciti es as entit ies we now
need to discuss the capacities for int eraction which these complex
events may be expected to exhibit. Eacb of the singularit ies defin ing a
mult iplicit y must be thought as pos sessing the capaCity to be extended or
7 2
I"."IIIIHI",J th tin '''lUlU " I J I» I. III. 'I" r III rlu \u ll",1pl Ul l' ''' ., .1 \0111 1111 111.111111 IIllullllll. II I I IJr h i 11\1 Jl
nwt .lpl.ori l .11 c1" 'llpllllll lit till p'Utl .11 11 1 dU ll ~I \ I' i" " ',hIlH,,1d l' lill il io ll, TIlt" I1 H't .1ph lll I Ihl Ot c ut rr lilt 01 ,I ph.l ( t r a nvi fio u in .111
a Clll.11 n1.lh.'rial SUI h .1' " .1"''' . \\'hl n It '.IIn i ' H IOI"d clown 10 .1 lTith .,1poi nt (about 100°C at !'oe.1 It·H·I) it will spo ntaneously dlang" natun
and condense into a liquid, b ut as we co ntin ue to dl·ITt'.lSe tlU'
tl·mpcratllrc . the singular e vent wh ich occurred at the crit ical poinl
will be foll owed by a series '!fordm ory el't~·nts (each additional low l'ring
of tcmp('rature will have only a linear cooling effect on the !i<luid
water) , a ser ies which exte nds up to the neighbourhood of another
singulari ty (O°C, w here the nex t critica l e vent, free zing, occ ur s) . Asim ilar idea would apply to the virtual: the singularit ies dl'iining .1
multiplicity would become the origin of se ries of ordinary idea l evc-n tx
extending up to the vicinity of other singularit ies bel on ging to o thvr
multipliciti es , Unlike the metaphor, however, these series of ide-al
e vents would not form a sequence in time but rather a se ries of
coexist ing clements, (I will expand on this in the next chapte r wh en Idiscuss the form of tem po rality of th e virtual. " )
To get rid of the metaphorical content and to show in what sense
the ser ies cxtc nding from singularit ies arc nonmetric (thus capable of
forming a virtual co ntinuum) I will need to introduce one more
techn ical term, that of an irifJnite ordinal series. Unlike an infinite ser ies
of cardinal numbers (o ne , tw o , three . . .) an ordinal seri es (firs t ,
second, third . .. ) does not presuppose th e ex istence of fully indi
viduated numerica l quant ities , To be defined an ordinal series demands
only ce rtain asymmetrical relations betw een abst ract clements, rcl a
tions like that of beina in between two other clem ents, In other words,
it is only the order in a sequence that matters, and not the nature
(numer ical o r otherwise) of the eleme nts so orde red . Bertrand Russell ,
wh ose thought in th ese matters has influenced Deleu ze, argues that
mu ch as non metric geomctries eventually provided the foundation for
the older metric ones , so ordina l ser ies became the foundation for our
vcr)' noti on of numeri cal quantity.54 There is, in fact, a direct
relation ship between metric spaces and cardinal numbers, on the one
hand, and nonmetric spaces and ordinal numbers, on the other. Two
metric ent it ies , two lengths, for example . can be d ivided in a simple
73
\\ l\ 1/11 0 1>.1" 1111/111 I " d 111111 1111 II " \ II,. /II , ,, I.. \ .11I"(Illllll.lred silll I \\ I 1,IIl .. sl.. I>h h U/I,\I/lh"IIOU " 11.. III"'" 1/, .,1 ..1<'1111 / ,
of the two Il'nglhs, Ordi/l.l l SI'I iI's, on till' orlu -r hand , hI h.1\ ' ilion
like topologica l snaccs , whe-re WI' can rigorousl) I'stahlish th: I .1 poinl
is nearby another , but not by cxa .t1 y how mu ch (givcn that their
separa tio n may b stretched or co mpr .ssccl) .
Russell introduced the term distance (or int ensit y) to define relati on s
of proximity betw een th e elements of an ord inal se r ies . 5'; As a relation,
an ord inal di stance canno t be divided , and its lack of div isibi lity int o
identical units implies that two ordinal distances can never be exactly
compared altho ugh we can rigorously establish th at one is greater or Ie s
than another. The d!lJerence between two distan ces, in o the r words,
cannot be cancelled through numerical identity, so th e resul ts of th ese
co m parisons are always anex act ye t rigo ro us . In sho rt, o rdinal distances
ar e a nonmetric or non-quantitative co nce pt. Dcleu ze adopts these
ideas from Russell but break s with him at a crucial point : he do es not
co nce ive of th e priority whi ch th e ordinal has over th e card inal as
bein g purely logical or conceptual, but as bein g ontological. In othe r
words, Dcleu ze establishes a genetic relationship between se rial o rde r
and its defining nonmetric distances, on one hand, and numerical
quantities , on th e othe r. An ord inal se ries which is den se (that is,
where between any two clements there is always another one) would
form a one-dimensional continuum out of whi ch cardina l numbers would
emerge through a sym metry- bre aking discontinuity.56
L t' s return to th e problem of assembling virtual multipliciti es into
a plan e of consiste ncy. As I said , each on e of th e singular idea l events
defining a multiplicity need s to be imagin ed as being extende d into a
series of ordinary events which are st ill virt ual o r ideal but that , unl ike
sing ularities, alre ady possess a minimal actualization .57 Each of th e
series which emanates from a singulari ty sho uld be im agined as being
den se and defined exclusive ly by ord inal distan ces, thus constituting a
one- d imensional co ntinuum. A heterogen eou s co nt inuum co uld th en
be woven from the many se rial co ntinua spring ing from each member
of th e population of mu ltipli citi es. To ensure that multipl icit ies are
meshed together by th eir di fferen ces, Deleu ze arg ues th at the relations
amo ng th ese se ries mu st be both convergent and diverqent, In othe r
wo rds, the series m ust be mad to co me togeth er and communi cate but
74
d ., I , . , "'''/' '/1111"'/'/"" II II" 1.0 ,I.,. 1.1111" 11 , ,,
" " 1\\ 1'\ 1111 /1.1 .11\ 11 " Ill ' dll /101 1" \ 111'1'... /I " I 1111 11'\ '011'
III \\ 1 111 10 .l\ olel ( lell ll ll l\, 1I11t11111 ,.11 ,.11.,.\ .1111 1 \' '' 111 11 111 111 ",).11 11 1
/11,1) Ill' us,·d 10 ' 1 111'1.1"' ,.1 1'1 fllldol l \ 1 " /1 ' 'IU,'/I" , tl1l' lllod ,tI
(" t(' Jo ril'S <pClSsibilil y) Ill' \\ bill'S 10 n -pl.u« . ' j
At Ihis poin t .1Il import.lnt qua lificarion should I" made . Multipli. it
iI'S , houk] nol b,' co nce-ived • S possl'ssi ng tlH' '. pae it to activ , I '
interact with one • nether th rough these s .r ies, De-le uze thinks about
them as en dow d \ ith only a me re apacity to be affected , since t1 ll'y
ar c, in his words, 'impass ive en tit ies - impassiv results , ,(,() TIll
ne utrality or steri lity of mu ltip licities may be ex plained in th e fo llowing
way . Although their divergent universalit y ma kes them indepe nde nt of
any particular mechanism (the same mult ip licity may b actua liz ,d b .
several causa l mecha nisms) they do depend on the empiricalJact that orne
causal mechanism or another actua lly exists," T his is merely to say that
th ey arc not transcendent but immanen t ent it ies. But beyond thi s ,
unlike ete rna l and fixed ar chetypes whi ch have no hist orical o rigin,
Deleu ze views multipliciti es as incorporeal 1Jects c!f corporeal causes, that
is, as historical resul ts of act ual causes possessing no causal powers of
their own. O n th e othe r hand , as he writes, ' to th e exte nt that they
diffe r in nature fro m th ese causes, th ey ente r, w ith one ano ther, into
relations of quasi-ca usality . T ogether th ey ente r into a relation wi th a
quasi-cause whic h is itself incorporeal and assures th em a very special
ind ependen ce . .. ' 6 2
I said before that th e co nst ruction of a vir tual cont inuum invol ves
co nside ring not only th e role of singularit ies but also of affects . Unlike
actua l capacities, which are always capaci t ies to affec t and be affected ,
vir tual affec ts ar e sharply divided into a pure capaci ty to be affec te d
(d isplayed by impassible multipliciti es) and a pure capacity to 4fect. T his
capacity, as I hinted above, is ex hib ite d by ano ther incorporeal enti ty
which Delcu ze refers to as a 'quasi-ca use' . At thi s point , int ro duci ng
more entities may strike us as ar t ificial, o r at least as inflati onary,
enc umbe ring an already unfamiliar onto logy with further un famili ar
features. But thi s int ro d uctio n is far from being artificial . A key
co ncept in th e definition of a mult ip licity is that of 'i nva r iant', but
invari an ces are always relati ve to so me tra nsforma tion (o r group of
transformations) . In othe r words, whe never we spea k of th e invar iant
P lllP" lllt ' ol.lII "11111 ) \\t o .110 Iltnllo .II ' llillt .1I1 1l1'l l lor , 01 1'OUpo f 0 p.' r. ltllr!'\, ,.11),11.11 ' II I Pt' I IO l"ll1l llg lot.,lIOIl . 11 .111 I.Itious, IHOII'll io n'i.
fc'ldings .1Ild .1 \'.l d,·')' o f o ther t''.ln ,fo rm.lt ioll ''i on It..lt cutitv. So the
o nto log ica l co nten t o f the virtual mu st also IH' l'Uri, hl 'd wi til at le.lsl
one ope ra to r. Th e qu asi-cause is. indeed, thi s 0p"'rato r and it is dcfjncd
not by its giving rise to multipl iciti es but by its capaci ty to affect them .
'T he qua si-cause does not create , it ope ra tes", as Dclcu zc says.v'
T his new entity must be as care fully co nstructe d as multiplicit ies
were : C\ 'CI1' ste p in the const ruct ion mu st meet the co nst ra int o f
avoiding esse ntialist and typological categories , and all the conce pts
involved in its definition must be shown to be pre- indi vidual. Roughly,
the task which the quasi-causal ope rato r mu st acco mplish is to create
am ong the infinite seri es springing from each singularity ' resonances or
echoes', that is, the most ethe real or least corpo real of rclati ons.t" The
techni cal aspects of this task may be specified using conce pts from
abstract co mmunication theor-y, In co mmunication theory , the actual
occurrence of an event is said to provide information in proportion to
the probabilities of the event 's occurrence: a rare event is said to provid e
more information on being actualized than a commo n one.es These
events , each with its own probability of occ urrence , may be arranged
in a se ries. \Vh en two separate series of events are placed in
communication, in such a way that a change in probabilities in one
series affects the probability distribution of the other, we have an
iriformation channel. A telegraph, with its coupled series of events
(electrical events defining letters in Morse code at both sending and
receiving ends of the transmission line), is an example of an informa
tion channe l. But in the abst ract version of communication theory
nothing whatsoever is said about the physical realization of a channel ,
such as the length of the transmission line, or the type of code used .
Simil arly, no mention is mad e of information flowing through a
channel : an emission of a 'quantum ' of information is associated with
an)' change in probabilities in one series relative to the othe r series.
(Technicallv, the tw o series are 'connecte d' onlv through a conditional" "
probability matrix. )""
T his definition of an infonnation channe l appea ls to Delcuzc pre
cisely because of its highly abstract nat ure , presupposing no thing abo ut
det ails of physical impleme ntation,"? But ma the ma tica l models using
dlllt-lIllll,llld.lllllll II HIII'lh 1'11111 tid I I \'o l 1\ \ 1111 \ III \llllu
)( ' " lI11pl ) llllil ' 1IC1(IlIIl ~ \,hhl. Il 111 11 pl l 1lIll l\ lcllI.ll, \ ollll IlII"I 'pl
01 .HI .Ih , tr.l( t 111 101111.1111111 (l. ,lIUli I ( I I ,11111' ' Ollll1l 11 lli( .Ilioll" .1111011
~ l' r i ('~ of idt'.ll ('\(' lIt s 1I1t 1"i1 IH huthl I 1I .1Il 1IIIIIwd to hlTOI1l(' tr ill)
pro -individua l. I w ill 11ll'l1li0 I1 h" I(' (111)' the 1110"t im por-tan t rrq uin
nu -nt , ah l10ugh Dl·leu /.l' d i..(u~.,,·~ S(·\t·r,l l more: the idea l ('\( 'n1'oo
for ming a virt ual se-ries must not bl' conn'in'd as having nUn/t'n ca /
probabilit il's of occ urrence associated with them ; th l')' mu st b,·
.uranged in se ries using only ordinal distances, and he distingui sht'd
from onc ano ther exclusively by the di fference between the singular
and the ordinary , the ra re and the common, without furt her specific-a
tion . In other words, the co upled changes in distribution s wh kh
co nstitute an informati on transfcr sho uld not be co nceived as chang"s
in conditional probabilities, but simpI)' chanaes in the distribution of th e
sina ular and the ordinary within a series .b K
I will return in the next chapte r to a more complete characte rization
o f the rela tions between th ese three elements of the virtual (multipli
cities, qua si-causal operator, plane of consiste ncy) . But to conclude the
present chapter I would like to address a possible obj ecti on to thi s
scheme, What motivates the postulation of a qua si-causal ope rato r?
After all, we feel confide nt postulating th e existe nce of multipliciti es
to the exte nt that we can study in the laboratory certain phen omen a
(such as the se ries of flow patterns conduction- convect ion- turbulencc)
which embody such a progressively determinable entity , Moreover, we
can also check empirically that a portion of the same symme try
breaking cascade is exhibite d by other processes (embryological pro
cesses , for ex ample) which depend on such different causal mechanisms
that they almost demand we postul ate a mechanism -ind ep endent entity
as part of th eir explanation . Hut what evide nce do we have that there
arc int en sive processes which can spontaneously pciform iriformation
transmission operauonsi I will argue in a moment that the answer to thi s
qu estion is that th ere arc in fact such processes, and that they provide
the justificati on for thinking that such ope rations may indeed be
performed virtually. But before doing that let me add that thi s reliance
on 'evide nce ' fro m int en sive processes (more exac tly, a rel iance on
traces left b), the virtual in the inten sive) would co nstitute one of the
main characteristics differentiating a theory of thc virtual from a theor-y
77
01 r-tet u ••1 ,111.1 11I1II11It.ihll 1"'1 III I , lI l1 llk 1111 d 1'''011 11'" I' 01 I 'I 'm I '
in human thought po,tlll.lll'd Ii ) lho ,~ ' ,, 1.0 hl,lIn, III ' ''I ii " lI l1li,'s.
there wou ld lu- <.In rml'",C'hlJl '!J th l' ' /fIU.,I, 'I Ill' lOI I~ 'l· Pb. of " in ' lol l
multiplicit y, quasi-causal opt 'rat ur and pl.lIH· of ul lI:o; ish"lC)' would he .
in thi s sen se, concrete empirtco-ideal notions, not abst ract catl'gor ies. f,9
Is there any e vidence motivating the postul ation of a qu asi-cau sal
operator? There is, in fact, a relati vely new field o f nonlinear scie nce
dedicated to the study of 'emerge nt co m putation' , that is, to the stud)'
of physical processes in 'which th e int eracti on s am on g co mpo nl'nts can
exhibit the capaci ty for non-trivial informati on processing ."? Th e
mean ing of the term 'com putat ion ' in the co nte xt of natural phenom
ena is relat ively easy to grasp if we th ink about DNA and the ce llular
machinery for its transla tion, since thi s invo lves the rel atively unprob
lematic idea that bio logica l mechanism s have been evolved for the
purpose o f sto ring , transferring and processing information . But I want
to focus my discussion on a more gen era l se t of physical phenomena
that do not involve any specialized hardware and ye t can be said to
transmi t informati on . W e need to keep in mind that informatio n
transfer need not involve any com pute r- like mec hani sm, bu t only the
establishm ent (by whatever means) of a correlat ion between the
pr obabilit ies of occur rence of two seri es of events. As the phil osopher
Ken neth Sayre puts it, we can conceive 'as an instance of information
transmission any process in which the prob abili ty of one or more
members of an ensemble of event s or sta tes is changed as the result of
a change in pro babili ty of an eve nt or state outside the ensem ble . Thus
conceived , information transmi ssion occurs with every physica lprocess , ' 71
The simplest non -biological instance of spontaneous correlation
betw een the probabilities of events is the behavi our of materials near
phase transitions. In th is case the tw o se ries of e vents forming the
information channel are, in a way , co llapsed into on e, since the
co rrelations arc established between th e probabilities of occurre nce of
spatially separated events in on e and the same syste m .72 More exactl y,
mat eri al syste ms can be characte rized thermodynamically by certain
variab les wh ose values are not fixed (even at equilibrium) but rath er
fluctuate (w ith definite probabilities) around a given sta te. It is these
/IUCtlltlllt11U1".tIIUIIII1UII th, t .111 '''," hhl.'IIII,I .llllln 11101\ 1,('
(""...1- 11 "1"1. ,'I 1'1 1111 11"1111111 ,11 .. t1uo 1",111111 HI It. 11.111\ 1·(ltliIHOI.,thlc' ,
or put dill.... n ·ntl), thn .11' 1110 II 11It( lit II I,.tl .! It intortu.tuon
t r.m sm iscion occur". Bil l .1 ., \ I. III "I'IHO.I' II,· .1 ph.IS'· t r .m su io u .
t1 Il' S ~' lluctuati ons I)(.'gill to di"pl.l ) ,orn·l.llion" till' corre latio n lenglh
{the d istance across wh ich ,'\','n l" iulhu-ucv ".Iell o ther's pr ob ahilit icv)
incfl'asing the closer the s)'Slt:m gl'ts 10 the crit ical point. In the ricinU)
oj the bifurcation the capacity to transmit iriformation is maximized. Thi:o;
pheno me no n does not de pe nd on the ph ysical mechanism s und erl ying
the phase tra nsition: the same idea applies to a metall ic mat eri al
switching fro m the magn et ized to the unmagnetized state , o r to .1
material switching from th e gas to th e liquid state . In o ther words , 111l'
phenomen on of strong co r rela tio ns between fluctuation events in till'neighbourhood of a ph ase transit ion displa ys divergent uni versalit y." !
To scientists working in the field of emerge nt co mputa t ion thi s
univ ersality is highl y significan t. Some even think that thi s univ ersal
capacity for information transmission is accompanied by com plement
ary capacities to store and process information associated with other
characterist ics of phenomena ncar phase transit ions.?" This has led to
the hypothesis that the specialized hardware which living organisms usc
to process information may have required that evo lut ionary forces kept
early organisms poised at the ed8e of a phase transition , or 'what amounts
to the same thing , away from any stab le altractor. Chr istopher
Langto n, a pioneer in th is field of research, puts it this way:
Living systems are perhaps best characterized as systems that
dynamically avoid att ractors . . . O nce such syste ms emerged near
a cr it ical t ransition, evolution see ms to have discov ered the natural
information processing capacity inherent in these near-critical
dynamics, and to have taken advantage of it to further the abi lit y of
such s}"stems to ma intain themselv es on essentially open -ended
transients . . . There is ample evidence in living ce lls to suppo rt an
intimate connection between phase transitions and life . 1\13n)' of the
processes and structures found in living ce lls are being maintain ed
at o r near phase transit ion s. Examples include the lipid membrane,
whi ch is kept in the vicinity of a so l-gel transition; th e cytoske leto n,
79
III \ 111.11 II" . 11 01 0 1 11111 I o IIII.III. " ' lit 101 11 Ih . 1'"1111 I.. 1\ " n
~IO \\ l h .lIld dl ~~ollli io ll : .l lld II" 1I.111II.11 101I .11 II I d. n .ll lII l ll l lI l (III'
ping and unzipping) of' tlu - ('lIl11 p ll·1I11·nl.lr · sl r.lIl1 l. IIf I) ,
Kauffman 's networks of' regulatory ge nes \ hich, as I discussed
above, may form th e basis of processes of difl cr int iat ion in populations
of ce lls, arc also poised systems of this typ . T hat is, in this as ', to o,
the maximum information transferring capacity is achie ved when the
network is poised at the brink of a threshold, a threshold beyond
which this capacity melts away . It is much too early in th e development
of this research programme to assess the full significance of these
claims . Some of th e early formal results (using cellular automata) have,
in fact, be en challenged .?" But the basic claim that th e vicinity of phas e
tran sitions is a specia l place wh en it comes to th e emerge nce of
spontaneous information transmi ssion (as opposed to processing or
storage) is st ill valid. And it is the existe nce of this emergent capacity
in systems which come vel)' close to but do not actualize the phase transition,
which justifies us in pos tu lating such an entity as a quasi -causal
op erator.
In conclusion I would like to add that, as un familiar and ap parently
complicated as De leuze 's scheme for the prod uct ion of a virtual
continuum may seem, he must at least be given cre dit for working out
in detail (however speculat ively) th e req uirements for th e eliminat ion
of an immutable world of transcendent ar chetypes. Giv en that essences
are typica lly po st ulat ed to explain th e existence of individua ls or of
natura l kinds, eliminating them involves giving an alt ernat ive explanation,
not ju st reducing th ese individuals and kinds to social conventions.
First , we must give a detailed description ef the int ensive processes efindi viduation which generate aetua lJorms. econd, we must sho w in detail
in what sen se the resources involved in individuation pro cesses ar e
immanent to th e world of matter and ene rgy , that is, we mu st not
sim ply deny transccndentality in gcn eral but describe concrete mechanisms
ef immanence to explain how th e virtual is produced out of th e actual. The
two halves of this chapter ar e merely a ske tc h of how these two task s
are to be performed . The third and final requirement will invol ve
discussing th e temporal dimension of Deleu zc 's ontology. This will
com plete the elimination of essences we have bcgun here, ensuring
80
Ih.11 111111111'111 III'" p''''' ''' tlWII 0\\11 111,10111 11\ ,11101 1""\I 'lIll1lg tI" '11I
horn IH'ill~ l'OlIhl,, ·d \\111i (·... 1"11.11 .1I·.Ill'I\I"·" 1111' I' .1 ''''III'I' ·IIll'1I1.11'\
task III Ih~)sl' pl'l'lt)l'IIlI·d ill Ihis Ih.ll'lI'r : dn·..I0I'"lg .1 lIll'ol'y of t iuu:
with .ll'lu.\1 .lIld virtual IMl'l.s , thl' t wo dissinul.u: h.11 v", Iillkl'd Ihrough
a properly intensive form of temporalit y. It is to this other task that I
now turn .
8 1
CIIAI'IIH ~
The Actualization ~f the Virtual in Time
T here is a conflict at the heart of physics, a confli t between two
for ms of scie ntific temporality. O n one hand, the re is the conception
of t ime that develop ed in the most prest igiou s branches of phy ics ,
classical mechanics and later the special and general theories of
relat ivity. On the o the r, the concept of time born in humble areas of
applied physics, such as engineering and physical che mistry, a conce pt
wh ich eventually becam e the time of classical thermod ynami cs. T he
main difference between these two forms of time , beside their di ffer ent
degrees of int ellectual prestige , is that whil e in classical and re lat ivist ic
physics th ere is no arro w of time, th e time of th e scie nce of heat
co ntains a fundamental as)'mmetry betw een past and future . T his
asymmetry is exe m plified by the fact that thermod ynami c syste ms have
a preferential direction always tending to approach thermal equilibrium
as their final sta te . As lon g as these two co nce ptions of time simply
coexiste d side by side , as they did for most of the nin eteenth ce ntury ,
their contradic tory relations did not cause any major foundational
co nflicts in the scientific co m m unity . But wh en the physicist Ludwig
Boltzm ann attem pte d to unite classical physics and thermodynami cs
into one un ified th eory (stat istical mechanics) , the co ntradiction
between reversibility at the microscopic level, at the level of the interac
tion s between the molecul es that make up a gas, for ex ample, and
irreversibility at the macroscopic level, at th e level of co llective qu antities
like temperature or entro py, co uld no longer be avo ide d .'
Th e term ' revers ibility of time ' has nothing to do with the idea of
time flowing backwards, that is, with a flow of time go ing from the
future towards the past. Rather it refers to the fact that if we took a
certain process, see n as a ser ies of eve nts, and reversed their sequential
o rde r, th e relevant properties of the process would not change. 2 A
sim ple ex am ple fro m classical physics would be the moti on of an
object in a frict ionless medium , such as a ball thrown up wards in a
82
, 111111111 1011 0 lof It II .Ill II II of Illll 'I II I 11111 III II til II 111111 II
l'" 111011 :\ 111011 011 1"<l U' 1 of 1111 1" IlII ouiol 1,,0 I <lh ti lt ,1111'
if pi 0 lll ,, ·d 111 II" l' r" , ()II II" 011", 1..11,,1, 1110 I p r ll" 't' III
1I11'I'IIHlll)II .II11ics, :-;1 1( h .1 dillusiou 0' lit .i t «OI" hl' 111111 , .11', ' not 1'1" "I
ibk in this svnsc. Diffusion, for «x.uuph-, tl'lId. to homo ', 'niz,' sm.rll
difference s or lIu .tuat ions, that is, u-nds to damp them. But if w, '
revers,' the sequcnce of event ' we gl' t th ' oppo 'it e effect . a clampin I
pron' " turn ing into a proces ' of amplificat ion of fluctuations, I Math
c m atically, these ideas abo ut processes are expresse d in terms of t lu
invariance if the laws governing a process: while the laws of .lassica]
and relat ivisti physics remain invar iant un der a time-reversal t rans
formation, the laws of thermodynamics do not."
I will arg ue in the foll owing chapte r that most of the object ive
co ntent of class ical physics can be recov ered in an onto logy without
laws. But in the traditional ontology of physics, laws ar e clea r ly th ..
single most important enti ty . Thus, given their ontological ce nt ra lity
and their invariance under time-reversal, it is not surprising th at for
mo st physicists the resolution of the conflict has tak en the form of
keeping the sym me try of the laws whil e explaining irrev rsibility
away. " On the other hand , the emerge nce of new co nce pts in the
nonlinear branches of classical physics, as well as the ex te nsion of
thermod ynam ics to situations far fro m equilibrium, has added new
mod els and new phen omena displ aying irreversible temporal behav
iour, forcing a re-evaluation of the co nflict's resolution . lIya Prigogin e ,
a leading pract it ion er in both th ese fields, has been one of the most
vocal cri tics of the atte m pts to elim inate irreversibility. As he argu e ,
if reversing the seque nce of eve nts which makes up a process has no
effect what oever on the nature of time , then tim e becomes a mere
co ntaine r for eve nts happening in it:
Conseque nt ly, as Henri Bergson and othe rs em phasized , everything
is given in classical physics: change is nothing but a denial of
becoming and time is onl y a parameter unaffected by the trans
formation that it describes, The image of a stable world , a world
that escapes the process of becoming , has remained until no w th e
very ideal of theoreti cal physics . . . Today we know that ewto
nian dynami cs describes only part of our physical expe rience ...
[hut It·l.tti,it .lI lt l l jl l.t ll t UIII pIa, 1t "lllI llC'llhcl till 1.1. 1 111 1\\ 111111.111
php .il"s: ., static- uuiv erst ', ., univ ('rs,' of bt·"'.'1 \\ Itlll HII I-c j l'tI"",t/."
T he Dclcu zian ontolugy I have dc scr lhed in tlw sl' IMgl's is, on thecontra ry, one charac te r izing a uni verse o f bl-'comins without bdnH. Ormore exact ly a universe where individual beings do e xist hut only asthe outcome of bccomings, that is, of irreve rsible pr ocesses ofindividuation . This is, of course, not a coinc ide nce , since Dclcuze wasgreatl y influen ced by those philosophers (such as Henri Bergson ) whowere the harshest critics of the reversible and uncreative temporalit yof classical scien ce. Ne vert he less, th e theory of tim e created byDeleuze, a theory which I will attempt to reconstruct in thi s chapter ,goes beyond the conflict between reversibility and ir reversibility . Theproblem of time in a Deleuzian ontology needs to he approached inexactly the same terms as that of space: we need to conce ive of anon metric tim e, a temporal continuum whi ch through a sym mc trybreaking process yields the familiar, divi sible and measurable time ofe veryday exper ience. In particular, we cannot take for granted theexiste nce of a linear flow of time alread y divid ed into identical instantsbearing such clos e resemblance to one another that the flow Illay beregarded as essentially hom ogen eous.
In the first part of this chapte r I will introduce the ideas need ed tothink about extensive and int ensive time. The term 'exten sive' may beapplied to a flow of time already di vided into instants of a s h'enextension or durati on, instants whi ch may be counte d using any devicecapable of performing regul ar sequences of oscillat ions. These cyclicsequences may be maintained mechan ically, as in old clock-w orks, orthrou gh the natural osci llatio n of ato ms , as in newer versions, but inei ther case sequences ifcycles of different extens ion arc used to measurestretches of time of different sca les : seconds, minutes, hours, days.Thi s idea , on the other hand , may be extrapo lated from the measuringprocess to the very process which gives birth to tim e. I will discuss atheory by the nonlinear physicist Arthur Iberall according to which themeasurable flow of tim e of our everyday experi ence is in fact a productof a metrirat ion or a quantization of tim e int o instants. Between thefastes t vibra tions of subato mic particles and the e xtre mely long lifeeyell 's o f stars and other cos mic bodi es, Iberall imagin es a nested set
01 II l l l b t lt lll pul .1111 1 ' I 1111 I I I III ,I 1"" '1 I 111111 , .11 , · .. IJl Il\ Ul lll '111m' \\Itll its nu-tru u u, 1111 1 1111 1.11 1, 1.1 IU UI I, .' .... uuu-s Ih.lt urn,i.. n jl( Iik,' th.tt 01 ll., .. Il.11 ph\ It • Ih.lt I ~. t111ol1l"llt 'd h th ,' pron ·....t ·
.lIld tr.U1sIIJrl1lation ... ml.UrllII' \\1111111 I I.
Afu-r rl'vi('wing II H'I"'lI's tlwo r} .1111 1 showing how it rvlau -s 10Ik lt-UZl" s, I will move on to discuss some of till' Inten sive l·h.lraclt ·ristics of rime, those relating to the individuation of the stable osci llator..which co lh-ctivclv create a metric temporali ty . I will describe th« wo rk, ,o f the nonlinear biologist Arthur Winfree who pioneered a method tost ud)' the birt h and death of oscillati ons, o r more exactly, a method tolocate the sensitive point in an oscillation at which an ex te rnal shoc k ofthe right int en sity and duration can completely annihilate it. 11 <, h.lSalso inn'stigat cd the opposite phen om enon, how a stimulus of the rightint en sity and timing can givc birth to se lf-sustained osci llat ions. WIMtWinfree ' s work sho ws is that the sequences of osci llat ions at d ifferentscales making up metric tim e cannot be view ed as co mposed ofidentical instants. Rath er, each seque nce will e xhibit a distribution ofsinsular and ordinary instants bearing witness to their intensive origin.W infree 's conce pts of critical l imina , durat ion and intensity will play acrucial rol e in defining the int en sive or nonmetric aspects of time ."
Let' s begin then with th e qu estion of exte nsive tim e. A nested setof cycles or different temporal scales would see m to offer the rightform of temporalit y for the flat ontology of individuals I proposedbefore . In thi s ontology, individual organisms are component parts ofspecies, mu ch as indiv idual ce lls are parts of the organisms themselves,so that ce lls, organisms and species form a ne sted set of individuals atdifferent spatial scales . But clearl y, each of these individuals alsoope rates at a different temporal scale so that somc thing like a nest edset of cycles would he need ed to complete the picture . On the otherhand, to think of species, organisms or ce lls as possessing a sing lccharacte rist ic spatial scale is too simplified . As I said, between the ce lland the organ ism there are a variety of spatial st ruct ures (tissues ,organs, syste ms of organs) bridging the two scales. A species, in turn,is typically co mposed of severa l reproduct ive communities (de mes)inhabiting different ecosystems, each community constit ut ing an indi vidual ope rating at a inte rmedia te spat ial scale between that oforganism and spe cies.
11111111 1' "1/11 11'1'1.. 10 10 1111'"1 d 11111 I 1'" ,III
dl\pl.l) 1/1 1 .1 I', l/ WIII oj /111I "II 1.111 IIldl 1, 111 I 0 1 ' 111 1 III , I"" . .un p l. p O SM ' SS iut vrna] 10, b \\ hi, h " 1.,1111 h 1111 " I tI" II 1"lIlpOI •.IS 'ales (th .ir sln' l> awake cycle), but 111.1" ,llso h. " , 1II01llhh ,md vi-ar l ., ..cycles and e ven longer Diles , like lilt' 1"IIgih o r t im, 1ll',·d,'d 10 achieve
sex ual maturity (r product ive ycl .s) . They also pll"sess lllallY short ' r
cycles displa yed in d iffere nt types of rhythmic be haviour: br eath ing ,
masti cati on, locomotion . Thi s means that act ual time , rath er than bein g
a simple nesting of cycles , may include overlaps between th multi
plicity of temporal scales asso ciated with each le vel of ind ividuality. In
th e present conte x t , however, it wi ll be more expe dient to assume a
simple embe dding of time scales. For thi s purpose we can assign (by
convention) a particu larl y prominent time scale to each indi vidu al
level, such as th e cycle wh ich measures the maintenance if th eir identity:
th e length of time after which all (o r most) of th e individual ce lls in
an organism have been rep laced by new on es without affecting the
organism ' s identity, or the length of time after whi ch all th e indi vidual
organism s that form a species have died and new ones have taken th eir
place, th ereb y preserving th e co nt inuity of th e species' own identity.
This simplified nested set of cycles wi ll constitute my working model
of ex te ns ive or actual time . The qu estion now is whether thi s metri c
temporality can be accounted for in th e sam e way as metric space , that
is, as the product of a sym metry-bre aking event .
Nonlinear dynamics, in fact , allows a natural approach to th e
quantization or metrization of time in terms of spontaneous broken
sym metry. In particular, there is a well -studied bifurcati on , the Hop'!
bifurcation, which converts a ste ady state attract or into a periodic one ."
To see in what sense this bifurcation implies a broken time sym metry
we can use a spatial analogy, I said before that th e phase transiti on
from a gas to a crys talline state offere d an ex ample of a loss of
invariance under spat ial displacement. While th e pattern of distribution
in space for th e gas remains basically th e same under all di splacements
(if we imagin e th e gas store d in an infinite containe r) a regular
arrangement of crys tals loses so me of thi s invariance and remains
visually un changed only for a specific number of displa cements (those
matching th e length of individual crys tals, or multiples of th at length) .
Sim ilarly , the time distribut ion of a proces ' caught in a steady state
86
It, ,,1 111 d, pi I III III I 1111 II I , hlll It . I
11,,1" II,hll ' ,IIIOII 0 111 0111 III lilt11'1. " I ti ll 1>tIIl HI ( 0 1
dur.iuon) 01 till' ," I. III I h 111111 dl II r1'1111011 11I1l h.in ,,·d, ,.II
olh ' I S \\ ill en',ll, ' ,I I 'I"' II ' \ " I ,', Ii Ih,ll IS 0 11I (If I}h(/~l' \\ ilh Ih,
ori rina ] one , As I' r i O'llll ,llId i, "I.s plll it, ,1 process 'i n tilt' n - 'i nl!
of uniform stcac lv stall' . . , ignores t imc . But once in till' pl'r iod i,
re 'ime, it sudd 'n ly "dis overs" time in th phase of the periodic
motion ... We refer to th is as the breakinB if temporal symmetry. ' 10
Unl ike linear osci llato rs (those most pr valent in c1assi al and
rel ati visti c physics) , a non linear oscillator born from a Hopf bifurcati on
d ispl ays a characteristic period (and amplitude). By co ntrast, the peri od s
and amplitudes of linea r osci llators (typically modell ed as sinusoi da l
osci llat ions) are not intrinsic but dep end on co ntinge nt det ails abo ut
thei r initial co nditions. 'I Arthur Iberall uses thi s ide a of an intrinsi c tim
scale not de pe nde nt on extrinsic co nst raints as a basis for his th eory of
th e quantizat ion of time . As he puts it , suc h a th eory should be bas 'd
on
. . . th e m ath ematics of sequences if pulses urifoldinB in time as
dist inpuisbed from sustained sinusoidal oscillations. The basic idea is that
each pul se of acti on , in a nonlinear syste m embe dde d in a real
univ erse, emerges as a new cre at ion out of its past . It is th
sustained linear instability in th e local env iro nment [which cause d
th e Hopf bifurcati on in the first place) th at ensures th e rep etiti ve
quality of th e acti on. On the other hand , in th e ideali zed lossless
[i.c . co nse rv at ive ) linear isochron ou s syste m, with its characte ristic
susta ined sinusoidal oscillation, causality for th e acti on would be
yoked irrevocabl y to th e endless past and to an un ending fut ure . 12
Iberall argu es that , given that nonlinear osci llato rs have a characte r
istic tim e scale, ranging from th e very sho rt cales of atomic oscillators,
to th e intermediate scales of biological osc illato rs, to th e very long
Iifecycles of stars and othe r cos m ic bodi es, we may view th em as
forming a nested set of levels. This em bedde d set would ensure 'the
unfolding of time , pul se by pulse ... Time is not a uni versal unity for
all levels of organizatio n . Yet le vel s are nest ed within one another and ,
with in limi ts , are re ferable to each ot he r .' 13 In other words, ra ther
I N T E N S I V E SCIENCE AND VIRTUAL PHILOSOPHY
than assuming th at t ime ex ists as an alre ady quanti zed flow (divide d
into uniform, iden tical instants ) we should accountJor th is metric structure
using the embedde d set of differently scaled osci llat iuns . In a sense ,
each oscillation would 'synthesize a pul se of me tric time , many nest ed
sequences of th ese pul ses yielding the famili ar form of time which we
hum ans can measure using a variety of chrono me te rs. This conccpt of
time is remarkably clo se to that of Deleuzcs for whom each of th ese
pulses of acti on would constitute a synthes is of ' present tirn e ' (the
'lived present ' of atomic, biological and cosmic osci llato rs) , a synthes is
that wo uld work by contracting an immediate past and futu re into a
living present. He refers to thi s metric or ex tensive time by th e name
of 'Chronos ", and writes:
In acco rdance to Chronos, onl y the present exis ts in time. Past ,
prescnt and future are not three dimen sion s of tim e; onl y the
presen t fills tim e, whereas past and future are two dimensions
rel ative to the present in time . In other words, whatever is future
or past in relation to a certain present (a certa in ex te nsion or
durati on ) belongs to a more vast present whi ch has a gre ate r
extension or duration . There is alway s a mo re vast present which
abso rbs th e past and the future. Thus, th e rel ativity of past and
future with resp ect to the present entails a relativity of the presents
themselves in relation to each oth er . .. Chronos is an encasement , a
coW ng up tifrelati ve presents . . . 14
Let mc ex p lain in wh at sense each cycle would constitute onl y a
present, and not a past or a future . Given an osci llato r at a particul ar
scale (a biological clock , for instan ce) , what is immediate past and
future for such an enti ty would still be part of the ' lived' present of
..In osci llato r ope rating at longer tim e scales , at the level of geo log ical
or ste llar dynamics, for exa mple . Co nve rsely, the minimum living
present for a biological oscillator already includ es man y past and future
eve nts for osci llators operating at atomic and sub-ato mic scales. Metric,
ex tensive time would the n be fundame ntally cycl ical and 'composed
only of interlocking pr esents' .I 'i I mu st emphasize at th is point that,
desp ite the refe rence to a 'Hvcd pr esen t ' , this acco unt of tim e has
nothing to do with psychological tim e. It is t ru e that Dclcuz c
TH E A CT UA LIZATI ON O F T H E V I R TU A L IN T IME
some times presen ts his theory of the syn thesis of th e presen t by
contraction of immedi ate past and future , as a psychological theory,
bu t thi s is simply a matter of convenience of presentation and not
fundamental to his account. 16
The idea that it is not subject ive ex perience but the objective t ime
scale of osc illato rs th at matt ers may be further illustrated with a well
known example fro m rel ativity theory 1 an example which has some
times led to confusion du e to a mistaken psychological interpretation.
The example concerns two twin brothers one of which stays on earth
whil e the twin travels in a spaceship at a speed clo se to that of light.
The rel ati vistic conclusion that th e twin on the spaceship would age
much less than the one who staye d on earth has some times been
challenged on the gro unds tha t th e differen ce between the two
situations is a matter of subjective conve ntion : while the twin in th e
spaceship may be said to be moving forwards rel ati ve to th e one on
the ea rth , it is also possible to say th at, taking the spaces hip as our
frame of reference, it is the ear th that is mo ving backwards relative to
the ship , so that the sit uation is strict ly symme tric. Given this
symme try, the shrinkage of time wo uld be an illusion, similar to th e
appare nt shrinkage in size which observ ers expe r ience as they ge tfurther away from each othe r. 17 This conclusion is, of co urse , false. As
the philosopher Han s Reichenbach argued lon g ago, the sit uation for
the two twin s is not symmetric. To see this, however, we must go
beyond the psychol ogical t ime of the observ er to the time scale of the
osci llators tif which the observer is composed , not only the biological
oscillato rs defining metabolic cycles at the cellular scale , but also the
ato mic oscillators of wh ich the cells th emselves are made. It is th ese
osci llators that ar e objectively ~cled in the case of the rapidly moving
twin, slowing down and hence retarding the aging process, but not in
the case of his eart hbound counterpart. IS
A better way of explaining in what sense we may speak of the 'lived
present ' of a particul ar osci llato r is through the relati ons between
objective time scales , on one hand , and the resulting capacit ies to
alTect and be affecte d, on the othe r. I said in Chapte r 2 that what one
individual may afford another may depen d on their relative spat ial
scales: the surface of a lake affords a wa lking medium to a small insect
hut not to a large mammal. A similar poin t applies to time scales. Each
I NTE NSIVE SCI EN CE AN D VI RTUAL PHILOSOPHY
level of temporal scale defines wh at oscillators at that level 'perceive .'
as relevant chance: certain cycle s are simply too slow for them to appear
as changing or moving relati ve to a faster level , and vice versa , certain
oscillations arc much too fast for them to even count as existing for
oscillators ope rating at longer time scales . Subject ive human time, our
psycho logically lived present with its expe rie nced duration , would
become in this interpretat ion a particular case of these objective
rel at ions of mu tual relevance between the affordances of osci llators,
Indeed , we may generalize this po int to include physical phenomena
which cannot be character ized as periodic . W hat matters for this
arg umc nt is th e existence of characteristic time scales, whether on e thinks
of these in terms of the intrinsic period of cyclic attractors or, more
gene rally, in terms of the relaxation time associated with any kind of
attractor.An example of what is meant by 'relaxat ion tim e' is the time taken
by a radio transmitter to settle into a stable pe riodic sta te after being
turned on , what engineers refer to as ' t ransient behaviour ' . These
transients occur in man y phenomena and in each case they disp lay a
charac te rist ic t ime scalc .!" In state-space terminology this can be
ex plained as follows . As I said before , all trajectories within a particular
basin of attraction will be deterministica lly drawn to the att rac to r .
O nce there they may be temporar ily dislodged from the attractor by
an ex ternal shock but as long as th e shock is not int en se enough to
expe l them from the basin , they will return to the attractor-. In thi s
case , the tim e taken for the trajectory to return to its att ractor is its
re laxa tion time . How this relates to the qu estion of affordances may
be illustrated with an example adapted from Arthur Iberall . There arc
some solid materials, refer red to generically as 'glasses' , wh ich un like
their crystalline counte rparts, do no t have a well-defined phase transi
tion from the liquid state. In a sense, glasses are 'arres ted liquids' , that
is, they retain thc amorpho us spatia l arrangement of molecul es that a
liquid displays but flow much more sJo wly. Roughly, the distinction
bet ween the glass and liquid states can be mad e in terms of relaxation
times: these arc relatively long for glasses and rel ati vely short for
liquids.lhcrall argUl~s that whet her a particular hody dppcaTs solid or liquiJ to
d Hil'cn observer w ill depend on the rati o lu-twccn re laxat ion and
THE A CTUALIZATION OF T HE VIRTUAL I N T I M E
obse rv ational tim e scales , in the sense that for sufficie ntly long
obse rva tional tim es th e glass will appear t o the observer as a flowing
liquid. "? Thc inclusion of the observ er in thi s description may give the
wrong impression that some thing psycho logical is being discussed , but
this impression disso lves once we realize that ' observation' is sim ply
on e particular instance of ' inte raction ' , In other words, what counts
here is th e ratio ef relaxation tim e to int eraction time, a ratio that can be
defined witho ut inclu ding a human observer in the picture. In particu
lar , we can let the liquid and glass interact with each other and spe ak
of how solid th e glass 'a ppears' to the liqui d , and vice versa. The glass,
given its long relaxation time scale relative to the scale of interaction
with the liquid, will be have as a solid , affording the liqu id, for instance,
an obstacl e to its flow, or affording it a channel in which to flow . The
flowing liquid , in turn, wi ll afford eros ion to the glass. In short , what
capacit ies the glass has to affect and be affect ed by the liqu id will
depend on their rela tive time scales , the characteristic durations of
their relaxation to equilihr ium .
T he objectiv e relativity of afTordance s with respect to te mp oral
scales mak es them the ideal candidate to define the ' lived present ' of a
particular indi vidual, that is, what this individual 'perceives' within its
o wn time scale as th e rel evant capaci t ies of the other individuals
interacting with it. It is in th is sen se that Deleu zc allirms , quite
literally , that even ino rganic things 'have a lived ex perien ce ' . 2 1 To
summarize th e ma in conclusion of this sec tion: materia l and ene rge tic
processes give time its metric and measurable form by their possession
of a characte rist ic time scale , specified either through relaxation times,
or as I will do in the rest of thi s section, through the intrinsic per iod
of nonlinear oscillations. To phrase thi s conclusion in Deleu zc' s words,
at anyone of these embedded time scales the present is 'cyclical ,
measures the movement of bodies and depends on the matter that
lim its it and fills it out ' .22
Having ske tched ho w exte nsive time should he conce ived in a
Dcleuzian ontology I would like to move on to discuss the ideas
needed to think abo ut the int ensive aspects o f temporality , In th is book
qu estions of intensity have been mo st ly related to the problem of the
ge nes is of individuals. In the case of the non linear osci llato rs wh ich
<Iuanti i",e tim e Arthur Winfn 'c 's uxpc rinu-ntal and theoretical work
I N T E N S I V E SCIEN C E AND V IFIT UAL PH I L OS OPHY
gives us, as I said, the means to explore the intensive propertiesinvolved in the birth and death of osc illat ions. Winfree ' s best -known
work deals with populations of uiological osc illato rs (the internal clocks
of fruit flies or mosquitoes , for instance) which he isolates from their
surroundings to perform controlled ex periments on their reaction toshocks of different timing, duration and intensity. Winfree ' s main
result is, basically, that a stnquiat, critical st imulus applied at a smqu lar,
sensit ive moment has a destructive effect on the sleep- awake cycle of
organisms , giving a popul ation of mosquitoes, for example, pennanentinsornnia .:" The stim ulus itself need s to be of the right duration and
intensity in order to act as an annihilating shock , but it neverthel ess
acts not as a direct cause of the death of an osci llation but merely as a
trigger. What effect the shock will have will dep end on the internal
in tens ive structure of the osci llator itself.For exam ple, if the osci llation is go verned by a period ic attractor
which contains within it a stable steady-state attractor (what Winfreecalls a ' black hole ' ) then the crit ical st imulus will co mpletely annihilate
the osc illat ion .H On th e other hand, th e result of the st im ulus may be
not steady-state, atemporal behaviour but arrhythmic, ambiguous
temp oral behaviour, if the periodic attractor is associated with a set ofsta tes (called a 'phaseless set') bounded uy a phase singu!arity ." In
addition to these results related to the extinction of oscillations,
Winfree has studied the complementary problem of wh at gives rise to
these osci llations in the first plac e. Basically, he has found tbat bychanging the expe r-imental conditions he can transform an annihilating
stimulus into a conj urina stimulus, that is, a critical shock that can create
osci llations, the phase singularity in this case becoming an organizingcent re for temp oral str uc turcs .?" Winfree' s results display many of the
traits that we have found characterize intensive processes, in particular,
mechani sm-independent tendencies. The tenden cy to be annihilated by acritical shock, for example , is not limit ed to the temporal behaviour of
animals with nervous systems but is also exhibited by thc behaviour of
much simpler oscillators , ranging from )'east cells to inorganic chemical
rca crions .?"
O ther aspects of W infree ' s work on osci llators illustrate a difi'erent
fC.' .ltUfl· of the Intensive: the ability o f nonlinear osci llators tu synchromre
or entrain one anothcrs tem poral be haviour, I said in Chapter 2 that
T HE A CT UALIZAT I ON OF TH E V IFIT UAL I N TI M E
the definition of 'intensive ' may be expanded to include capacit ies, and
in particular, the capacity of an individual to form assemblages withindividuals very different from itself, Unlike the quantitative or
qualitative properties of an individual, which as emergent properties
refer to an individual' s inside (that is, to the interactions among the
lower scale individuals which co mpose it) , an intensive property in theex panded sense refers to 'an adequate outside with which to assemble in
heterogeneity' , as Deleu ze puts it. 211 The capacity of nonlinear osci llators to entrain one another's temporal behaviour is a particularly
striking example of this other aspect of the intensive, allowingbiological osci llators, for instance , to synchronize their sleep-awake
cycles with cycles outside themselves, sucb as the day-night cycle of the
planet. Entrainment is another phenom enon which Winfree has studied
in det ail, partly because of the need to prevent it from happ ening
whil e studying the effects of annihilating st imula. O nly if mosquito or
fruit fly populations are isolated from the effects of the Earth 's rotati on
will their int ernal clocks di splay their intrinsic duration or period . This
period varies for different animals , from twenty-three hours for
mo squitoes to twenty-five for humans, explaining the name 'circadian'
given to these clo cks, a term meaning 'nearly a day's length' ,When not in isolat ion, circadian docks becom e entrained with
the planet's own rotational period of tw enty-four hours, a synchro
nizing capacity with obvious adaptive value since it allows a flexiblecoord ination of internal rhythms and seasonally cbanging day lengths.
Thanks to entrainment, biological oscillators can mesh, or form
a het erogen eous assemblage , with the daily and seasonal rh ythms
of their ex ternal environment. Entrainment displays the typicalcharacteristics of an intensive process, st im ulus- independence and
mechanism- independence . Synchronization o f temporal behaviour is
t rigge red rather than caused uy relati vely weak co upling signals
which may be optical, chemical or mechanical. The exact nature of
the signals serving as stimuli is not as important as their intensity:
these signals must be maintained at a critical threshold of strengthelse the synchronization will abruptly stop.?" A similar indifference is
displayed towards the mechanisms implementing osci llating behaviour:
e ntrainmen t occurs in populations of purdy physical oscillators. suchas the vihrating <.'omporwnts of lase r light. in inorganic chemical
INTENS IVE SCIENCE AND V IRTUA L PH ILOS OPHY
reacti ons, and in a large varie ty o f hiological osci llators , including themen strual cyc les of humans. ?"
The theory of metric time in terms of a nested set of cycles which I
sketched above involves a kind of temporality whi ch is inherent ly
sequential , each individual life being a linear sequence of osci llat ions.
T he first part of W infr-ee ' s work sho ws th at th ese linear seque nces arc
not , in fact , homogen eous series of identical mom ents or instants.
There are, in eac h series. a distribution of singula r and ordinary moments
and this distribution implies that there exist relati ons of critica l tim ing
be tween the sensitive points of osci llato rs and exte rnal shoc ks . The
second part of his work displa ys a different aspect of int en sive time.
an aspect which tak es us beyond sequential and int o parallel temporal
structu res. The phenomen on of ent rainme nt allows many indep endent
sequences of oscillatio ns to act in unison, to become in e ffect a sing le
para llel process. The most dramatic and well -studied example of thi s
phenom enon is perhaps the slime mould Dicty ostetium. The lifecycle of
this crea ture involves a phase where the organism s act as individual
amoebae , the behaviour of each constituting an independent sequential
proCl~ss . At a cr it ical low point of availability of nutrients, however,
we witness the spontaneous aggregation of an entire population of
these amoeb ae into a single field of parallel osc illato rs , eventually
leadin g to their fusing together into a single organi sm with differenti
an-d parts. As on e scientist has remarked, witnessing this phenomenon
'one may reall y be watching a replay of the basic kind s of events
responsible for the appe arance of the first mult icellular organism s. ' JI
In the next sect ion of thi s chapte r I would like to extend these ideas
about cri tica l duration and timing as well as parallelism to more
co mplex processes of individuation than those exe mplified b), the slime
mo uld . But let me first summar ize what I have said about the birth of
metric or ex te nsive tim e. I gave before an exam ple of how eac h of the
embedde d cycles making up thi s form of temporality may be said to
I", bo rn th rough a symmetry-brea king event (a Hopf bifurcati on ). Thi s
was , howe ver, a purely forma l example leaving out the details of
prol'l'ss wh ich constit ute the subs tance of the intensive. Addi ng to th is
forma l model W infree ' s experime ntal results mitigatc but do no t
l'ompll'tt' ly so lve tilt' problem , \ Vc can compare this simplified model
of tilt' birth of me tric tlrnc to the me taphor I lISt,1! in the J.lst chapter
THE ACTUALIZATION OF T H E V IR TUAL IN TIME
to illustrate the birth of metric space. T he neat picture of a symme try
breaking cascade transforming a topological space into a metric one
had to be co mprehensively reworked to make it physically plau sible :
the non metric aspects of int ensive processes t urned out to be subtle
and co mplex, as did the metric aspects of the ex te nsive products;
mo reover the least met ric level of the embedded set had to be,replaced with a virtual continu um whose description required yet
another set of co mplex concepts.A sim ilar complcxification is now in order to put some Ilesh on the
rath er skeletal formal model of a Hopf bifurcation . I will return to my
two e xamples of individu at ion processes (the genes is of organisms and
species) not only to add detail to W infree 's ideas abo ut cr it ical t iming
and parall elism , but more importantly, to show how int ensive tempo
rality may be crucia l to the eme rge nce of novelty in biological evo lution .
The process of embryogenes is, for instance. involv es the parallel
development of many simulta neous seque nce s of events , the relations
between these sequences det ermined in part by the relative duration
of these processes with respect to one another, and by the relative
timing of the on set or cessation of onc process relative to another. At
this scale, as I will argue in a moment, the eme rgence of brand new
design s may come about through relative accelerations in these parallel
processes. A different source of novelty may be illustrated by moving
up in scale to a discussion of ecos ystems, which as individuation
enviro nme nts may be said to play rel ative to spe cies the role which an
egg or a womb play for individual organism s. In thi s other case too ,
relative accelerations in the tempo of evolution may lead to radi cal
innovations. Unlike the temporality of the embryo, however, where
the term ' inte nsive' has its original mean ing, ecosyste ms will involve
the e xpanded meaning, that is, the source of accelerati on and inn ova
tion in thi s case is the assemblage of het erogen eous species in the
process known as symbiosis.Let me begin with the temporal aspects of the genes is of organisms .
In the last chapter I emphasized the role of rates of change and co uplings
be tween sepa rate rates as key to understanding embryologica l devel
opml·nt. Altho ugh a rate of change docs not need to involve time (we
may he interested in the rate of change or pressurt' relat ive to oceanic
depth or at mos pheric ht'ight . for example}, tinu: docs en te r into the
I NTEN SIV E SC IENCE A ND V IRTUAL P H ILOSO P HY
formulation of many important rates. These rates of change display th e
same int erplay between characte r istic time scale and alTordances which
I mentioned before in connec tion to rel axation times (the latter ar e , in
fact , nothing but rates of appro ach to equilibr ium) . A process may
change too slo wly or too fast in relation to another pr ocess, the
relationship between th eir temporal scales det ermining in part their
respective capacities to affect one another. Even when two processes
ope rate at similar scales, the result of their interaction may dep end on
their coupled rat es of change. For example , the graphic patterns which
man y organisms display in their skins (e .g. zebra stripes or leopard
spots) may be explained as the result of the var iable conce ntration of
che mical substances, a conce ntration which depends on the rat es at
which substances react with each other relative to the rates at which
the products of such reaction diffu se through an embryo's sur faces.
Different patterns may be achieved by contro lling th ese relative rates,
a task performed by gen es and gen e products (enzymes).
As the physicist Howard Pattee has conv inc ingly argued , in the
developing organism we find an int erplay between rate-dependent
phen om ena (like che mical reaction and diffu sion effects) and rate
independent phen om ena . While the formation of sel f-o rganize d patterns
of chemical concentration do es dep end on th e relati ve rates of diffu sion
and reacti on , the information contained in genes does not depend on
the rate at whi ch it is decoded . On the other hand , thi s rat e
indep endent information, once translat ed into enzymes , act s by control
line rates. t? Enzymes are catalysts , and the latter are defined precisely
as chemical elements capable of accelerating or decelerating a chemical
r 'actio n. The fact that embryo logical development is all about rat es of
change which are coupled or un coupled through th e action of ge nes
and gene pr oducts, sugges ts that th e processes underlying embryologi
cal developmen t may be view ed as a kind of 'compute r program ' . But
this met aphor sho uld be used care fully because th ere are different
kind s of compute r programs pr esupposing diffe rent forms if time, some
lIsing seq ue ntial or serial tim e , others departing sharp ly from these
linear forms of temporal ity. As Stuart Kauffman puts it :
It is a major initial point to realize that, in whatever sense the
gen omic regulatory system constitutes something like a develop-
T HE A C TU ALI Z A T ION OF THE V I R TU AL IN T I M E
ment al program , it is almost ce rtainly not like a serial-processing
algorithm . In a ge nomic syste m, each ge ne responds to th e various
'products of tho se ge nes whose pr oducts regulate its acti vity. All the
different genes in the network may respond at the same time to the
output of th ose genes which regul ate them . In other words , the
ge nes act in parallel. The network, in so far as it is like a compute r
program at all , is like a parallel-processinp network. In such net works,
it is necessary to conside r th e simulta neous activity of all th e genes at
each moment as well as the temporal proeression if th eir activity
patterns. Such progression s constitute the integrated behaviors of th e
parallel -processing genomic regulatory syste m . 3.3
Thinking about th e temporality involv ed in individuation processes
as embodying the parallel op eration of many different sequential
processes throws new light on the question of the emergence of
novelty. If embryological processes followed a st rict ly sequential order,
that is, if a unique linear seguence of eve nts defined the production of
an organi sm , then any nov el structures would be const rained to be
add ed at the end if the sequence (in a process called 'terminal addition' ) .
On th e contrary , if embryo nic development occurs in parallel, if
bundles of relatively inde pe nde nt pr ocesses occ ur simultaneously, th en
nell' desiens may arise ]rom disenBaBinB bundles, or more pr ecisely, from
alt ering the duration of one process relative to another, or th e relative
timing of the star t or end of a process. Thi s evo lutionary design
strategy is known as heterochrony, of whi ch the most exte nsively studied
case is the pro cess called 'n eoten y'. 34
In neot eny the rate of sex ual maturation is disengaged from the rate
of development of the rest of th e body, ind eed, accelerated rel ative to
somatic development, resulting in an adult form which is a kind of
'g ro w n-up lar va ' . 35 Neoteny illustrates that nov elty need not be the
effect of terminal additi on of new features , but on th e contrary, that it
can be the resul t of a loss of certa in old features. Humans, for example ,
may be regard ed as ju venalized chimpanzees, that is, primates from
wh ich a developmental stage (adulthoo d) has been eliminated . More
ge ne ra lly, the loss of a feature mad e possible by the uncoupling of
rates o f change may pr ovide an esca pe rout ' from morphologies that
have becom e too rigid and specia lized allowing organisms to ex plore
IN T E N S I V E SC IENCE AND V IRTUAL PHIL OS OP HY
new developmenta l pathways.J" T o Del eu ze thi s aspcct of indiv iduation
pro ces, es (an aspect which must be added to populati on thinking to
co mplete the Darwinian revol ution) is highly significant because it
el iminates the idea that e volutionary processes possess an inherent
dr ive towards an increase in co mplexity, an idea which reintroduces
teleology into Darwinism . As he writes, "re lat ive progress . , , can
occ ur by formal and quantitative simplification rather th an b)' compli
cation, by a loss of co mpo nents and syn theses rath er than by acquisit ion
. . ' It is through populations that one is formed, assumes forms, and
through loss that one progresses and picks up speed .' 17
The flexibi lity with whi ch parallel processes endo w e mbryo logical
dev elopme nt may be said to come to an end once the final organism
acquires a more or less fixed anatomy, That is, at this poi nt the
inten sive becomes hidden under th e exte nsive and qualitative . Yet ,
anato mical feat ures arc never fully fixed even in adulthood . Many parts
of the body retain their capacity to sel f-re pair , and in some animals
eve n thc capacity for complete regeneration . Additionally, even if
relative to the flexibili ty of an embryo the anatomical propert ies of a
finished organism are ind eed rigid , its behavioural properties may no t
he, parti cul arly if such an organism is endowed with flexibl e skills
beside its hard -wi red reflex es and behaviou ral routines . At any rate,
even the mos t anato mically and behaviou rally rigid individual , eve n the
1110st extensive of finished product s, is im med iately caught up in larger
scale indiv iduat ion processes where it becomes part of othe r int en sities,
such as the inten sive properties characte rizing ec osystems.
O ne of the most importan t facto rs conside red in studies of ecos)'s
terns is changes in the population density of each of the interact ing
species. Populat ion den sity, like temperatu re or pressure , is an inten s
ivc property th at canno t be divided in e xte nsion. Hut like other
intensit ies it ma y be divid ed by phase transiti ons. In particular, there
are critical th resholds at which the sta te of a populat ion changes in
kind, such as min imal values of den sity (so me times called 'nucleation
thresholds") below whi ch a populat ion goes extinct. '· Similarly, mu ch
as a populat ion of molecul es will spo nta neously te nd to relax, afte r a
certain characte r-istic timc , to an eq uilibrium valu e for its temperature,
so populat ion d"'nsity will exhibit a characte rist ic re laxat ion tim e afte r
he ing suhjected to an envi ronme ntal shock, sudl as a part icularly harsh
THE ACTUALIZATION O F T H E V IR TUAL I N TIME
win ter. The ecologis t Stuart Pimm argues tha t this rate of return to
equilibrium characterizes a population ' s resilience to shocks: sho rt rates
of return to equilibrium signal a robust population, that is, one capable
of recovering rapidly afte r a shock, wh ile lon g rel axation times betray
poor resilien ce and hence , vu lnerability to ex tinction . Given that
extinct ion mean s the death of a species as an indiv idual , and that the
ex tinct ion of one species may mean the rapid birth of others to occupy
the vacant niche , these int en sive properties may be said to partly
characte rize processes of individuat ion at thi s scale .
Ecosys te ms involve processes opera ting at several simultaneous time
sca les. O ne factor affect ing population den sity is int ernal to a species,
that is, det ermined by the birth and death rates of a population . This
fact or disp lays a relatively short time scale of re turn to equilibr ium.
When the densit ies of several populations are co upled in parallel, as
,...·hen a population of plant s, hervibo res and carnivores is coupled into
a food chain . relaxation times become longer: when the density of a
predator popul ation affects tha t of its prey, and thi s, in tum, the
dens ity of the plants it consumes , re-equ ilibrat ion afte r a shock may be
delayed until the cascading effects stop. This lon ger t ime scale of
recovery is determined by the degree of connectivity whi ch one species
has to other species, that is, by the length of thefood chain to whi ch the
species belongs. Finally, there are even lon ger-term pr ocesses det er
mined by non-biological factors such as the rate of availability of
mineral nutrient s in an ecosystem during reco vcry fro m a catas trophe,
such as the effects of the onset or cessation of an Ice Age . 39 Given th e
importance of resilien ce as protection against extinction , and given the
key ro le which the degree of connect ivity plays at intermed iate time
sca les, an ecosyste m may also be conside red a parallel-processing
net work in which changing relationships offi cness (between predators
and prey, or hosts and parasites) propagate at different rates th roughout
the network influencing both the eme rge nce of new , and the disap pearance of old, individual specics .t"
Relations betw een population den sities, however, give us a ni)' a
rough ide a o f the co mplex temporal structu re of an ecosyste m , Co n
sidered as a network in which the fl esh (o r hiomass) of plant s and
anima ls circulate, an CCOs)'stc m will display a varict )' of temporal
rhythms d larackrizing eac h of its alime ntary co uplings, these rh yth ms,
I N T E NS I V E S C IEN CE AN D VIR TU AL PHI L O S OP H Y
in turn, associated with the spectrum of osci llatory behaviour at
d ilTerent scales ex hibited by every organism . But considered as an
individuation environment there is a particular rhythm which must be
sing led out : the evolutionary rates of each of the co upled species.
Evolutionary rates used to be thought as basically uniform, characte r
ized by a linear and gradual accumulation of gene t ically code d beneficial
traits . This rat e of accumulation would vary from species to species,
du e to their different generation times, but within each species it was
supposed to be basically un iform. Today we know that thi s picture is
incomplet e given that for a variet)' of reasons there occur accelerations
and decelerations in these evolutionary rates. (The very large time
scales invo lved in evo lution means, ho wever, that even an accelerated
rate will st ill characterize a very long process, one between 5000 and
50,000 years, for e xam ple!')
As in the case of em bryological developmen t where loss of a
part icu lar process or com ponent may lead to the emergence of novel
features, in an ecosystem losses may also lead to accel eratio ns in
evo lutionary rates and rapid spread of novel designs. An extinct ion
event , for exam ple, may eliminate a set of spec ies and vacate their
niches, leadin g in t urn to an ex plosion of new design s by ot her spe cies
(an ada ptive radiation) to occupy the vacant positions in the food
chaln .:" A di ffe rent exam ple of events leading to accelerated evolut ion
and rap id emergence of new capacit ies is 9'mbiosis. Altho ugh tradit ion
ally the term 'sy mbiotic relat ion ship ' refers to a partic ular kind of
alimentary co upling (one in which both partners ben efit from the
assoc iatio n) the difficulty in defining and estahlishing mutually beneficial
re lat ions has led to a new view of its nature and function. Today
symbios is is defined as an assemhlage of heterogeneous species which
persistsJor lona periods, relativ e to the generation times of the int eracti ng
organisms, and wh ich typically lead s to the emeraence if norel metabolic
cdpelbili t ies in at least one of the partners. r" The em phasis on long
dura tion is due to the need for coel'olulion between the partners. both
of wh ich need to have e xerted selection pressures on eac h othe r biasing
till' long-term accumulat ion of the ir genes and bodily traits . (Given
that some membe rs of an ecosyste m rna)' have arrived through recent
invasions or co lonizat ions. not all interact ing co uples in a food chain
nt·t·e! to han ' ('(u.·voln·d.)
THE A C T UALI Z AT I ON OF T H E V IRTUAL IN T I M E
Symbiosis as a source of evolutionary innovation oc cu rs at many
level s of scale . At the ce llular leve l, for exam ple, two of the key
capacit ies at the basis o f food chains may have eme rge d through an
assembly of heterogeneities. Phot osynthesis , the ability to 'bite ' int o
solar rad iation to produce che mical ene rgy sto red in sugars , and
respiration, the abi lity to tap int o a reservoir of oxygen as fuel to burn
these sugars, are both thought to have emerge d through cellular level
symbioses with micro-organisms.'" At larger scales, examples include
the auton omous com munities of micr o.organi sms which line the guts
of hcrviborcs allowing th em to digest ce llulose , the bact eria that allow
legumes to fix nitrogen , and the fungi wh ich permit man y plant roots
to ge t access to phosph orous . In all these cases, novel capabilit ies to
e xplo it otherwise unavailable resources have co me about no t through
a slow and gradual accumulation of favourable mutations but through
an accelerated process: mes hing the capabilit ies of tw o or more
het erogeneous populations of organ isms followed by th e subseq uent
coe vo lution of the partners.H
W hen discussing inten sive processes Dclcuze usually di vides the
subject into singu larit ies and affects , but some times he uses an
alte rnativ e and equivalent formulati on in terms of spee ds and affects :
speeds if becoming and capacities to become.": The many parallel processes
which define a developing em bryo, fo r example , are defined by their
rel ativ e speeds , and by the accelerati ons and decelerat ion s th ese may
und ergo resulting in the product ion of no vel forms . In Delcuzian
terms, such an indiv iduatio n environm ent wou ld be characterized in
part by relation s of 's peed and slowness, rest and movemen t , tardiness
and rapidity ' .4 7 As I said , changes in these relative spee ds may be used
as an evolutionary strategy (he te rochro ny) allowing an organism an
escape route from an over-specia lized design . Eco s)'stems also display
relations of relative speed between para llel processes but in this case
the emerge nce of novelty depends more on the capacity to Join in with
a heterogen eous partner in a com mon cocvolutionarv line of flight.
o r as Delcuzc puts it . on ' a co mpos ition of speeds and affects involv
ing entire ly different individuals, a sym biosis ' .'HI To phrase this in
Prigogin e ' s term s of being and h(.~coming : whereas emlJryogenesis is
a process through wh ich a yet unform ed ind ivid ual becomes what it is,
acqu iring a well -defined inside (the intr insic propt·rti t·s defining its
I N T E N SI V E S C IEN CE A N D VIRTUAL PHILOSOPHY
being), symbiosis represents a process through which a fully formed
bein g may cease to be what it is to become somethinq else, in association
with something heterogen eous on th e outside .
This description of more com ple x forms of inten sive temporality
was intended as a comple me nt to th e simpler formulation in terms of
th e ind ividuation of oscillations. Questions of cr itical timing and
duration, as w ell as of parallelism , are st ill prominent but have acquired
a subtle r form . Similarly, the problem of th e metrization or quantiza
tion of time, which also had a sim ple formulation in terms of a nest ed
set of seque nces of oscillation s, need s now to lose so me of that
simplicity. In particular, for th e sake of ease of presentation I have
arti ficially se parate d issues related to time and spa ce, but in reality we
arc always confronted with complex spatio- ternporal ph enomena. Even
the sim ple oscillators st udied by Winfree ar e nonlinear spatia-te m poral
osc illators where th e spat ial and temporal aspects interact. For thi s
reason, the qu estion of th e emerge nce of metric o r exte nsive prop erties
sho uld be treat ed as a sing le process in which a continuous virtual
spacetime progressively differentiates itself into actual discontinuous
spatio- tc mporal st r uc tures operating at different scales. In other words,
the emerge nce of a metric space t ime involves the ent ire flat ontology
of ind ividuals, each nested level of scale co nt ribut ing to th e metrization
of space and time simultaneou sly.
I would like to conclude thi s chapte r with a more detailed discussion
of this virtual space time . In Chapter 2 I described th e elements whi ch ,
accord ing to Deleuze, constitute th e content of a nonmetric contin
uum : changing populations of virtual multipliciti es (co nce ived as
complex ideal eve nts) and a quasi-causal op erator whi ch asse m bles thi s
heterogeneous population into a plane of consiste ncy. This particular
breakdow n of th e co nte nts o f th e vir tual is, of co urse, speculative, and
as su h, it may very well turn out to be wrong. There is, as I said , an
.m piricism of th e virtual, even if it does not (and should not) resemble
the e mpirical study of th e act ual. But whil e th e specific solutio n which
Dclc uzc prop ose may turn out to be inadequate , he sho uld ge t cre di t
for having aclcquat ' Iy posed th e problem. In o rder to ge t rid of essent ialist
and t Ipo lug ical thinking it is not eno ugh to den oun e th transcendent
and aflirm the imm: ncnt , R ' pia ing Plato ' s transc mdc nt ss inc ' S with
risto l1l" · im ma ne nt natural sta tes, for c ample. gets us out of
THE ACTUALIZ ATION OF THE VIRTUAL IN TIME
esse n tialism but not of typ ological thought. One mu st also give
mechanisms c1 immanence (however speculat ive) to explain th e ex iste nce,
relat ive auto no my and ge ne t ic power of th e virtual."? Let me first
sum ma rize wh at I said before abo ut the quasi-causal ope rator, the
mann er in whi ch it meshes multiplicities by th eir differen ces, since thi s
co nstitutes th e first immanen ce mechanism. I will th en describe the
second task whi ch Deleuze ascribes to thi s virtual ent ity : to Benerate the
multipliciti es by ex tracting th em from actual inten sive processes.
T ogether, th ese two tasks ensure that th e resulting virtual space t ime
does not have th e form of a transcendent space filled with tim elessesse nces .
I described th e first task of th e quasi-causal ope rato r as that of giving
vir tual multipliciti es a minimum of actualization by prolonging th eir
sing ularities into se ries of o rd inary ideal events , and establishing
relations of co nve rgence and divergen ce between th ese se r ies . I said
that to specify how th ese immaterial linkaBes between se ries arc
established Dcleu ze borrows from th e mo st abst ract vers ion of co m
munication theory th e concept of transmission of information in a
channel (a sign / signal syste m , in his terms,). An information channe l
(signal) exists when ever two heterogen eous se ries of events ar e
co upled by chang ing probability distributions. No referen ce need s to
be mad e to eithe r a causal mechanism or to anything actually flowing
in th e channe l. Quanta of information (signs) ma y be said to pass from
one ser ies to another wh en ever a change in the probability distribution
in on e seri es is correlat d to a change in th e other on e . Such a linkage
of se ries of events through signs occurs spontane ously in some intensive
syste ms , suc h as syste ms poised at the edBe of a phase transition. Even
whe n suc h poi sed syste ms are inorganic, that is, even in the absen ce of
specialized biological hardware , th ey can cohe re n tly transmit informa
tion as long as th ey manage to remain in the vicinity of th e crit ical
event without actually crossing th e threshold.
The em bryo logical and ecolog ical indi viduation processes I have just
discussed , at least when modelled as parallel -p rocessing networks,
dis play th is emergent abi lity in th e neighbourhood of a critica l point c1connectiVity. Stuart Kau ffman argues, for example, th at the many food
.hains that form an ecosyste m mu st not exceed a ce rta in riti al length
(typica lly o f four sped": a plant , a hcrvibore , a pred ator , and a
I NTENSIVE SCIENCE A N D VIRTUAL PHILOSO P HY
predator of th e predator) for th e parallel network to display complex
behaviour. 50 This sensitive valu e ma y be achieved via the coevo lut ion
of th e members of a food chain . Similarly, th e parallel network form ed
by gen es and gen e products whi ch const itutes th e informational
backbone of a developing embryo also need s to keep its degree of
co nnect ivity near a crit ical value . Kauffman explicit ly compares thi s
crit ical value (not too low but not too high ) to th e singular zone of
inten sity exist ing at th e phase transiti on between a gas and a so lid (that
is between states with too little and to o mu ch orde r , respecti vely) and
argues that embry os and ecosys te ms may need to be poised at the edge
in orde r to maximize th eir emergent co mputational capacities.F'
Unlike actual poi sed syste ms , however, where information trans
m ission takes the form of co rrelations between the numeri cal probabi li
ties of occurance of two ser ies of eve nts , virtual ser ies must exclusively
involve changing distributions of th e singular and the ordinary, given
th at vir tual se ries and th e space th ey form cannot presuppose any
me tric o r quantitative notion without begging th e qu esti on. In particu
lar , vir tual se r ies mu st be conce ived as den se ordinal ser ies whi ch, as I
argued, arc logi cally and ge ne t ically prior to alr eady quantized numer
ical se ries and can be regarded as on e-dimensional nonmetric cont inua .
In addit ion , th e requirement of not presupposing any notion to whi ch
th e virtual is suppose d to give rise implies that th e statist ical distribu
tion s invo lved in an information channe l canno t be conceived as fixed
(or 'sede ntary' ) like th e famous Gaussian or bell -shaped distributions
characterizing th e stat istical properties in many actual population s.
Unlike these familiar equilibrium distribution s whi ch refer to alread y
individuate d populations occupying a metric space , Deleu ze designs th e
quasi -ca usa l operator to produce mobile and ever-changing (' no mad ')
dis tribut ions in th e virtual series, establish ing both conve rge nt and
div 'rgent relation s between them. 52
In sho rt, th e first task of th e quasi-cau sal op erator is what Deleuze
calls a condensation if singularit ies, a process invo lving th e continuo us
creatio n of co mm unicat ions between th e se ries em anat ing from every
singularity, linking th em together through non-ph ysical resonances,
while simu ltaneously rami fying or di ffere nt iat ing th e se r ies, ensuring
they are link ed together only by th eir differen ces .5 I T he mesh of on '
dimens ional co nt inua that results would co nst itute th e spatia l aspl' t o f
THE A C T U A LI Z A T I O N O F THE VIRTUAL IN TIME
th e vir tual. To thi s, a temporal dim en sion , whi ch Deleu ze call 'Aion",
should now be added . As he wr ites, th e specification of th e virtual
implies, on th e one hand, a space of nomad distribution in which
singulari ties ar c d istributed (T opos) ; on th e other hand , it impl ies a
tim e ifdecomposition whereby th is space is subdivided into sub-spaces. Each
one of th ese sub-s paces is successively defin ed by th e adjuncti on of
new points ensuring th e progressive and co m plete determination of
th e domain under conside rat ion (Aion) . There is always a space
which co nde nses and precipitates singularit ies , just as there is always
a time whi ch progressively co mpletes th e event through fragments
of future and past eve nts.v'
Deleu ze borrows th e term 'adjunc t ion ' from th e mathematician
Evariste Galois, th e cre ato r of gro up th eory. I will return in the next
chapte r to th e work of thi s pion eer, but at thi s point it is enough to
say that th e ope rat ion Gal ois defined as 'adjunc tion of fields ' is an
abstract ope rat ion ve ry clos ely related to the idea of th e progressive
differentiation of a space through a cascade of sym me try -bre aking
transiti on s. In othe r words, th e successive det ermination of sub-s paces
to whi ch Deleu ze refers is sim ply th e progressive unfold ing of multi
pliciti es through a se ries of symmetry-bre aking events. T he form of
temporality involved in thi s unfolding, however, sho uld be conceived
in a very different way from tha t in whi ch actual bifurcati on events
occur . The latter invol ve a temporal sequence of events and stable states ,
th e seque nce of phase transiti on s whi ch yields th e se ries of stable flow
patterns co nduct ion-eonvection-turbulence, for exam ple . Moreover,
as eac h bifurcati on occurs, only one of th e several alt ernatives available
to th e syste m is actualized. For example , in th e transiti on to th e
co nvection regime , eithe r clo ck or anti -clockwise rotating convec t ion
ce lls may emerge , but not both . Indeed, at eve ry bifurcati on th ere ar e
alt ernatives th at ar c phy sically unstable (unlike the two options for
co nvection ce lls both of whi ch are stable) whi ch means that even if
they are actua lized th ey will not la t very long and wi ll be destroyed
by any de tab ilizing fluctuation. ss In a virtua l un folding , on th othe r
hand , the symm etry-brcaking events not only Jullj' coexist with one
ano ther (as opposed to follow eac h othe r), but in add it ion, eac h brok en
INTENSIVE SCIENCE AND VIRTUAL PH ILOSOPHY
symmetry produces all the alternatives simulta neously, regardless of
whethe r th ey are physically sta ble or not.
T his virt ual form of t ime, involving the idea of absolute simulta neity
(or abso lute coexistence) would see m to vio late th e law s of re lativi ty.
In relati vist ic physics two events cease to be sim ultaneous th e moment
they become se parated in space, th e dislocati on in tim e becoming all
the more evident th e larger the se pa rating distan ce .56 There are two
reasons, however, why thi s sho uld not be an object ion to De leuzc's
conception of vir tual time . The first and m ost obvio us reason is th at in
vir tua l space there are no metric distances, only ordinal distances which
join rather than separate events . Mu ch as the noti on s of spat ial ' length'
or 'a re a' lose th eir meaning wh en we move away from Eucl idean
geometry to othe r ways of specifying th e relati ons of proximity
defi ning a space , so sho uld the notions of 's t re tch' or ' lapse' of time
separating non-simultaneous events be meaningless in the co ntext of a
no nmctric temporalit y. But there is a second and more import ant
reason why relativisti c co nstraints on absolute sim ultane ity, suc h as th e
constraint on th e maximum speed at wh ich causal signals may travel ,
sho uld not apply to th e vir tua l. T he temporality of th e virtual sho uld
not be co mpare d to that of th e processes governed by th e laws of
relat ivity, but to the temporality if the laws themselves. Unlike ex pe r imen
ta l laws (like Boyle ' s law of ideal gases) whi ch simp ly record laborato ry
reg u larities , fundamental law s (such as Newto n's or Einstein's) arc not
m re mathematical re -descript ions of ex pe r ience .57 Although physicists
do not usually speculate abou t the onto log ical sta tus of fund amental
laws, to philosophe rs th ese laws ar e supposed to be eternal, and to be
valid simultaneously through out th e un iverse . In other w ords, in phil o
sophical discussions fundamenta l laws enjoy th e same form of timeless
ness as immutabl e essences. And it is th is form of time th at th e virtual
is supposed to re place.
Nevertheless the ques tio n re ma ins, what form of temporality would
allow the absol ute coexiste nce of virtual events? Or what amo unts to
th e same thing, how should we co nceive of a non m etric fo rm of time?
It clearly can no t be any presen t tim e , however long , since the very
concept of' a present assumes that of' a stretch or lapse of' ti me of a
particu lar chara tc rist i sca le. But it cannot be a t imeless d imen sion
either if we an' to avoid the tr ap pings of essen tia lism. The so lutio n
THE A CTUALIZATION OF T HE VIRTUA L I N TIME
which Deleu ze prop oses to esca pe th ese alterna tives is inge nious .
Unlike a transcendent heaven inhabite d by pure beings without becoming
(unchanging esse nces or law s with a permanent identity) th e vir t ual
needs to be po pulate d excl usively by pure becomings without being .
Un like act ual becomings wh ich have at most an intensive form of
temporality (bund les of seque ntia l processes occur ring in parall el) a
pure becoming mu st be characte rized by a parallelism without any trace
if sequentiality, or even directionality , Deleuze finds inspi rat ion for thi s
co nception of time in phase transit ions, or more exactly, in th e cr itical
events defi ning unactualized transi t ions. W hen see n as a pure becoming ,
the cr itic al point of temperature o f o-c, for exa m ple, m ar ks neither a
melting nor a freezing of' wate r, both of which are actual becomings
(becoming liquid or so lid) occurring as the cr itica l threshold is crossed
in a definit e direction. A pure becoming, on th e other hand , would
involve both dir ecti ons at once, a m elting- fr eezing event which never
actually occurs, but is ' alwaysforthcoming and already past. ' 58
The events invol ved in th e constructio n of vir t ual space, th e
progressive un folding of virtual multipliciti es as well as the stretc hing
of th eir singularities into series of o rdinary points , need to be th ought
as pure becom ings in thi s sense. In th is co nst ruction, as Deleu ze says,
•Time itself urifolds . . . instead if things urifolding within it . . . [T imeIceases to be cardinal and becomes ordinal, a pure order of time ."!"
Unlike actual time, which is made exclusive ly out of presents (what is
past and future relat ive to one t ime scale is still th e living present of a
cycle of greater duration) , a pure becoming wo uld imply a temporality
which always sidesteps the present , since to ex ist in the present is to be ,
no longer to become. This temporality must be conceive d as an ordinal
conti nu um urifolding into past and f uture, a t ime wh ere nothing ever
occurs but where every t hing is endless ly becoming in both unlimited
d irections at once, always 'already happ en ed ' (in th e past direction )
and always 'a bout to happen ' (in th e future di rect ion). And unlike
actua l t ime which is asymmetric relat ive to th e direction of relati ve pasts
and fut ures, a pu re becoming would imply a temporality which is
perfectly symmetric in thi s respect , the d irectio n of the arrow of time
em rgi ng as a brok en symmet ry on ly as the virtual is act ua lized. "?
I said in Chapter 2 that multiplicities, being in o rpo rca l ffc ts of
mate ria l causes, are impassi ble or ausa llv steri le enti ties. T h· t ime of
INTENSIVE S C IEN CE AND V I RT U A L PHIL O S OPH Y
a pure becoming, always already passed and ete rnally ye t to come ,
forms the temporal dimension of thi s impassibility or ste rility of
mult lplicitl es."! But I also said that the quasi-causal ope rato r, far from
be ing impassible , is defined on the contrary by a pure capacity to
alTect, act ing in parallel with physical causality in the production of th e
virtual. In particular, th e quasi-cause mu st be capable of weaving
m ult iplicit ies into a het erogeneous co ntinuum and to do so co nstantly
so as to endow th e latter with a certain auto no my fro m their co rporeal
causes.b1 \Vhat temporal aspect would co rrespo nd to the exe rcise of
this capacity? Here again , we cannot presuppose any metric co nce pts ,
that is, we cannot assume that thi s performance occurs in an)' present
st retch of time , however short . This othe r time must ind eed be
conceived as instanta neous, As Del euze writes :
Corpo real causes act and suffer throu gh a cos mic mixture and a
uni versal present which produces the incorporeal event , But the
qu asi-cause operates by doubling thi s physical causality - it em bodies
the event in the most limited possible present which is the most
precise and the most instantaneous, the pure instant grasped at the
point it d ivides itself into future and past ."!
In wh at sen se would a temporality charac te rized by a instant which
unfolds itself into past and fut ure he nonmetric? Actual time , as I said ,
ma)' he sec n as the product of a metrization or qu ant izat ion performed
hy a nested set of presents with characte rist ic t ime scales. Whether
one views the latt er in terms o f relaxation times or in terms of the
intrinsic peri od of nonlinear osci llat ions , th e processes occur ring in
actual time always have a time scale of limited Juration and )'et are
potentia lly irifinite, in the sense that a particular seque nce of cycles
l11a)' go on pu lsing for eve r . Virtual timc , on the o the r hand, would
he.' nonmct r ic in the sense that it is unlimited in the past and future
directions in wh ich it unfolds, bu t alwaysfinile like the insta nt without
thickness tha t pe rforms the un folding ." T he time of the virtual would
he.' consti tuted cn tirelv bv wh at , from the point of view of metr ic, ,tu nc , canno t he hut !Oingular ities : a maximum and a minimum, events
of unhmllcd Juration (the unfo ldi ng of mu ltiplicit ies) and events of cero
THE ACTUALIZATI ON O F THE V IRTUAL IN T I M E
duration (t he operat ion of the qu asi-cause ) . The quasi-causal operato r
would have to
bring about the corresponde nce of the minimum time wh ich can
occur in the instant with the maximum time whi ch can be thought
in accordance with Aion . To limi t the act ualizat ion of the event in a
present without mixture, to make the instant all the more intense ,
taut , and instantaneou s since it ex presses an unlimited future and an
unlimited past. bOO
No doubt, this description o f the temporal aspect of virtualit y lacks
the precision of its spa tial co unterpa rt. The latter has the advantage of
ove r a century of mathematical work on the nature of nonmctric
spaces and their broken sym me try relations to metric ones , whereas
similar formal treatments of time do not really exist. Moreov er, even
if we disregard time and focus only on space, Deleuzc ' s description of
the virtual co ntinuum goe s beyond the resources available from those
formal theories and may therefore see m mu ch too speculative and
com plicate d. Why, on e may ask, go through so mu ch trouble to
speci fy the immanen ce mechanisms through which a virtual continuum
is const ructe d when it is simpler and more natural to assume that the
entit ies revealed by nonlinear mathematics (att racto rs , bifurcations) ar e
of the same t)'pe as our more familiar Platoni c entities? A leading
figure in the theory of dynamical syste ms , th e mathematician Ralph
Abraham , for example , phrases his evaluation of the merits of the field
this way:
The ben efits of using dynami cal co nce pts at the present stage of
devel opment of sclf-organi7..atio n theory fall in two classes: perman
ent ones - the acquisition of conce pts to be em bedded in morpho
dynam ics , guiding its development ; and temporary ones - the
prac tice of new patterns of thought. In the first ca tegory I would
place the att rac to rs , the stable bifurcations, and their global bifurca
tion diagrams, as esse ntia l features of morphod ynami cs. These rna)'
he.' fl'ga rdcd as guidel ines, exclusio n rul es and topolog ical rest rict ion s
on the full complex ity of morphodynamic se.''1uences , .. I Sl'C' [the
INTENSIVE SCIENC E AND VIRTUAL PHILOSOPHY
importance of dynamicism] for self- organizing syste m th eory as
temporary and preparatory for a more co mp lete morphod ynamics
of th e future . And yet, dynamicism eve n now promises a permanent
legacy of restrictions, a taxonomy of lega l, universal restraints on
morphogen etic processes - a Platonic ideali srn .t"
Deleu ze would agree with much of what is ex pressed in thi s
passage, particularly th e characte r izat ion of th e rol e of virtual entit ies
as to po logical restricti on s or co nstraints, that is, as quasi-causal rela
tions whi ch com plement causal ones in th e determination of a given
sel f-organizing or inten sive process. On th e other hand , to view th e
set of top ologi cal restrictions discovered so far as forming some kind
o f fixed , ete rn al taxonomy, would seem to him to defeat th e very
point of po stulating such const rain ts in the first place . No doubt, it is
much simpler to assume the existe nce of Platonic en tit ies than to
define a co mplex ope rat ion through whi ch th ese entities ar e meshed
into a co ntinuum th ereb y acquiring a ce rtain auton omy from actual
events. T he preferen ce for simplicity here , however, has less to do
with the elim ination of redundant features (the legitimate use of
simplicity arguments, as in O ccam's razor) and more to do with
f amiliarity . Arguments based on th e latter , as physicist s conce rned with
the co nce ptual foundations of their sub ject ar e aware, make an
ilkgi timate use of simplicity."? In th e present conte xt , it see m s to me,
to espo use a Platonic ideali sm on th e basis that it is a more familiar
thesis would be misguided . Given that no philosopher (o r scie ntist) has
ever before specified mechanisms of immanen ce, our lack of familiarity
with th e latter sho uld be seen merely as a co nt inge nt fact about
int e llect ual history not as a basis to reject a new theory.
I emphasize thi s point about sim plicity because however complex
th e description of th e virtual may see m so far, it is onl y half th e story.
In parti ular , we ma y grant that th e above description is a reasonable
specificat ion of how a nonmetric space t ime continuum ma y be built
NiI-cn a populat ion of virtual multipliciti es and st ill demand to know
where these multiplicities come fro m. Clearly , they canno t be simply
.iss umcd to exist on thei r own since this would make th em into ent it ies
hard ly d ist ingui shabl e fro m immutabl essenc . Ther is , in fact,
anot lu-r task which th e quas i-ca usa l ope rato r mu st perform, anothe r
THE ACTUALIZAT ION OF THE V IRTUAL IN T IME
immanen ce mechani sm which accounts for th e very ex iste nce of
multiplicities. As Deleu ze ays, th e quasi-cause 'extracts sinBularitiesf rom
the present, and from indi viduals and persons which occupy thi s
present ' . 68 This extraction operation, recovering a full multiplicity from
a partial spat ia- te m poral actualization, defin es th e second immanen ce
mechanism. Del eu ze so metimes usc a geometric characte rizatio n of
thi s operation, describing it as th e ext ract ion of a section or slice.
O rdinar ily, thi s mathematical ope rat ion sim ply reduces th e dimension
ality of th e object to whi ch it applies . A slice of a th ree -dimensional
vo lume , for exam ple , is a two-dimensional surface, whil e th e vo lume
itsel f ma y be viewed as a slice or sec t ion of a four-dimensional
hypervolume. The anal ysis of attractors in state space, particul arl y
strange or chao tic attr actors , makes exte nsive usc of thi s op eration (a
' Po incare sec t ion') to ex t ract information from a complex topological
shape and displa y it in a way wh ich is easier to study."? Deleuze,
however, has a more elabo ra te ope rat ion in mind, one that docs not
have a co unte rpart in mathemati cs.
T o see wh at thi s or iginal slicing ope rat ion am ounts to let 's re t urn
to th e example of th e sequence of flow patterns co nduc t ion-convec
tion-turbulen ce. Let ' s imagin e a co nc re te physical syste m in a state of
convec t ion , that is, actualizing one of th e available flow patterns (a
periodic attractor) . In thi s case , th e virtua l com ponent (the attractor)
exists merely as an effec t of actual causes , such as re lations between
temperature and density differen ces or compe tition between gravita
tional and viscous forces, causal relation s whi ch account for th e
emerge nce and maintenance of co nvection ce lls . Dc lcuze 's hypothesis
is that suc h an actual syste m ma y be 's ampled' or 's liced through ' to
obtain its full qu asi-causal co mpone nt, th e entire set of attractors
defining each flow pattern and th e bifurcati on s whi ch mediate between
patterns. In other words, a Deleuzian sec tion would not consist in a
mere reduction o f th e original dimensionality, but in an elimination of
every detail o f th e actual event except its topoloqi cal im'arian ts: th e
distribution of its singularit ies, as well as th e full dimensionality o f its
sta te space .
Let me spe ll out th e details of th is important idea. I aid in Chapt r
1 that Del uze bor ro ws fro m Riemann th co n ept of an - (limen
siona l mani fold whi h doc ' not n ccd to be mb cdded in a 'pace of
I NTE NS IV E S C IENC E AND VIRTUAL PH I L O S O P H Y
+1 dimensions to be studied , but that constit utes a space on its own ,
each one of its dimen sions defining a rel evant degree of freed om of,
or a relevant way of changing for, a given dynamical system . Each
multiplicit y extrac ted or sampled from actual inten sive processes would
possess a definite dimen sionality (a specific value for the N variable)
since the process it governs is capable of changing in only a finite
number of relevant ways. This finite number of dimensions would
co nstitute a key characte rist ic defining the virt ual multiplicit y as a
conc re te universal entity, and this finite number would vary for
diffe re nt multipliciti es extracte d from different processes. In other
words , the population of multiplicities would be dimensional ly hetero
geneous. Given that the plane of consiste ncy mu st assemble multiplici
ties together by their differences, thi s 'plane' cannot be conce ived as a
two-dime nsional surface but as a space of variable dimensionality,
capable of bringing a dimen sionally diverse virtual population into
oex iste nce . As Del euze writes:
It is only in app earance that a plan e of this kind ' reduces' the
numbe r of dimen sions; for it gathers in all th e dimen sion s to the
ex te nt that fiat multiplicities - which nonetheless have an increasina or
decreasinp number if dimensions - are inscribed upon it ... Far from
redu cing the multiplicities' number of dimen sion s to two, the plan e
I?I consistency cuts across them all, inters ects them in orde r to bring
into coexiste nce any number of multiplicities, with any number of
di me nsions . Th e plane of consiste ncy is the intersecti on of all
concre te forms . .. The only questi on is: Does a given becoming
reach that point? Can a given multiplicity flatten and co nse rve all its
dime nsio ns in th is way, like a pr essed flower whi ch remains just asalive dry?70
Dclcu zc so me times phrases his description as if the qua si-causal
operator was th e agent performing th e extraction or sec tion ope ration,
some other times ascribing this agency to the plane of consi ten cy
itse lf, The di ffercn e bet ween the two formulations is, I believe,
unimportant . What is impo rtant, on the othe r hand , are the det ail of
till' operat ion and thei r justification . In particul ar , th e fact that eac h
mu lt iplicity dd llH's a space of its o wn, that is, the absence cj' a space if
THE ACTUALIZATION O F THE VIRTUAL IN TIME
N+ I dimensions wh ere they would be embedde d, is key to the task of
conce iving a virt ual space which does not unify multipliciti es, that is, a
space composed by th e coe xisting multipliciti es them selves in their
het erogeneity. Similarly , the qua si-causal operator is oft en referred to
as a ' line' but not because it would be a on e-dimen sional entity .
Rather, th e qua si-cause would ope rate at N- ] dimensions, unlike a
transcendent source of unity which mu st op erate from a suppleme ntary
(e .g. N+ 1) dimension . In Deleu ze 's wo rds:
Unity always operates in an empty dimen sion supplementa ry to that
of th e syste m co nside re d (overco ding) . . . [But a) multiplicity never
allows itself to be ovc rcodc d , never has availabl e a supplementary
dimen sion over and abov e its number of lines [or dim en sions) . . .
All multipl iciti es are flat, in the sense that th ey fill or occupy all of
their dimen sions: we will therefore speak of a plane of co nsiste ncy
of multiplicities, eve n though the dimen sions of this ' plane' increase
with the number of co nnec tions that ar e mad e on it. Multiplicities
ar e defined by the outside : by the abstract line , the line of flight
. . . according to whi ch th ey change in nature and connect with
other multiplicit ies . . . The line of flight marks: the reality of a
finite number of dimen sion s that the multiplicity effectively fills; the
impossibility of a supplem entary dimension, unless the multiplicity
is transformed by the line of flight; the possibility and necessity of
flatt ening all of the multiplicities on a single plan e of consiste ncy or
ex te rior ity, regardless of their number of dimen sions.7\
Let me summar ize what I have said about th e two immanen ce
mechanisms. Th operator's first task, to assembl e multiplicities together
by cre ating converge nt and divergent relations among th e ordinal seri es
emanat ing from th em, may be conside re d a pre-actualization . It would
endow multiplicities with a minimum of actuality and, in thi s sense, it
would represent the first broken sym me try in th e cascade that culmi
nat es in fully formed actual beings. The second task of the quasi-causal
ope rator, to ex t rac t vir tual events from inte nsive pr ocesses may, in
turn , be see n as a veritable counte r actua liza tion since it wo uld follow a
direct ion oppos ite to that which goes from the vir tual to till' inte nsive,
and from there to the ex te nsive and qualita tive. 71 'o untc r-actualizat ion
I N T E N SIV E SCI E NC E AN D VI R TU A. L PH IL OS OPHY
would, in fact, compleme nt pre -actualization : while the former e xtracts
flat (o r folded) m ultiplicit ies from act ually occurring events , the latter
would take these and 'unflatten ' them, that is, it would allow them to
progressively unfold and differentiate without fully actualizing them.
Each of these two operations would possess a temporal dimension : the
quasi-causal operator would sample or section all actual events, at alldilTerent time scales, instantaneously; then, each flat multiplicity would
be immediately unfolded in two unlimited directions at once, past and
future, distr ibuting the singularit ies which define each of the unfo lding
levels on both sides of the instant at once, 'in the manner of a pod
which releases its spo res' . 7 l
The operation of pre -actualization would give multiplicities not only
a ce rtain autonomy from the intensiv e processes acting as their realcauses , it wo uld also endow these impassive and stcrile effects with
whatever mo rphogenet ic power they enjoy .7 4 In other words , pre
actualization would not on ly explain how an unactualized singularity
bel onging to a physical system with multiple a!tracto rs wou ld subsist
as a potential alternative state , it would also explain how the singularity
that is actualized gets its power to attract in the first place . To the
ex te nt that linking multiplicities together and endowing them with
productivity foreshadows the intensive processes which follow down
the symm etry- breaking cascade , the quasi-causal operator is referredto as a 'dark precursorT" The operation of counter-actualization, on
the other hand, would operate in the oppos ite direction, up the cascadefrom the inte nsive towards the virtual. I said in Chapter 2 that some
areas of the world, those defined by processes which are nonlin ear and
which operate far from equ ilibrium, do not conceal the virtualunderneath extensities and qualities but rather reveal it, or allow it to
express itself.?" These areas woul d represe nt a spontaneous movement
toward s the virtual wh ich is st ill physica l and co rporeal but whi ch may
I" , given a boost making it reach the level of a pure virtuality. T o th e
e xtent that co unter-actualizatio n accelerates an escape from actuality
which is already present in some intensive processes, the qua si-causa]
0pt'rator is referred to as a 'line of flight '. 77
In conclusion, I wo uld like to repeat that whatever the merits of
Dl'lcuzc 's particular proposals for the implementation of the quasi
t',l U~.l l Opt'rator, we should at least credit him with having e lucidated
THE A C T UA L IZ A. TI ON OF T HE VI R TUA L I N T IME
the overall constraints that any implem entation wou ld have to meet. If
we arc to get rid of essent ialist and typo logical thought we need some
process through which virtual mu ltipliciti es are derived from the actual
world and some process through which the results of this derivation
mal' be gh'en eno ugh co here nce and autono my . Deleu ze himself gave
seve ral different model s for each one of these tasks, a fact that shows
that he did not think he had achieved a final solut ion to the problem,
on ly its correct formulation . On the other hand , he clea rly thought
that the problem itself was worth posing, regardless of its particularsolutions . That this is indeed the case may be glimpsed from the fact
that Dclcuzc ' s description of his co nstruc tivist method in philosophy
close ly mat ches the two tasks whi ch the ope rato r is supposed to
accomplish : creat ing virtual events (m ultiplicit ies) by extrac t ing them
from actual processes and laying them out in a plane of consistency .711
This methodology, moreover, is what in his view would distinguishphilosophy from science . As he writes :
It co uld be said that science and philosophy take oppos ed paths,
because philosophical concepts have events for consistency whereasscientific functions have states of affairs or mixtures for ;eferencc :
through con cepts, philosophy cont inually extracts a consistent event from
the states if eif!airs . .. whereas through functions, science continually
actualizes the event in a state of affairs, thing, or body that can bereferred ro. ?"
It matters little whether we describe thi s method as involving two
separate operations (to extract ideal events and to give them co nsist
ency) o r as a single one (to extract a consistent e vent). The importa ntpoint is that Dcleuze con ceives of pre-actualization and counter
actualization, howe ver implemented, a') defining an object il'e movement
wh ich a phil osopher mu st learn to grasp . As he puts it, we phil oso
phers must invent devices to allow us to becom e 'the quasi-cause
of wh at is produced within us, th e O pe rato r"."? Spelling out the
details of Dcleuvcs methodology will involve co nnecting the resultsof his onto log ical analysis with ques tions of' epistemology . In cp istc
mological terms to extract an ideal event from an actually oecuningone is, hasicall v, to dcfilU' what is problemauc about it , to grasp what
INTENSIVE SC IE NCE AND VIRTUAl. PH Il.O SOPHV
abo ut the eve nt objectively stands in need oj explanation . This involves
discerning in the act ual event what is relevant and irrelevant for its
e xplanation , what is important and what is no t. T hat is, it involves
correctly grasping the objective distribution ojthe singular and the ordinary
defi ning a well-posed probl em. T o give consiste ncy to these well
posed probl ems, in turn , mean s to endow them with a ce rtain aut on
omy fro m th eir particular solutions , to show that probl ems do not
disappear be hind th eir solutions , just like virtual multiplicit ies do not
disappear behind act ualized indiv idu als. T he ep iste mo logical side of a
Dc lcuz ian ontology is co nstitute d by such a philosophy of problems
and this will form the subject matter of the followi ng chapter.
C H A PTE R 4
Virtuality and the Laws if Physics
In a flat ontology o f indi vidu als, like the one I have tri ed to de velop
here , there is no room for rcificd to talit ies . In particular , th ere is no
room for ent ities like ' society ' or 'culture ' in ge nera l. Institutional
organi7..ations, urban centres or nation states are in th is ontology not
abstract totalities but concre te social individuals, :\"ith th e same o~~ologica l sta tus as indiv idu al human bein gs but ope rating at larger spatio
temporal scales . Like organisms or species these larger socia l
indiv idu als are products of co ncrete historical processes, having a date
of birth and , at least potentially, a date of death or ex tinction . And
like organ isms and species, th e rel ations betw een individuals at each
spatia -te mpo ral scale is on e of parts to wh ole , with each individual
eme rging from th e causal int eractions am on g the members of popula
tions of smaller scale individuals. Althou gh the det ails of each individu
ation process need to be described in det ail, we can roughly say that
from th e interactions among individual decision -makers, institutions
eme rge ; from int eractions among institutions, citi es t~merge; and from
urban interacti on s, nation states eme rge _1 The population serving as
substra tum for the eme rg ence of a larger whole may be very hetero
ge neo us or, on the contrary , highly homogen eous. But even in those
cases where the degree of homogen eity at different scales is high
eno ugh to suggest the existe nce of a single 'culture' or 'society' , the
temptation to postulate such to talit ies mu st be resisted, and the degree
of homogen eity whi ch motivated such postul ati on must be given a
co ncrete histori cal explanation .
Thus far I have used the term 'science ' as if its use was unprobl em
atic, but given the requirements of a flat o nto log)' it is clear that this
te rm shou ld no t be used since it refers to an abstract totalitv and
moreover, to a to ta lity defined by an essence . Instead , we mus; : trh-e, ,to ide ntitY the specific processes which have gin>n rise to inJiI"idual
SCIentific fi elJs, which like an) other indivi dual , mus t he conceived as
I N T E N S I V E SC IE NCE AND V IRTUAL PHILOSOPHY
com posed of populations of entit ies at a smaller scale . In the case of
the field of classical mechanics, for ex ample. these co mponents are ,
ro ughly: populations of mathematical models and techniques for the
indi viduation of predi ctions and e xplanations ; populations of phenom
ena produced in laboratories and population s of machin es and instru
me nts wh ich individuate and measure those phen om ena; populati on s
of ex pe rime nta l skills , theoretical conce pts and institutional pract ices.
Like an organic species , the degree to which an individual scie ntific
field has a well -defined identity will dep end on co ntinge nt historical
facts such as its degree of int ernal hom ogen eity and its degree of
isolation from othe r fields. Simil arl y, the degree to whi ch several fields
resemb le each othe r should be given a historical explanation , such as
one field serving as exemplar for the const ruction of ano the r . or the
ex port of inst ruments and techniques from one field to another, or th e
shari ng of institutional compone nts among different fields. This way
the qu estion of whether th ere is such a thing as 'scien ce ' in ge neral
becomes an empirical question, one which, I believe. sho uld receive a
nega tive answer . Many conte mporary ana lysts do ind eed seem to thin k
that, as a matter of empirical fact, science d isplays a deep and
characte rist ic disunity .2
In the first par t of thi s chapter I wo uld like to develop the ideas
needed to think abo ut individual scienti fic fields, using classical mech
anics as a concre te example, but also to review some of th e traditional
philosophical obstacles which have historically prevented a correct
assessme nt of the di sunity, heterogen eit y and divergent development
of 'sc ience". At thi s point it should come as no surprise that in my
view the main obstacle has be en th e entre nchme nt of essentialist and
ty po logical thought in philosophical st udies of scientific practice . Many
philoso phe rs in the past have tak en the essence of classical mechanics
to be its exceptionless laws. Thi s is particularly true when fundam ental
laws, such as Ne wton's law s, ar e view ed as gene ral truths from wh ich
"\'l'rything else[ollows mechanically , tha t is, by simple logical deduction.
\VI1l'n species arc view ed no t as indi vidual entit ies but as gene ral
c.ltegorit."s, the productive or ge net ic processes whi ch yield these
indi vidu als te nd to he ignored . Similarly. the view of law s as general
t ru ths has tended , historically, to eli minate from philosop hical discu s-
V IRTUALITY A ND THE LAWS OF PHYSICS
sian the productive or Benetic connections invol ved in the physical
processes go verned by those law s.
More specifically , the esse ntialist view of laws has co ncealed the
producti ve po we r of causal connections, that is, the fact that events
act ing as causes actually produ ce their effects. Co ntrary to a popular
miscon ception , ph ilosophical approaches to scie ntific practice have
thrived, from the seve nteenth century on , in a world devoid of causes
and ruled e xclusive ly by laws sta ting constant regu larities. J Part of what
made possible the re placement of causes by laws was a view of
causa lity as an inherently linear relation, such that , give n a particular
cause, the same effect was bound to be produce d. Clear ly, if causality
always exhibited thi s simple form , if effects always followed mechani
cally and necessarily from their causes, postulating a separate produc
tive power of causes distin ct from the exce ptionless laws governing
their ope ration would be redundant. But more co mplex forms of
causality do exist , nonlinear and sta tistical causality, for instance, and
these arc involved in all the inten sive production processes which I
have described in previous chapte rs. Hen ce a crucial task for a
Dcleuzian episte mologist invo lves rescuing these ge netic links between
events from the limbo wh ere general laws have cast them.
Besides concealing productive relations behind static categories , the
traditional philosophical approach to laws may be crit icized for subor
dinat ing mathematica l models to Iinaui stic statements . Much of what I
have argued in this book depends on treating mathematical models in
thei r specificity, that is, as disp laying a certain beha viour which is crucial
for thei r successful application to scientific tasks . The most obv ious
example is the tendency of solutions to an equation to approach an
attrac to r , a tenden cy which is not displayed by linguistic translati ons
o f the conte nt of the equation but which dep end on the speci fic
math ematical form of both the equation and the ope rato rs that act on
it. Thus, a second task for a Deleu zian episte mologist is to rescue
models and their dynamic behaviour from static linguistic renderings
of law s. These tw o related errors , elimination 1" causes and subordination
to lanau one (and deductive logic) arc the basic characte rist ic of essen
tialist approaches to classica l physics , and their criticism will form the
suhj('ct matt er of the first sec tion of this chapt e r , Let ow begin with
I NT E NSI V E S C I EN CE AN D VIRTU A L P HI LOS O P HY
tlu- dismi ssal of productive causes in favour of constant regularities. As
th l' ph ilosopher of scien ce [an Hacking puts it:
Ilume notoriously taught that cause is only constant conjunct ion .
To say that A caused B is not to say that A, from some power or
characte r within itself, broupht about B. lt is only to say that th ings
uf type A are req ularlv followed by things of type B ... Hume is in
1:1 t not responsibl e for th e widespread philosophical acceptance of
a co nstant -co njunction attitude towards causat ion. Isaac Newton did
it, unintentionally. The greatest triumph of th e human spirit in
l lu rnc ' s day was held to be th e Newtonian theory of gravitation
. .. Immediately before Newton , all progressive scientists thought
that th e world must be understood in terms of mechanical pushes
anti pulls. But gravity did not see m ' mechanical' , for it was action
at a distance .. . For em pirically minded peopl e th e post -N ewtonian
alt itude was, th en, this: we should not see k for causes in nature,
hut only regularities . .. The natural scie ntist tries to find universal
state me nts - th eories and law s - which cover all ph enomena as special
cases. To say that we have found th e explanat ion of an event is only
to say that th e event can be deducedfrom a aeneral repul aruy,"
Ii a king argues that this elimi nat ion of producti ve causes in favour
of state ments o f regularities (and deducti ve relations between those
si.ucmc nts) is characte rist ic not of physics in general, but only of
phi loso phies of physics whi ch concentrate exclusively on th e th eoretical
t ompo nent of a field at th e expe nse of its expe rimental compone nt .
T he day to day practi ce of expe r imental physicists, consist ing as it does
in specific causal int erventi ons in reality , is much too rich and complex
10 he rcdu cd to logical relations between state ments. The expe riment
alist is d irectl y invo lved in productive relation s, whether th ese involve
till creat ion o f an apparatus to individuate ph enomena or th e use of
inst rum -nts to produce individual measurements of properties of those
ph -no rncna . It is only th eory-obsessed phil osophies, whether held by
physicisls o r professional philosophers, that can afford to forget about
1..ur sa] co nnec tions and co n e ntrate ex lusively on logical relation s. The
ult imatc ex press ion of thi s esse nt ialist stance is a model of s icntific
c-xplnnnt ion de velop ed in the twentieth c >ntury whi ch takes th e
VIRTUAL ITY A N D TH E LAW S OF PH Y SI C S
Humean reduction of cau ses to linguistic statements of regularities to
an extreme.
In this episte m ological th eory, known as th e deduai ve-nomolopical
approach , scientific explanat ions ar e treated as logical arguments consist
ing of several propositions, on e of which must be an exceptionl ess
law . The term 'proposition ' refers to th e meaning of declarative
sentences , that is, to what two sentences in different languages,
expre ssing th e sam e state of affairs, have in common. In this model,
to explain a particular laboratory phenomenon is to deduce it from a
set of propositions: from a linguistically stated law (su ch as 'two bodies
are gravitationally attract ed to each other in direct proportion to th e
product of th eir mass es, and in inverse proportion to th e square of
their distance ') and a set of propositions describing initial (and other)
co nditions, we derive further propositions which may be treated as
predictions to be tested for th eir truth or falsity in a laboratory. [I' th e
behaviour of th e ph enomenon conforms to th ese predictions we can
cla im to have explained it , not, of course, by having given causal
mechanisms for its production, but in th e way one explains things in a
ty pological approach: subsutninp it as a particular case under a aen eral
catea0T)'. Although hardly any w orking physicist would accept that his
or her co m plex explanatory practi ces are captured by this simplistic
theory , th e deductive-nomological approach has dominated much of
twentieth-century philosophy of scien ce and continues to have many
defenders in thi s field. 5
When on e accepts thi s model of explanation th e structure of th e
th eoretical component of a scientific field takes th e form of an
axiomatic: from a few true statements of general regularities (the
ax io ms) we deduce a large number of consequences (theorems) which
are th en compared to th e results of obs ervations in th e laboratory to
check for their truth or falsit y . Given that deduction is a purely
mechanical way of transmittina truth or falsity, it follows that whatever
truth on e ma y find in a th eorem mu st hav e already been contained in th e
ax ioms. It is in thi s sense that axioms are like esse nces. To counter
thi s essentialist concept ion , a new gen eration of philosophers has
d eveloped an alt ernative charac te rizat ion of what a th eory is, reintro
du ing productiv causa l rel ations as an int egral part of explanat ions,
as well as rejecting th e Iingui sti haracterizati on or explanator
I N T E N S I VE S C IENC E AND VIRTUAL PHILOSOPHY
practices . In th e view of these phil osophers, ex planations , rath er than
being simply logical arguments, involve a complex use of mathem atical
mo dels of different types: mod els of gene ra l relation s, models of
partic ular experimental situations, as well as sta tistical models of th e
raw data gathe re d in laboratori es. O ne of the de fenders of this new
view, Ronald Giere , puts it this way:
Even just a brief examination of classical mechanics as pr esented in
modern textbooks provides a basis for some substantial conclusions
about the overall structure of this scientific th eory as it is actually
understood by th e bulk of th e scientific community . What one finds
in standard textbooks may be described as a cluster (or cluster of
clu sters) of models , or , perhaps better, as a population if models
consisting if related famili es if models. The various families ar e
constructed by combining Newton's laws of motion, particularly
the second law , with various for ce functions - linear functions,
inverse square functi ons, and so on . The models thus defined are
then multiplied by adding other force functions to th e definition .
T hese define still further families of models. And so on."
Giere emphasizes the point that, despite the fact that some members
of this population of mod els (Newto n's laws of moti on ) serve to
generate the various branchin g famili es , the relation bet ween a funda
mental model and those deri ved from it is not like that between axioms
and theore ms. Far from being a mechanical process of deduction , the
com plex modelling practices which have historically gene rated th ese
families invol ve man y judicious approximations and idealizati ons,
gUided by pri or achievements serv ing as exe mplars ." I will return to
this ques tion in a moment but for now I would like to add that th e
basic idea of th inking of a physical theory as a population of models
fits we ll with th e onto logical stance I am defending . Such a population
is easi ly conce ived as the product of a historical accumulation, subject
to all the co ntinge ncies of such histori cal pr ocesses, and hence with no
pr ·t n e that it represen ts a co mplete or final set of mod els. At any
rate, the completeness or clos ur of the set becomes an empirical
matt 1', not .orncthing to b assumed at the outs t as in axiomatic
trva t rncnts . ~ rtai n popul ations (like those of the sub -fi lei of classica l
V IR TUAL IT Y AND T H E L A W S OF PHYS I CS
dynamics) may see m to have achieved closure at a ce rtain point in
history only to be reop en ed later giving rise to a new round of
accumulation, as when compute r- dr iven developments in nonlinear
dynami cs reop en ed what was widely conside re d a closed field . As I1ya
Prigogin e puts it : ' Unfortunate ly, many co llege and uni versity text
bo oks pr esent classical dynamics as a closed subject .. . [but] in fact ,
it is a subjec t in rapid evo lution. In the past twenty years , [physicists]
have introduced important new insights , and further developments can
be ex pected in the near future. ' 8
The philosopher of scie nce Nancy Cartwright has pr op osed a set of
distinctions that may be used to describe the non-axiomatic stru cture
of this population of models. Som ewhat parado xicall y, she argues that
the fundamental laws of physics, those laws which in axiomatic
treatmen ts ar e assume d to be the highest truths, are indeed false. Th e
laws of physics lie , as she puts it . What she means is that a fundamental
law achieves its generality at the expense if its accuracy. A fundamental
law , such as Newton 's law of gravity, is strict ly speaking true onl y in
the most artifi cial of circumstances, wh en all other forces (like
electromagne tic forces) ar e absent, for instance , or when there is no
fr iction or other nonlinearities. In other words, the law is true but
on ly if a very large 'a ll othe r things being equal' clause is attached to
it .? W e can compe nsate for the sho rtcomings of fundamental laws by
adding to the basic equation other equations representing the action of
other forces or th e complex causal interacti ons between forces. But
then we lose the gene rality that made the orig inal law so appealing to
essentialists. Th e mod el becomes more true, describing with increased
accuracy the structure of a given expe rime ntal phen omenon , but for
the same reason it becomes less gen eral. In short , for Cartwright th e
objec tive conte nt of physics do es not lie in a few fundamental laws,
but in a large number of causal models tailored to speci fic situations.
(Giere does not speak of 'causal models' but of 'hypo these s ' linking
th e abstract models and the world , but th e overall thrust of his
argument is very close to that of Cartw right .10)
The esse ntialist may obj ect that , given that the speci alized causal
mod els are d rive d fro m the fundamental laws , th y mu st inherit
what ver degree of truth they have fro m thos law s. But Ca rtwright
(l ike Giere) rcpl ic that this oversim plifies the description of the
INTENSIVE SCIEN CE AND V IRTUA L PHI LOSOPHY
modelling practices of real physicists. The causal models are not
logicall y deduced from th e general laws, but construct ed from them
using a complex set of approximation techniques which cannot be
reduced to deductive logic. As Cartwright says, the content of th e
causal models 'we derive is not contained in th e fundamental laws that
exp lain them. ' II In short, the population of models whi ch con stitutes
the theoretical component of classical mechanics may be roughly
divided into two sub -populations: a large number of causal models
closely adapted to particular expe r imental situations, and a few
fundame ntal models corresponding to basic laws from which branching
families of other abstract models ar e derived . This breakdown of the
ontents of the population leaves out a different class of models,
statistical models if the data, which is also very important. Positivist
philosophers used to think that the predictions deduced from axioms
and auxiliary premises (those describing initial conditions) were con
fro nted directly with observations in a laboratory, that is, with raw
data. But for at least two hundred years physicists have used statist ical
mo dels to organize the raw data, and, in particular, to attempt to
capt ure the distribution ifmeasurement errors in th e data. 12 Beside ignoring
th is important kind of model , th e po sitivi st emphasis on 'the observer'
is misleading because it reduces to a subjective phenomenon what is in
fact a complex practi ce of data gathering, involving not passive
observations but active causal interventions.
Leaving aside the expe rimental side for a moment, what ar e we to
think of the few fundamental laws ? Is it correct to say that they lie, or
is it more accurate to say that they are not the kind of mathematical
objects that can be true or false? Cartwright suggests that the function
of these laws is to un!fj and oraanize th e rest of the population. 13 This
is, I beli eve, a ste p in the right direction but we cannot simply take
this unifying capability for granted; we must at least try to account for
it. Histori cally, the unification of the different branches of classical
mechanics was achieved by a ser ies of physicists and mathematicians,
start ing with the work of Leonard Euler in th e mid -eighteenth century
• nd culminating a hundred years later with that of William Hamilton.
It may be said that, togeth I' with other important figures (Maupc rtuis,
I ,lgr.1I1gl'), th esc scie ntists transformed classical me han ics from a
VIRT UA L I TY AND TH E LAW S O F PH Y S ICS
science of forces to one of sinq ularities. In the words of the historian
Morris Kline:
Hamilton 's principl e yields the paths of falling bodies, th e paths of
projectil es, the elliptical paths of bodies mo ving under th e law of
gravitation, the laws of reflection and refracti on of light, and the
more elementary phenomena of electricity and magnetism. How
ever , the chief achievem ent of the principl e lies in showing that the
phenomena of all these branches of physics satisfy a minimum
principle. Since it relates these phenomena by a common mathemat
ical law, it permits conclusions reach ed in one branch to be
reinterpreted for another. Hamilton's principle is the final form of
the least -action principle introduced by Maupertuis, and because it
embraces so many actions of nature it is the mo st powerful single
principle in all of mathematical physics. 14
Th e history of minimum principles, the idea that, for example , light
moves along the path that minimizes travelling distance, is ind eed a long
one having roots in Greek antiquity and medi eval philosophy. IS In th e
seve nteenth century , Pierre de Fermat cre ated th e first application of
this idea in th e conte xt of ear ly modern physics, the Principle of Least
T ime govern ing the behaviour of light in geometrical optics. For much
of its history the principl e carried strong theol ogical overtones as it
was assoc iate d with the beli ef that it reflect ed the econo my of thought
of a Cre ator. Maupertuis eve n went as far as to state that his Least
Action principl e was the first scientific proof of the existe nce of God.
Event ually the theological connection was lost , as scientists realized
that what mattered was not th e ideological interpretation but the
math ematical technology that was create d around these ideas: the
calculus if variations. Thi s was the first technology ever to deal directly
with singular it ies and it rivals in importance, as far as its effects on
ninet eenth- and tw entieth -century physics, the other mathematical
fields I have discussed in this book (differential geometry, group
thcorvj.!"
One way of looking at the calculus of variations is as a novel way of
I'0sinfl mechanical problems. Instead of looking at a problem in physics as
I N T E N SIVE S C I E N C E AND VIRTUA L PHI LOSOPHY
a problem of the causal effccts of forces, one looks at it as a pr obl em
of finding, am on g the many possible pr ocesses that ma y change a
physica l systc m from one sta te to ano ther, the actual process. More
exactly, the techniques develop ed by Euler and Lagrange allow th e
construction of a set of possibiliti es (fo r exa mple, a set of possible
paths which a light ray might follow) and supply th e resources needed
to sort these possibilities into two gro ups, on e of ordinal)' and one of
sinpulat cases . As it happen s, the results of expe riments show that the
singular cases (a minimum or a maximum) are the ones that are in fact
act ualized ."? Although the singularities uncovered by the calculus of
variations are not, st ric t ly speaking, attractors, its cre ators did see m to
thi nk that they played a similar role . Attractors are described as
defining the long-term tendencies of a syste m , that is, the state the
syste m will adopt if we wait long eno ugh to allow it to settle down.
Thi s emphasis on th e final sta te sugges ts that one way to look at the
difference be tween attractors and causes is through the old distinction
mad e by Aristotl e between final and qpcient causes. Euler him self,
when introducing his var iatio nal techn ology, used this Aristotelian
distinction:
Since the fabr ic of the universe is most perfect , and is the wo rk of
a mo st wise Cre ator, nothing whatsoever takes place in the universe
in whic h some relat ion of maximum and minimum does not appear.
W herefore there is absolutely no doubt th at eve ry effect in the
univ erse can be explained as satisfacto rily fro m final causes, by the
aid of the meth od of maxim a and minim a, as it can from the
effective causes the mselves .. . Therefore, two method s of studying
effects in Nature lie ope n to us, one by means of effective causes,
which is co mmo nly called the direct method, the other by means of
fina l causes . . . O ne ought to make a speci al effort to see that both
wuys if approach to the solution if the problem be laid ope n ; for thus
not only is one solutio n greatly stre ngthe ned by th e othe r, bu t ,
more than that , fro m the agreement between th e tw o so lutions we
sec ure the very highest satisfaction. 18
In a Dc lcuz ian ontology final causes wo uld have to be replaced by
quasi -causes in order to avoid ascribing teleological or goal-seeking
VIRTUAL IT Y AND THE LAWS OF PHYSICS
behaviour to physical systems . But the impo rtant poin t for my
argument is that it was precisely the ability to pose a pr obl em not in
te rms of specific efficient causes (forces) but in a way which by-p assed
causal det ails, that allowed the variational version of classical mechani cs
to play a unifying and organizational role in th e population of models.
The singulari t ies whi ch th e calculus of variatio ns uncovered rep res
en te d, in my terminology, a mechanism -indep endent reality. On th e
other hand, as Euler himself acknowledged , thi s method was comple
mentary not exclusive to the causal one . O ne may know that a given
classical mechanical pr ocess will tend to minimize some quantity, but
the full ex planation of the process will also invol ve a correct descrip
tion of the causal mechani sm s that achieve such minimizati on . Thi s
other task, how ever, must be performed by other mod els, less ge neral
and more specifically tailored to th e details of an ex perime ntal
situatio n .
To summar ize the arg umc nt of thi s sec tion , far from being mere
mathematical express ions of linguistic truths, laws must be viewed as
models fro m which th e m ath em atical form cannot be eliminated . Th e
un ificatio n brought about by the calculus of variat ions, for ex ample,
cannot be understood otherwise since its techniques do not appl y to
ling uistica lly stated laws. These irre ducibly mathematical models form
a gro wing and heterogen eous populati on , some members of which
carry causal informatio n about pr oductive relati ons between events,
ot hers embody quasi-causal relations between singular itics . In other
words, the populat ion of mod els making up the th eoreti cal compone nt
of classical mechanics contains a large number of speci fic causal models
which are th e vehicles for truth (the part of the population that
inte1aces with the actual world), and fewer models whi ch do not refer
to the act ual worl d (hence are neither true nor false) but wh ich
nevertheless do inteJjace with the virtual world by virtue of being well
posed problems. For Deleu ze a probl em is defined precisely by a
distribut ion of the singular and thc ordinary, th e important and the
un importan t , the relevant and the irrelevant. A we ll-posed probl em
ge ts these distribu tion s right, and a solut ion always has th c truth it
des erves acco rding to how well specified the co rrespo nding pr obl em
is .!" In these t rm s N iwton' s a hi vem nt wo uld co n ist not in having
discovered gene ral tru ths about the uni v ersc , but in having correct ly
INTENS IVE S C IEN CE AND VIRTUAL P HIL O SOPH Y
posed an objective problem defined by the simplest distribution of
singular ities (unique minima or maxima). This interpretation preserves
the obj ectivity of Newton ' s law s but it deflates his achievement
somewhat, in th e sense that , if th e insights of nonlinear dynamics
abo ut multiple attractors ar c correct , th e single minimum problem is
not th e most general on e .
T his conclusion assumes, however, that the traditional axiomatic
appro ach to physics can be replaced by a problematic approach, that is,
that problems can replace fundamental law statements. But this
replacement needs more justification given that it go es again st th e grain
of th e traditional ontology of physics . Hamilton 's Least Action prin
ciple , for exam ple, is still interpreted by most physicists as an axiom
ex press ing a general truth from which many particular truths in physics
follow mechanically. As Morris Kline puts it:
To th e scient ists of 1850, Hamilton's principle was th e realization
of a dream ... From th e time of Galileo scientists had been st r iving
to deduce as many phenomena of nature as possibl e from a few
fundam ental physical principles ... Descartes had already expressed
the hope that all th e law s of science would be derivable from a
sing le basic law of th e universe. i?
And, I sho uld add , thi s hope for a single law state ment from which
every thing else follows has displayed a consid erable resilien ce and
long vity, st ill animating th e dream for a final theory among some
onte mpo ra ry physicists. Therefore the task for th e next section of this
chapter will be to describe in more detail the extra-propositional and
sub-representative nature of th ese distributions of th e important and th e
unimportant which arc supposed to replace law statements as well as
esse nces . In Dcleuzc 's words:
It will be said that th e essence is by nature th e most 'important'
thing . This, however, is precisely what is at issue: whether notions
of' importance and non -importance ar c not precisely notions whi ch
co nce rn events or accidents, and arc mu ch more 'important' within
accid ents than th e crud, oppos it ion between esse n e and acc ide nt
itself. Th e probl em of th ought is ti d not to esse n cs but to th
VIR TUALITY AND THE LAW S O F PHY S ICS
evaluat ion of what is important and what is not, to th e distribution
of th e singular and regular, distinctive and ordinary points, whi ch
takes place entirely within th e un essential or within th e description
of a multiplicity, in relation to th e ideal events that const itute the
conditions of a probl em. 21
I will focus first on a particular kind of problem, explanatory problems,
to show th e rol e which th e cau sal and th e quasi-cau sal play in th e
explanat ion of physical phenomena. As Ian Hacking has argued, th e
same positivist biases which promote th e beli ef that causality is not an
obj ective relation also promote th e downplaying of explanation as an
epistem ological activity, that is, promote th e po sitivist thesis that
'explanations may help organize phenomena, but do not provide any
deep er answer to Why questions ... ' 22 To th e non-positivist philo
sophe rs who arc reviving th e study of causality, on th e contrary,
questions as to why a phenomenon occurs are crucial since they require as
answers more than a mere description of regularities. Answering a
Why question typ icall y demands supplying a causal explanat ion , per
hap s in th e form of a causal model of a mechanism. In addition, I will
argue that th ese qu estions sometimes require supplying a quasi-causal
factor to explain whatever regularity there is in th e beha viour of th e
mechanism s, that is, to capture the m echanism -indep endent aspect of
th e ph enomen on .P Despite th e fact that qu estions and answers ar e,
indeed , linguistic enti ties , Why qu estions involv e as part of th e
co ndit ions that make th em answerable, or well -posed, a non-linguisti c
or ex t ra-propositional aspect whi ch is properly problematic: a distri
bution of the relevant and th e irrelevant. Let me begin this new
sec tion with a quote from th e philosopher Alan Garfinkel who has
developed an original approach to these matters:
When Willie Sutton was in prison, a pri est who was trying to
reform him asked him why he robbed banks . 'Well , ' Sutton repli ed,
' that's whe re th e money is.' There has been a failure to connect
here , a failure of fit. Sutton and th e pri est are passing each other by
. . . Clea rly th ere arc different values and purposes shaping th e
qu stion and answer. Th 'y take different thing to be problemat ic or
stand in n cd of ex planat ion. For th pri e t , wh at sta nds in need of'
IN TEN S I VE SC IENCE AND V I R T U AL P H ILOSOP HY
explanation is the decision to rob at all. He do es not really care
what . But for Sutton, that is the whole question . What is problem
at ic is the choice of what to rob. 24
Garfinkel suggests that requests for explanations may be modeled as
questions having the form ' W hy did event X (as opposed to Y or Z)
occ ur?' with the clause in parenthesis constituting what he calls a
contrast space. The misunderstanding between the thief and the pri est in
his example is du e to the fact that each is using th e same question but
with different contrast spaces . While for the thief the question is 'W hy
rob banks?' (as opposed to gas stations or retail stores) for the priest
the question is 'Why rob banks ?' (as opposed to making an honest
living). The thief's answer is ind eed a true answ er, but as far as th e
pri est is concerned, it is an irrelevant answer, a fact that suggests that
the rel evancy and valid ity of an explanation is relative to a particular
contrast space . These spaces capture both what is presupposed in a
question (Gi ven that one must rob , why banks?), and hence considered
to be not in need of explanation, as well as the rel evant explanatory
alternatives . Garfinkel argues that characte rizing contrast spaces invol ves
going beyond the resources of language , even in cases (like the thief
and pri est exam ple) wh ere the situation is mostly linguistic. As he puts
it:
T hese contrast spaces are still not well-understood obj ects. Th eir
structure is not readily identifiable with any of th e traditional objects
of logic, for example . They have some similarities with ' possible
worlds', for instance, but th ey are not simply spaces of possible
wo rlds . Th ey are more like equivalence classes of possible worlds
(unde r the relation 'differs inessentially from') with almost all
possible worlds excluded altogether from th e space. (Contrast spaces
are typically quite small.) .. . Basically, these spaces are similar to
what physicists call stat e spaces. A state space is a geometric
representation of th e possibilities of a system; a parametrization of
its states , a display of its repertoire .25
I have already discu ed why linguisti cally sp Hied possible worlds
f: iI to break with esse ntialism, and how bringing in math matical
V IR TUA L ITY AND THE LAWS OF PH YS I CS
entmes (such as state spaces and th eir attractors) can eliminate the
need to characte rize rel evant alte rnatives (equivalence classes) through
relations like I differs inessentially from ' . In a typical nonlinear state
space , subdivided by multiple attractors and their basins of attraction ,
the structure of th e space of possibilities depends not on som e
extrinsically defined relation (specifying what is an inessential change)
but on the distribution if sinpularities itself. Th e traj ectories in state
space , defining possible sequences of states, are spontaneously broken
into equivalence classes by the basins of attraction : if the starti ng point
or initial condition of two different traj ectories falls within a given
basin both traj ectories are bound to end up in the same state, and are
equivalent in that respect. Garfinkel, in fact, acknowledges the rol e
which attractors may play in structuring the contrast spaces of physical
and biological explanations. As he says, 'What is necessary for a true
explanation is an account of how the underlying space is partitioned
into basins of irrelevant differences, separated by ridg e lines of cr iticalpoints. '26
How does a distribution of singularities obj ectively define th e
correctness or truth of a problem? Th e answer is that, as Del euze says,
'there are problem s which are false through indet ermination, others
th rough overdetermination'. 27 In other words, a problem may be false
or badly posed if the alternatives which str ucture a contrast space are
roo sharply difined , since in that case th e validity of the explanation
becomes too dependent on the occurrence of precisely those events
(overde te rmination). On the contrary, the problem may fail to be true
if it is so vapuely difined that it is impossible to tell whether an actually
occ ur ing eve nt belongs to on e or another of the relevant alternatives
(inde te rminat ion) . Let me give an example of a problem which is not
w II posed du e to its conditions being overdetermined. Garfinkel
illustrates this case with a well-known ecological phenomenon, the
rhythmic or periodic changes in the overall numbers of coupl ed
populations of prey and predators (rabbits and foxes, in his example) .
As the population of rabbits increases th e foxes' numbers also increase
du to the ex tra available food. But at some point, there are too man y
fox 's 0 that th population of rabbits is reduc d. This, in turn, brings
down the number of fa xes, which allows th rabbit population to
recover and sta rt the cycl again. Thi s cycl ic beha viour of the ouplcd
INTENS IVE SCI ENCE AND VIRTUAL PHIL OSOPH Y
populations is what is ecologically problematic about th e situation, that is,
what demands an explanation . 28
W e may pose the problem in two alternative ways, on e at the level
of intera ctions between individual rabbits and foxes, which gives an
ove rde te rmined contrast spac e with too many alternatives, and another
at the level of the overall density of th e populations yielding a well
posed problem. To put this in linguistic terms, if we posed the
pr obl em 'Why was this rabbit eaten?', one answer may be framed at
the population level (because of th e large number of foxes) and another
at the organism level (because it passed through the capture space of a
speci fic fox at a specific time) . In other words, on e problem is 'Why
was thi s rabbit eaten (as opposed to not eaten)?' while the other is
I Why was this rabbit eaten (by this particular fox as opposed by this
or that other fox)?'. The second contrast space includes much that is
irrelevant to the question since , given a high enough density of foxes, if
this rabbit had not been eate n by this fox it would have been eaten by
ano ther. In other words, there is a ce rtain degree of redundant causality
ope rat ing at the micro-level, so that framing th e question at that level
is bound to yield the wrong distribution of the important and the
unimpo rtant. 29 The second way of framing th e question is, as Garfinkel
says, explanatorily unstable:
The gene ra l crite rion in th e cases we are dealing with is that an
objec t of explanation should be chosen whi ch is stable und er small
perturbations of its conditions. In the whole microspace of the foxes
and rabbits syste m there is a point corresponding to the death of
that rabbit at th e hands of that fox, at that place and time, and so
forth. Now imagine a kind of mesh laid over the space, which
det ermines what is to count as relevantly the sam e as that event.
[This is, in effect , the contrast space of the explanation.] If the mesh
is very fine, the resulting causal relations will be relatively unstable .
P .rturbing the initial conditions slight ly [say, making the rabbit pass
not so near that fox] will result in a situation which is different,
in '(Iuival -n t . [The rabbit not being eaten by that fox.] If however,
w · choose a mesh large enough (and cleverl y nou gh ) we can
·apturt· a sta ble rel ation , lik the on b twc m high fo population s
VIRTUALI TY AND THE LAWS O F PHYS ICS
and high likelihoods of rabbit deaths. [Where changing the path of
the rabbit st ill results in its being eate n but by another fox. 30]
Using the notion of explanatory stability , Garfinkel develops an
application of contrast spac es to differentiate the validity of expla
nations operating at different scales of reality. In the context of a flat
ontology of individuals this differentiation is crucial since we would
like to have objective criteria to tell when an explanation is valid at
the level of individual organisms , for example , and when we need an
explanation at the spatio-t cmporal scale of an individual species. In th e
example just mentioned, a population-level intensive property (density)
can furnish a more stable explanation of the cyclic behaviour of the
pr ey-predator system than an organism-level on e. Similarly for expla
nations of social phenomena, some will be adequate at th e scale of
individual subj ects, others will serve to answer Why questions at the
scale of individual institutions, and yet others will capture the relevant
causal effects of individual citi es or nation states.
In short, causal problems should be fram ed at th e correct level
given that each emergent level has its own causal capacities, th ese
capacit ies being what differentiates these individuals from each other.
But what about quasi-cau sal factors, how do they affect the success or
failure of explanations? To return to our example , if the properties of
the cyclic dynamics of the prey-predator syste m , the duration of the
cycle , for example , are not stable, that is, if exte rn al shocks can easily
change this duration, th en there is no need for quasi -causal factors.
But , on the other hand, if such shocks only temporarily change the
dura tion and th e cycle spontaneously returns to its original period,
then there will be an aspect of the dynamics not explained by the
causal model, a mechanism-independent aspect which still demands
explanation. Population biologists have in fact observed such stable or
ro bust cycles both in the field and in the laboratory, a fact that has
influen ced the introduction of attractors as part of their explanatory
mod cls.! '
I sho uld cmphasiz that, despite my choi ce of example , there i
nothing speci fically biological about this argument. Th e ex act same
ideas apply to syst im s of causally int ern ting populations of inorg. nic
I NTE N S I V E SC IEN CE AND V IRT UAL P H I L O S O P H Y
entities. I have mentioned seve ral tim es th e regim es of flow of
convection and turbulence . When explaining such phenomena one has
to frame the problem at the correct level so as not to introduce
irrelevant differences. Given a convec t ion cell and its cohere nt cyclic
behav iour, for example , there are a large number of micro-causal
descript ions (of indi vidual mol ecules colliding with on e another) which
are irrelevant to its explanation. In other words, there is a larg e causal
red undancy at the micro-level , with many collision histories being
.ompatible with the same macro-level effect: a coherent cyclic flow
pattern. Here th e proper level of explanation will involve macro-causal
factors: temperature and density gradients, competition between grav
itational and viscous forces, and so on. Moreov er, th e existence of
ritical thresholds recurring at regular values for th e gradients (struc
tural instab ilities) and th e robustn ess of the recurring flow patterns to
shocks (asymptotic stability) will call for additional qua si-causal factors:
bifurca tio ns and periodic attracto rs. (O r, in the case of turbulence ,
chaotic attractors.)
Let me pause for a moment to bring th e different lines of the
argument together, and th en link the conclusions to those reached in
previous chapte rs . I argued first that the axiomatic approach to classical
mechan ics, exem plified here by th e deducti ve-nomological model of
ixplanation , views laws as the main car rie rs of objective truth , a truth
which is then mechanically transmitted to th eorems via deduction.
Exp laining a given phenomenon is modelled as a logical argument ,
subsuming the truth of a theorem describing the phenomenon under
the trut h of a law. An alte rnative approach, a probl ematic approach,
I' jects the idea that fundamental laws express gene ral truths and views
th ern instead as posing cor rec t problem s. Problems are defined by
th ir presuppos itions (what is not being explained) as we ll as by their
contrast spaces (defining what the relevant options for explanation
ar i) . In the particular case of ex planations in classical physics, where
the laws are expressed by differential equations, th e presuppositions
arc th physical quantities chose n as relevant degr ees of freedom
(w hich makc up th different dimensions of a state space) while the
co nt rast spa is defined by a distribution of singularit ies in sta te space,
that is, by a part i ular partition of possibiliti es into disti nct basins of
attraction. As the xam pl of hydrod ynamic reg imes of 1I0w shows,
VIR T UA LI T Y A ND T HE L A WS OF PHYSICS
how ever, a contrast space may have a more complex structure: a
cascade of symmetry-breaking bifur cation s may link several such spaces
in such a way th at a problem may aradually specify itself as the different
cont rast spaces it co nta ins reveal themselves, one bifurcation at a time .
These conclusions are directl y connected with the onto logical ideas
I explo red before, but to see this connec t ion we must expand the
conce ption of probl em s beyond those involving scientific explanations.
In Deleuze 's approach the relation between well -po sed ex planato ry
problems and their true or false solutions is th e episte mological
co unte rpart of th e onto logical relation between the virtual and the
actual. Expl anatory problems would be the co unte rpart of vir tual
multiplicities since, as he says, 'the vir tual possesses the reality of a
task to be performed or a probl em to be solved' . 32 Individual solutions,
on th e other hand , would be the counte rpart of actual individu al
beings: 'An organism is nothing if not th e solution to a problem, as
are each of its differenciated organs, such as the eye which solves a
light probl em .P ? Let me illustrate th is idea with a simple example I
used before : soap bubbles and salt crystals, viewe d as the emerge nt
result of int eractions between their constituent molecules. Here th e
problem for the population of molecules is to fi nd (or compute its way
co) a minimal point of ene rgy, a probl em solved differently by the
molecules in soap films (which collec tively solve a minimization
problem state d in surface-te nsion terms) and by th e molecules in
crystalline structures (w hich co llectively solve a bonding ene rgy prob
lem). It is as if an ontologica l problem, wh ose conditions are defined
by a unique singular ity, 'explicated' itself as it gave rise to a variety of
geometr ic solutions (spherical bubbles, cubic crys tals). 34
This intimate relation between episte mo logy and ontology, bet ween
problems posed by humans and self-posed virtu al problem s, is charac
teristic of De1euze. A true problem, such as th e one which Newton
posed in re latively obscure geometric terms and which Euler , Lagrange
and Hamilto n progressively clarified , would be isomorphic with a real
virtual problem . Similarly, the practices of ex pe rime ntal physicists,
whi h includ e amo ng other thin gs the skilful use of machin es and
instruments to individu ate phenomena in the laboratory, wo uld b
isomorphic with the intensiv proc sses of individuation which solv
or .xpli at' a virtual problem in rea lity. This co nce ptio n of th task of
INTENSIVE SCIENCE AND VIRTUAL PH ILOSOPHY
theoretical and experimental physicists runs counter to the traditional
realist picture which views it as that of producing a corpus of linguistic
propositions expressing true facts which mirror reality. In this old and
tired view, the relation between th e plan e of reality and that of physics
wo uld be one of similarity . Yet, as Deleuze says, there is 'no analytic
resemblance, correspondence or conformity between the two plan es.
But their indep endence do es not preclude isomorphism . . . ' 3S Indeed,
as I said in the conclusion of the previous chapter, there is a further
isomorphism which must be included here: the philosopher must
become isomorphic with the quasi -causal operator, extracting problems
from law-expressing propositions and meshing the problems together
to endo w them with that minimum of autonomy which ensures their
irreducibility to their solutions.
In the second part of this chapter I would like to discuss the details
of these isomorphisms, one involving the experimental, the other the
theoreti cal component of classical physics . This will imply dealing with
both sides of the relation, that is, not only the laboratory and modelling
pra cti ces of physicists, but also the behaviour of the material phenomena
and machinery which inhabit laboratories as well as the behaviour of the
mathe matical models with which the theorist makes contact with the
virtual. I will begin with a discussion of how the capacity of material
and ene rge tic systems to self-organize and self-assemble, a capacity
which reveals a properly problematic aspect of matter and energy , is
co ncea led when physici sts or philosophers focu s on linear causality at
the xpe nse of more complex forms. Yet, I will also argue that even if
a material system under study has been fully linearized and domest
icated, the causal relations between experimentalist , machines, material
phenomena and causal models are still nonlinear and problematic. Indeed,
the physics laboratory may be viewed as a site where heterogeneous
assemblages form, assemblages which are isomorphic with real intens
ive individuation processes.
I will th in move on to questions of quasi-causality and compare
Dclcu z ·'s episte mological approach to state space, an approach that
emphasizes the singularit ies that define the conditions of a theoretical
pr obl em, to thos of analytical philosophers who stre ss the solut ions
to the problem, that is, who sec not the singularit ies but the
trajectori es in state spa c as the conv 'yors o f theoreti cal knowl edge .
VIRTUALITY AND T H E L A W S O F PHYSICS
While trajectories bear a relationship of geometric similarity to
quantities measured in the laboratory, the singular it ies defining a
problem in physics are isomorphic with those defining the conditions
of a virtual multiplicity. Here too, I will argue that it is the behaviour
of linear equations that conceals the problematic aspect of mathematical
models. In short, whether we are dealing with causes or quasi -causes,
with experime nta l or theoretical physics, the crucial task is to avoid
the subordination if problems to solutions brought about by th e search for
simple linear behaviour. Let me begin with a quote from the philo
sopher of science Mario Bunge on the conception of matter brought
about by excessive concentration on linear causes:
Before atoms, fields and radioactivity became pieces of common
knowledge, even scientists could be found that shared the belief that
' brute matter' is a homogeneous, unorganized and quiescent strif! entirely
lackinn spontaneity - th e matter, in short, dreamt by immaterialist
philosophers. From th e fact that every experiment is an encroach
ment on matter, they jumped to the Aristotelian conclusion that
matter is nothing but the barren receptacle ifforms - a beli ef still held
in esteem by those quantum theorists who hold that it is the
experimente r who produces all atomic-scale phenomena. I"
And, I could add, st ill held in esteem by those cr itics of scien ce
who think that all phenomena are socially constructed. This conception
of matter as basically inert is directly linked to the defining character
istics of classical causality, the most important of which is the simple
additivity of the effects of different causes. This apparently innocent
assumption is indeed full of consequences, some of which are fatal for
the philosophical project which I have sketched in these pages. In
particular, a flat ontology of individuals assumes that, at every spatio
temporal scale, there are properties of a whole which cannot be
ex plained as a mere sum of the properties of its component parts, but
which emerne from their causal interactions. Without stable em ergent
pr operties, and the novel causal capacit ies these, in tum, give rise to,
the co ne pt of a larger scale individual collapses .
T he id a of additive causes becam dominant in physics for th
appare nt Simplicity \ ith which it endo w a syste m lind r study."? In
IN TEN SIV E SCIENCE AND VIRT UA L PHI LOSOP H Y
from the Aristotelian concept of efficient cause: externality . In this
view , causes are taken to be exte rn al agents operating on relatively
passive targets, hence being solel y responsible for whatever effects are
produced . The previous four traits of linear causality presuppose
exte rnality to the extent that they break down precisel y when the
body being acted upon ceases to be a mere patient . A failur e of uniqueness
occ urs whenever one cause can produce several effects depending on
the tendencies of the body it acts upon, and similarly for the case in
which th e same effect can be triggered by a variety of cause s. The
elimination of necessity in favour of enhanced probability and the
different probabilities of achieving an effect which a causal process may
transmit also depend on the probabilities to be affected carried by the
target of th e cause . And, of course, the failure of uni -di rectionality
and proportional ity are directly linked to the fact that the bodies acted
upon by causes are not passive but can rea ct back and exe rcise theirown causal powers. t "
Th e flat ontology of individuals I have defended in these pages
dep ends crucially, as I said, on th e elimination of linear causes , or,
at least , on cutting them down to size by showing them to be
speci al limiting cases. In this ontology individuals alwa ys exist as part
of populations in which the most meaningful and rel evant causal
r elations are of the statistical or probabilistic kind. None of the e
indiv iduals is ever a passive receptacle for extern al causal influences
sin e their int ernal causal structure always plays a part in determining
th final effect . Th e lack of uniqueness and uni-directionality is further
stre ngthened by the existence of quasi-causal relations. If the internal
dynamic of an individual is such that several alternative stable states
arc availabl e to it, it is hardly surprising that the same effect (a switch
bctwe n two attractors, for example) may be brought about by a
vari ty of causes , and conversely, on e and the same exte rn al cause
may trigger different effects depending on how close an individual is
to a bifurcation , or to th e border of a basin of attraction.
In short , whil e linear causality makes th e response of a material
syste m to an external cause basically unproblematic (given the cause ,
the re is nothing lse in th e effect that demands explanation), nonlinear
ami statisti al cau ality re-problemoti ze material syste m, showing th em
capable of s ·If-organization and self-assembly, with man y thin gs left
V IR T UALI T Y AND TH E L A W S O F PH YS I CS
un explained in the effect afte r the mere citation of an exte rn al cause .
In addition , linear and nonlinear causality impl y two different models
for the relationship between matter and form. Additivity and ex te rn al
ity presuppose, as I said , a matter obedient to laws and constitu ting an
inert receptacle for forms imposed from the outside. Matter under
nonlinear and non-equilibrium conditions is, on the other hand,
intensive and problematic, capable of spontaneously giving rise to form
drawing on its inherent tendencies (de fined by singularitie s) as well as
its complex capacities to affect and be affected. As Deleuze says, the
first model :
assumes a fixed form and a matter deemed homogeneous. It is the
idea of the law that assures the model' s cohe re nce, since laws are
what submits matter to this or tha t form, and conversely, rea lize in
matter a given property deduced from the form . . . [But that]
model leaves many things , activ e and affective, by th e wayside. On
the on e hand , to the formed or formable matter we must add an
entire ene rgetic materiality in mo vement, carrying sing ularities . . .
that are alread y like implicit forms that are topological, rather than
geome trical, and that combine with processes of deformation : for
example, th e variab le undulations and torsions of th fibers guiding
the operations of splitting wood. On th e other hand , to th e essential
prop erties of matter deri ving from the formal essence we mu st add
variable int ensive eifJect5, now resulting from the operation , now on
the cont rary , making it possible : for example , wood that is more or
less porous, more or less elasti c and resistant. At any rate, it is a
question of surrendering to the wood, then following where it leads
by connec ting operations to a materiality instead of imposing a form
upon a matter . ..44
Although Del euze is referring here to artisans (carpenters in this
exa mple , but also blacksmiths) sim ilar conclusions appl y to exper i
mental physicists . As Ian Hacking has forcefu lly argu ed, experime ntal
physics, far from being a mere app endage of theoretical phy ics
(supplying tests to confirm or disconfirm prediction s from formal
mod ' Is), has in fact a lif,· of it own . For example, the exp rim entalist
must indi viduate in a stable and r peat abl way laboratory phenomena .
INTENSIVE SCIENC E AND V IRTUAL PH ILOSOPHY
Rath er than being a mere by-product of theoretical knowledge of laws,
the indi viduation of phenomena involves, as Hacking says, 'a keen
ability to get nature to behave in new ways' .4-5 In the traditional
interpretation, thes e material and ene rgetic phenomena were supposed
to be unintelligible outside a th eoretical framework , but Hacking
shows that , on the contrary, laboratory phenomena (such as polariza
tion of light, the photoelectric effect , Brownian motion) typically
survive the birth and death of new theories, or what amounts to the
same thing, the switching from on e to another incommensurable
theoreti cal paradigm. Many times the individuation of a phenomenon
not only precedes the development of a theory that will explain it, but
it remains in this problematic state, crying out for an explanation, for
man y decades. r"
Beside individuating phenomena that mayor may not occur nat
urally , experimental physici sts must develop techniques and procedures
to isolate, identify and manipulate entities which have been individu
ated by obj ective processes occurring outside the laboratory. In this
as too, it is a question of connecting op erations to a materiality
instead of deducing the form of the entities in question from a
theoret ical law . As Hacking argues, physicists individuate entities like
elect rons by int ervening causally in the world, int eracting with real
electro ns so as to determine their mass (as was done by Thompson in
1897), or th eir charge (as performed by Millikan around 1908 ), as
w II as othe r of their properties. t? The individuation of electrons (as
we ll as other formerl y theoretical entities) is even more complet e
when exper ime ntalists move beyond their properties to study their
apaci ties. W e learn from electrons, we acquire expertise about them,
by making them part of heterogeneous assemblages where they affect
• nd are affecte d by other ent it ies , and it is this causal know-how more
than anything related to general laws, which gives us confidence that
these individuals actually exist . As Hacking writes:
T he re are an enormous number of ways in which to make instru
ments that rely on th e causal properties of electrons in order to
produ c d ired effects of unsurpassed precision . . . W e do not
mak instrum nts and th n infer the reality of the lectron , as
when we test a hypothesi , and th 'n b Ii -ve it hccaus ' it pass d th e
V I R TU A LI TY AND THE LAWS O F PHY SI C S
test. That gets the time-order wrong. By now we design apparatus
relying on a modest number of home truths about electrons, in
order to produce some other phenomenon that we wish to investi
gat e . .. W e spend a lot of time building prototypes that don 't
work. W e get rid of innumerable bugs . . . The instrument must
be able to isolat e, physically, the properties of entitie s that we wish
to use, and damp down all the other effects that might get in our
way. We are completely convinced if the reality if electrons when we
reoularly set out to build - and iften eno up ]: succeed in buildino - new
kinds ifdevice that use various well -understood causal properties ifelectrons
to inteifere in other more hypothetical parts ifnature.4 8
It is in the context of these complex laboratory practices that the
causal models I mentioned before (th e part of the population of models
that interfaces with the actual world) are deployed. As the sociologist
of science Andrew Pickering has argued, experime ntalists , machines,
causal models and electrons (or other material entit ies) form, in the
conte xt of a particular experime ntal project, a heterogeneous assem
blage . Each of the se distinct components retains its heterogeneity but
they are meshed to on e another in a complex process in which causal
mod es are fine tuned to better adapt to the results of an experime nt ,
machines and procedures redesigned to change the way they affect and
are affected by phenomena, and skills sharpened to cope with 'unfore
seen difficulties. In this assemblage each of the component parts plays
a role interactively stabiliz inq the whole. As Pickering writes, 'Scientific
knowledge should be understood as sustained by, and as part of,
inte ractive stabilizations situated in a multiple and heterogen eous space
of machines, instruments, con ceptual structures, disciplined practices,
soc ial actors and their relations, and so forth. ' 49
Following Del euze we may think about these complex assemblages
as the epistemological counterpart of the intensive in ontology. Much
as virtual multiplicities (view ed as self-posed ontological problems)
depend on intensive assemblages like ecosystems to progressively give
rise to ontological solutions, so expe rime ntal problems must first be
embodied in an int ensive assemblage prior to their being solved. In
I 'a rn ing by doing, or by interacting with and adjusting to material s ,
machin . and mod els, xperime ntalists proorcssivcly discern \ hat is
IN T E N S I V E SC I E NC E A ND V IRT U A L P HI LOSOP HY
relevant and what is not in a given experiment. In other words, the
distribution of the important and the unimportant defining an expe r i
mental problem (what degrees of freedom matter, what disturbances
do not mak e a difference) are not grasped at a glance the way one is
supposed to grasp as esse nce (or a clear and di stinct idea), hut slow ly
hrought to light as the assemhlage stabilizes itself th rough the mu tual
accom modation of its heterogeneous components. In this assemb lage
the sing ularities and affects of the ex perimentalist 's body are meshed
with those of machines, mo dels and material processes in order for
learning to occ ur and for embodied exp ertise to accumulate. so On the
other hand , besides th is expertise (w hich may he applied in the design
and performance of other experiments and which, therefore, remain sintensive) there are also extensive or formal products of laboratory
practices: individual pieces of data, individual facts, individual solu
tio ns , whi ch take th eir pla ce in th e corpus of accumulated knowledge.
As Deleuze writes, 'Learning is the appropriate name for the subject ive
acts carried out when one is confronted with the objectivity of a
probl em . . . whereas knowledge de signates on ly the gen erality of
concepts or the calm possession of a rule enabling solutions. ' 5 1
To summarize , there are two different ways of subordinating
problems to solutions in the causal realm . One involves the eli mination
of the nonlinear causal capacities of the material systems under studyeithe r by homogenizing them or hy focusing on low-int ensity cquilib
rium situations. In either case, one studies a matter so obedient to
laws that the productive aspect of causal connections may be disregarded and he reduced to a constant regu larity. What makes a material
system problemati c, what continuously demands new explanations, is
precisely the opcn -endedness of the assemhlages it may form, or the
multiple stable states in which it may exist and the abrupt transitions
it may undergo . But if we assume that there is always a unique stable
state, or that a cause always produces one and the same effect , wemay forget about the problem and focus on the sol ut ion: the constant
n'gularity itself as descr-ibed hy a law . O n th e other hand , one
subordinates problems to solutions when the complex causal interven
tions in reality which the ex perimentalist must perform, as we ll as themutual adjustments between machines, skills and 'a large number ofinh'r!ocking low level generalizations', S2 arc relegated to a secondary
VI RTU ALITY A ND THE L A W S OF P H Y S I C S
place and the formal cogn it ive products of thi s assemhlage are tak en as
the only worthy objects of phil osophical reflecti on . O nce detached
from their intensive individuation co ntext, where the experimental
learning of relevances and irrelevances takes place, these individual
items of knowledge become significant only hy reference to a theoret
ical framework of laws and abstract concepts.
Let me turn now to the subordination of problems to solutions in
the realm of the quasi-causal. As I said before, the par t of the
population of models which interfaces with the virtual is not the one
composed of detailed models of causal mechanisms but the one
including the mu ch simpler on es expressing fundamental laws. Unlike
the case of co mplex causal models, the relation of problems to
solutions in the case of basic law s (and models directly derived from
them) may he approached using the results of Deleuze' s ontological
analysis of state space . State-space ideas do not apply to causal models
for two reasons. On e is their sheer co mplexity: the mathematical
techniques need ed to ana lyse sta te space are typically valid only for
models with a few degrees of freedom, defining a state space with a
low dimensionality, and are not at present sufficiently developed toapply to more com plex cases. This lim itat ion may be lifted on e day as
these techniques improve but there is a more important reason why
they w ill sti ll he of limited valu e to the experimenta list : state spaces do
not capture any iriformati on about causal processes.
Let me exp lain. In som e interpretations of state space the series of
poss ible states whi ch populate it (that is, the trajectories or sol ution
curves) are erroneousl y endowed with causal significance, with each
successi ve state viewed as the cause of the following one (or in some
interpretations, the initial state is taken as the cause while the final
state is the effect) . This is, indeed. a mathematical expression of thepos it ivist redu ction of the productiv e or genetic aspect of causes to a
process if" uniform succession (another version of Humc ' s regular conjunc
tion). But as critics of positivism have pointed out. only actual events
can perform the genetic role of causes. As Mario Bunge argues . 'states
cannot have a productive virtue of their own. The state of a material
system is a syste m of qualities , not an eve nt or a string o f events .Evcry state is the outcome of a set of determiners . . . Consequently
then' can he no action of one state upon another state of a given
I NT E NS IVE SC I E NC E AND VI RT U AL PH I L O S O P H Y
systemj in particular, there can be no causal links among states, nor among
any other system of qualities . -ss
On the o the r hand, whil e the analysis of the state space of a model
may not provide us with causal information, it can be made to )ield
insight about quasi-causal relations . This epistemo logical result , how
ever, depends on a particular ontological interpretation of the contentsof state space. Deleuze , as I said , does not view the differential
relations defining a model as expressing a law go verning the generation
of the serie s of sta tes that make up a trajectory, but as defining a
vec to r field whi ch captu res the overall tendencies o f th e syste m as a
distribution of singularities. 'Beneath the general operation of laws' as
he says 'there always remains thc play of singularities.' 'i'' These
singular ities define the conditions of the problem, ind ependently of its
so lutions, while each solution curve is the product of a specific
individuation process guided at every point by th e tendenci es in the
vector field :
Already Leibniz had shown that the calculus . . . expre ssed problems
which co uld not hitherto be solved or, indeed, even posed ... One
thinks in particular of the role of the regular and the singular points
which enter into the complete determination of the specie s of a
curve . No doubt the specification of the singular points (for
example, dips, nodes, focal points, centres) is undertaken by meansof the form of integral curves, which refers back to the solutions of
the differential equations. There is nevertheless a complete deter
rnination with respect to the existence and distribution of these
points which depends upon a completely dilTerent instance, namely,
the field of vectors defined by th e equation itself . . . Moreover, if
the specificat ion of the points already sho ws the necessa ry imman
ence of the problem in the soluti on, its involvem ent in the solution
which covers it , along with the existence and distrihution of points,
test ifies to the transcendence of the problem and its directive rolein relation to the organization of the solutions themselves.ss
To bring out the originality of Deleuze ' s ana lysis it will help to
co ntrast it with the analyses perform ed by analytical phil osophers wh o
focus exclusively on the epistemo logical role pla)'{'d hy trajectories . In
V IR T UALITY AND T H E L A W S OF PH Y S I C S
one approach, for example , the role of the trajectori es is to be used as
predictions about the specific sequence of values "vhich the relevant
properties of the syst em being mod elled will follow. The first ste p in
the procedure , according to this approach, involves making measure ments of the properties of a real system in a laboratory and plotting
the resulting numerical values as a curve. If the laboratory system isprepared in such a way that it starts its evolution in the same initial
conditions as the model, thcn this curve and the co rresponding state
space trajectory should be oeome<ricolly similar. A perfect match
be tween the two, with the state-space trajectory exact ly tracking the
plotted values, co uld then be inte rpre ted as meaning that the model is
true to the modelled system. Given that, du e to empirical limitations,
we cannot prepare a laboratory system to start at precisely the sameinitial conditions as an abstract mod el , the relation between plotted
values and predicted trajectories will not be a perfect match, so that
their relation will be one of approximate truth . Neverthel ess, it is the
geometrical similarity, or approximate similarity, between the twocurves that matters for epistemological purposes.56
An alternative view would disregard this extrinsic resemblance
between metric objec ts, and emphasize instead the common possession oftopological invariants. As one physicist puts it,
For present purposes, a system may be viewed both as a field of
phy sical phenomena in which a class of elements exhibits its
functions or behaviors in space and time , and as an abstract
description which presumably may be isomorphic with the physical
field . . . Two systems will be viewed as fun ctionally isomorphic
over a dynamic range if they have the same sing ularit ies of motion, inthe stability sense, over that range . 'i7
This would be the co rrect stance to adopt in a Deleuzian anal ysis. The
episte mo logical valu e of state space would be to reveal a topoloqicol
isomorphism between singularities in the model and singularities in the
physica l system bein g mod elled . T his isomorphism , in tum, would be
ex plained by sho wing that the model and the physical system are co
actualizations of the same virtual multipli city (or o f part of the same
mult iplicity t given that the isomorphism is valid only within a rangl') .
I N T E N SI V E SCIE NCE AN D V I R T UA L P HI LO S O P H Y
Dcleuze's approach does not exclude the possibility that there can be
sim ilarities between traj ect ories and plotted values, but this resemb
lance must itself be explained as a result of the common topological
properties of the systems producing the curves . The repl y that
possession of common properties is what makes a mod el and a real
ystem similar is, as th e phil osopher Nel son Goodman argued long
ago, redundant. As he put it, 'to say that two things are similar in
having a specified property in common is to say nothing more than
that they have that property in common'. 58
There is another way of stating th e differen ce between th ese tw o
philosophical approaches to the episte mology of state space . In the
analytical appro ach , the main episte mological relati on is that between
laws (ex pres sed by differential equations) and the traj ectories obtained
as solutions to tho se equations. This relation is one of Beneral to
particular. In other words, if we ignore the rol e which the vector field
plays in the individuation of traj ect ori es, it see ms natural to view laws
as stat ing a Beneral rule govern ing the volution of series of states , and
to see eac h t rajectory as th e result of appl ying that rul e for a particular
initial condition . In th e Deleuzian approa ch , on the co ntrary, th e
particu lar sta te at which a trajectory starts becomes irrelevant, given
that ma ny different start ing points within the same basin of at traction
end up in th e same place , the at tracto r. In othe r wo rds, it is th e
distribu tio n of sing ularities itself that det ermines what changes in initial
conditions are relevant (relative to the end state) and wh ich are
irrelevant , O n the other hand, the gene rality of the law (of which a
giv'n tra jec tory and plot of real values are particular instances) is
r 'p laced by the uni versality of virtual multipliciti es of wh ich both model
.1IIt! real syste m are divergent actualizations. As Deleuze wr ites,
'S ingularity is beyond particular proposition s no less than uni versality
is h 'yond ge ne ra l prop ositions. '59
The subse rv ience of problems to so lutions in the analysis of state
SIl.\CC is but one example of an error with a rather long history, a
' long perversion ' wh ich Deleuze traces back at least to Aristotl e ."?
O riginally, the subo rdinat ion deri ved fro m the habit of thought of
think ing abo ut problem s as if th ey were proposition s, that i , from
missin J the non -lingui st ic and ex tra-proposi t ional nat ure of th ir
VI R T UALI T Y A ND TH E L A WS O F PHYSICS
co nditio ns (contrast space). But in more recent times, in the historical
peri od when classical mechani cs developed , the surrende r to solutions
took a more specific, more mathematical form . To Del eu ze , math
ematical problems are subordinated to their solutions when ever the
we ll-posedness of a problem is approached in terms of its solvability
(the possibility of finding a solution) . In the final sec tion of th is chapter
I would like to discuss two episode s in the history o f mathematics
where this traditional subordination was inverted , with solvability
becoming a consequence of the well -posedness of a problem . As I will
discuss in a moment, thi s inversion has for Deleuze revolutionary
consequences whose impact has not been gen erally appreciated . One
episode invol ves the history of algebraic equations, and the reversal of
th e subordination had, as one of its con sequences, the birth of group
theory. The other episode is more familiar, relating to the history of
differential equations , having as a result th e birth of the th eory of
dynamical syste ms, whi ch is th e source of the modern approach to
sta te space.
Let me begin by describing in very rough form the techni cal issues
invo lved in questions of so lvability in the case of algebraic equatio ns .
T here are two kind s of solutions to equations, particular and Beneral. A
partic ular solution is given by numerical values whi ch, whe n used to
re place an equation's unknowns, make the equation come out true .
(Fo r ex ample, an algebraic equation like x2 + 3x - 4 = °has as its
nu me rical so lutio n x = I .) A gene ral or exact so lution , on the othe r
hand, does not yield any specific value or set of values but rather the
Blobal pattern if all particular solutions. Thi s gen eral pattern is typi cally
give n by another equation or formula. The above example , which
may be written as x2 + ax - b = 0, has th e gen eral solution
x = V d h + b - ~ . When mathematicians speak of the solvability of
an equation th ey usuall y mean its exact solvability, and the subordi
nation of problems to solutions ste ms from th e demand that a well
posed problem have an exact solution , not just numeri cal ones. By th e
sixteenth -ce ntury mathematicians knew that exact solvability was an
achi vable goa l, at least with equations wh ere the unknown variable
was raised up to th fourth pow er (that is, thos including x2 , x I and
x"). But th n a crisis ensued. Equations raised to th fifth power
INTENSIVE S CIENCE AND VIRTUAL PH ILOSOPHY
sed to yield to th e previously successful method . Was this lack of
:t solvability indicative that there was something wrong with the
olern as it was posed by the fifth degree equation?
'he answer cam e two centuries later when it was noticed that there
a pattern to the solutions of th e first four cases, a pattern which
ht hold the key to understanding the recalcitrance of the fifth,
wn as the quintic. First Joseph -Luis Lagrange and Neils Abel, and
I Evariste Galois, found a way to approach the study of this pattern
g resources that today we recognize belong to group theory . In a
.hell we can say that Galois ' showed that equations that can be
cd by a formula must have groups of a particular typ e, and that
quintic had the wrong sort of group' .6 1 I cannot go here into the
mical details of Galois's work but what he achieved was to invert
subordination of problems to solutions: rather than general solv
ity defining the correctness of a problem, the form if the problem
me the explanation ifBeneral solvability. In other words, whil e before
exact solvability of the first four ca e was tak en for granted (as a
Jerty which problems mu st have) it now became something that
ld be explained by a uni versal feature of th e problem which these
, cases posed. This is what Del euze means when he says that 'it is
th e solution which lends its generality to the problem, but the
rlem which lends its universality to the solution ' , 6 2 a universality
ured in this case by a group of transformations. But how exactly
> a group of transformations capture th e universal conditions that
ne a problem as a problem, that is, independently of its solutions?
'0 answe r this question let me first take a different example, th e
of transformation groups to study the invariants of physical laws.
) of the mo st typical transformations in this case are displacem ents
pace or time. Given a law -governed physical process that can be
'oducc d in a laboratory, if we simply move it in space (for instance ,
-cproducing it in another, far away laboratory) we can expect the
liar aspects of its behaviour to remain invariant. Similarly, if we
)Iy hange th tim e at which we begin an experime nt , we can
ct this tim e displacem ent to be irrelevant as far as the regularity
he pro ess is on c rned . It is onl y the difference in time b tween
first and final states o f the process that matt rs, not the ab olute
at which the fir t tat e 0 ur s. Thus, via transformation appli ed
V IRT UA L I T Y AN D T H E L A W S OF P H Y S I C S
to the equat ions express ing laws, we can discover those types of
change to which th e law is indifferent, that is, the types of changes
whi ch do not matter as far as the law-like process is concerned . The
sense in which the group of an equation captures th e conditions of a
probl em is th en that it reveals distributions of the rel evant and the
irrelevant, th e irrelevance of using absolute time or absolute position
as inputs to a law for instance. It may be asserted without exaggeration
that understanding this connection had profound implications in the
history of physics playing a crucial rol e , for example, in the develop
ment of the general theory of relativity. 63
Similarly, Galois's analysis of algebraic equations relied on the use
of certain transformations (substitutions or permutations of th e solutions)
which, as a group, showed what changes were relevant to the validity
of the equation (or more exactly, to th e validity of the relations
between solutions), More specifically, wh en a given permutation of
one solution by another leaves the equation valid, the two solutions
become, in a sense, indi stinpuishable as far as this validity is concerned.
The equation is indifferent to the switch. As Morris Kline writes, 'The
gro up of an equation is a key to its solvability becau se th e group
expresses the degree of indistinguishability of the [solutions]. It tells us
what we do not know about the [solutions]. '64 Or as Deleuze would
put it , the group reveals not what we know about the solutions, but
the object ivity if what we do not knoll' about th em, that is, the obj ectivity
of the problem itself. 6 5 Moreover Galoi 's method involves the equi
valent of a symmetry-breaking cascade in that the solutions to the
equation become increasingly 'more accurately defined as the original
group gives rise to sub-gr oups which proBressively limit th e substitu tions
leaving the relations invariant. In other words, through a cascade which
unfold s th e original group, the problem itself becomes progressively
better specified and, as a by-product if this se!f-specification, individual
solutions emerge . As Dcleuze writes:
We cannot suppose that , from a technical point of view, differential
calculus is the onl y mathematical expression of problem s as such
... More r cent ly othe r procedures have fulfilled this rol e better.
Recall the cir I in which the th ory of problem s was caught : a
problem is .olvabl only to the ex te nt that is is 'tru ' but we always
I N T E N S I V E SCIENCE A ND V IRTUAL PHI LOSOP HY
tend to define the truth of a problem by its solvability . .. The
mathematician Abel [lat er followed by Ga lois) was perhaps the first
to break thi s circle : he elaborated a whole method according to
which so lvability must follow from the form of a problem. Instead
o f see king to find out by trial and error whether a given equation isso lvable in general we must determ ine the conditions of the problem
which progressively spe cify the field s of solvability in such a way
that the stateme nt contains the seed of the solution . This is a radical
reversal of the problem- solution relation, a more considerablere volut ion than the Copernican .w
T he reversal of the problem - solution relation also had revolutionary
co nseque nces in the case of differential equat ions . Although very
different from their algebraic coun terpart, equations in the calculus
alsn have particu lar and gen eral solut ions, both produced by th e
inh.'gration opcrator. As it happens, mo st differential equations cannot
he so lved by integration in a general or e xact way . Today we get
around this limitation by using computers to generate a population of
m,lny numerical so lutions , a popu lation which may be used to discoverlh,' general pattern . In the eighteent h century, wh en the physics whi ch
Ne wton and others had created was first given differentia l form, this
way out of the difficulty was not, of course , available . One conse
qucnce was the neglect of models who se constituent equations could
1I0t be solved exactly, given that without a way of knowing the ove rall
pattern of particular so lutions, physicists could not learn very muchfrom a model. Thus, in a vcry real sense, the solvability of a problem
was what made it worthy of study . As the mathematician Ian Stewartwrites :
Th e math emati cians of the eighteenth century ran headlong into a
problem whi ch has plagu ed theoretical me chanics to this day: to set
up the equations is one thing, to so lve them quite another . . . Thet'ightcl'nth ce ntury's main achievements were in setting up equations
to model physical phen omena . It had much less success in solving
them ... A process of se lf-selection set in, whereby cquations thatco uld not he solved were automatically of less interest than thosethat could."?
VIR TUALI TY AND THE LAWS OF PHYSICS
One can hardly blame these ma themati cian s and physicists for falling
prey to this process of self-s election, since they were operating within
the limits imposed by the mathematical technology of their t ime. On
the othe r hand , the long. tern l effccts of subordinat ing th e cho ice of
problems to their solvability did influen ce their (and their successors")
wor-ld view, biasing it towards a clockwork picture of reality. Thereason for this was that the equations that could be exact ly so lved
happened to be the lin ear equations. The mathematical difference
between linear and nonlin ear equations is ex plained in terms of thesuperposit ion prin cipl e. which states that g iven tw o d ifferent so lutions of
a linear equation, their sum is also a valid so lution. In oth er word s.
once we have discov ered a fcw solutions to an cquation many more
can he obtained for free via the superposition principle. In an eracharacte rized by the gene ral scarcity of exact solutions, such a principle
mu st have see me d like a gift from th e optim izing rationality of God .
Conversely , failure to obey thi s prin cipl e promoted the negle ct of
nonlin ear cq uationst" In the term s I have been using in this chapter
we may say that superposition, that is, a property of the behaviour ofsolutions, biased the process if accumulation that created the population
of models making up the theoretica l component of classical mechanics.
The requireme nt of exact solvability prom oted the accumulation of
linear model s at the expense of nonlinear ones , and even the fewnonlinear models allowe d to become part of the population were used
only in a linearized form . ( Linearization is achieved by using non linear
models only for very low int ensities of the recalcitrant variables. ) AsStewa rt puts it :
Classical mathematics co nce ntrated on linear equations for a sound
pragmatic reason: it co uld not solve anything else . . . So doci le arelinear equatio ns, that classical mathematicians were willing to
co mpromise their physics to get them . So the classical theory dea ls
with shallow waves, low-amplit ude vibrations, small temperaturegradients [that is, linearizes nonlinearities). So ingrained became the
linear habit that by the 1940s and 1950 s many scientists and
enginee rs knew litt le el se . . . Linearity is a trap. The behaviour of
linear equa tions . . . is far from typical. But if you decide that only
linear equations are worth thinking about, se lf-ce nso rship sets in.
I N T E N S I V E S CIENC E AN D V IRT UA L PHILOS OPHY
Your textbooks fill with triumphs of linear analysis, its failures
buried so de ep that the graves go unmarked and the exis te nce of
the graves goes unremarked . As the eighteenth century beli eved in
a clockwork world , so did the mid -twentieth in a linear one ."?
T he co unte rpart to Abel' s and Galois' s reversal of the problem-.
solution relation is represented by the work of Henri Poin care on the
qualitat ive (or topological) study of differential equatio ns . His was a
novel approach crea te d, like the group-theo re tic approach to algebraic
equa tions, to break through the barrier of a recalcit rant pr obl em: the
three body problem, the problem of modelling the mutual int eractions of
three solar system bodies (such as the sun , the earth and the moon).
Altho ugh other mathematici ans had already approached the st udy of
so lutions by analysing their behaviour in the neighbourhood of singular
points, Poin care approached the wider questi on of the way in which
the existence and distribution of singularities organized th e space of all
so lutions. In other words , like Galois, Poincare by-passed exact
so lvability as a way to get global information and instead used a novel
method to investigate the space difining the problem itself, that is, he used
the distributions of singular points as a way to gain qualitativ e
information about the tendenci es in the behaviour of all solutions. "?
Poin care' s phase-portrait approach to state space has, of course,
been the basis of mu ch of what I have said in this bo ok abo ut the
on to logy of the virtual and the problematic. But Galois' s approach has
also been crucial since it provided the idea of a progressive spe cificat ion
of virtual multiplicities through symme try -breaking cascades . In short ,
a theory of virtuality as has been pursued in these pages depends
fundamentall y on the results of th e reversal of th e problem- solution
relat ion , and conversely , subordinating problems to solutions may be
see n as a practi ce th at effective ly hid es the virtual , or th at promotes
tln- illusion that th e actual world is all that must be explained. Thus
construed , this subordination joins th e axiomatic treatment of classical
physics as a barrier to a more satisfactory probl emati c approach." In
.uldi t ion , there are th e obstacl es posed by the linearity of causes in
ex pe rime nta l physics, and the linearity of models in theoretical physics ,
both of which arc intimately related since the former' s addit ivity is
t'Cj ui\',lll'n t to the latt er' s supe rpos it ion. Additivity and supe rposi tion
VI R T UA LI TY AND THE L A W S O F PHY S I C S
characterize an unprobl emat ic world , or at best, a world whi ch is only
temporaril y probl ematic or in need of ex planation, but whi ch will
eventually yield to a supe r-law or a th eo ry o f everything whi ch will
leave nothing unexplained . On th e othe r hand, nonlinear model s and
the ir multiple at t ractors, as well as nonlinear causes and their co mplex
capacit ies to affect and be affected , define a world capable of surprising
us through th e emergence of unexpect ed novelty, a world where there
will always be some thing else to explain and whi ch will therefo re
re main forever problematic . As Mario Bunge writes :
If the joint acti on of severa l causes is always an exte rnal juxtaposi
tion, a supe rpos it ion , and in no case a synthesis having traits of its
own, and if the hypothetical pati ents on which the causal agents act
arc passive thin gs incapabl e of sponta neity or se lf-activity ~
incapable, in short, of adding some thing o f their own to the causal
bond - then it follows that, in a sense, e}Jects preexist in th eir causes.
According to thi s extre me but consistent doctrine on the nature of
causation, only old things come out of cbanpe; processes can give rise
to objects new in number or new in some quantitative respe cts, not
however new in kind ; or again , no new qualiti es can emerge . A
,v-orld running on a str ictly causal pattern [i.e. a linear pattern] is
such as yog is, Thomists and eighteenth-ce ntury Ne wto nians ima
gined it, namely, a uni verse without a history . . .71
Unlike this linear world, the ontology I have devel oped in this book
is fully histori cal. Each of the individuals which populates thi s other
world is a product of a definite historical pro cess of individuation and ,
to the extent that an individual ' s identity is defin ed by its eme rge nt
propert ies and that these properties depend on th e continuing causal
inte ractions among an individual' s parts, each individual is itself a
historical causal process. The realm of the quasi-causa] is also fully
histori cal but, as I ex plained in the previous chapte r , it possesses its
own o riginal form of temporality and thus bear s no resemblance to
causal history. In ot her wo rds, in a Deleu zian onto logy there ex ist tw o
histo ries, one actual and one virtual, having complex int eraction s with
on e ano the r. O n one hand there is a historical series of act ual eve nts
gt'lwticall)' involved in the producti on of other events , and on the
I NTENSI V E SC IE NC E AN D V IRTU AL PHILOSOP H Y
other, an equally historical series of ideal events defining an objective
realm of virtual pr obl ems of which each actualized individual is but a
speci fic solution. To conclude with Del euze 's own words ,
It is correct to represent a double series of events whi ch develop in
two planes, echoing without resembling each other: real events on
the level of the engende re d solutions, and ideal events embedded in
th e conditions of the problem, like the acts - or, rather, the dreams
- of the gods who double our history. 7 ~
Appendix: Deleuze's Words
Gilles Del euze changes his terminology in every one of his books.
Very few of his conce pts retain their names or linguistic identity. The
poin t of thi s terminological exuberance is not merely to give the
impression of differ ence through the use of syno nyms , but rather to
develo p a set of different theories on the same subject, th eories whi ch
are slightly displaced relative to one another but retain eno ugh over
laps that they can be meshed together as a het erogen eous assemblage.
Thus, the different nam es wh ich a given conce pt ge ts ar e not exact
synonyms but ncar synonyms, or sometimes, non-synonym ous terms
defin ing closely related conce pts . In th is book I delib erately homo
genized th e tenninology for the sake of clarity but giving a list of
ncar synonyms will now prove useful to the read er as he or she
mov es back from my sim plified presentation of Deleuze' s ontology to
his original ones . In fact, beyond providing a mere list I will try to
map the connections between the different terminologies and discuss
the different ways in whi ch the onto logy is conce ptualized and artic
ulated in each of the books. As I map th ese tenninological connec
tions I will use the following abbreviations of Deleu ze ' s books,
followed when necessary by a page number (chapte r numbers refer
to the pr esent book):
Anti-Oedipus
A Thousand Plateaus
Difference and Repet ition
l.oOic if Sense
What is Philosophy ?
AOATP
D&R
LOS
WIP
The main sou rces used in my reconst ru cti on were D&R, where
the theory of multipli it ies and the virtual co nt inuum they form is
most clearly articul ated , and L S which pr es .nts the most detailed
A P P EN D I X : D E L E UZE 'S WORD S
description of the qu asi-causal op erator. I will begin this app endix with
a list of the compone nts of Deleuzc 's ontology (D&R, 277-8) . [ will
then expand the description of each of the seven compone nts of this
'o nto logical list ' , not onl y to relate them to the terminology used in
Illy pr esentation, but also to add details wh ich I left out for the sake
of simplicity but which are now necessary in order to rel ate the items
in thc ontological list to those in other books. Finally, [ will take three
hooks, ATP, AO and W [P, and map each component of the list to
their co unte rp arts there .
T HE ONTOLOGICAL LIST
( I) the depth or spatium in which inten sities ar e organizcd;
(2) the disparate series th ese form , and the fields of individuation that
th 'y outline (individuation factors) ;
(3) the ' dark pr ecursor ' which causes them to communicate ;
(4) thc linkages, internal resonances and forced movements which
result ;
(5) the constitution of passive selves and larval subj ects in the system ,
and the formation of pure spatio-ternporal dynamisms;
(6) the qua lities and exte nsions . .. whi ch form th e double differen
ciation of the syste m and cover over the preceding factors;
(7) the cent res of envelopment which neverthe less testify to th e
persisten ce of these factors in th e developed world of qualities and
ex te nsit ies .
l , Inten si ve Spatium
This term refers to th e virtual continuum formed by multipliciti es. In
this hook I used the term 'plane of consistency' to refer to it , a term
used throu ghout ATP. Other near synonyms include ' plane of imman
l'll 'c' (W [»), ' body without organs' (AO , ATP) , 'rnachini c phylum '
(AT I'), and ' ide al or met aph ysical surface' (LO S). A possible sourc of
co nfusion h .rc is the term ' inte nsive ' which in my presentation was
lIsed ill relati on to indi viduation processes, not the virtual co ntinuum .
I)l, leu:l.l' uses the term in thr 'e se nses :
AP PEN D I X : DE LEUZE 'S W O RDS
a) Its original , thermodynamic sense in which it refers to inten sive
properties, like pr essure, temperature or density. Differen ces in
these quantities have a morphogen etic effect (they drive fluxes
of matter or ene rgy, for example) and when not allowed to get
cance lled (as in non-equilibrium physics) display the full potential
of matter-en ergy for sel f-o rganization.
b) A second derived sense in which it refers to the assembly of
different compone nts as such , that is, the cre at ion of het ero
geneous assemblages in which the compone nts ' differences are
not canc ell ed through homogenization.
c) A third derived sen se in whi ch it refers to the properties of
ordinal series , Th ese ser ies ar e con sti tuted by the differences
between their terms, that is, by asymmetrical relations such as
'in between' . When we conside r more than one term between
two other s, this ser ial relation is called a 'distance ', although this
term must be qualifi ed (Deleuze speaks of 'non-de composable
distances' ) to distingui sh it from its non -te chni cal meaning where
it refers to a metric concept (such as ' length ' ) . Finally, there are
th e uncance llable differences, or constitu tive inequalities , which
ordinal ser ies present when compared to one another (on ly
judgments of greater or lesser are possible, not of exact equal
ity) . lt is mainly in this third sense that the term is used in the
expression 'intensive spatium ' as the following quote shows:
Differe nce , distance and ineq uality are th e positiv e charact eristics of
depth as intensive spatium (D&R, 238),
2. Multiplicities and Divergent Series
Altho ugh the term 'm ult iplicity' is not used in the list above, it is clear
that it belongs in this entry since the 'disparate series' mentioned are
no thing but the effect of expanding in a serial form the singularities
d 'fining each unfolding level of a mu ltiplicity. The term has some near
s Ilonym s: 'partial objects ' (AO ); ' philosophical co nce pts' (W [P);
' idea l eve nts' (LOS). Some times Dcleuze refer s to multipliciti es
indirectly via their compo nents , suc h as ' nomadic singularit ies' and
APPE NDI X : DELEUZE 'S W O R D S
'noe matic attr ibutes ' (LOS) , or 'vague essences' and 'becomings'
(AT P) .
Th e term 'disparate ' means ' d ifference of difference' (D&R, 241 ) .
To speak of 'disparate ser ies ' is another way of expressing the idea
that the ordinal ser ies which form the nonmet ric continuum mu st be
rela te d to one another via '!ffirmative divergence, so that not only are the
se ries mad e up of differences, their divergent relations further differen
tiate these d1Jerences:
Di fference mu st become the element , the ultimate unitv; it must,therefore refer to other differences which nev er identify it but
rath er differentiate it. Each term of the series , being already a
differen ce, must be put into variable relations with other terms,
th ereb y co nstit ut ing other se ries devoid of cente r and convergence.
Divergen ce and decentcring must be affi rme d in the se ries itself.
(D&R, 56)
3. Dark Precursor
T his term refers to what in my reconstruction I called the 'quasi-causal
operator '. Its near synonyms include: 'quasi-cause", 'al eatory or para
doxical point' and ' nonsense ' (LOS); 'line of flight ' and ' abstract
machine ' (ATP); 'desiring machines' (AO) ; 'conceptual person ae'
(W IP); 'o bject = x ' (D&R, LOS) .
4. Resonances and Forced Movements
This entry includes the effects which the quasi-causal operator has on
the multipliciti es and their series . In my reconstruction I used an
info rma tion-theore tic model for these effects (in terms of emissions of
signs or informati on qu anta) but Del euze also uses an alt ernative physical
model in terms of resonances (D&R, LOS , WIP) . The te rms ' resonance'
and ' forced movem ent.' should not be taken as mere physical met aphors.
Rath er , we sho uld think about resonance as positi veftedback, a gene ric
pr on'ss which implies one or o the r form of mutually stimulating couplin8s
k.g. .autocatal ysis} inducing resonance s amo ng het erogeneous elem ents,
as well .lS the ampljfica t ion iforiginal djfferences (forced mov em ents) .
AP PEND IX : DELE UZE 'S WORDS
The crucial idea is that th e qua si-causal op erator m ust couple the
ordinal series ema nating from multiplicities so as to weave these into a
nonmctric co ntinuum. Resonances are the means to effect couplings,
while the resulting forced movem en t produces the continuum (LO S,
239-40). As I have just said, th e couplings between ser ies must ensure
the ir affirmative divergen ce , keeping the continuum op en and in
co nstant variation . But also , as a separate operation (what I called ' pre
act ualizatio n ' in Chapter 3) , it must induce some convergences in the
se ries, since it is in these cent res of co nvergence that th e process of
actua lizat ion begins :
To be actualized . . . mean s to extend ove r a ser ies of ordinary
points; to be selecte d according to a rul e of converge nc e ; to be
incarnated in a bod y; to become th e state of a body; and to be
ren ewed locally for the sake of limited new actualizations and
ex tensions . (LO S, I 10)
5. Passive Selves and Spatio-Temporal Dynamisms
T his entry contains the two components of what in m y reconstruction
I referred to as 'inten sive individuation processes' . The first meaning
of the term 's patia-tem poral dynamism ' is straightforward, referring
to the phenomena of sel f-organizat ion which occur in many non
equilibrium systems. Self-organizing dynamics ar e typically governed
by the singularities (at t racto rs and bifurcations) which chara cteri ze
differential relations (that is, coupled rates of change or relations of
re lat ive rapidity and slowness.) In thi s se nse, th e term relates to the
first sense of the word 'intensive', as in a non-equilibrium material
where inten sive differen ces have not been cancel led . But the term also
refers to 'a ffects', or th e second sense of 'inten sive ' , that is, to the
capaci ties and dynamisms whi ch produce heterogeneous assem blages.
That the tw o senses are intimately connected is clear from the
''''lowing:
It is no longer a qu estion of imposing a form up on a matt er but of
d ahorating an increasingly rich and co nsistent materi al , the better
to tap incr easingly int en se forces . What makes a matt' rial incrcas-
A PPEND IX: D ELE UZE 'S WORDS
ingly rich is the same as what holds heterogen eiti es togcther without
the ir ceasing to be heterogeneous . (ATP, 329)
Unlike spa tio- te m po ra l dynamisms, the terms passive sel f' and
'larval subject' recei ved very little elaboration in my recon struction ,
mostly because I wanted to keep the description of Deleu zes ontology
as free from anthropocentrism as possible . The first term is related to
the 'passive synthes is' whi ch forms the core of Dclcu zc ' s theory of
time , the synthesis of 'living presents' which metricize or give measure
to time . In his theory, this synthesis is directly related to the gen esis
of subjectivity (it is a co nte mplative subject who contracts instants into
a present ) but, as I ex plained in Chapter 3, these 'co ntemplations'
occur everywhere, in the form of proto-perception s and proto-feelings
wh ich even microscopi c individual entities may be said to have . Hence,
we not only contract instants to synthesize our psychological sense of
present , we are made out of micro-contractions and their presents:
W c arc made of contrac ted water , earth , light , and air - not onl y
prior to the recognition or representation of these, but prior to
their being sensed . Every organism, in its recept ive and perceptual
el em ents, but also in its visce ra, is a sum of contractions, of
retentions and expec ta t ions . (D&R, 73)
The term 'larval subject' is clo sely related to these ideas, referring
to the 'vo luptuous consumption' of the intensities which drive spa tio
tempo ral dynamisms . The best ex am ple here is the developing embryo
as it experiences the intensive fold ings , migrations, and oth er transfor
mations which will eventually turn it into a fully formed organism .
Indeed, unlike my recon struction where the term 'individual' refers to
the final produ ct (o rganisms, species, etc.) in Deleu zc 's work it refers
to the lar val subjec ts themselves. It ofte n has the mean ing of a
Lc ibnizian 'mon ad', and it is said to be born during pre -actualization ,
that is, from the ce nt res of co nvergence whi ch occur in the virtual
series :
A world already envelops an infinite syste m of singularities selected
through co nve rgence. Within this world, howe ver, individuals are
APPEND IX : DELEUZE 'S WORDS
co nstituted which select and envelop a finite numb er of the singular
ities of the system . . . An individual is therefore always in a world
as a circle of co nvergence, and a world ma y be formed and thought
on ly in the vicinity of the individuals whi ch occupy or fill it. (LO S
109-10)
To avoid co nfusion, I will usc the term 'intensive individual ' to
refe r to these monad s, and 'individual' without qualification to refer
to the ex tended and qualified actual entities whi ch form my flat
ontology of indi viduals .
6. Extensities and Qualities
These are the two characte rist ics which define the realm of the actual,
the fully co nstituted world of ex te nde d and qualified individuals. In
ATP these two characteristics are referred to as 'substances ' and
'forms' respecti vely . To sec the connec tion one needs to think , on the
one hand, of a substance without any oth er characte ristic than its
manner of occupying space (its extension), and, on the other hand, of
the form s or structures wh ich endo w this substance with specific
qua lit ies (such as its mechanical or optical properties) . Given that no
act ua l substance is ever purely ex tensional, these two characteristics
arc 'not really distinct. They are the abstract components of every
articu lat ion ." (AT P, 50 2)
7. Centres of Envelopment
This co nce pt was not discussed in my reconstruction. I introduce it
here not only bec ause it appears as the last item in the listing of
ontological components under discussion , but also because its defin ition
relates to aspects of the theory of the actual whi ch bear on questions
of te rmino logy. The different spheres of the actual (roughly, the
physico-chemical, organic and cultural spheres) need to be conce ived
witho ut presupposing a teleological devel opment or ' any kind of
rid iculous cos mic evolutionism' (AT P, 49) . There are , on the other
hand, very real distinction s bet ween these spheres . In particular, unlike
the physico -che m ica l sphere wh ere the 'co de' that underlies form s o r
A P P E N D I X : DELEUZE 'S W O R DS
q ua lities is distributed throughout the three-dimen sionali ty of a struc
ture , in the organic sphere this code becomes detached as a separate
one -dime nsional structure: the linear seque nce of nucleic acids consti
tuting the gene tic code, The ge net ic code, in Deleuze 's view, repres
ents an interiorization if the intensive individuatinq factors whi ch in
physico-chemical st rata remain ex te rn al to indi vidu als. Thi s int eri ori
zation, which characte r izes the increase in complexit y of living syste ms ,
is what is referred to by th e term 'centres of envelopme nt' :
Th e functi on of these centres may be defined in se veral ways
we claim that complex systems increasingly tend to interiorize their
co nstitutive differen ces: the centres of envelopment carry out this
interiorization of the indi viduating factors. (D&R, 256)
Summary
Let me now summarize what I have just said about the co nte nts of the
ontological list. Items 1, 2 , and 3 co nstitute the eleme nts of the virt ual:
the co nti nuum , the multipliciti es and the quasi-causal ope rato r. Items
4 and 5 may be made to cor res pond, with a bit of tw eaking , to th e
intensive . The reason why some tweaking is necessary is that it invol ves
s 'parating the divergent and the conve rgent relati ons between the
xer ic , the former belonging to the virtual and the latter (as a kind of
pre-actualization) to the int ensive. Centres of conve rgence would
orrcspond to wh at some scie ntists call ' mo rphogenetic fields ' , o r what
Dc leuzc calls ' fields of indi vidu ation '. Although Deleu ze includes as
part o f Item 2 'fie lds of indi viduation ' , and the resonances of Item 4
also produce di vergen ces, it will prove useful to keep the tw o Items
.lpart and define the inten sive both by the fields of individuation and
th . spa tio -tc rnpo ral dynami sms that perform th e actualization of these
field s. Pinally, Items 6 and 7 form the conte nts of the actual. Precisely
because the vir tual, the int en sive and th e actual are aspects of one and
til(' same process, or th e different mom ents o f a cascade of progressive
d iffer mtiation , some Items (4 and 7) represent areas of ove rlap
(so l!wthing of th virtual, onv rg nc , within the inten sive ; so me thing
of the inte nsive, cnv .lo pmc nt ntrcs, in th actual). Let me now show
hm\ tln- virt ual, the int en sive , and th actual arc trc: ted in other books.
APPEND IX : DELEUZE 'S WORDS
A THOU SAND PLAT EAU
In ATP the different spheres which make up the actual world (physico
chemical, orga nic , cultural and so on) are called 's trata' . The term
's tratification' is near syno nymo us with 'actualization'. The different
extensit ies and qualities which characte rize the actual world are
referred to as 's ubstances' and ' forms' , and also as ' te r ritorialit ies' and
' codes ' . Thus, Del eu ze writes that strata ' proceed simultaneously by
code and by territoriality ' (AT P, 40) . The inten sive processes which
give rise to strata, and which become hidden und er strata, ar e th erefore
called 'territorializat ion ' and 'coding' . Given that some parts of the
wo rld may be pu shed away from their equilibr ium state , ther eby
revealing the hidd en inten sive factors, the terms 'de ter ritor ialization'
and 'decoding ' ar e used to refer to th ese departures from the rigidity
of strata, or rather, to the inten sive movem ents which animate strata
fro m within. In D&R, Deleuze had alre ady introduced the notion of
'dc-differe nciation' (D&R, 249) but it is onl y later that thi s notion
aC'luires its full importance and that it is divid ed along the two
components of actualizati on.
Indeed , as I argued in Chapter 3, th e quasi-cau sal ope rato r may be
said to accelerate th ese departures from actuality in an ope ration called
'counter-actualization ' . In ATP , Deleu ze speaks of ' re lative det errit
orializations' to refer to moveme nts away fro m the actual toward s th e
intensive , and of 'abso lute det erritorializati on ' to refer to counte r
actualization , the acce leration of these movemen ts allowing them to
reach all the way int o the virtual. The three compo ne nts of the virtual
(the continuum , th e multiplicities that co mpose it and th e quasi -causal
ope rator which effects the composit ion) have exact counte rparts in
T P as the following ex tract illustrates:
There was a first gro up of notions: the Body without Organs or
dcstrat ified Plane o f Co nsiste ncy ; the Matter of the Plane, that whi ch
oc .urs in the bod y or plane (sing ular, non segm ented multipliciti es
composed of int en sive cont inuums , emissions of particle-signs, con
junctio ns of flow ); and the Abstract Machine , or Abstract Machines, in
Sl) far. s they co nst ruct that bod y or draw the plan e or 'diagram' what
oc urs (line. of flight , or • hsolut det errit orialization) . ( T P, 72)
APP E NDIX : O E LEUZ E 'S W OR D S
Multiplicities ar e said to ' occ ur ' in the plane of consiste ncy because,
as I argued , they are ideal events or becomtnqs. The term ' no nseg mente d '
should be read as near synonymous with ' nonmetr ic", and ' inte nsive
continuum ' as 'ordinal co ntinuum' . The ' emi ssion s of particle-signs '
arc the resonances that couple the multiplicities, and the 'conjunctions
of flows' correspond to mutual amplifi cations or forc ed movements,
The quasi-causal operator, here called the 'abstract machine', is
characterized in terms of 'lines of flight' which refer to th e process of
counte r -actualization, and is said to 'draw th e plane ', that is, to extract
ideal eve nts from what actually occu rs and to mesh these multipliciti es
into a heterogeneous cont inuum . As Deleu ze writes 'the plane of
consiste ncy does not preexist the movements of deterritorialization
that unravel it, the lines of flight that draw it or cause it to rise to the
surface , the becomings that compose it ' (AT P, 270). Finally, the
' centre s of envelopme nt ' are not given a special nam e but they are
referred to indirectl y when it is asserted that ' the abstract Machine
exits simultaneously devel oped on the destratified plane it draws, and
enveloped in each st ratum whose unity of composit ion it defines . .. '
(AT !' , 70 ; my emphasis) .
This is, roughly, the mapping from one set of terms to another .
But in ATP we witn ess an elaboration of the original ontological
components and thi s introduces new terms and ideas . In particular,
in ATP the actual world is not defined simply in terms of extensit ies
.1nd qualities, but of vcry spe cific articulations of th e exte ns ive and
the qualitati ve . As I discu ssed in my reconst ruction, the actual consists
exclusively of individual entit ies , eac h individual at a given level of
scale emerging from the interactions of populations of smaller scale
individ uals. Deleu ze refers to these two scales of every stratum as the
' mo lecular ' and the ' molar ' . Stratificati on co nsists in producing popu
lations of ' mo lecules ' and organizing them into ' molar' , or large
scale, aggregates . (Clearly, 'molecul es' may be cell s or even organ
isms, when the molar scale is that of the organism or the species,
rcspcctivelv.} Thus, every stratum needs a double articulation, a
doub] o play of substances and form s, of exte nsit ies and qu aliti es, one
at the level of mol ecu lar populations and another at the level of molar
.lgg reg.lt es :
AP P EN DI X : DE L EU Z E 'S WO RDS
The first artic ulatio n chooses or deducts, from unstabl e particle
flows, meta stable mole cular or quasi-m olecular units (substances)
upon which it imp oses a statistical order of connect ions and
successions (forms) . The second articulation establishes functi onal ,
compact , stable structures (/orms) , and constructs the molar com
pounds in which structures arc simultaneously actuali zed (substances) .
(AT P 40 -1 )
This process is called a 'double articulation ' . Although the term
'double differen ciation ' alr eady occurs in the ontological list , it refers
only to the pair substance and form , not to thi s more elaborate
inte rpl ay of territorialiti es and codes . A similar elaboration is evident
in Del euze 's treatment of th e inten sive . As I argued in Chapter 2,
eve n the most rigidl y metric (or ' most stratified ') indi vidual still has
unactualized capacit ies to affect and be affected, and ma y not be
limi te d to a sing le stable equilibrium but have a vari ety of unactualized
stable states available to it. Th ese two aspect s of the int en sive , 'affects'
.1I1d 's ingularit ies' , become further developed int o 'parastrata ' and
'cpist rata' in ATP . On one hand, affects endow individuals with the
capacity to establish novel connections with alien mili eus, as with the
evolution of the capacity to tap into a reservoir of ox ygen, or other
non -alimentary ene rgy sources . Organisms may also have the capacity
10 act ively shape thei r environme nt , as spide r webs or beaver dams
illustrate . These capacities are what Delcuze calls 'parastrata", the
(·.lpad ty to connect with an 'anne xe d or associated milieu' (ATP, 5 1) .
O n the other hand, a fully formed individual may be capable of a
\ ,u'kty of stable states whi ch may be actualized by crossing crit ical
points and give rise to ' variat ions that ar e tol erated below a certain
threshold of iden tity ' (AT P, 50) . Th ese ' intermed iate states or milieus'
.m- what Deleuze calls 'epistrata". As he writes, even ' a single chemical
substance (sulfur or carbon , for example) has a number of more or
h'ss dcu -rrit orializcd sta tes' (AT P, 53) . The relations of the different
t('l'ms for int en sive factors can then be summa r ized like this :
Fo rms re late to codes and proccsst.~s of coding and decoding in the
IMf.l-"t rata; substances , l»..·ing formed matt ers, relate to tcrr'itorialities
APPENDIX : DELEUZE 'S W O R D S
and mov em ents of territorialization and deterritorialization on the
epistrata. (AT P, 53 )
Finally, there is a term which refers to the actualizatio n (or
effectuation) of th e quasi-causal ope rato r itself. I did no t discuss this in
d .tail, but I did give an ex ample in Chapte r 2 of the neighbo urhoo d of
a pha e transiti on (or 'e dge of chaos'). Deleu ze 's own example is not
crit ical points in a line of values, but crit ical sUIfaces in objects with
volume (LOS , 103) . (In both cases the quasi-cause opera tes at an N-Idim ension, as discussed in Chapte r 3) . In ATP, the organic membrane
as a critical surface is kept as an instance of the quasi-cause as it exists
effect uated in the actual , organizing the division of epist rata and
paras trata (AT P, 49-50). But now a spec ial term is coined for this
actualized qua si-causal ope rator : ' machinic assemblage' . As he writes:
'The most important probl em of all : given a machini c assemblage ,
what is its relation of effectuation with the abstract machin e? How
do cs it effectuate it, with what adequation ?' (AT P, 71 ) .
Much as th e quasi-cause o r abstract machine endows the vir tual
continuum with consiste ncy , the machinic assemblage endows actual
ent ities with consiste ncy , 'What we term machini c is precisely this
synthesis of heterogen eiti es as such' (ATP, 330) . The machini c assem
blage performs the different ope rations invo lved in stratification, suc h
as articulating a stratum with whatever serves as its subs tratum (e.g.
the pr e-biotic soup for organic strata), as well as doubly ar ticulating
the different extensit ies and quali ties, substances and forms, which
defin e a give n stratum (AT P, 71 ) . But also , as an actualized quasi
cause , the machini c assemblage is the agent behind co unter
actualizatio n:
T he assemblage has two poles or vectors: on e vec tor is oriented
toward s the strata, upon which it distributes territorialities, relative
dct c rr itori alizati ons and rct erritori alizations: the othe r is oriented
toward s the plane of consiste ncy or destratificati on , upon wh ich it
co njugatcs pr ocesses of det crrit orialization , carry ing them towards
till' abso lute of the eart h. (AT P, 145)
APPENDIX : DELEUZE 'S WORDS
A TI -O EDIPU S
In th is book the mapping of the items of the onto logical list is less
straightforward . In particul ar , the virtual and the intensive are gro uped
togethe r in a pro cess whi ch is referred to as ' mo lecular' (in the sense
just mention ed ), while the actual is referred to as ' the molar '. Unlike
ATP, wh ere all kinds of strata ar e co nside re d, in AO only the
actualization of human soc ieties is dealt with , so the molar see ms to
beco me synonymous with ' large social aggregat es', such as stable
persons, govern me ntal or economic insti tution s, agricult ural or indus
trial machin es. But it sho uld be kept in mind that thi s narrowing of
the meaning of ' the molar ' is a matter of focus and not a change in the
underl ying theory.
W ith some care , in fact , th e different elem ents of the onto logical
list can be paired with their counterparts in AO. The virt ual and the
intensive processcs of actualizati on are referred to as 'des ir ing produc
tio n' and defined as consist ing of three separate ' passive syntheses'
(AO , 26). These are referred to as ' the connective', ' the disjuncti ve '
and ' the conjunctive' syntheses. (This three-p art classificati on first
appears in LO S, 174. ) Th e disjunctive synthesis involves th e crea tion
of divergent relations among se ries , and it is said to occ ur on the bod y
without organs (AO, 13). It therefore re fers to th e virtual continuum,
'a pure fluid in a free sta te, flowin g without inte rruption, streaming
over the surface of a full body' (AO , 8) . The conjunctive synthes is, in
turn, invo lves the cre ation of conve rgent relat ions am ong series, an
operation whi ch as I said above , forms ' individuation field s ' which
already prefigure the intensive (pre- actualization). Thi s synthes is cap
tures one of the aspec ts of th e inten sive , the emergence of a larval or
passive subjec t, 'a strange subject with no fixed identity, wandering
abo ut ove r the body without organs .. . being born of the [int en sive]
states that it consumes , . " (A0, 16) . Finally, th e connec t ive synthesis
captures ano the r aspect of th e inten sive, the machini c assemblage. It
connects or couples together het erogcn eous 'partial objects or organs'
thro ugh the emiss ion of 'e ne rgy flows' (AO, 323) . Here the term
' partial' is not used in its xt in ive sense but in the sense of matter
filling spa • to a give n d 'g r of int ensity. 'The ye, the mouth , the
.lJlII S degrees of matter" ( 0, 309) .
APPENDIX : DELEUZE ' S WORDS
This interpretation of the three syntheses gives us one of the
elements of th e virtual (the plane of co nsistency or bod y without
organs), and tw o of the intensive (larva l subjects, assembl ages), but
leaves several things out. In particular, the other two elements of thevirtual, mult iplicities and the quasi-causal operator, don 't see m to be
included. Multipliciti es appear in AO as ' partial objects' when these
'a tt ach themselves to the body without organs as so man)' point s of
disjunction between which an entire network of new syntheses is nowwov en marking the surface off into coordinates , like a grid ' (AO, 12) .
This co rresponds to the idea that multiplicities exist in the sphere of
the intensive embodied in self-organizing processes, but may beextracted from these as 'Rat multiplicities' or 'pure events' and
deployed as such on the plane of consistency. The quasi-causal operatoris, in turn, referred to as a 'desiring machine':
Insofar as it brings together - without unif}'ing or uniting them
the body without organs and the partial objects, the desiring
machine is inseparable both from the distribution of partial objects
on the body without organs, and of the leveling [i.e. flatt ening]
e ffect exerte d on the partial organs by the body without organs ,
wh ich results in appropriation. (AO, 327)
The desiring machine is said to have 'chains' as its apparatus oftransmission (AO, 327). The term 'chain' is used instead of 'series'. It
has the meaning of a 'Markov chain' (AO, 39), a series of events in
which the probability of occurren ce of an y event depends only on the
previous one in the series . In other word s, a 'chain' is a partially
aleatory series. This co rresponds to one of the effects of the quasi
cause, bri efly discu ssed in Chapte rs 2 and 3, of injecting chance in the
distrihutions o f virtual singularities to create 'nomadic' distributions,
,1S opposed to the 'sede ntary' probability distributions which character
ize population s in the actual world. This is also expres sed by saying
that the quasi-cause must affirm all of chance with every throw of the
dk-c ( LO S, 59 -60) . The term 'chain ' is also used as in the expression
'signil),ing chain' hut without any reference to a fixed code , linguisticor otherwise . Rather these heteroge(wotls chains arc mad e of 'Hying
br-ic-ks , . . containing within [them] not only an inscription with signs
APPENDIX : DELEUZE'S WORDS
from different alphabets , but also various figures, plus one or seve ral
straws, and perhaps a co rpse' (AO , 40).
There is onc more detail to be discussed which provides an
important bridge to the nex t book to be deciphered (W IP). Much as
multipliciti es are woven into a virtual continuum through their divergences, but also form individuation fields when their series converge,
' the points of disjunction on the body without organs form circles that
co nverge on the desiring machines; then the subject , , . passes through
all the degrees of the circle, and passes from one circle to another'(AO , 20). The term ' passing' is used here as synonymous with
'becoming', and the 'degrees of the circle' are 'intensive quantities in
their pure state ' (AO , 18). The idea here is that this larval subj ect
w ith out identit y can move about the plane , from one individuation
field to another , becoming now this and now that intensive individual
depending on the intensities it consumes . This is the key idea behindthe process which in AO, ATp and WIp is referred to as 'becoming
animal' (as well as 'becoming-w oman", 'becoming-mo lecule' , etc .).
The co nce pt app ears first in D&R , 254:
W e sho uld not say tha t individuals of a given species arc distinguished
by their participation in other species: as if, for exampl e , there was
ass or lion, wolf or sheep, in every human being, There is indeed
all that and metempsychosis retains all its symbolic truth. However,
the ass and the wolf can be considered species only in relation to the
fields of individuation . . . lit is true that sorncone's soul) never
change d bodies, but its bod y could be re -env elopcd or re -irnplicated
in order to enter, if need be, other fields of individuation . . .
In other words, becoming-animal is an operation which cannot be
perfo rmed within the actual , by a transformation from a fully consti
tuted individual of one species to another of a difTercnt speci es. But if
we move towards the virtual, towards those circles of convergence orfields of individuation where there are still communications between
not-yet -actualized species, one can become 'rc -cnvcloped' in another
field . T his theme is elaborated in AO, 86 and in ATP , 238 and
becomes a key component of Dcl euzc's theory of artistic practice as
dis{'uss('d in WIP.
APPEND IX : DELEUZE 'S WORDS
WH AT IS PH ILOSO P H Y?
Muc h as AO narrows the focus of the onto logy and deals only with
the act ualizat ion of social structures, WIP deals ex clusively with the
rela tions bet ween the virtual , the int ensive and th e actual, on one
hand, and the different forms which thou8h t assumes in certain societies
(p hiloso phical, artistic and scientific forms of thought). The virtual
appears here as ' the plane of immanen ce' explore d by philosophical
thought; the int ens ive as ' the plan e of compos it ion' as it app ears in
artis tic tho ught ; and th e actual as ' the plan e of referen ce ' as it is
investigated by scientific thought. Let me discuss each one of these
'planes' star ting with the actual world,
One way of thinking about the plan e of referen ce is as a flat
ontology of indi vidual s. Th e subject matter of scie nce would be , in
th is interpretat ion , the world of fully consti tuted individuals and the
metric and measurable spacetime they form. In other words, actual
indivi duals would form the referen ce of scientific statements, and all
ref rents wo uld form a 'plane' precisely in the sen se that, ontologically
at least, they do no t have a hierarchical str ucture but remain a ' flat'
set , \'arying only in spatio-temporal scale . In Chapters 1 and 2, where
I discussed the philosophical conce pt of 'multiplicity', I emphasized that
the scientific ideas involved (differe nt ial relations, singularities) had to
he detached fro m th eir original context wh ere they are related to
mathematicalJ unctions. Th e justification I gave for this transformation
was that functions, as they are ordinarily used , presuppose indi vidu a
tion . Indeed, in some of their uses (as in their use to create state or
phase spaces) they define procedures for the individuation of states
within these spaces . T hese states of affairs constitute a re ferent, and
I he use of functions the re fore foll ows the line wh ich goes fro m the
virtua l to its act ualization , retaining only the final product.
T his is part of what Deleuze mean s when he asserts that the objec t
of science is 'functions which are presented as propositions in discursive
sysu-ms' (W IP, 117). I will return below to th e question of wh ether
one can cha racterize scie nce in th is way. As I said in Chapte r 4, 1 do
no t think there is such a th ing as 'science ' in genera l, so I reject many
of the det ails of the characte rizat ion given in WIP . Nevertheless, the
IMrt of it that I do kee p is the assertio n that most s icntilic fields tend
APPEND IX : D ELEUZE 'S W OR D S
to study the world in the direction of act ualization , sometimes
co nce ntrating on the final pr oduct and disregarding the pr ocess (e .g .
equilibrium th ermod ynamics) , sometimes studying the process but
always in the directi on of the final product.
Art, on the other hand , may be said to study, or engage with, the
inte nsive itself. The term ' inte nsive' is used in a varie ty of senses
only some of whi ch are relevant to th is characte rizatio n. One of the
co mpo nents of th e inten sive given in th e onto logical list was the
lar val subject who consumes int en sities as such, and is born and
reb orn of th ese voluptuo us consum ptions . In thi s case , the inten sive
state co mes first or it is prior to the individual that lives it (AO , 20) .
In other words, obj ecti ve inten sities do not constitute psychological
sensations bu t the very 'be ing of th e sensible' (D&R, 140) , a being
which is itself imperceptible psychologically given that intensities
become hidd en underneath qualities and exte nsities (D&R, 230), In
W IP this being of the sensible is divided into two co mpo nents ,
'perce pts' and 'affects';
By mean s of the material [e .g . paint , canvas, brush], the aim of art
is to wrest the percept from perceptions of objects and th e states of
a perceiving subject , to wrest the affect fro m affecti on s [e. g .
feelin gs] as th e transition fro m one state to another: to ex tract a
bloc of sensations , a pure being of sensat ions . (W IP, 167)
Simplifying somewhat , we may say that 'pe rcepts' are related to the
passive selves involved in the synthesis of living presen ts at all scales of
reality, in th e organi c and inorgani c world. Even though these presents
are constituted by 'conte m plations' or 'contracti ons of past and future
instants', they do not refer to a psychological reality . As Deleu ze
writes :
Th e plant conte mplates by co nt racting the ele me nts from whi ch it
originates - light , carbon, and the salts - and it fills itself with
olors and odo rs that in each case qualify its var iety, its co mpos ition:
it is sensatio n in itsel f. It is as if flowers sme ll themsel ves by
sme lling what composes them .. . before being perceived or eve n
sme lled hy an agent with a ner vous system and a br ain . (WIP, 2 12)
APPENDIX : DE LEUZE'S WORDS
On the other hand , affect s re fer to sta te transm ons whi ch mu st be
und erstood as ' becomings', in the sense of a becoming-animal or
becoming-plant discussed above . Th e artist must reach th at inte nsive
state where one can leave one individuation field to enter ano ther,
where one can reach 'a zone of indet ermination , of indiscerni bilitv , as
if things, beasts , and persons . . . endlessly reach that point that
immed iately preced es their natural differentiation ' (W IP, 173) . Finally,
having reached the very being of the sensible , the artist mu st place
these percep ts and alTects in their own plane , a plane of compos it ion,
a bloc or compo und of sensations whose 'only law of creation is that
the co mpo und mu st stand on its own' (W IP, 164) .
T hus, in a very lit eral sense , art is conce rned with making perceptible
the usuall y hidd en realm of the intensive . Similarly, philosophy mu st
ma ke the virtual intelligible. Philosophy must go beyond the centres of
convergence wh ere th e lar val subjects of percepts and affect s und ergo
intensive becomings, to reach the virtual in its full divergen ce and
difference, its continuous or ' inse parable variations' (W IP, 126).
Philo sophy cannot perform this task via a set of propositions whi ch
rifl er to the virtual, but rather, it must const ruct a thought wh ich is
isomorphic with th e virtual. The re fore, any philosophy mu st be con
structed out of the three compo ne nts of th e vir tual: multipliciti es,
qua si-causal ope rato r, and the continuum . In WIP these three compo
n .nts are referred to as 'conce pts', 'conce ptual persona e ' , and ' plane
of immanence', respecti vely.
Th term 'concept ' do es not refer to a semantic entity , that is, to
cone pts in th e ordinary sense, a sense in which there would also be
s i mtific conce pts (e .g. entropy) . Rather, it is defined as an enti ty
whi h wo uld be isomorphic with virtual multipliciti es.
[A concept is] a multiplicity, an absolute su rface or volume [e .g . a
ma nifold I ... mad e up of a ce rtain number of inse parable inten sive
vari: tions according to an orde r of neighborhood, and traversed by
a po int in a sta te of survey . (W IP, 32)
To say that a conce pt 'orders its co mpone nts by zo nes of neighbor
hood ' (W IP, 20) is to say that the relations it invo lves ar nonmetric
or ordinal. This re fers to the third sense o f 'i nte nsive' as defined
APPEND IX : DELEUZE 'S WORDS
above , and to the definition of to po logical spaces in Chap ter I , and is
also ex pressed by saying that a concept 's co mponen ts are 'i nte nsive
ordinates' (WIP, 20) . Concepts, therefore, are not to be thought of
sema ntica lly, bu t literally as sta te or phase spaces, that is, as spaces of
possibili ties st ructured by singularities and defined by their dime nsions
or intensive ordinates. As Deleu ze writes, 'Every conce pt therefore
has a phase space, altho ugh not in th e same way as in science' (W IP,
25) . For example , the Cartes ian conce pt of ' the Cogito' wo uld be a
space with three dimen sion s (doubting, thi nkin g and being) each
divide d by singularities into phases (e .g. perceptual , scie ntific, obses
sional doubting , as different phases of doubt, as oppose d to different
species of the genus doubt) .
The idea of a 'point in a state of survey' refers to an op eration of
the quasi-cause whi ch I did not describe in my recon structi on. Much
as multipliciti es mu st be meshed together into a continuum whil e
preserving th eir dilTeren ces ('exo-consiste ncy'), so the heterogen eou s
components of a multiplicity must th emselves be meshed by a 'po int
of abso lute survey' (W IP, 2 1) which continuously traverses them at
infinite speed ensuring th eir 'endo-consistency' . Exo-consiste ncy is
explained in WIP in terms of resonances between divergent series:
Co nce pts which have only [endo-]con sisten cy or inten sive ordinates
outside of any coordinates, freely ente r into relation ships of non
discursive reson ance . .. Con cep ts ar e cente rs of vibrations, each in
itself and everyone in relat ion to all the othe rs . This is wh y they
all resonate rather than cohere or corres pond to each othe r . ..
They do form a wall, but it is a dry -stone wall, and everyth ing
holds together only along diverging lines. (W IP, 23)
The qua si-causal ope rator behind these effects of endo- and exo
consistency is referred to as a ' con ceptual persona ' . Thus, Deleuze
writes: 'The conce ptual persona is need ed to cre ate concepts on th e
plane , just as the plan e need s to be laid out. But these two ope ratio ns
do not merge in the persona , which itsel f app ears as a distinct ope rato r'
(W IP, 76). Co nce ptual person ae are endo wed with all the characte r
ist ics of the quasi-cau 011 operato r. Mu ch as th latter mu st inject as
mu h hancc into the distribution ' of th singular and till' ordinary in
APPEND IX : DELEUZE 'S WORDS
virt ual series, ' the persona establishes a corresponde nce between each
throw of the dice and the inten sive features of a conce pt ... ' (W IP,
75 ) . And mu ch as the ope rato r is said to ex tract ideal eve nts fro m
what act ually occ urs (that is, to perform co unte r-actualizat ions or
' counter-effectuations'), in philosophy 'i t is precisely th e conce ptual
persona who co unte r-effectuates th e event' (W [P, 76).
But why the term ' pe rso na'? A clue to the meaning of thi s
expression may be glimpsed fro m some rem arks in LO S. As [ have
just said , in the circles of conve'8ence defined by pr e-actualized multi
plicities an int ensive indi vidual develop s (larval subject) , an indi vidual
which ex presses the world which conve'8ent ser ies form. Similar ly, in
the divergent series a ' virtual person ' develop s, a person who ex presses
what is common to man y different worlds (LO S, 115 ). A more
eI tailed ex planation , however, emerges from a discussion in D&R .
Much as a larval subject is born from percepts and affects which do
no t refe r to psychological phen om ena, but ar e the very being of th e
s msiblc, so personae are intimat ely connecte d with what constitutes
the very being of the int elligible (D&R, 141 ). Differen ce in inten sity
is the being of the sensible (sentiendum') and simultaneously that
which cannot be sensed (by fully actualized indi viduals) since it is
normally covere d by ex te nsities and qualiti es (D&R, 144). imilarly,
the being of the intell igible (cogitandum') is what can only be tho ught
and at the same time that which marks the impossibilit y of though t
(again, imposs ibility from the point of view of a fully actualized
think ' 1') , Hen ce the need to invent a conce ptual person a to capture
these cogitanda or ' thought-events', a persona who ' lives inte nse ly
within the thinker and forces him to think ' (W IP, 70 ).
Finally, there is the third compone nt : the virtual co ntinuum itsel f
or the 'plane of immane nce ' of a philosophy. Thi s refers to the
presu pposit ions of a philosophy, the main one of whi ch is an assumed
' im: ge of thought' (W IP, 37), in other words, a pre-con ceptual
intuitio n of w hat it is to think: 'E very philosophy dep ends upon an
int uitio n that its co nce pts constantly develop through slight differen ces
flf intensity .. . ' (W IP, 40). O ne way of und erstanding what thi s
means is to think of the relation b tween co nce pts and the plane of
immanc n l' as that between so lut ions and problem s. As I discussed in
'hapu-r 4, problems ar not reelucibl ' to th ir so lutio ns bu t rather arc
APPENDIX : DE LEUZE 'S WORDS
defined by their conditions : a give n distribution of the singular and the
ordinary , the impo rta nt and the unimportant. As such, pr obl em s are
inherently 'obscure yet distinct' and only acquire clarity in the pr ocess
which progressively specifics each of their so lutions . The intuition
referred to above wo uld refer to the grasping of a pr oblem as such, as
d istin ct and obscure (as oppo ed to grasping an esse nce, or a clear and
distinct idea) , an intuition which can only reveal itse lf progressively as
co nce pts are create d as cases o f so lutio n :
If th e conce pt is a solution, the conditions of the philosophi cal
problem are found on th e plane of immanence presupposed by th e
conce pts . . . and the unknowns of th e problem are found in the
conce ptual persona e that it calls up . . . Each of these three instances
is found in the others, but they are not of the same kind, and they
coexist and subsist without one disapp earing into the other . . .
[T]he three acti vities making up [the philosophical method] continu
ously pass fro m one to the othe r, support one another, some times
pr eced e and sometimes follow each other, one creating co nce pts as
a case of solutio n , ano the r laying out a plane and a mo vement on
the plane as the conditions of a prob lem , and the othe r inventing a
persona as the unknown of the pr obl em . (W [P, 8 1)
In my reconstruct ion of Deleuze ' s onto logy I used as a guiding
constraint the avoid ance of the categories of typological thou ght:
resemblance, identity, analogy and contradiction. But [ co uld have as
we ll said that what guides this construct ion is the avo idance of the
image of thou ght implied by these categor ies: ' a natu ral capacity for
th ought endo we d with a capaci ty for truth or an affinity with the
true ... ' (D&R, 131) . This image which, Dcl euze argues, haunts the
history of philosophy, has the result of turning the plan e of immanen ce
int o a plane of transcenden ce, Or what amounts to the same thing, to
trap philosophy within the plan e of referen ce, linking it to linguistic
propositions whi ch are either true of or false of their referents. This
manoeu ver, of course, closes the road to th e virtual or the problematic.
[f, on the co ntrary, the image of th ought leads to a plan e of
im mane nce, th n phil osophy 'docs not consist in knowing and it is not
inspired by truth. Rath er it is categories like Int I' sting, Rem ark able ,
APPENDIX : D E LEUZE 'S WORDS
or Important that deter-min e success o r failure ' (W IP, 82 ). The image
of tho ught th at has thi s problematic e ffect is on e in whi ch thought is
horn from the violent shock of an encounter with pure intensiv e
difl ercn ces (being of th e sensible), a shock whi ch a philosopher may
th en be capable of comm unicating to his or her other faculties, leading
all th e way to pure virtual dilTeren ces (being of th e int elligible) (D&R,
140).T his is not th e place to argue for or against thi s view of philosophy.
Whether o r not all phil osophical systems may ind eed be analysable in
terms of the three components o f the virtual remains an open question.O n the o the r hand, I must take issue with the imagc of science which
WI!' develops, particularl y because my disagreement with it bears not
just on narrowly scientific quest ions but on deep ontologica l matters .
Speci fically, my main divergen ce from Deleuzc 's ontology occurs atth e level of th e flat ontology of individuals. I m entioned above that I
broke with Dclcuze 's terminology by using the term 'individual' for
extended and qualified actual beings, while he reserves it for intensivebei ngs (larval subjec ts) . But the break is more than just terminological.
Altho ugh a flat ontology meshes well with many of Delcuze' s ideas
(his th eory of actual time as a nested set of cycli c presents of different
durations, for example) , it is unclear to what extent he subscribed to
suc h a view. In part icular , in a flat ontology as I have developed here
th ere is no room for totaliti es, such as 'society' or 'science' in general.But Dclcuze does not seem to mind such entities . For example , while
I would never speak of a virtual multiplicity co rresponding to all of
society (i.c, a 'social Idea' or 'social multiplicity ' ) he does so without
hesitat ion (D&R , 186).
In the case of 'science ' as defined in WIP, that is, in term s of
functions working as discursive propositions, the problem is that the
image invoked is one too clos e to that created by Anglo.American
philosophers of science of the first hal f of the twentieth century. All
the examples o f ' func t ives" (the components of functions) given in
\VIP co me from classical mechanics . No mention is made, for instance,
of the operators of quantum physics , which use functions themselvesas inputs ,1I1d outputs . And, of course, the question of what chemical
or biological functions arc is left most I)' unspecified . This amount s toclt'flning scie nce as if its ' e ssen ce' was classical mechanics . Furthermore,
APPEN DIX ; DELEUZE'S WORDS
mu ch as o ld-scho ol anal yti cal phil osophers disregarded th e actual
mathematical models used by ph ysicists and focused exclusively on se t
theory, so Dcl euze view s set theory as the too l which constitutes the
plan e of reference of scien ce (WIP, 121). My analysis in Chapter 4- o f
classical me chani cs (as an indi vidual field) broke with all this . It
preserved the idea that clas sical physics (as many other scie nt ific fields)
is mostl y co ncerned with the plane of reference (actual beings , metricspaces) but it uses a very different co nception of how referen ce (o r
the fix ing of reference) is achieved, placing more em phasis on caus al
interventions than on representati ons. Similarly for my treatment of
mathematical models, which are not reduced to linguistic entiti es(func t ions as propositions) hut tackled in th eir specificity.
On th e other hand , my ana lysis of classical physics meshes we ll with
Dclcuze's views on scien ce as developed elsewhere . The requirement
of avo id ing th e categories of typological thought to prevent th e plane
fro m becoming a plane of transcenden ce may also be expressed by
saying that we must avoid the ' classical im age of thought , and th e
st riating of mental spa ce it elTects' (AT P, 379) . Th e term ' striate d
space ' refers to a metri c space , while non metric spaces , 'vectorial,
proj ective, or topological' (AT!', 361 ) are referred to as ' smooth ' .
The transformation of thought itself into a metric space is not,however, an internal affair of philosophy , but on the contrary, it's
directly linked to th e relations between individual phi losophers (e .g .
Hegel) and indi vidual State or Royal institution s. It is these intitutions
whi ch first st riate or metricize real space (e .g . agricultural lands, urban
areas), and later perform the same operation on mental spaces . The
opposite transform ation, to create a nonmetric space for thought is
pe rform ed by philosophers (e .g. Spinoza) wh o operate outs ide of th eState .
A simi lar distinction is made between scientific fields or even,among the different practices (theoretical as opposed to expe rimental)
within one field, We have. on one hand, 'Royal science ' (the science
of th e great Ro yal Societies o r Academies at th e se rv ice of th e Stat e),
and. on the other, the 'minor sciences' operating in less prestigioussurroundings. Roughly, the distinction is between scientific practices
which arc axiomatic or theoremat ic , as opposed to problematic; that
0lll'ra l l~ within metr ic and exactly measurabl e spaces, as opposed to
APPENDIX : D E L E U Z E ' S W O RD S
d >aling with anexact yet rigorous nonmetric on es; that focus on the
simple behaviour of matter, as in ideal solids or gases, as opposed to
confronting the complex behaviour of liquid s (e .g. turbulence); and
that st ress constant and homogeneous laws, as opposed to becomings
and heterogeneiti es (AT P, 361) . My account of classical physics, which
is clea rly at odds with the Royal and legalisti c image which that field
has of itself, ma y be seen as an account from the point if view if min or
science. But for the same reason, it mak es the distinction whi ch WIP
establishes between science and philo soph y pass right through the
middle of science itself. Thi s, it seems to me, is the 'more Deleuzian '
approach to the subj ect.
Notes
TH E MATHEMATI CS OF TH E VIRTUAL:MANI FOLDS, VECTOR FIELDS AND
TRANSFORMATION GROUPS
1. The term 'multiplicity' makes its first appearance, as far as 1 can tell , in
1966 in Dcleuze 's book on Bergson, Gilles Deleuze, Berpsotiism (Zo ne Books,
New York, 1988), p. 39 . Its final appearance occurs in Deleuze' s last book
in collaboration with Felix Guattari, Gilles Deleuze and Felix Guattar i, What
Is Philosophy? (Co lumbia University Press, New York, 1994), p. 15.
2. Morris Kline, Mathematical Thouqh: fro m Ancient to Modern Times, Vol. 3
(Oxford University Press, New York, 1972), p. 882. (My emphasis)
Making surfaces into spaces, by eliminating the supplementary dim ension ,allowed the differentiation and study of different metric geometries . As
Morri s Kline wri tes:
Thu s if the surface of the sphere is studied as a space in itself, it has
its own geo metry , and even if the familiar latitud e and longitude are
used as the coo rdinates of points, the geo metr y of that surface is not
Euclidian ... However the geometry of the spherical surface is Euclidian
if it is regarded as a sur face in three-dim ensional space. (p. 888)
For the details on Gauss coordinatization pro cedure, which is what
guarantees th is absence of a supplementary dim ension or embedding space,
see Lawrence Sklar, Space, Time, and Space-Time (University of CaliforniaPress, Berkeley, 1977 ), pp. 27-42.
3. Kline, Mathematical Tboupbr, p . 890 .
4 . Gilles Deleuze, D!lJerence and Repetition (Columbia University Press, New
York , 1994), p . 182. O n page 183, for example, he says: ' In all cases the
mult iplicity is intrinsically defined, with out externa l reference or recourse
to a uniform space in which it would be submerged .' See also Gilles Deleuze
and Felix Guattari, A Thousand Plateaus (University of Minnesota Press,
Minneapolis, 1987), pp . 8-9,
Unity always operates in an empty dimension supplem ntary to that ofthe system considered (ovcrcoding) . .. [But aJ multiplicity never allows
r
NOT ES
itself to be overcoded, never has available a supp leme ntary dim ension
over and above its number of lines, that is, over and above th e
multiplicit y of numbers attached to those lines.
.5 . Deleuze and Guattari, II Thousand Plateaus, p . 266 . Th e rem ark quoted is
made about th e 'plane of consiste ncy ' not about multipliciti es. But th e
form er is nothing but the space formed by the multipliciti es th emselves , as I
will exp lain in detail in the next chapte r .
6. Wh en Dcl euze defines his multipliciti es he always see ms to be referring to
manifolds whose dimensions are used to represent degrees of freedom (or
indep endent variabl es) of some dynamic, and not to manifolds as mere
geo me tric objec ts . Thus, in his first introducti on of th e term he says,
Riem ann defined as 'm ultiplicities ' th ose things that could be det ermined
by th eir dimen sion s or their independent variables. He distinguished
between discrete multiplicities and continuous multipliciti es. Th e former
contain the principle of th eir own metrics . . . Th e latter found a m etrical
principle in some thing else , eve n if onl y in ph enom ena unfolding in th em
or in th e forces acting in th em. (Bcrasonism, p. 39)
And else where he says, using the word ' Ide a' to refer to concre te univ ersals
or multiplicities as repla cem ents for esse nces ,
An Idea is an n-dimen sional , continuous, defined multiplicit y. Colour
or rath er, the Idea of colour - is a three dim ensional multiplicit y. By
dim en sion s, we mean th e variables or coordinates up on whi ch a phenom
eno n depends; by continuity , we mean th e set of relations between
changes in th ese variables . . . by definition, we mean the elements
reci procally determined by th ese relations, eleme nts which cannot change
unless the multiplicit y changes its order and its metric. ( D!iJerencc and
Repetit ion , p. 182)
7 . I take th is rather Sim plified description fro m Ian Stewart. Does God Play Dice?
The Mat hematics ifChaos (Basil Blackwell, Oxford, 1989), Chapter 6 .
H. Loo king for relation ship s between th e differe nt solution curves [i.e .
tra jectories ] of the same differential equation, Poin car e began with a local
analysis and examined th e beha vior of these curves in th e neighb orhood
o r a singular point . . . He sho we d that there were four possible different
types or singular points and classified them by the behavior of the nearby
so lutio n cu rves: nccuds (no des), through which an infinite number of
sol utio n curves pass; eols (sadd le points), th rough which only tw o so lutio n
curves pass ... .foyers (fo i) , which th e so lution curves approach in the
NOTES
manner of a logarithmic spiral; and centres (ce nte rs), aro und whi ch th e
so lutio n curves are closed, envelo ping one another. Having used direct
algebraic co mputat ion to sho w that th ese four types necessaril y exist, he
studied th eir distribution. He found that in the gene ral case only three
types pr evailed - nod es, saddle points and foci - with cen te rs arising in
only exceptio nal circ umstances. (June Barrow-Green , Poincare and the
Three Body Problem [American Mathematical Societ y, 1997J, p. 32)
Roughly, we can say that Poincar e discover ed not only the existence of ce rta in
recurrent ' to po logical forms' which are bound to app ear in a large class of
differ ent physical models, but also that some of th ese forms are 'more
gene r ic' than othe rs, that is, th at if we study the distr ibution of singularities
in many different models some of them (cente rs) are less likely to occu r
than oth ers. See also discussion of th e term 'gene ric' , a technical term
whose meaning is still evolving , in Ralph Abraham and Chr isto phe r Shaw ,
Dynamics: The Geometry if Behavior, Vol. Three (Aerial Press, Santa Cruz,
198.5) , 1'1' . 19-34.9. Deleuze and Guattari , A Thousand Plateaus, p. 40 8.
10. 'To rever se Platoni sm ' , as Del eu ze says, we need ' firs t and for em ost to
rem ove esse nces and to substi tute events in th eir place, as jet s of singu lari
ties ' (Gilles Deleuze , Loqic c1 Sense [Columbia Uni versity Press, New York,
1990], p. 53) .11. Speaking of the image of the light of reason (or of rationalit y as a faculty
capable of graspin g the essential truth of thin gs) Deleuze says,
Th e very conce ption of a natural light is inseparable from a ce r tain value
supposedly attached to the Idea - namely, 'clarity and distinctness' ...
Th e restitution of the Idea in the doctrine of th e faculties requires th e
explosion of the clear and distin ct , and th e discovery of a Dion ysian value
according to whi ch th e Idea is necessarily obscure in so fa r as it is distinct, all
the more obscure th e more it is distinct . ' (Em phasis in th e original;
Gilles Deleu ze , D!iJerence and Repetition , p. 146)
Th e term 'Id ea ' here refers to multipliciti es , and th e fact that Deleuz e uses
that Platoni c term shows he mean s to repla ce essences with multipliciti es,
Ideas are by no means essences . In so far as problem s are th e object of
Ideas, probl em s belong on th e side of events, affecti ons, or accidents,
rather than o f theorematic essences . . . Co nsequently the domain of
Ideas is that of th e inessential. (I" 187)
12. Self-ass mhl y during [the ear ly stagcs of) em bryo nic development is not
mediated by direct gene int erv ention . Wh en all the tran scriptions have
NOTES
been prevented [thro ugh the use of an inhibitor] the regular cleavage
patt erns are re tained. However, the polarity of molecular organizatio n of
both the egg's cytoplasm and its nucleus ... are essential for normal
development. Hence the main features of [earl y] embryogenesis - ce ll
differentiation, indu ction, det ermination of pattern form ation - all ste m
from the ooge netica lly originated, spatial distribution of preformed
informatio nal macrom olecules. Th e initial conditio n of embryogenesis is
ooge nesis. The epigenetic.~ of embryo nic development is built on the
topo logical self-organization and orienta tion of macromolecules of the
total egg. (Vladimir Glisin , ' Molecular BioloBJ in EmbryoloBJ. The Sea Urchin
Embryo", in Se!f-0rsanizins Systems. The Emerpence eif Order, ed . Eugene
Yates [Plenum , Ne w York 1987], p . 163)
The term 'oogenesis' refer s to the pro cess which creates th e egg in the first
place .
13. Joe Rosen, Symmetl) ' in Science (Springe r- Verlag, New York, 1995), Chapter
2.Besides closure, a collec tion of enti ties togeth er with a rul e of comb i
nation needs to display associativity, and possession of identity and inver se
elements. The set of positive integers (including zero, and using addition as
a comb ination rule) displays associativity because the result of adding two
numbers first, and then addi ng a th ird one is the same as that of adding the
first to what results from adding the last two. It also conta ins an ' identity
cle ment' , that is, an eleme nt whi ch added to any other leaves the latt er
unchanged (in this case the identity elem ent is the number zero) . But it fails
to be a gro up because it lacks inverse elements , those which when compose d
with certain othe rs yield the identity element. For instance , the number
'-3' when composed with the number '+3' does yield zero (w hich is the
identity eleme nt) but '-3' is not part of the set of positive integer s. Thus,
for the integers to for m a group we must also include negati ve numbers in
the set.
14. T his dyna mic aspect of sym me try- based classifications is obscured in standard
presentations of the subject by the fact that the emphasis is not placed on
the t ransfor mation as an event , but on its input and output. That is, the
t ransformatio n is a pro cess but all that matter s math ematically is the init ial
and final states of the object transformed. See Ian Ste wart and Martin
Go lubits ky, Fea1ul Symmetry (Blackwe ll, Ox ford , 1992), PI" 32-3.
15. JIM, p. 97.Besid s assuming ideal solids and gases, th is illustra tion of broken
s)"lnn1l'try assumes that the gas containe r and the crysta l lat ti care infinit
ill all direct ions. The use of an 'obs rve r' to define invar iancc is just a
NOTES
convenience. The subjective po int of view can , in fact , be avoided. See Joe
Rosen , S)'mmetl)' in Science, PI" 173- 4 .16. Stewart and Golubitsky, Fea1ul Symmetl)', Chapter 7 .
17. Ralph Abraham and Christopher Shaw , ' Dynamics: A Visual Int roduction ' ,
in Se!f-Orsanizing Systems, ed. Yates, p. 576.
18. Stewart and Golubitsky, Fea1ul Symmeuv, Chapte r 5. See also , Gregoire
Nicolis and lIya Prigogine , Exp/orins Complexity (W. H . Freeman, Ne w York
1989), pp . 12-1 5.
19. Brian C. Goo dwin, 'The Evolution of Ge neric Forms', in Orsanizational
Constraints on the Dynamics eif Evolution, ed. J. Maynard Smith and G. Vida
(Mancheste r University Press, Manchester 1990), PI" 11 3- 14 .20. Dele uze, Difference and Repetit ion, p. 187.
Altho ugh Deleuze does not ex plicitly use the term 'symme try- brea king
cascade ', he docs refer to an 'e mbedding of groups' (p. 180) precisely in
the contex t of explaining how a multiplicity may be pr ogr essively deter
mined . Unfortunately, his brief discussion of gro ups uses a very obscure
aspect of Galo is's meth od , the originato r of group theory, called the
'ad junction of fields'. The two formulations are, nevertheless, equivalent,
fields of numbers and groups being two related ninet eenth-cen tury abst ract
objects. An algebraic problem , specified progressively as its field is com
pleted by successive adjunctions, is the eq uivalent of an abstract smooth
space being specified by a progr essive series of broken symmetries, yielding
increasingly mor e differe ntiated, more striated spaces . Deleuze 's discussion
of Galois is correct technically, but it is not as clea r and intuitive as the
equivalent formulation in terms of 'embedding of !,TfOUps' . Hence in this
reconstruction I will stick with the clearer alte rnati ve. But wheth er one uses
fields or groups, it is clear that some form of prosressil'e differentia tion is a key
component of the concept of a Deleuzian multiplicity.
2 I . What distingui shes a pace as opposed to a mere set of points is some
concept that binds the points togeth er . Th us in Euclidea n space the
distance between points tells how close points are to each othe r . . . As
Frechet [a pioneer in the development of topol ogy] pointed out, the
bind ing pr oper ty need not be the Euclidea n distance function . In
particular he generalized the noti on of distance by int roducing the class
of metric spaces. In a metric space, which can be a tw o-dim ensional
Euclidean space, one speaks of the neighborhood of a point and means all
those points whose distance fro m the point is less than some quantit y
. . . However , it is also possible to suppose that the neighb orh oods,
certain subse ts of a gh'en set of poin ts, are speci fied in some way, even
without the introduction ~f a metric. uch spaccs are said to have a
NOTE S
neighborhood topology. (Mor ris Kline , Math emat ical Thouqht ; p. 1160;
my em phasis)
1 will use the term ' me tric space' and 'no nmetric space' throughout th is
book in the sense in which th ey are defined in this quote but 1 will take
some liberties. I will spe ak of top ological spaces , for example, as th e ' least
metric ' and of Euclid ean as th e 'most metric' , even thou gh it would be
more techni cally co rrec t to differentiate fla tures if spaces that do or do not
depend on an)' strictly metric property.
22. Dclcuze usuall y speaks (follow ing Bergson ) o f tw o different t)'p es of multi
pliciti es, metric and nonmetric, whi ch he calls ' striated' and 's mo oth'. For
the purposes of ensuring th e co r re ct int erpretation of Delcu ze 's position
her e it would have been ver y useful if he had ever discussed Felix Klein 's
work, thereb y clarifying the relations between the metric and the nonmetric
as one of group inclusion . Unfortunately, as far as I can tell, Dcl euz e never
discusses Klein . On the oth er hand, Deleuze is perfectly aware of th e
ex iste nce of several nonmetric geo me t ries and uses a sinnle term (' smooth
space ') to refer to all of th em:
It is the difference between a smooth (vectorial , projecti ve, or topolonical )
space and a striated (metriC) space: in th e first case 's pace is occupied
without co unting' and in the second case 's pace is counte d in orde r to be
occupied'. (De lcuze and Guattari , A Thousand Plateaus, P: 361 ; my
emphasis)
T he definitions given in the extract are his own, but are linked to th e
more orthodox definitions. A metric space is counted in order to be
occupied in th e sense in which sede ntary cultures divide the land int o
measured (or counte d) plots in orde r to inhabit it:
Good sense is . .. agricultural , inseparable from the agrarian problem,
th e establishme nt of enclosure s, and the dealings of middle classes th e
part s of whi ch are supposed to balan ce and to regulate on e another. Th e
ste am engine and livestock , but also properties and classes, are th e living
sources of good sense , not onl y as facts that spring up at a particular
peri od, but as ete rn al archetypes. (Deleuze, Loqic if Sense, p. 76 )
To the sede ntary way of metricizing space, of dealing with it as esse ntially
exte nsive, Dcleuz opposes an int ensiv e way of oc upying space the way a
liquid do cs, that is, occupying it without Jividing it or co unting it. Thi s
alternative h· calls a ' no madic d istribution ' . Th e distin cti on bctwc n scdcnt
.lTV .md nomad ic distribution s is first mad , in DilJ;'rence and Repetit ion ,, .
N O T E S
pp . 36-7, in relation to questions of typolog ical thinking, but is taken
further in an actual co mpariso n of nomad and sede ntary cultures
. . . eve n though th e nomadic traj ect ory may foll ow trails or customary
ro utes , it do es not fulfill th e function of the sede ntary ro ad, wh ich is to
parcel out a closed spaee to people, assigning each person a share and
regulating the com munication between shares . Th e nomadic traj ect ory
does th e opposit e : it distributes people (or anima ls) in an open space .. .
sede ntary space is striate d [i.e. metricized], by walls , enclosures and
roads between enclosures, whil e nomadic space is smooth [i.c. non
metric], marked on ly by 'traits ' that ar e effaced and displa ced with th e
traj ect ory. (De lcuzc and Guattari , II Thousand Plateaus, p. 380; emphasis
in the original)
23. Morris Kline, Math emat ical Tboupht, p. 917.
24. David A. Brannan, Matthew F. Esplen, Jerem y J. Gra y, Geometry (Cambridge
University Pre ss, Cambridge, 1999 ), p. 364.
25. This way of describing the subject oversimplifies things some what. First of
all , th e actual relati ons between the different geome tr ies are more complex
than the Simplified hierarchy 'topological-differential-projective-affme
Euclidean geome tries' may sugges t. For th e detail s of Klein 's orig inal
classificati on see ibid., P: 919.
My friend the math ematician Andreas Dress (pe rso nal com munication)
summarizes Klein 's programme (called th e Erlange r Program) like this,
Th e Erlanger Program by Felix Klein is based on the fact that depending
on whi ch (bijective) transformations you need to deal with (isome trics
keeping distances invariant , similarities scaling all distan ces by th e same
fact or and, hen ce, keeping rati os of distances invariant, affine maps
keeping rati os of distances of points on parallel lines invariant, proj ectiv
ities keeping cro ss-ratios of distan ces invariant , differential transforma
tions respecting infinitesimal straightness, hom eomorphisms respecting
nothing but infinit esimal closeness) , it always makes sense to ask ( 1)
which features of configurations within th e space of int erest do remain
invariant , and (2) wh ether a basic famil y of such features can be found so
that every other such feature can be expressed as a function of those basic
ones.
26 . Morris Klin " Math ematical Thouqlu , p. 9 21 . Th er e are imp ortant exce ptions
to this state me nt. Some mathematicians, like Riemann himself, but also
\ illi: m ' IilTord, did see an ollto logica l connec tion between the metric and
N O T E S
nonme t:ric prop erties of spaces. As one historian of twentieth-century physics
writes,
[RiemannI asserted that space in itse lf was nothing more than a three
dim ensional mani fold devoid of all form: it acquire d a definite form only
through the mat erial co nte nt filling it and det ermining its metric relations
. . . Riemann' s anti cipation of such a dep end en ce of the metric on
physical data later provided a justifi cation for avoiding the noti on of
absolute space wh ose metric is ind epend ent of physical forces . For
example , more than sixty years later, Einstein took Riemann 's em pirical
conce ptio n of geome try using it as an important justification for his
gene ral theory of relati vity.
(Tia n Yu Cao, Conceptua l Development if Twenti eth -Century Field Theories
[Camb rid ge University Press, Cambridge, 1997], P: 373)
27 . Gordo n Van W ylen, Thermodynami cs (j ohn Wil ey & ons, New York , 1963) ,
P: 16.
28 . Wh at is the significance of these indivisible distances that are ceaseless ly
transformed and cannot be divid ed or transformed without their eleme nts
changing in nature each time? Is it not th e int ensive characte r of this type
of multiplicity' s elem ents and the relations betw een them ? Exact ly like a
spee d or a temperature, which is not co mpose d of oth er speeds or
te mperatures , but rath er is envelo ped in or envelops othe rs , each of
which marks a change in nature. The metrical principle of these
multiplicities is not to be found in a homogen eous milieu but resides
elsewhere , in forces at work within them , in physical phen om ena
inhabiting them . . . (De leuze and Guattari , A Thousand Plateaus,
pp . 3 1-3)
T he term 'd istance' is used as if it was a nonmetr ic property, though in its
usual meaning it certainly den otes something metric. Deleuze takes this
specia l inte nsive mean ing of 'distance' from Bertrand Russell as I will discuss
in de tail later in the next chapte r . O n distances as int ensive magnitudes, or
as 'i ndivisible asymme tr ical relations ' see Deleuze, Difference and Repet it ion,
p. 237 . Deleuze does not explicitly give phase transitions as examples of
'c hanges in kind ' . But one of the very few illustrations he does give is indeed
a symmet ry-brea king transition , 'For exa mple , one can divide movem ent
into the gallop, tro t, and walk , but in such a way that what is divided
changes in natu re at each moment of the di vision ... ' (Dc lcuzc and
C uauari, /1 Thausatul Plat eaus, p. 483).
NOTES
O n phase transitions in animal movem ent as broken symme tries see,
Ste wart and Golubitsky, FeOIjul Symmetry, Chapte r 8.
29 . Cao, Conceptual Development ,?! Twentieth-Centu rJ Field Theories, p. 283.
30. Th e essen tial idea of grand unified theories . .. [is] the general form of
hierarchical symme try br eaking: an und erl ying large gauge symme try of
all int era ction s is brok en down in a success ion of ste ps, giving a hierar chy
of brok en symme tr ies . (ibid., p. 328)
31. It is beyond the sco pe of this chapter to analyse Einste in's use of differential
mani fold s in technical detail. But I sho uld at least mention the way in which
his usage differs from that of Deleu ze. In Einste in's theory a gravitational
field const it utes the metr ic struc ture of a four-dimensional manifold
(spacetime), and to thi s exte nt, the metric properties of space (rathe r,
space time) are ind eed connected to the physical processes wh ich occ ur
within it. However, as the philosoph er of science Lawre nce Sklar reminds
us, despit e the fact that Einste in's field equation does rel ate the metric of a
manifold to the distribution of mass and energy, the relation between the
two is not genet ic: the metric is not caused by the mass-energy distribution ,
it is only associated with it in a lawlik e way. ee Sklar, Space. Time, and
Space-Time, pp . 50- I .
32. Th e mo ve away fro m metamath em atics (set theory) and back to the actual
mathem atics used by scientists was initiated by the philosopher Patrick
Suppes . Yet the credit for the introducti on of state space into mod ern
analytica l philosoph y, as we ll as the cr ed it for em phasizing physical mod ality
in the analysis of that space , goes to ano the r philosoph er, Bas Van Fraasen .
See Bas Van Fraasen, l.aws and Symmetry (C lare ndo n Press, O xford , 1989) ,
Chapte r 9.
33. Ralph Abrah am and Chris to phe r Shaw, Dynamics: The Geometry cd' Beha vior,
Vol. 1 (Aerial Press, Santa Cruz, 1985 ), pp. 20- 1. My description is merely
a paraphrase of the foll OWing description:
Th e modeling pro cess begins with th e cho ice of a particular state space
in which to represent the system. Prolonged observations lead to man y
tr ajectories within the state space. At any poin t on any of these curves, a
veloci ty "ector may be deri ved [using the differentiation operato r ]. It is
useful in descr ibing an inherent tenden cy of the system to move with a
habitu al velocity, at part icular po ints in the state space. Th e prescription
of a veloc ity vect or at each point in the state space is called a velocity
vector .fielJ. T he sta te space , filled with trajectories, is called the phase
p"r/mit of till' d -narn ical system. The velocity vecto r field has been
NOTES
derived from the phase portrait by d!fTerentiation . . . Th e phrase dyruunical
Sj'stem will specifically denote thi s vector field . (Emphasis in the original)
l4. Albert Lautman, quoted in Gilles Deleuze, Loqic if Sense (Columbia Univer
sity Press, ew York , 1990) p. 345. (My emphasis)
Lautman 's Le Probleme du Temps (fro m which thi s ext ract is taken) and
'Essai sur Ie otion de tructure et d ' Existence en Math ematiques ', are
Dclcuzc 's main sources on the ontological analysis of stat e space. Deleu ze
paraphrases Lautrnan 's description in other books, but given the ce ntrality
of these ideas in his work 1 prefer to qu ote Lautrnan ' s own words.
15. Abraham and Shaw, Dynamics: The Geomeuy ifBeha vior, pp. 35-6.
36. Nicolls and Prigogine, Explorina Complexitv; pp. 65-71 .n . Abraham and Shaw, Dynamics: The Geometry ifBehavior, pp . 37-41.38. Abraham and Shaw, Dynamics: A Visual Introduction, p. 562.
~9 . Deleuze, D!lJerence and Repetition, pp. 208-9. (Emphasis in the original. )
Deleuz e borrows the ontological distinction of the actual and the virtual
from Bergson . See Deleuze, Berpsonism, pp . 96-7.
40. Willard Van Orman Quine , quoted in Nicholas Rescher, 'The Ontology of
the Possible', in The Possible and the Actual, ed . Michael J. Loux (Cornell
University Press, Ithaca , 1979), p . 177.
41 . For a brief account of the recent history of modal logic, see Michael J.Loux , ' Introduction: Modality and Metaphysics', in Loux, The Possible and th e
Actual, pp . 15-28 .4 2. Ronald N. Giere, 'Constructi ve Realism ' , in lmap es if Science. Essays an
Real ism and Empiri cism with a Reply by Bas C. Van Fraasen, cds . Paul M.
Churchland and Clifford A. Hooker (University of Chicago Press, 1985),
p. 84 .4 1. Bas Van Fraasen, Laws and Symmetry', p . 223. Van Fraasen discusses the tw o
standard typ es of laws, laws of succe ssion (which gov ern the evolution of
trajectories, and are exe mplified by Newton' s laws) and laws of coexiste nce
(which restri ct position in state space, and are illustrated by Boyle 's law for
ideal gases) .
44 . Exactly mat ching initial conditions in the laboratory and the model is not
possible, so we normally deal with bundles ?f traj ectories in state space. Th e
statistical distribution of a small population of initial states in the model is
mad e to mat ch that of the errors which the exper imente r may have made in
pr 'paring the real syste m in a parti .ular initial condition. In what follow s
thi s point will not make mu ch differen ce so I stick to the simpler case of a
single trajectory.
4S . t ;i(,rt, rgues that the regularities exhibited by the possible histories reveal
'om, thin g about the w USCl I reqularitles in the real ph)'Sical s 'sl .m:
NO TE S
For the modal realist, the causal stru cture of the model, and thu s, to
some degree of approximation , of the real syste m, is identical with the
modal structure. For any real syste m , the functional relation ship among
the actu al values of [the degrees of freed om] are causal not because they
hold among the actua l values in all such real systems but because they
hold for all possible values of thi s particular system . (Consrrucrrre Realism ,
p. 84; emphasis in the original)
See also Ronald N . Giere , Explaintnq Science. A Coanit i l'e Approach (Univer
sity of Chicago Press, 1988), Chapte r 4. Gier e is, in this case , wrong. State
space, as I will argue in Chapter 4 , provides no causal information about the
modelled processes.
46 . One's attitude towards modalities has a profound effect on one's whole
theory of science . Actualists . . . must hold that the aim of scienc e is to
describe the actual history of the world . For [modal realistsI . . . the aim
is to describe the structure of physical possibilit y (or propensity) and
necessity . Th e actual history is just that one possibilit y that happ ened to
be realiz ed .. . (Giere , Constructi ve Realism, p. 84)
47. Deleuze, Loaic if Sense, p . 54.48. Considering that Deleuze 's analysis hinges on the differen ce between the
differ entiation and int egration operators of the calculus, it will be necessary
to remove on e traditional obj ection to the very idea of giving an ontological
dime nsion to these operato rs. Thi s objection is that the output of the
differentiation op erator (instantaneo us rates of change or infinit esimals)
cannot be thought of as anything but mathematical fictions . ot to do so has
led in the past to man y ste ri le speculat ions and controversy . However,
alth ough a vector field is ind eed com posed of man y of these instantaneous
rates of change, what matters to us here arc not the 'instants ' themselves,
taken on e at a tim e, but the topoloqical in variants which those instants displa y
collect ively , that is, the singularities of the field .
49 . Ste phe n G. Eubank and J. Doyne Farmer, ' Intr oduction to Dynamical
ystems ' , in Introducti on to Nonlinear Physics, ed . Lui Lam (Springer-Verlag,
New York, 1997), p. 76.50. Abraham and Shaw , Dynamics: The Geometry if Behavior, pp. 7- 11.51. Attractors ar e indeed defined as a 'limit se t ' with an open inset (its basin).
But the word 'limit' in the definiti on mak es all the difference in the world ,
since it refers pr ecisely to the tendenci es of traj ectories to approach the
att rac tor in the limit. See ibid. , p. 44.
S2. 'Intuitively, according 10 Russell, a syste m is det erminist ic exactly if its
N O TES
pr evious stat es determine its later states in the exact sense in which the
argum ents of a function determine its values. (Van Fraasen, Laws and
Symmetl)' , p. 251)
See Van Fraasen 's discussion of the relation between the modal category
of physical necessity and deterministic laws in Chapters 3 and 4 of Laws and
Symmetry :
53. Nicolis and Prigogine, Explorinq Comple xity, p. 14. (Emphasis in the original. )
54. For example , the way Deleuze approaches the question of necessity is by
splitt ing the causal link : on one hand , processes of individuation are defined
as sequences of causes (every effect will be the cause of yet anoth er effect)
while singularities become pure incorporeal ifJeas of tho se series of causes; on
the oth er hand, these pure effects are viewed as having a quasi-causal capacity
to affect causal processes. By splitting causality this way, Dcleuze manages
to separate the det erminism which links causes to causes, from strict
necessity . See Lopic t?f Sense, p. 169 .
Deleuze uses the word 'de te rminism' as synonymous with ' necess ity',
and uses the word 'des tiny' instead for the modified link between causes . I
keep the word 'de te rminism' to avoid introdu cing neologisms, but empha
size the break with strict necessity. Anoth er way of expres sing Delcuzc ' s
conceptualization of this modality is from D!lJerence and Repetition , p. 83,
Destin y never consists in step- by-step deterministic relations between
presents which succeed one another . . . Rather, it impli es between
successive presents non-localizable connections , actions at a distance, systems
of replay, resonances and echoe s . . . which transcend spatial locations
and temporal successions.' (My emphasis)
Th e idea of 'non-localizable connec tions' is the key conce pt her e and can
be und er stood by refer ence to convec tion cells. Whil e the causal intera ctions
between the cell 's components are localizable collisions (billiard- ball style
causality) , the source of cohere nce in the flow pattern (the periodic attractor)
is, indeed, nowher e specifically in space or tim e. Th e attractor establishes
connec tions (e lse there would be no coherence in the flow) but not
localizable ones.
')5. Willard Van Orman Quine , ' Reference and Modality', in From a Loqical Point
'!f' Viell' (Harper & Row , New York, 1965 ) , p. 155. Even though most
modal analyses deal with purely linguistic phenomena, such as counte rfactual
sente nces , the mom ent one approaches such sente nces as referring to the
real world (tec hnically, the mom ent we quantify over possible entities) we
arqu ir« an onto logical commitme nt to the existence of ess mces. In othe r
NO T ES
words, we commit ourselves to affirm that objects possess some of their
pr operties necessarily while others only contingently .
56. The first option (ensuring transworld identity through particular essences or
hacceiti es) is exemplified by Alvin Plantin ga, 'Transworld Identity or
Worldbound Individu als?', in Loux, The Possible and the Actual, pp. 154-7 .
The seco nd option (co unte rparts linked through general essences) is
illustrated by David Lewis, ' Counte rpart Th eory and Quantified Modal
Logic', in The Possible and the Actua l, pp. 117- 21.57. Delcuze , D!lJerence and Repetit ion , pp . 211- 12. See also Deleuzc , Berqsonism,
p. 97 . Deleuze does not, in fact , refer to the virtual as a physical modality,
but the fact that he explicitly contrasts virtua lity and possibilit y (following
Bergson ' s lead ) does indicate that he is thinking in modal terms.
58. I take this description of Arist otelian philos oph y from Elliot Sober, The
Nature t?f Selection (MIT Press, Cambridge , 1987), pp. 156-6 I .
59. Deleuzc, Difference and Repetiti on, p. 29. To avoid falling prey to the dangers
of representationalism (or as I call it typological thinking) Deleuze follow s
Michel Foucault 's analysis of classical representation, wh ich according to the
latter forms an episte mo logical space with four dim en. ions or 'degrees of
freedom ' : identity, resemblance, analogy and opposition, P: 262 .For a discussion of this aspect of Foucault 's thought from the point of
view of an analytical philosoph er see Gar y Gutting , Michel Foucault 's Archae
oloBY rif Scientific Reason (Cambridge Univer sity Press, 1993), Chapter 4 .
In what follows I Simply take the idea that there are recurrent features in
these classificatory practices (rese mblance, identity, etc .) but not that these
form a global entity called an 'e pisteme' . I do not believe such global entities
or totalities exist as will becom e clear in the followi ng chapte rs .
60 . 'The first formula posits resemblance as the condition of differ ence . It
ther efore und oubtedl y demands the possibility of an identical concept for
the tw o thin gs that differ on condit ion that they are alike . . . According
to the other formula, by contrast, resemblance, identity, analogy and
opposition can no longer be conside red anyth ing but effects of a primary
difference or a primary system of differences . (Dc lcuzc , D!fJerence and
Repetiti on, p. 117)
Dclcuze, in fact , does not speak of 'c onstraints guiding a construct ive
project ' . He rath er affirms his desire for creating a ph ilosophy '!f' difference,
and then denoun ces the categories of typological or represent ational thinking
as obstacles to reaching that goa l. Th e differences he has in mind are not the
e ucrnal diffe rences between thinq« that are part and parccl of classificatory
pract ices, bUI productive differcnces perhaps best illustra ted by inccmil'e
NOTES
d!fJerences, dilTerences in temper ature, pr essure , etc. within one and the
same system, which are mark ed by thres holds of intensity determi ning phase
tra nsitions. See p. 222 .61 . Ronald F. Fox , Eners)' and the Evolution if Life (W . H. Free man, New York,
1988), p. 8.
T he mechanisms by which the chemical clem ents come into existe nce is
stella r nucleosynthesis. The processes involved are an example of how ener8Y
./1011' pr odu ces complex states of matter from simpler constituen ts. A
combination of gravitational energy and nucl ear energy converts vast
quant ities of hydr ogen gas, the simplest ele me nt, into the nuclei of other
more complex cleme nts . Nucleosynthesis involves nuclear reaction cycles
and happ ens in stages that corre late stro ngly with changes in ste llar
structure . (Emphasis in the original)
62 . Philosopher s tend to imagine that a piece of bulk material is simply a
collect ion of individual crystals arranged so perfectl y that , for all practi cal
pur poses, th e properties of the bulk sample are simply a sum of the
properties of these crystals. In oth er words, they imagine we can divide the
hulk sample in extension and , given the packing arrange ment of the crysta ls,
we will alwa ys end up with a similar if smaller sample . But in realit y, we
do not have perfectly regular crystal lattices (the irregularities playing a
crucial ro le in the stability of the stru cture) and we canno t divide a bulk
sample beyond a given size without losing some eme rge nt pr operties:
Like the biologist , the metallurgist is conce rned with aggregates and
assemblies in which repeated or ex te nded irrepularities in the arranged
atoms becom e the basis of major structural features on a larger scale ,
eventually bridging the gap betw een the atom and things perceptibl e to
human senses . (Cy ril Stanley Smith , 'Structure, Substructure, and Super
structure ', in A Search for Structu re [MIT Press, Cambridge, 1982), p. 54 ;
my em phasis)
See also, in the same volume , Smith, ' Grain Shapes and other Metallur
gical Applicat ions of To pology'. O n the eme rgence of bulk prop erti es at
di fferent critical scales, see Michael A. Dun can and Denn is H. Rou vray,
tIIicroclu.ltw (Scientific Ameri can , Dece mber, 1989), p. 113.
2 THE ACTUALIZATION OF THE VIRTUAL IN SPACE
I . Michael T . Ghisclin, Metaphysics and the Oriqin l?I Species (State University of
New York Press, Albany, 1997), p. 78.
NOTES
2. A good history of this de bate, explaining the ro le which Michael Ghiselin
played in it , can be found in David L. Hull , Science as a Process (University of
Chicago Press, Chicago , 1988) , Chapter 4.
3. Ghiselin , Metap hysics and the Oriq in if Species, pp . 37-41.4. It is unclear to what extent Deleuze subscribes to th is idea of a flat onto logy
of singu lar individuals. Some parts of his theor y (for example, his theory of
tim e involving a nested set of larger and larger temporal scales) seem to
dem and such an onto logy. Yet , elsewhere , he does seem to talk of tot alities.
Thus, while I view the rea lm of the socia l as a flat onto logy (made of
individual decision -makers, individual instituti onal organizations, individual
cities, individual nation states) and thu s would never speak of 'society as a
whole' or 'culture as a whole ' , Deleuze does talk of 'society as a whole '
and spec ifically, of a virtual multiplicit y of soc iety . See, for example, Gilles
Dele uze, D!lI erence and Repetit ion (Co lumbia University Press, Ne w York,
1994), p . 186. There are also terminological problem s that need to be not ed
give n that Dcleuze uses the term ' individual' in a very idiosyncratic way. In
parti cular, he does not use 'actu al entity' and 'individual' as synonyms as I
do. For Deleuze the term ' individual' refers to an entity in the process '!f
actu alizati on , that is, before it acquires its final qualiti es and extensities . For
example, a fully develop ed hum an being would be an actu al entity , but the
embryo as it is being unfolded and develo ped wo uld be an individual. On e
would be an ex tensive being, the othe r an intensive one. (See , for example,
pages 247 and 250.) I will use the word ' individual' in the sense in which it
is used by Ghisclin to link it to anti-essenti alist thought, but this should not
cause mu ch distortion to Dcleuze.
O n the other hand, I do break with Deleuze 's use of the term 'species '
which does not seem to impl y that species are also individuals, and hence ,
the produ ct of an indi viduati on pro cess disti nct from the one that gives rise
to organic individuals during embryoge nesis . He does no t see m to keep the
tw o levels of scale separate (as I think they should be) and speaks of 'species'
and 'parts ' as the organic expression of qualities and exte nsities respectively
(p 25 1). Yet , he does acknowledge in passing the role of rep roducti ve
isolation in th e individuation of species. He writes,
A kineti cs of population adjoins, without resembling, the kinet ics of the
egg; a geog rap hical process of isolation may be no less formative of
species than intern al genetic variations, and sometimes precedes the
latt er . (p. 2 I7)
5. Ernst Mayr, quoted in Elliot Sober, The Nature of Selection (MIT Press,
Cambridge, 1987), p. 156.
NOTES
6. Ibid. , p. 159. Sober makes some corrections to Mayr 's way of explaining the
reve rsal of Aristotelian esse ntialism . He believes it is incorrect to compare
averages and essences, as Mayr do es in the extract , since averages may be
taken to be real properties at the populationa l level. So the reversal is
characte rized in terms of the rol e of variation : while for Aristot elians
hom ogeneity is the natura l state and variation is what needs special
explanation , for population thinker s it is variation which is nat ural , while
homogeneity, when it exists, is what needs to be explained .
7. lbid. , p. 160.
8. Gilles Deleuze and Felix Guattari, A Thousand Plateaus (University of
Minnesota Press, Minneapolis, 1987), p. 48. (My emphasis)
9. Niles Eldredge, Macro-E"olutionary DynamiCS (McGraw- Hill, New York ,
1989 ), pp. 155- 7 .
10.1 . D. Murray, Mathematical BioloBY (Springer-Verlag, Berlin 1989), pp . 1-4.
II. Ibid., pp. 8- 1l.
12. In both organism and cellular populations, for example, we are concerned
with rates of birth (rates of cell division ), rates of death , as we ll as migration
rates. These rates of change , in turn, define in both cases a dynamical
pro cess which disp lays threshold effects as we ll as asymptotic stabl e states.
Divergent uni versa lity also implies that these organic phenomena may share
dynamical feat ures with inorganic ones. Some processes, like the formation
of concentration patterns due to an interaction between the rate at which a
chemical react ion proceeds and the rat e at which the pr odu cts of that
reac tio n diffuse, occur in both embryological processes and non -biological
chemical processes (like the famous Belou sov-Zhabotinsky reaction), a fact
which suggests that a virtual multiplicity can be divergently actualized in
both organic and inorganic mo lecu lar populations. Indeed , the mathematical
techniques and analytical methods which are used to model intera ction s
between animal and plant populations (such as predator-prey systems) are
direct ly appli cable to reaction kinetics, that is, to the dynam ical models of
interacting populations of mol ecules, organic or inorganic. Sec ibid., p . 63 .
13. For a discussion of population -level qualiti es see Sober, Nature ef Selection,
p. 167 .
14. How does aetuali zation occur in things thems elves? Beneath the
actu al qualities and exte nsities [of things them selves] ther e are spatio -
tempor al dynami sms. Th ese arc the actualizing, differ enciating agencies .
Th ey must be surveyed in every domain , eve n though they are ordinarily
hidd en by the const ituted qualiti es and exte nsities. Embryology shows
that the division of the egg is secondary in relati on to more significant
Illorphogeneti c rno vcm mts: the augm ent ation of free surfaces, stre tching
N O TE S
of cellular layers, invagination by foldin g, regional displacement of
groups . A whole kinematics of the egg appears which implies a dynamic.
(Deleuze , D!lJerence and Repetition, p. 214)
IS. Gerald M. Edelman, Topobioloqy, An Introduction CO Molecular Emb'J'oloBY (Basic
Books, New York, 1988) , pp. 22- 4 .
16. As a result of epithe lial-mesenchymal transformation , two kinds of motion
can arise that differ to some degree in scale . The first invo lves the
obvious cel l migration that can take place after conversion to mesen
chyme, as well as its cessation following cond ensation of mesenchyme
into rounded epithe lial masses. Th e second . . . is the folding, invagina
tion or evagination of whole tissue shee ts to form various st ruc tures ,
including tubes . In both cases, new ce llular environments are created ,
leading to the possibility that different inductive Signals will be released.
(Ibid., p. 70)
17. lbid., p. 94 .
18. lbid.; pp. 80 -1.
19. The phras e 'an exact yet rigorous ' is used on several occasions by Dele uze to
refer to a style of thought, but also to a characte ristic of topological
manifolds themselves. O ne occasion is the discussio n of Bertrand Russell's
concept of 'ordinal distan ces ' which I will discuss later in the main text.
See, Dele uze and Guattari, A Thousand Plateaus, p. 483. Another use of the
phrase occurs while discussing Husserl's notion of ' vague and material
essences ' , topologieal essences which are assimilated to singularities (events)
and affects (p . 407) .
20. Arthur T . W infree, When Time Breaks Down. The Three-Dimensional DynamicS
ef Electrochemical JVa ves and Cardiac Arrhythmias (Princeton University Press,
Prin ceton, 1987), p. 253 . (My emphasis)
21. Stuart Kauffman , The Orioins ef Order. Se!f0roanizat ion and Selection in
Evolution (Oxford University Press, New York , 1993), p . 461 .
22 . lbid., p. 44 2.
23. Th e expec ted network connec tivity features exhibit stro ng self-organiza
tion properties analogous to phase transitions in physics, as the number
of regu latory connec tions , M, among N gen es increases. If M is small
rel ative to N, the scrambled geno mic system consists of many small
genetic circuits , each unconn ected to the remainder. As the number of
regulatory connect ions, M, increases past the number of gen ' S, N, large
connec ted circuits form . The crystallization of large circuit s as M increases
is analogo us 10 a phase transition. ( tuart Kauffman, 'Sc lf-O rganizatlon.
NOTES
Selective Adaptation and its Limit s', in Evolution at a Crossroads, eds .
David. J. Depew and Bruce H. Weber [MIT Press, Cambridge , 1996),
pp . 180)
24. In Deleuze's philosop hy th e connec t ion between multiplicities, on one hand,
and quali ties and extensities, on the other, is more intimately defined , with
differential relations corresponding to qualities and singularities to
extensities.
[A) multiplicity such as that of co lour is constituted by the virt ual
coe xiste nce of relati ons between genet ic or differential eleme nts of
a part icular order . Th ese relations are actualized in qualitatively dis
tinct colours, while their distinctive poin ts are incarna ted in distinct
extensit ies , which correspond to tho se qualit ies ... W e have see n
that eve ry pro cess of actualizatio n was in this sense a double differencia
tio n, qualitative and exte nsive. (Deleuze, D!iJerence and Repetit ion ,
p. 245)
25. K. Eric Drexler, ' Biological and Nanomechanical Syste ms : Contrasts in
Evolut ionary Capacity', in ArtifiCial L!fe , cd. Christo pher G. Langton (Addi
son- Wes ley, Redwood City, 1989) , p . 510.
26. Dele uze, D!iJerence and Repetit ion, P: 223.
Intensity cre ates the extensities and the qualiti es in which it is ex plicated;
these ex te nsities and qualities are differ enciat ed . .. Crea tion is always
the production of lines and figures of difler enciation. It is neverth eless
true that intensity is explicated only in being cance led in this differ en
ciated syste m that it cre ates. (p. 255)
27. Van Wy len, Thermodynamics, p. 16.28. Bert rand Russell , Principles if Mathemati cs (W. W. Nort on , New York) ,
p. 104 (for remarks on pleasur e) and p. 171 (for remarks on colour) .
Dc lcuze wo uld not co unt pleasure as an intensive quantity part of mental
irulivuluatinq processes . He see ms to view pleasur e as an effect of the cance lling
of intensive differe nces:
Bioph ysical life imp lies a field of individuati on in wh ich differ ences in
intensity are distributed her e and there in the fo rm of excitations. Th e
quanti tat ive and qua litative process of the resolution of such diffe rences
is what we call pleasure . (Deleuze, D!iJerence and Repet ition , p. 96)
29. Marti n II . Krieger , DoinS PhySiCS. How PhySicists Take /-101.1 C!l th e World (Indiana
l.ln ivcrsity Press, Bloom ington and Indianapolis, 1992), p. 130.
10. n,·J..uz<', D!I]crmce lind Repa it ion , p. 222. (My emphasis)
NOTES
In this extrac t, 'd ivers ity' refers to the wo rld of actu al phenomena and
their exte rna lly defined differe nces (that is, to difference as subordinated to
rese mb lance) while intensive differe nces define the in-itself (nuo mena) of
the world, the positive and prod uct ive differences which create or generate
phenom ena.
31. lIya Prigogine and Isabelle Stengers , Order out l' Chaos. Man 's Nell' Dialoque
with Na tu re (Bantam Books, Ne w York, 1984) , p. 135.32. Deleuze explains the relation between intensive differences and gene tic
differences b), saying that 'complex systems increasingly tend to interiorize
thei r constituent di ffere nces ' , that is, thei r individuating factors (D!iJerence
and Repetit ion , p . 256). See also Deleuze 's discussion of Darwini an differ
ences on pp . 248-9.33. Wh en discussing the virtual and the intensive, Deleuze usually divides the
subject into two areas, although the terminology varies . Someti mes he speaks
of 'singu larities and affects' , other times of 'speeds and affects', yet in other
places he speaks of 'events and att r ibutes' . All these formulat ions are , I
believe , equivalent . See furthe r discussion and references in Chapte r 3,
footnote 46.
34. O n this new class of formal spaces which complements state space, see
W alter Fontana, ' Functional Self-Organization in Complex , Syste ms' , in
1990 Lea ures in Complex Systems, eds, L)'nn Nadel and Danie l Ste in (Addison
W esley, Redwood City, 1991); and, in the same volume , Stuart Kauffman,
' Random Grammars: A New Class of Mod els for Functional Integration and
T ransfor mation in the Biological , Neura l and Social Sciences' .
35. W e know nothing about a bod)' until we know what it can do, what its
affects are , how they can or cannot ente r into composition with other
affec ts, with the affects of ano the r body, either to des troy that bod y or
to be destroyed by it , eithe r to exc hange actions and passions with it or
to join with it in composing a mor e pow erful body. (Deleuze and
Guatta ri, A Th ousand Plateaus, P: 257)
36. James J. Gibson, The Ecoloqical Approach to Visual Percept ion (Houghto n Mifllin
Company, Boston, 1979), pp. 15-1 6.37 . lbid., p. 132.38. orne of the rec ur rent assemb ly patt erns that have been discover ed (and
which may tu rn out to be universal) are of the type that articulates
hete roge neous elements. Stuart Kauffman has coined the term ' meshwo rk'
to refer to th is type of assem hlage.. ee Stua rt Kauffman, Random Gramma rs,
p.428.I haw mad' -xtcnsivc usc of Kauffm an 's meshwork s, and of the ir
op posite, hierarchies, as recur ren t assernhly patt erns for the analysis of
NOT ES
human history in Manu el DeLanda, A Thousand Years if Nonlinear History
(Zone Books, New York, 1997). A similar distin ction (or a spec ial case , that
of centralized and decentralized decision -making systems) as well as a relat ed
set of recurrent assembl y patterns (clockworks, motors and networks) is
d iscussed and appli ed to history in Manuel Dcl.anda, War in the Age ifIntelligent Machines (Zon e Books, New York , 1991).
~9 . It is no longer .a question of imp osing a form upon a matter but of
elaborating an increasingly rich and consistent mat erial , the better to tap
increasing ly intense f orces. What makes a mat erial increasingl y rich is the
same as what holds heterogeneiti es topether with out their ceasing to be
heterogeneous. (Del euze and Guattarl , A Thousand Plateaus, p . 329; my
emphasis)
40. Delcuze , D!fference and Repetition, P: 22 3.
There is an illusion tied to intensive quantities. This illusion, however, is
not int ensit y itself, but rather the movem ent by which difference in
intensity is canceled. Nor is it onl y apparently cancele d . It is really
canceled , but outside itself, in extensity and underneath quality. (p . 240;
my emphasis)
41 . It is now an easy matter to extend our discussion to nonequiltbrium states
. . . They can be transient . . . But they can also be permanent if we
establish and maintain appropriate conditions, which we refer to as
constraints. Thus , a temperature difference appli ed between two sections
of a slab .. . will result in nonequilibrium situations in which the syste m
is never allow ed to identify itself with its enviro nment. We should not
conclude from these examples that non equilibrium is an artificially
imposed condition ... we see non equilibrium states in mu ch of our
natural environme nt - for example , the state of the biosphere which is
subjected to an energy nux that arises from the balance of radiation
between th e sun and the earth . (Emphasis in the original; Gregoire
Nico lis and lIya Prigogine, Exploring CompleXity [W. H. Freeman, New
York 1989], p . 56)
42. lbul ., p. 59.4L lbid. , p. 60.
44. David Acheso n, From Calculus to Chaos. An Introduction to Dynamics (O xford
University Press, Ox ford , 1997) , pp. 54-6.
45. Delcuze and Guatta ri, Whar ;s Philosophy ?, p. 140. (My emphasis)
4fl. Richard Hin 'hlilTe , T oward a H omol ogy 01' Pro cess: Evolut ionary lmpli ca-
N O T ES
tions of Experime ntal Studi es on the Generation of Skeletal Pattern in Avian
Limb Development ', in Organi zational Constraitus on the Dynamics ifEvolution ,
cds. J. Maynard Smith and G. Vida (Mancheste r University Press, Man
cheste r 1990), p . 123. (Emphasis in the original)
The biologi st Brian Goodwin, who has taken the br oken symmetry
approach to classification to its extreme, argu es that these insights about
specific organs may be generalized to explain the dynamical origin of all the
morphological features behind our static classifications:
There are several consequences of this view of morpbogenesis. First , it is
evident that morphology is gen erated in a hierarchical manner, from
simple to complex , as bifurcations result in spatially ordered asymmetri es
and periodicities, and nonlinearities give rise to fine local detail. Since
th ere is a limited set of simple broken symme tries and patterns that are
possible (e.g. , radial, bilateral, periodic) , and since developing organism s
must start off laying down these elem ents of spatial orde r , it follows that
these basic forms will be most common among all species. On the oth er
hand, the finer details of pattern will be mo st variabl e between species ,
since the pattern-generating process results in a combinatorial richness of
terminal detail , and specific gen e products in different species stabilize
traj ectories leading to one or another of these . . . Th e fact that Virtually
all the basic organismic bod y plans were discovered and established
during an early evolutionary period, the Cambrian, is oft en remarked
with surp rise , but it is just what one would expect on the basis of the
above argument. (Brian C. Goodwin, 'The Evoluti on of Gen eri c Form s' ,
in Organizational Constraints on the Dynamics clj' Evolution, eds. Maynard
Smith and Vida , pp . 114-[5)
See also Brian Goodwin, How the Leopard Changed its Spots (Simon & Schuster,
New York 1996), Chapter 5.
47 . When I introduced dilTerential geometry in Chapte r 1 I said that one of
Gauss 's achievements was to get rid of an embedding space by coordinatizing
the manifold itself. This allow ed him to define the equivalent of metri c
lengths (and oth er properti es) in this differ enti al space. This coordinatization
is an example of what I mean when I say that a nonmetric space is
metricized . DeIeuze also refers to this operation in his discussion of the
relation betw een metric (str iated) and smooth spaces in Deleuze and
Guatta r i, A Thousand Plateaus, p. 486.
48 . Co nsiste ncy necessarily occ urs between heterogeneit ies, not because it is
the hirth of a diff erentiation. hut because heterogeneit ies that were
limnerl )' conte nt to coex ist or succcc l one another becom e hound up
NOTES
wit h one ano the r through the 'conso lidatio n' of th eir coexiste nce or
succession . . . W hat we term machinic is pr ecisely this synthesis of
het erogen eities as such . (ibul.; p. 330)
T erms like "self-consistent aggregate' and ' mac hinic assemblage' are used
synonymously in this book.
49. Although there are a few math em ati cal functions which produce seve ra l
outputs, the majority of them have a sinole outpu t. That is, some functions
map inputs and outputs (ar!:.'1Iments and values) in a one- to-one fashion,
others in a man y-to -on e fashion , and a few in a one- to-many form. See
Russe ll, Principles rif Mathematics , pp . 265-6. Deleu ze 's reciprocal det ermina
tion , I believe , would imply a many -co-many mapp ino, and a mapping such as
this would be useless as a functi on . On th e othe r hand , this ' useless '
mappi ng wou ld capture th e desir ed idea for a multiplicit y, an organization
of the ' many' as such, without the need for th e 'one ' .
50. Dcleuze , D!lJerence and Repetiti on, pp . 172 - 4 .
In ' W hat is Philo sophy' , th e distinction between virtual multiplicities
(there referred to as 'conce pts') , and functions is mad e the basis of Dclcu zc' s
critique of science's inability to gra~p the virtual. Unfortunately, the analysis
there is obscu re d by his introduction of unfamiliar terms like ' functive'. Sec
Dcleuze and Gu attari , What is Philosophy ?, pp . 117-1 8.
I think its is clearer to see his rejecti on of functions as mod els for the
virtual in terms of th e pr e-indi vidual nature of the virtual coupled to the
fact that functi on s may be taken to represent ind ividu ation processes. Thi s
way it becomes clear why function s without this individuation aspect
(wi thout a distinction between dep endent and inde pe nde nt variables) can
indeed be mad e part of th e virtual , Reference to ' form less func tions' as a
defining ele ment of concre te uni versals (o r as th ese are some times referred
to, 'a bst ract machin es ' ) can be found in Deleuze and Guattari, A Thousand
Plateaus, p. 141.
51. De leuze, Loqic rif Sense, P: 52. Let me elabo rate thi s point (eve nts as pr e
indi vidu al ent it ies) by first conside r ing the typ e of individ uality of actual
crcnts. Co mpare d to th e individuality of an organi sm or a species (to mention
only the two entit ies for which I have given individuation pr ocesses) an
actua l eve nt has a more fleeting and changing individuation . Del iuze argues
that eve nts have th e ind ividu ality of a haecceity , exe mplified by the ' thisness'
or un ique Sing ulari ty of a mom en t. As he says
T h re is a mode o f ind ividu ation very differe nt fro m that of a person,
subject, thing , or substance. W e reserve the name haccceity for it. A
season, a Winter, a summer, an hour , a date have a perfect individua lity
1'lCking nothing, even ulOugh th is individua lity is di ffere nt from that of a
NOTES
th ing or a subject. T hey are haecceities in the sense that th ey consist
entire ly of relation s of movement and rest between molecules or
particles , capacities to affect and be affecte d. (Deleuze and Guattari , A
Thousand Plateaus, p. 26 1)
Some of th e het erogeneou s assemblages I mentioned before, such as th e
assemblage of a wa lking anima l, a piece of ground and a grav itational field,
have this individuality. Thi s is particularl y clear if we do not picture an
abstract case but thi nk instea d abo ut a concre te eve nt: this animal walking
on this hot and humid summe r day. (' This should be read without a pause :
th e animal-stalks-a t-five-o' cloc k' , p . 263). Thi s event consists of affects, not
only the affordances of anima l, gro und and fie ld , but also the capacities of
the othe r individuals involved, including degrees o f heat and humidity. (' A
degree of heat is a perfect ly individuated warmth distin ct from th e substance
or subj ect that receives it . A degree of heat can ente r int o com position with
a degree of whiten ess, or with another degr ee of heat , to form a third
unique ind ividuality . . .' , p. 253) T he event also consists of relations of
rapidity and slowness : th e gro und affords the animal a so lid sur face only
because relative to the spee d or temporal scale of change of the animal, the
ground changes too slow ly. At geological t ime scales th is piece of so lid
ground would indeed be mu ch more Iluld .
T o apply th is to ideal eve nts . Th e singularit ies wh ich populate the virtual
are also haecceities , but the tw o definin g features (spee ds and affects) are
distributed differe ntly: a singulari ty is nothing bu t an accidental feature in a
field of speeds (o r velocit y vectors) its indi viduality consisting enti rely of its
inva riance , that is, its capac ity of not being affect ed by ce rt ain transforma
tio ns which affect th e rest of th e field .
52. The term •conde nsat ion of singu larities' to refer to the expansion of
singulari t ies into series, and th e establishme nt of converge nt and divergent
relation s betw een series, is used for exa m ple in Deleuze, D!1Jerence and
Repetit ion, r- 190 .
53. Multipliciti es (o r Ideas) are referred to as 'complexes of coe xiste nce' in
ibid ., p. 186.In othe r wo rds , unlike th e singu larit ies which define an int en sive process
which may be actualized only one at a time (e ither because th e bifurcation s
need to be crossed sequentially, or because only one am ong alternati ve
attracto rs may be occupied) virtual singularit ies all coe xist within their own
special temporalit y. Wi thin th e int ensive ' the Ideas, relations, variati on s in
these rela tio ns [embed ded levels] and dist inct ive point s [singularitiesl arc in
a sense separated: instea d of coexisting they enter states of simultanei ty or
succession' (p. 252).
N OT E S
54. Th e impo rtance of orde r , from a purely mathematical standpo int, has
been immeasurably increased by man y modern devel opments. Dedekind,
Canto r , and Pean o have shown how to base all Arithmeti c and Analysis
upon series of a ce rtain kind . . . Irration als are defined . . . entire ly by
the help of orde r . . . Proj ectiv e Geom etry [has] shown how to give
points, lines and plan es an orde r indep end ent of metrical considerat ions
and of quantity ; whil e descriptive Geom etry proves that a very large part
of Geom etry demands only th e possibility of serial arrangem ent. (Russell ,
Principles ifMathematics, p. 199)
55. lbid.; pp . 157-9 . Actually, Russell uses the more gene ral term 'magnitude '
to refer to th ese indi visible int ensiti es , and 'distance' as a speci al case of a
magnitude. A terminologi cal confusion should be avoid ed here. Russell uses
the term 'magnitude' to oppose that of 'quantity' (one involves onl y serial
orde r, the othe r cardinal number) . But wh en Deleuz e co mme nts on Russell' s
wo rk (as wcll as Mein ong 's ), he uses ' mag nitude ' as synonym with 'quanti ty'
and opposes both to 'distance' . See Deleuz e and Guattari , A Thousand
Plateaus, p. 483.
This terminological conflict should not be a probl em here since I will not
be using th e term 'magnitude ', and I will always use the term 'ordinal
distance' instead of just 'dis tance ' to distingui sh th e latter from ' me tric
d istances' or length s. Although Russell introduces distances as inte nsive, that
is, as indivisible in exte nsion, he then devises a sche me which allow s him to
speak of distanc es as divisible (by reducing them, via a conve ntion, to
extensive 's tre tc hes') and thu s abandon s any hope of linking th e int ensiv e
and the ex te nsive morphogenctically. (Russe ll, Principles if ,tlathematics,
pp . 180 - 2)
')6 . Ordinal constr uction do es not imply a supposed same unit but only .
an ir re ducible noti on of distan ce - th e distances implicate d in th e depth
of an intensive spatium (orde re d distan ces). Identical unity is not pr esup
pose d by ordinatio n ; on the co ntrary, this belongs to cardinal number
. . . 'We sho uld not , therefore , believe th at cardinal number results
unaIyticalIy from ordina l, or from th e final terms of finite ordinal series
. . . In fact, ordinal number be com es cardina l only by ex te nsion , to the
exte nt that the distances [are] develop ed and equalized in an ex tensity
established by natu ral number. W e sho uld therefore say th at , from th e
outset, the concept of number is synthetic. (Deleuze , D!fference and
Repetition, p. 233; m), emphasis)
Russell , on the other hand , establishes between magnitudes and numbers
on I)" .1 Iogic'lI re lation, that bet ween the ge ne ral and the part icular : qu antities
NOTES
are magnitudes which are particularized by spatio -te mpo ral positi on (Russe ll,
Principles ifMathematics, p. 167) .O ne way of bringing up the differen ce between Deleuzc ' s and Russell' s
appro aches to series and numbers, is by contrasting th eir analyses of the
th eory of ir rational numbers of Dedekind . Arguing that there were gaps in
th e co mpact series of ratio nal number s, Ded ekind introduced th e noti on of
a 'cut', a way of segmenting a dense cont inuum int o two , mutually
excl uding, parts. His idea was to define th e conce pt of number in terms of
such cuts performed on purely ordinal continua. Some of these discontinui
ties yield rational numbers, but others, he po stulated , mu st yield irrationals.
Russell , for whom the den sity of the rati onal s see ms to be eno ugh, objects
to this merely postulated existe nce of irrational cuts, and equates irrational s
with one of th e classes of rati onals crea te d by th e cut . Thi s, in effect ,
ex plains one exte nsive conce pt (num ber) in terms of anothe r, equally
exte nsive one (class or set). Deleuz e, on th e cont rary, sees in th e conce pt
of a cut a way to ex press the genesis of numeri cal quantity out o f int ensive
non -numeri cal co ntinua: ' In thi s sense , it is the cut which constitutes th e
next genus of number, the ideal cause of continuity or the pu re eleme nt of
quantitati vity' (De leuze , D!fference and Repetition, p . 172).
57. Dcleuze , LoOic cj"Sense, p. 109.
58 . Divergen ce and disjun ction are , on th e cont rary , affirm ed as such . But
what does it mean to make divergen ce and disjun ct ion th e objects of
affirmation ? As a gene ral rul e two th ings are simultaneou sly affirm ed only
to the exten t that their differen ce is deni ed . . . W e speak, on the
contrary , of an ope rat ion according to which tw o thin gs . . . are affirm ed
through th eir difference . . . to affirm their distance as that wh ich relates
one to the other insofar as th ey are different . . . Th c idea of a positive
distance as distance (and not as an annulled or overco me distan ce) appea rs
to us esse ntia l . . . Th e idea of positi ve distance belongs to top ology and
th e sur face . (Ibid., p. 172 )
59. Co nverge nt and divergent relation s defin e th e modal stucus of vir tual relation s.
Following Leibniz, Deleuze calls th ese virtual relati on s compossibility and
incompossibilitv:
T wo eve nts are com poss ihle wh en the ser ies which are organ ized aro und
their singularit ies extend in all direction s [that is, co ll\'erge ]; th e)' are
incornpossihle wh en th e ser ies diverge in the vicinity of constitutive
singularities . Converge nce and divergence are ent irely original relat ion s
which cover th e rich domain of alogica l com patibilit ies and incompatib il
irics. (Ihid., p. 172 )
N O TE S
The modal status of the virtua l may be more easily grasped by contrast ing
it with othe r modal rel ations, such as the relations which modal logicians
postulate to exist between possible worlds. Th e modern theor y of possible
worlds is also based on the ideas of Leibniz, but disregards these alogical
capacities or affect. Briefly, the key relation between possible worlds is that
of accessibili ty: one world is accessible from anoth er possible one , if eve ry
situation possible in one is also possible in the othe r. Given this relation,
possible worlds may be grouped togeth er into families or equ ivalence classes.
Wh enever situations in one class are imposs ible in anoth er one, that is,
when ther e exist logical or physical contradict ions between them , worlds
belonging to one arc inaccessible from those belongin g to the oth er (Michae l
J. Loux , ' Intro duction: Modality and Metaphysics ', in The Possible and the
Actual , pp. 20- 8).
Dclcuze would accept these ideas but argue that contradic tions between
possible worlds are a deriva tive phenomenon . In other words, that distribu
tions of possibl e worlds, and their fully individuated contents, depend on
deeper relations of compossibility and incompossibility between pre-indi vid
ual multiplicities: where the series emanating from multiplicities converge ,
a family of accessible possible worlds would be defined; wher e they diverge,
an inaccessible family of worlds would begin . See Gilles Deleuze , The Fold.
Lcibntz and the Baroque (University of Minn esota Press, Minn eapolis, 1997),
p. 60 . ee also Deleuze , D!fJerence and Repetiti on , p. 48, wher e he adds 'the
noti on of incompossibility in no way reduces to that of contradiction and
docs not even impl y real opposition: it impli es only divergence ... '
60 . Dcleuzc, Loa ic rif Sense, p. 5.
6 I . peaking of the particular case of catastro phe theory, wher e the limitation
to pot enti al-driv en syste ms with four degr ees of freedom makes a full
classification of attractor s and bifurcat ions possible, Alexander W oodcock
and Monte Davies write
In any syste m governed by a pot ential and in which the system 's behavior
is det ermined by no more than four differ ent factors, only seven
qualitatively differ ent types of discontinuity [bifurcation] are possible . In
other words, while ther e is an infinit e number of ways for such a system
to change continuously (staying at or near equilibrium) , there are onl y
seven structura lly stable ways for it to change discontinuously (passing
through non -equilibrium rates), Other ways are conce ivable, but un
stable; th 'y are unlikel y to happ en mor e than once . . . The qualitati ve
type of any stable discontinuity does not depend on the spec ific nature of
the pote ntial involved , merely on its existence. It do 's not depend on the
specific conditions reg ulating behavior , merely on their number . It does
N O T E S
not de pend on the specific quantitative cause-and-effect relati onship
between the conditions and the resulti ng behavior merely on the empirica l
Jaa that such a relationship exists. (Alexande r W oodcock and Monte Davies,
Catastrophe Theory (E. P. Dutton , Ne w York , 1978), p . 42; my emphasis)
Th ere are tw o imp ortant ideas expressed here. Th e first is related to the
question of uni versality: as long as differ ent equation ' or differ ent physical
syste ms share the same topological invariants (the same number of singulari
ties, the same number of dim ensions) the detail ed nature of the equations
or of the syste m (the speci fic type of intensive differ ence driving th e process,
or the speci fic quantities which define the process) does not make mu ch
difference in the spec ificatio n of their long -term tend encies. Th e second idea
relates to the question of immanence: the long-term (asympto tic) tend encies
of a process may he independ ent of specific causes , but they do depend for
their very existe nce on ther e being some causal process or another.
62. Deleuze, Loa ic rifSense, p. 169. (My emphasis) Deleuze adopts this approach
from the Stoics who wer e the first to split the causal link : on one hand,
processes of individuation are defined as sequences of causes (every effect
will be the cause of yet another effect) whil e singularities becom e pure
incorpor eal effects of those series of causes ; on the other hand, these pure
effects are viewed as having a quasi-causal capacity to endow causal processes
with cohere nt form. By splitti ng causality this way, Delcuze manages to
separate the determinism (or destin y) which links causes to causes, from
strict necessity.
63 . lbid., p. 147.64. Th e image of echoes and resonan ces as that which links multipliciti es recurs
throughout Deleuze 's work. See Chapte r 3, footn ot e 53 for an explanatio n
and examples.
65 . Kenn eth M. Sayre, Cybernetics and the Philosophy rif Mind (Rout ledge and
Kegan Paul, Lond on, 1976), p . 23.
66 . lbid., pp. 26- 30.67. Th er e is a close relation between communication theory and thermodyn
amics. Much as in the latter the equilibrium state (for an isolated system) is
defined as the one characte rized by maximum disorder (maximum entro py),
the state achieved once differ ences in int ensity have been cance lled, so in
the former equilibrium corresponds to a situation where the differ ences
within series have been cance lled , where all the events have becom e
equiprobable. In such state no information may flow in the channel (ibid . ,
pp. 38- 43).Deleuze uses th is connectio n between the intensive and the informational
to define the relations between the series of ideal vents. As I have said, he
NOTES
re fers to an information channel as a 's ignal' , and to th e information quanta
as 's igns' ,
Such systems, constit ute d by placing dispara te elements or het erogeneou s
series in cc:nmunicat ion, arc in a sense qui te common. They are signal
sign syste ms. The signal is a structure in which differen ces in po tentia l
arc distributed, assuring the comm unication of disparate co mpone nts: th e
sign is what flashes across the boundary of two levels, between tw o
comm unicating series. Indeed, it see ms th at all phen omen a respond to
th ese conditio ns inasmuch as they find th eir ground in a co nstitutive
dissymmetry, di fferen ce, inequality . All physical syste ms are signals, all
qualities are signs. (De leuze, Logic c1 Sense, p . 26 1)
Sec also De leuze, Difference and Repetition, pp . 20 and 222 .
68. If we examine th e singular ities co rres po nding to th e tw o im portant basic
se ries we see that th ey arc distin gu ished, in both cases, by their
distribution . From one to the othe r, certa in singular points disapp ear or
are divide d, or und ergo a change of nature and funct ion. Th e moment
th e two series resonate or co mmunicate we pass fro m one distri bution to
another. (Dele uze , Logic c1 Sense, p. 53)
69 . It is in difference that . . . phen om en a Hash th eir meaning like signs. Th e
inte nse world of differen ces . . . is pr ecisely th e object of a supe rio r
em piricism . This empiricis m teaches us a strange ' reason' , th at of
the mult iple, chaos, and differen ce . (Deleuze, Difference and Repetition,
p. 57)
T here is in addition a temporal dim ension of the virt ual, which I will
discuss in the next chapte r, whi ch also defines thi s othe r empiricism.
An Idea, in this sense , is neither one nor mu ltipl e , but a mu lt iplicit y
constituted of d ifferentia l ele ments, differe nt ial re lations betw een th ose
clements , and singularit ies co rrespo nding to those relation s .. . All three
are projected in an ideal te mpo ral dimen sion which is th at of pr ogressive
determination . There is, therefore , an empiricism c1 the Idca . . . (p . 278;
my emphasis)
On the concepts of mu ltiplicity and quasi-causal operator (and rela ted
ideas, like 'perplication' , 'complication', etc.) as ernpirico-idcal notions, see
r- 84 .70 . Stephanie For rest, 'E mergent o rn putationr .'elf-organizing, o llcctive and
Coop erat ive Phen om en a in Nat ura l and Art ificial om puting ctwo rks", in
NOTES
Emerpent Computation, ed . Ste phanie Forrest (MIT Press, Cambridge, 1991),
p. 2.71. .lyre, Cybemetics and the Philosophy c1 Mind, p. 30.72 . W hen th e two series of events are co llapsed into one we get what is called
a 'Markov process'. See ibid., p. 29.
73. David L. Goodstei n, States C!I Matter (Dover, New York, 1985), pp . 468 -86.
See also Nicolis and Prigogine, Exploring Complexity , pp . 168- 85.74 . T hese other charac te r istics arc a 'c r it ical slowing down ' (rel axation tim es
become lon ger as th e singularity is approached) and 'sensitivity to size ' (the
dynamics o f a syste m can take into account de tails abo ut boundary con
di tion s). However , the link between th ese phenomena and information
pro cessing and storage has been esta blished only with in the narrow field of
'cellular auto mata' mod els of comp uta tion. See Christophe r G. Langto n,
'Computa tio n at th e Edge of Chaos ', in Etnerpent Compu tation , ed. Forres t,
pp . 32-3.
75 . Chris to phe r G . Langton , 'Li fe at the Edge of Chaos', in Artificial Life II, eds.
Christo phe r G . Langton, Charles Tay lor, Doyne Farmer and Steen Rasmus
sen (Addison-Wesley, Redwood City, 1992), pp . 85-6.
76 . Melanie Mitchell , James P. Crutchfield and Pet er T. Hraber , ' Dyna mics,
Co mputat ion , and the "Edge o f Chaos": A Reexam ination ' , in Complexity:
Afetaphors. Models. and Reality, cds . George A. Co wa n, David Pines and David
Meltzer (Addiso n-Wesley, Redwood City , 1994), p . 5 10.
Th e resu lts present ed here do not disprov e the hypothesis that co mputa
tional capability can be co rrel ated with phase tr ansit ion s in [cellular
automata ] rule space. Ind eed , this general phenom ena has alrea dy been
not ed for other dynami cal syste ms . .. More generally , the co mputa tio nal
capacity of evolving syste ms may very well require dynamical properties
characte ristic of phase tr ansit ions if th ey are to incr ease their complexity .
3 TH E ACTUALIZATIO N O F THE VIRTUAL IN TIM E
1. O n the history of these conflicting conce ptio ns of tim e and a philosophical
discussion of the different ways in which th e conflict has been approached in
both physics and philosoph y of scie nce, see Lawren ce Sklar , Physics and
Chance. Philosophical Issues in the Foundations C!f Statistical Mechanics (Cambridge
University Press, Camhri dge, 1995), Chapte r 10. And Robert B. Lindsay
and Henr )' Margen au , Foundations tif Physics (Ox Bow Press, W oodbridge ,
1981), Chapte r 5.
2. Joe Rosen , Symmcrry in Scicnce (Springer. Ver lag , Ne w York, 1995), p. 141.
In add ition to r('vc rsing the o rde r of th e temporal seque nce, a time
NOTES
'reflection' transformat ion changes the sign of any variable (such as velocit y)
that depends on th e time variable . This introduces some subtle ideas that
matter in a se rio us analysis of th e symmetry properties of laws. ce
discussio n of this point in Sklar, Physics and Chance, pp . 246-8.
3. Gregoire Nicolis and lIya Prigogine, Explorinq Complexity (W. H. Freeman,
ew York 1989) , p . 52.
4 . Euge ne P. Wigner, ' Invariance in Physical Theory', in Symmetries and
Rifleetions, cds. W alter Moore and Michael Scriven (Ox Bow Press, W ood
bri dge, 1979), p . 4.
As the physicist Euge ne W igner rem ark s, if physical regularities had not
displayed this minimal am ount of invariance , we would probably never have
discovered th em at all simply because they would not app ear to us as
regularities. lnvariance und er transformations can also reveal subtle assump
tions behind a law . For instan ce , to say that a law is invariant und er spatial
or temporal displacem ent implies that, as far as the regulariti es describ ed by
th e law are conce rned, space and tim e are homoqeneous. Similarly, to say that a
law is invariant und er rot ation in space is to say that th e absolute or ientatio n
of the states of the pro cess mak es no differen ce in th e pro cess' s behaviour,
but it also means that we assume space to have uniform prop erties in all
directions (te chnically, we assum e it to be isotrop ic) .
5. T here arc several strategies for ex plaining ir reversibility away. Some
physicists, for example, think th e inh erent directionality of th e arrow of
time, so evident in macr oscop ic pro cesses, is merely a subjective effect (an
effect of our ignorance of all th e micr o details) . T o othe rs th e direction alit y
of time is not reducible to psychology but it is nevertheless den ied the status
of a t rue law , being merely a continge nt sta tistica l result. As the physicist
John Wheeler puts it, th e real molecular int eraction s arc 'time-symmetric
with on ly th e sta tistics of large numbers giving it the app earan ce of
asymmetry' (j ohn A. W heeler, 'T ime T oda y ' , in Physical Oriains if Time
A~mmet'J' , eds , Jon athan J . Halliw ell, Juan Perez-Mercader and W ojciech H.
Zurek [Cambridge Uni versity Press, Cambridge , 1996), p . 1) .
In gene ra l, th e authority of th e old reversible tim e has been pr eserv ed
and th e tim e of classical thermodynamics has disapp eared from the structu re
of the edifice of physics. As Wheeler puts it ,
The expansion of th e empire of tim e has elevated the conce pt, human
born as it is , to platform up on platform upon platform of authority.
R 'gularit ies of sun and seaso n raised the first foundation . On top of it
cw to nian dynam ics erecte d a second and tight er platform ; special rela
tivity a third , terraced further in and up; and gene ral re lativ ity stands at
ti ll' sum mit, the final level of authority. ot cxc pt out of the mou th of
NOTES
Einste in 's 1915 and still standard theory of spacetime can one hear th e
ge nerally agreed acco unt of all tha t 'time' now means and measures. (p. 6)
6 . lIya Prigogine , From Beina to Becomino (W. H. Freeman . ew York, 19S0),p. 19.
7, Arthur S. lberall , Towards a General Science if Viable Systems (McGraw- Hi li,
ew York 1972) .
Iberall' s onto logy is based on individuals which he calls ' atomisrns' (a
category of which atoms would be only one instance) . He co nce ives of these
in general as auto no mo us, nonlinear osci llators. Thanks to their nonlinearity
th ese atomi sms are show n capable of int eracti ve orde ring (via ent rainme nt,
for example) and capable of forming a continuum at a larger scale . Th ese
cont inua, in turn , are sho wn to und ergo symme try- breaking bifurcations
which fragm ent them (or quantize th em ) to yie ld super-a to misms, that is,
ind ividuals at a larger spatio- te rnporal scale. Ibcrall sho ws in detail how this
alternation of ato m ism and continuum can be used recursively to account
for man y features of physics, che mistry, biology and eve n sOciology. He also
shows, on th e other hand, how mu ch this picture breaks with those of
classical and, more importantl y, quantum physics, given tha t the latter docs
not give a morphogen etic acco un t of quantization .
S. Winfr ee does not use the terms ' intensive' or 'nonme tric', Yet, in the previous
chapte r I qu ot ed W infree's ideas abo ut top ological thi nking when appli ed to
biology and his ideas are ind eed very close to those of Deleuze . Using my
terminology, we can sa)' th at an anexaet yet riaorous approach characterizes
W infree 's research on the birth and death of oscillations, a process which
also exhibits divergent un iversality or m echani sm -indepen den ce . In his
words,
As a resul t of these co llec tive efTorts, th e reality of phaseless sets, phase
singularities , tim e crystals, and so on became firmly established. Th eir
physiological ' me aning' is less clear . . . But that deficien cy is in a way
the most int eresting aspect of these findin gs: because their pr edi ction was
in no way dependent on the mechanistic underpinn inqs of circadian physiology,
the same principl es mig ht find appli cati ons in other areas of physiology
and bio chemi stry . Th ese prin cip les m-e not 'mathematical', in the familiar
sense of 'mathematics ' as 'moving symbols around on paper' or ' mov ing
numbers aro und in co mpute rs' . Th ey are, rath er, [top ologica l] concepts
about conti nuity that could be used in diverse contexts with s!ifJicient riBor
to precisely infer biological or che mical eve nts that had not been
observed. (Ar thur T. Win free , II'hen Time Breaks Down. The Three
Dimensional Dynamics if Electrochemical lI'al 'cs and Cardiac Arrhyt hmias
[Princet on Unin 'rsit)' Press, Princet on , 19 71, pp. 64 - 5; m), emphasis)
N O T E S
9. Ian tewart and Martin Golubitsk y, Feaiful Symmetry (Blackwe ll, O xford,
1992), pp . 66 - 7.
10. Nicolis and Prigogine, Explorinq Complexity , p. 21.
11. Ibid . ,p.103.
12 . Iberall, Towards a General Science ?f Viable Systems, p . I S3.
J 3. lbid. , p. 161.
14 . Gi lles Dcl euze, Loa ic rifSense (Columbia Uni versity Press, Ne w York, 1990),
p. 162. (My emphasis)
1S. lbid., p. 62 .
16 . Gilles Deleuze , D!iJerence and Repeti ti on (Co lumbia Univ ersit y Press, ew
York, 1994), PI'. 70 -1. In these pages Deleuze , following Hume, does
indeed pr esent this contraction whi ch synthesizes pr esent time as a faculty
of the mind : a cont ractile power rif contemplation or imaqination which retains a
past and ant icipates a future. But a few pages later (p . 73 ) he says that ' we
are made of contracte d water , earth , light and air - not merely prior to th e
recognition or representation of th ese, but pri or to th eir being sensed ' .
Clear!y, this remark mak es no sense within a purely psychol ogical int erpre
tat ion , but it does if we think of thi s contraction as involving a metabolic
cycle with a characte rist ic tim e scale. He goes on to ascribe to habi ts (or to
the co ntraction of rep etitive , habitual behaviour) a similar power of synthesis
(I" 74) but again , this appli es not only to the habi ts of human beings but to
any re pe titive, cyclic beha viour at all scales.
A soul sho uld be attributed to the heart, to th e muscles, nerves and ce lls,
but a conte mplative soul wh ose entire functi on is to cont ract a habit.
T his is no mystical or barbarous hypothesis. On the co ntrary, habit here
manifests its full gene rality: it conce rns not only the senso ry -mo to r habits
that we have (psycho logically), but also, before th ese, the primary habi ts
that we are; the th ousands of passive syntheses of wh ich we are orqanica lly
composed. (My emphasis)
17. T he philosopher who argued again st th e relativisti c conclusions regarding the
co ntrac tion of tim e in th e twins' case is, of course, Henri Bergson. Bergson
was wrong in assuming that th e case for the two twins is sym me tric , or as
he put it, a purc 'effect of perspectiv e' similar to that of tw o observ ers
looking at eac h othe r at a distance and seei ng each other shrunk in space.
• l'l' li lT .xamplc his repl y to criticisms by Andre Metz in Henri Bergson ,
'Pici itiou s T imes and Real Time ", in Berqson and th e Evolution ?f Physics,
cd. P. . Y. Gunte r (University of Tennessee Press, Knox ville , 1969),
pp. 169 7 1.
This volume co ntains man y of the pice s written abo ut till' de bate
incl uding the exch.mge lx-tw ccn III rgson and Einstei n himsel f. If one focuses
NOTES
on Bergson 's excessive ly psychological int erpret ation of relati vity th en one
mu st grant that he lost thi s de bate. O n th e other hand, if instead one sees
him as arg uing for the need of an acco unt of me tric time (which must
emerge fro m a nonmet ric , virt ual tim e) then the outcome of th e debate is
less clea r. Th is is Deleuze 's own interpret at ion. He sees Bergson as
cr it icizing Einste in for not having und erstood th e difference between the
actual and th e virt ual, th e difference between metric and nonm etric
multipliciti es.
Bergson thu s brought to light tw o very different kind s of multiplicit y, one
qualitative and fusional , continuo us, th e othe r numeri cal and hom ogen
eo us, discret e ... Th e confro ntation between Bergson and Einstein on
th e topi c of Relativity is incomprehensible if one fails to place it in the
context of th e basic theory of Riemannian multipliciti es , as modified by
Bergson. (Gilles Dcl euz e and Felix Guattari, A Thousand Plateaus [Univer
sity of Minnesota Press, Minn eapolis , 19871, p . 484)
ee also Gilles Dcleuze , Berpsonism (Zone Books, New York , 1988), Chapter
4.
18 . Hans Reichenbach, The Philosophy rif Space and Time (Dover, Ne w York,
1958), p. 194 . An exam ination of the relations between the theori es of tim e
in nonlinear and relati vist ic physics is beyond the sco pe of this book, but
neve rt heless tw o concl usions follow rather directl y. One is th at there is no
inco mpat ibility between th e two and ind eed th e nonlinear theory may
compleme nt that of relativity by giving a morphogen eti c account (via
conce pts like the Hopf bifurcation ) of th e eme rge nce of the osci llato rs
(clocks, elec tromagnetic vibrations) used in the expos ition of relativity. On
the othe r hand , once we realize th at th e metric of tim e is eme rge nt , that is,
that oscillato rs operating at differ ent scales literally quantize time, th e
shri nkage o f time at veloci ties near th e speed of light becom es less counte r
intuiti ve: an eme rge nt metric, as opposed to an intrinsic one, is easier to
visualize as subject to int en sive transformations that do not pr eserv e certain
of its properties invariant.
19. In his care ful exam inatio n of foundational qu esti on s the philosopher Law
rence Sklar shows that besides th e need to deriv e the tim e-asymmetric
macro scopi c beha viour of a thermodynamic syste m from the tim e-symmetric
microscopi c laws, th er e are two additional fundamental question s in th e
[ound ation s of sta tistical mechanics: to show that th e final equilibrium state
of a syste m is indeed an attraetor for its initi al and all its other int ermedi ate
states, and that th t ime scales of approach to equilibrium in math em at ical
mo de ls r .Ilcct the tim e scales obse rved in the lahoratory. klar argu l's th at
tlu s tw o <llIcslions arc 0f'~n problems in equilibrium tlu-rm od 'namics:
NOTES
physicists have not yet rigorously demonstrated that equilibrium states
att rac t, nor explained why the relaxati on tim e exhibits a characte ristic scale
(Sklar, Physics and Chance, pp . 156- 8, 189 and 216) .
klar, how ever, neglects to mention that both of these open probl em s
have indeed been given a mor e precise formulation , if not solved, in far
from-e quilibrium thermodynamics. In this field on e gets th e asympto tic
approach to a particular state as an integral part of one's mod el, whil e in
conservative system s without attractors the asymptotic stability of the final
equilibrium state needs a speci al explanation. A similar point appli es to
relaxat ion tim es. Unlike the conservative system case , in non -cons ervative
syste ms we have an explanation , in terms of the 'area ' covered by the basin
of att raction , which is an integral part of the mod el. Sklar does discuss
Prigogine 's work to some exte nt , but not the specific points raised here
(pp. 269- 76) ..?O. Ibcrall discu sses this issue in more tec hnical terms (including terms like bulk
viscosity and bulk modulus needed to define the relaxation tim e of internal
mod es) which are beyond the scope of this book to explain. Yet I believe
his basic point is captured by my simplified example . ee his discussion in
Towards a General Science l!f Viable Systems, pp. 122-6.
2 I. . . . the inte ract ions between bodi es condition a sensibility , a proto
perceptibility and a proto-affecti vity . . . What is called 'pe rce ption ' is
no longer a state of affairs but a state of the body as indu ced by another
bod y, and affection is the passage of thi s state to another state as increase
or decr ease of potential -power through the acti on of othe r bod ies .. .
Even when they ar e nonliving , or rath er inorgani c, thinas have a lived
e.tperience because they are perceptions and affections. (Gilles Deleuze and
Felix Guatt ari, What is Philosophy ? [Columbia University Press, New York,
19941, p. 154; my emphasis)
Elsewhere he is even more explicit about this . W e saw before that
the actualization of the world relies on int ensive processes of self
organization (such as co nvection cells or the mi!,JT"ation and folding of
embryo nic ce lls). He refers to these phenomena as 's patio- te rnporal dyna
misms' and says
N O T E S
22. Deleuze, Loq ic ifSeme, p. 62.
23. W infree , When Time Breaks D OII'n , p. 22.
24 . Leon Glass and Michael C. Mackey, From Clocks to Chaos. The Rhythms if Life
(Prince to n University Press, Princet on , 1988), p. 94 . (This text contains a
discussion of Winfree 's work , and references to black holes.)
25. Winfree , When Time Breaks DOlin , p. 99 .
26. ue., Chapte rs 7 and 8.
27. Thro ugho ut we will discover again and again , in a sur prising diversity of
contex ts the same paradox ical enti ty : a moti onless, tim eless organizing
cente r called a phase singular ity . Th is is a place where an otherwise
perva sive rhythm fades int o ambiguity - like the outh Pole , where the
24 hourly tim e zon es converge and the Sun merely circles along the
horizon . (Ibid., p. 5)
Our [topological] inferenc es seldom involved speculation about adaptive
values , molecular mechani sms, or neural pathways. But they led us to
ever sharpe r focus on expe rime ntal condit ions in which somethinq stranae
was 8 uaranteed to happen: return of metamorphosing flies to the timeless
condition of the newl y fertilized egg, perpetual insomnia in mosquitoes,
abrupt suspension of pacemaking in othe rw ise perfectl y healthy and
capable heart mu scle, vortex ce nte rs of arrhythmia in elec trically rhythmic
tissue, chem ically tim eless rotors seque ncing reactions aro und their
perimeters, and che mical clocks made of shifting patt erns of color
topologically locked into three dim ensional organizing cente rs. (Ibid.,
p. 254; my emphasis)
28. Deleuz e and Guattari , A Thousand Plateaus, p. 24. (My emphasis)
29. O ne of the more ro bust and striking predictions of thc theory of mutual
synchronization was that it should fail abruptly below a critical coupling
stre ngth . John Aldridge and E. Kendall Pye tri ed this expe rime nt with
yeast and found exactly that : when the cells get more than about tw enty
diam eters apart , the amplitude of their collec tive rhythm falls abruptly.
(Arthur T. Winfree , Bioloqical Clocks [Scientific Ameri can Library, New
York, 19871, p . 128)
Actu alizati on takes place in three series: space , tim e and also conscious
ness. Every spatio -te mpo ral dynamism is accompanied by the eme rge nce
of an e1em ' ntary consciousness which itself tra ces dir ecti ons, doubl es
movem ents and migrations, and it is born on the threshold of the
conde nsed singular itks of th bod y or objec t whose conscio u 'ness it is.
(Dclc uzc, D!Dcrence and Repetiti on , p. 220)
30. Populations of cric kets entrain each other to chirp coherently. Populati ons
of fireflies come to cohere nce in flashing . Yeast cel ls display cohere nce in
glyco lytic oscillati on. Populations of insects show cohere nce in their
cycles of c1osion (e me rge nce from the pupal to the adult form) . . .
Popul ations of wom en living together may show phase entrainme nt of
their ovulatio n cvcles. Populations of secreto ry ce lls, such as till' p ituitar "
N O T E S
pancreas , and other organs, release their hormones in cohe re nt pul ses.
(Alan Garfinkel , 'The Slime Mold Dict yostelium as a Mod el of Self
Organization in Social Syste ms', in Self- Organizing Systems. The Emergence
i!I Order, ed . F. Eugen e Yates [Plenum Press, New York, 19871, P: 200)
\ 1. M. Cohen, qu oted in ibul. , P: 183.
P. Il ow ard H. Pattee, ' Instabilities and Information in Biological Self
O rganization' , in Sell-Organizing Systems, p . 334.
I \. Stuart Kauffman, The Origins eif Order. Self- Organizat ion and Selection in
Evolution (Oxford Univ ersity Press, New York, 199 3), p.442. (My
emphasis)
\.1. Rud olf A. Raff , The Shape eif L!Ie. Genes, Development and the Evolu tion eifAnimal Form (University of Chicago Press, Chicago, 1996), p . 260. Unlike
terminal addition, which implies that ear ly stage s of the development of an
embryo resem ble (or recapitulate) early stages of species (or higher tax a)
d .velopment , the type of heterochrony involved in parall el networks destroys
any similari ty between th e tw o.
IS . lbid.; p. 255 .
Ill. IIJ id . , P: 337.
Dissociation app ears paradoxical as a cre ator of developmental novelty
because nothing new is add ed. In the case of some het erochronic
dissoc iat ions, such as neoteny in th e axolotl, a novel developmental path way
and life history have result ed from th e loss eif a featu re of th e ances tral
system . (My emphasis)
n. Dc lcuze and Guattari , A Thousand Plateaus, p. 48 .
IS. W. H. Zure k and W . C. Schieve, ' Nucleatio n Paradigm : Survival Thresholds
in Population Dynamics' , in Self-Organization and Dissipative Stru ctures: Applica
li ollS in the Physical and Social Sciences, eds . William C. Schieve and Peter M.
Allen (University of Texa s Pr ess, Austin, 198 2) , pp . 20 3-22 .
It). . tu art L. Pirnm, The Balance eif Natu re. Ecological Issues in the Conservation eifSpecies and Communit ies. (University of Chicago Press, Chicago, 1991 ),
Chapte rs 2 and 3.
·10. Kauffma n, The Origins ?f Order, p. 256 .
·\ 1. Rud olf A. RafT and Thomas C. Kauffman ; Embryos, Genes, and Evolution
(Indiana Univ ersit y Press, Bloomington , 1991 ) , p. 40 .
42 . T he fastest evolutionary rat es fall in th e last, and perhaps most int eresting
category, tachytely .. . Tachytel y resembles th e punctuation of Eldredge
and Gould in that both rely on exce pt ional high rat es of evolutio n .
However whil e Eldrl· dge and Gould focused on a speci ation mod el .. .
N OT E S
[Simpson] suggest ed that the primary concomitant of tach ytely is a shift
in a population from on e major adaptive zone to another . . . Thus
tachytely is possible during early radiation s of new groups expanding into
vacant adaptive zon es. During th e rapid radiation all lineages are relatively
poorl y adapted and not mutually compe tit ive. Th e result .. . is the
production of divers e lines that qui ckly becom e ex tinct as other lines
conso lida te their position s in the adaptive zone at the expe nse of th eir
less-efficient cousins. (lbid., p. 44)
4 L Angela E. Dougl as, Symbiotic Interactions (O xford Uni versity Press, New
York , 1994) , pp . 7-9. This author em phasizes the eme rge nce of novel
met abolic capabilities related directl y to th e flow of biomass in food chains .
As she says, ' Nutritional int era ctions are fundamental to most symbioses,
beca use the metabolic capabilit ies most com monly acquired through sym bi
osis relate to nutrition ' (p. 56 , and see Chapter 7 for an evaluat ion of the
eco logical impact of symbiosis).
44. We rner Schwe m rnlcr, 'Symb iogenesis in Insects as a Mod el for Cell
Differe ntiation, Morphogenesis, and Speciation', in Symbiosis as a Source eifEvolutionary Innovation , cds . Lynn Mar guli s and Ren e Fester (MIT Pre ss,
amb ridge , 1991 ) , p . 195.
4') . Deleuze places great emphasis on symbiosis as a means of becoming.
Coevolutio n , as in th e aparallel evolution o f th e wasp and th e orchid it
pollin ates, is a well -known example . See Deleuze and Guattari, A Thousand
Platea us, p. 10. But more generally, the very definition of a het erogen eous
assemblage as a 'rhizome ' has its origin in symbiosis. Though his introductory
exa mple of rhizome is bulbs and tubers, that is, plants without an arborescent
root syste m , he immediately acknowledges that ' plants with roots or radicles
can be rhizomorphic in other respects alt ogether' (p. 6) . Thi s other respect
may be illustrated by the formation of th e so-called rhiz osphere, th e und er
ground food web composed of th e plant ro ot s of different species together
with the diverse micro-organisms that form symbiotic couplings with th em
and interface th em to th e flow of und erground nutrients .
46 . Throughout this book I have used his first formulation, singularities and
affec ts, hut he uses several others . Som etimes he says that in th e virtual
co nt inuum (plane of consistency) bodi es are charact erized by speeds and
affects (ibid .• p. 260).
Elsewhere , he says the virtual cont inuum (Aion) is 'the locus of incor
poreal events and o f attributes which are distinct from qualities ' (Dc leuzc ,
Log ic eifSense, p. 165) . Here , ' events ' refers to singularit i s , whil e 'attributes '
are capacities to affect and be affected (to cut and to be cut, to us ' his
example).
N O T ES
47. lbid., p. 255. Rapidit y and slowness, however , should not be conceive d
as involving merely quantitative or exte nsive dilTer ences. Speed is an
int ensive property subject to critical thresholds, as in the case of fluids
which, below a critical speed, have one pattern of flow (laminar) but which ,
beyond the threshold , display a completel y dilTerent pattern (turbulence) .
See p. 371.
48 . lbid.; p. 258.49 . Th e term ' me chanisms of immanence ' does not , to my knowl edge, occ ur in
Deleuze , but he expresses himself in similar ways.
Many mo vem ents, with a fraai /e and delicate mechanism, intersect: that by
means of which bodies, states of alTairs, and mixtures, conside red in their
depth, succeed or fail in the production of ideal surfa ces [plane of
consiste ncy ]; and conversely , that by means of which the events of the
surfa ce are actuali zed in the present of bodi es (in accordance with
complex rul es) by imprisoning their singularities within the limit s of
worlds, individuals and persons. (Deleuze , Loaic ef Sense, P: 167; my
emphasis)
50. Co nnec tance is, in fact , controlled in food webs. Good evide nce suggests
that the number of connections in food webs is adjusted such that each
species maintains roughly a constant number of connections to othe r
species, regardl ess of the number of species in the web . . . [as displayed
inJ data on more than 100 food webs - terrestrial, freshwater , and
marine. A number of properties - such as length of food chains;
connectance ; ratios of top, intermediate and bottom species ; and ratios
of predators to prey - appear stable and scale invariant , both with respect
to the numbers of species in the web and with respect to the aggregation
of 'guilds ' of similar species int o single ' tro phic species' or the aggrega
tion of similar species int o higher tax onom ic units. (KaulTman , The Oriai ns
C!f Order, p. 263)
51. Ibid . , p. 219.52. Although the famous Gaussian, or bell-shaped, distribution does represent
an important eme rgent property of widely dilTerent populations (that is,
ther e is someth ing recurrent or universal about it) it is nevertheless an
equilibrium distribution, and th e populations exhibiting this bell shap~ are
exa mples of distributions in extensity , fixed in their form and occ upymg a
metric, divisible space (much as sedentary cultures do). At the virt ual level ,
we must go beyond the e distributions, we mu st make a dilTerent use efhancc, Unlik e traditi onal games of chance (ro ulette, di c) in which fixed
rules lor rc th aleatory facto r to be r rained only at ce rtain points (the
NOTES
spin ning of the ro ulette, the throw of the dice) leaving the rest as a
mechanical development of the consequences , at the level of the virtual we
must allow the rul es to change with every throw and inject chance at eve ry
point, to yield truly nonmetric (or nomadic) distributions. In Deleuze' s
words
Each throw emits singular points . . . But the set of throws is includ ed in
the aleator y point [quasi-causal operato r ], a unique cast which is endlessly
displaced throughout the ser ies . . . Th ese thro ws are successive in
relation to one another , yet simultaneo us in relation to this point which
always changes the rul e , or coo rdinates and ram ifies the co rresponding
series as it insinuates chance ove r the entire length of the series . . . Each
thro w operates a distribution of singularities, a conste llatio n. But instead
of dividin g a closed space between fixed results which corre spond to
hypotheses [as in traditional treatments of probability], the mobile results
are distributed in the open space of the unique and undi vided cast. This
is a nomadic and non -sedentary distribution . (Dele uze , Loaic ef Sense,
pp. 59-60; emphasis in the original)
') l. Dcleuze olTers an alternative model for this task of the quasi-causal operator
which is based on the idea of entrainme nt, or more speci fically, the
phenome non of frequ ency entrainme nt. For tw o grandfather pendulum
clocks to entrain, weak siqnals must be transmit ted from one to the othe r to
couple them (in some cases , these are weak vibrations in the wo oden floor
on which the clocks are placed) . If the frequ encies of the two clocks are
close to each oth er they may resonate and the two clocks will lock into a
single frequ ency. Th e resulting entrainme nt of the two oscillators represents
a much stronaer linkaae (forced movem ent) between the tw o oscillators than
the weak signals which or iginally coupled them . In Deleuzc 's words:
A syste m must be constituted on the basis of two or mor e series , each
series be ing defined by the dilTerences between the terms which compose
it. If we suppose that the series communicate und er the impulse of a
force of some kind [e.g. the quasi -causal op erator), then it is apparent
that th is communication relates differences to other differences, constitu t
ing differ ences between differ ences within the syste m. Th ese seco nd
d gre' dilTerences play the role of 'dilTere nciator' . . . Thi s state of alTairs
is adeq uately ex pressed by certa in physical concepts: coup/ina between
hct erogeneous syste ms, fro m whi h is derived an internal resonance within
the syste m, and fro m which in turn is deri ved a fo rced morcmcnr , the
am plitude of which exceeds that of the basic scric them selves. (Del uze ,
D!ffercn e and Repcu tion, p. 11 7)
N O TES
Deleu ze uses this ' re sonance ' model for th e action of the quasi-causal
ope rator in other places. For example,
Conce pts [multipliciti es], which have onl y consiste ncy or inten sive ordi
nates outside of any coordinates , freely ente r int o relati onships of
nond iscursive resonan ce ... Concepts are centers of vibrations, each in
itself and everyone in relation to all others . Thi s is wh y th ey all resonate
rather than cohere or corres po nd to each other. (Dele uze and Guattari ,
What is Philosophyi; p. 23; my emphasis)
Clearly , if we interpreted th e term 'concept' as ' semant ic conte nt of a
term ' (or in any other lingui sti c way) this paragraph would be come
meaningless. Th e term ' inte nsive ordinates ' must be int erpreted in terms of
posit ive ordinal distances (which distinguishes it from any cardinal numerical
coordinate) and not as referring to on e of th e member s of the couple
'ordinates' and 'abscissas' which are simply th e nam es of two coo rdinates.
54 . De le uze , Loaic if Sense, p. 121. (My emphasis) Thi s is about th e specification
of the conditions of a problem, but problem s are , in Deleuzes onto logy ,
no thing but virtual multiplicities. I discu ss this relationship in Chapt er 4 .
55 . itewar t and Golubitsky, Fea1ul SymmetIJ', pp. 14-16.56 . Lawre nce Sklar, Space, Tim e, and Spa ce-Time (University of California Press,
B erkcley, 1977) , pp. 25 1- 86.57. T he reason why it is hard to find a physicist who would think of laws as
en ti ties in need of ontological analysis is that mo st of th em have an
instrumentalist or ope rationalist attitude toward th eoreti cal entit ies. Ever
since Ne wton refused to give me chani sms to explain the action of gravity
and settle d on describing how plan ets move, as opposed to explaining lYhy
they do so, man y physicist s have accepted a non-realist approach to laws, as
we ll as unobservable entit ies in gen eral. Thus, expe rime ntal laws (like
Boylc' s law) are defined as symbolic representations of laboratory regularities
or routines '?f experience, whil e fundamental laws become basic hypotheses
fro m which on e can derive experimental laws, and the validity of which is
not sett led empirically but through th e validity of their conse <j uence s. In
ncither case is the ontological status of the laws th emselves an issue . See
Lindsay and Margenau , Foundations if Physics, pp . 14-16 (for expe rime nta l
laws) and pp . 22-6 (for fundamental principles) .
While phil osophers can take this stance and argue that, if all speci fic
ex perime nta l laws may be deri ved from a set of fundamental ones, th en th e
latt e r may be see n as a set of axioms and treated as ete rn al truths, as in
Euclid's ax iomatic tr eatment of geome try . But as th e physicist Richard
I·c 'nman has argued, scientists cannot do this because they are awa re that ,
unlike essences, fund am ental laws may have seve ra l diffe r .nt forms. New-
N O TE S
ton 's laws of motion , for example , may be expressed in three ways wh ich
are, mathematica lly, completely different : the original force form, the field
form, and the var iational form . Th ese arc taken to express one and the same
law because they have th e same mathematical conse <juences and thu s we
canno t tell them apart expe rime ntally. But th e existe nce of a variet y of
forms docs eliminate th e temptation to adopt a Gr eek axiomatic approach,
forc ing physicists to adopt, as Feynman puts it, a Babylonian approach. See
Richard Feynman, The Character '?!. Physical Law (MIT Pr ess, Cambridge,
1995), pp . 50-3 .
Per haps th e only clear state me nt one can get from physicists as to what
funda me nta l laws are supposed to be co mes from the appli cation of gro up
theory to the law s th em selves , For example , th e well -known invariance of
cwton's laws under translation s in space and time impli es that
give n the same esse ntial initial conditions , th e result will be th e sam e no
matter when and where we realize th ese. Th is principle can be formulated
, . . as the state me nt that th e absolute po sition and th e absolute time are
never essential initi al conditions . . . If th e universe turned out to be
grossly inhom ogenous, th e laws of nature in th e fringes of th e universe
may be quite different from th ose we are studying .. . Th e po stulate of
invariance with respect of displacem ent in space and time disregards this
possibility, and its appli cati on on the cosmological scale virtually presupposes
" homoqeneo us and sta tionary uni verse. (W igne r , ln varian ce in Phy sical Theory ,
p. 4; my emphasis)
.lcarly, this is a more sophisticate d stance than naive essent ialism , since
1his post ulate of ln variance (which may imply that basic laws are simul
t,' Ilt'ollsly valid every where , and have been so always) can , in turn, be treated
.1' .111 ap proxi rnate hypoth esis , I return to the <juestion of laws in Chapter 4.
, S "or if it is a qu estion of knowing, . . ' why water change s its state of
qu" lity at 00 centigrade', the <juestion is po orl y stated insofar as 00 is
cOllsidereel as an ordinary point in the thermomet er. But if it is
('ollsider xi, Oil the contrary, as a singular point, it is inseparable from
tlu- event occurring at that point, always being zero in relation to its
re.r lization on th e line of ordinary points, always Jorthcomina and already
/'" t . (Dcleuzc , Loaic if Sense, P: 80; my emphasis)
li lt" (·. ·.K! same formulation recurs thro ugho ut Dcleuzc 's work:
ion : Ihe inr h-Iin iu- time of the eve nt, the floatin g line that knows onl y
Iw,-ds and co nti nua lly divides th.lt wh ich tran spires into an alrea dy -t here
11..11 is .11 the sallie time no t -yc-t-h cre , a simultaneous too -Ian- and too -
N O TES
early, a something that is both go ing to happen and has just happened .
(Deleuze and Guattari, A Thousand Plateaus, p. 262)
T he meanwhile, the eve nt, is always a de ad tim e; it is th er e wh ere
nothing tak es place , an infinite awaiting that is alread y infinitely past ,
await ing and reserv e. (Dele uze and Guattari, What is Philosophy ?, p . 158)
"'J, Dclcuzc, D!lJerence and Repetition , p. 88. (My empha is)
Th e joint . . . is what ensures th e subordina tio n of time to those pr op erl y
cardinal points through which pass th e peri od ic movements wh ich it
measures [e .g . th e nest ed set of cyclic pr esents] . . . By contras t, tim e
out of joint means dem ented tim e . . . liberated from its overly simple
circular figure, freed from th e events that mad e up its conte nt ... in
short , time presenting itself as an empty and pure form. Time itself
unfold s . . . inst ead of things unfolding within it . . . It ceases to be
cardinal and becomes ordinal, a pure order of tim e .
1>0. I have said before that each cyclic present is a contrac tion of past and future
instants at a given temporal scale . Hence it is a veritable ' synthes is' of
present time, a syn thesis whi ch Del euze calls ' passive ' because it involves no
activity .ither by th e world or by th e subject.
Passive synthesis or co ntraction is essentially asymmetrical: it goes from
the past to the future in th e pr esent, thus from the particular to the
general, th er eby imparting direction to th e arrow of time . (Deleuz e,
D!fJcrence and Repetit ion, p. 71)
(,1 , The infinitely divisible event is always both at once. [future and past , acti ve
and passive] It is ete rn ally that which has just happ ened and that which is
about to happ en, but never that whi ch is happen ing .. . Th e eve nt , being
itself impassive, allows the act ive and the passive to be int erchanged more
easi ly, since it is neither the one nor the other, but rather their common
result. (De lcuzc, Loaic ifSense, p. 8)
1> 2. Dclcuzc, Loqic ifSense, PI" 94-5.(d . Ihid . , p. 147.
64 . lbid. , p. 165.
65 . Ibid . , p. 147.
h h. Ralph II . Abraham, ' Dynamics and el f-O rganizatio n ', in Se!f-Oraanizina
'stems. Th Emerqence if Order, ed. F. Euge ne Yates (Plenum Press, Ne w
York , 1987) , p . 606.
(,7 . 0 11 questi on s of simplicity and famili arity in th e foundat ion s of physics, sec
U nds . and Margellau , Foundations ifPhys ics, p. 18.
NOTES
68. Deleuze, Loa ic if Sense, p. 166.
69 . Ian Stewart, Does God Play Dice? The Mathematics if Chaos (Basil Blackw ell ,
Oxford, 1989), pp. 114- 21.
70 . Del euze and Guattari, A Thousand Plateaus, p. 25 1. (Emphasis in the
original)
71. lbid. , p. 9. Th e term 'line of flight' , referring to th e quasi-causal ope rato r,
is defined else where (I" 488) as a fractal line. Precisely because th e ope rator
and the plane it constr ucts mu st cut and pr eserv e N-dime nsions for eve ry
multiplicity, Del eu ze co nce ives of it as necessaril y having a fractal number
of dime nsio ns, a number wh ich is not a wh ole number but a fracti on. For
example , a flat piece of paper is a two-dimensional entity , but one fold ed
into a ball has a dimension between tw o and three , that is, it is a fractal
dim ens ion. 0 do es a one- dime nsional string so fold ed that it begins to fill a
plane. T he op erator itself would not be a transcendent agen cy ope rating in
N+ I dimens ions but on the contrary , it would work on N-I dimensi ons (a
line forming a plan e , or an aleatory poine cir culating through one- dime nsional
series) . O n th e fractal dim ensionality of the plan e, see also Del euze and
iua ttari, What is Pbilosophyi; pp. 36 - 8.
7) . I c1cuze uses th e term 'counte r-act ualization ' for the ex traction of ideal
events from actual ones in Deleuze, Loqic if Sense, pp. 150- 2. Il l' does not
II S l ' th e term 'pre-actualizat ion' hut thi s term do es capture th e meaning of
till' oth er task th e quasi -cause mu st perform .
In gene ral, as we have see n, a singularity may be grasped in tw o ways: in
its existence and distribution [in th e vector field), but also in its nature,
ill conform ity with which it exte nds and spre ads itself out in a det ermined
direction ove r a line of ordinary points. This second aspect alr ead y
represents a certa in stabilizatio n and a beainn ina if the actualizati on ifin,qu/ariti cs. ( Ibid., p. 109; my emphasis)
71 . . . th insta nt extracts singular points twice project ed - once into the
future and once int o the past - forming by this double equation the
const itutive clements of the pure event (in the manner of a pod which
r .lcases its spo res). (Ibid. , p. 166 )
/., [W Ill'1I ,1 multiplicit yI is grasped in its relation to th e quasi-cause which
prod uces it and distributes it at th e sur face , it inherits , participates in,
,lIId even enve lops and possesses th e force of thi s ideational cause. W e
h.ivc s e n that th is [qua si-jcau sc is nothing outside its effect, that it haunts
lhi, llll'( t , and that it maintains with the effect an immanent relation
whic h tu rn s th, product, the mom ent thai it is produced , int o sonl'thing
prod urtivr-. (Deh-uzc , 1.0lI'c of Sense, p. 95)
NOTES
This extract is about 'se nse' not 'a multiplicit y' but the two terms are
closely related .
75. Once communication between het erogeneous series is estab lished , all
sorts of consequences follow within the syste m. Some thing passes
betw een the borders, events explode, phenom ena lIash, like thunder and
lightning ... what is this agent, this for ce which ensures communicat ion?
Thunderbolts explode between differen t intensities, but they are pr eceded
by an invisible, imp erceptible , dark precursor, which determines their path
in advance but in reverse , as though intagliated. (Deleuzc, D!lJerence and
Repetiti on , pp . 118-1 9; emphasis in the or iginal)
76. Dcleuze does not speak of nonlin ear , no nequilibri um areas of the world , but
he does distinguish specia l pro cesses (such as th e spo ntaneo us formation of
metas table surfaces) fro m those character izing full equilibr ium structures .
O nly the former have the pow er to give rise to the virt ual.
When we say that bodi es and their mixtures produce [the virtual ], it is
not by virtue of an ind ividuati on which would presuppose it. Individua
tion in bodies, the measure in their mixtures . . . pr esupposes . . . the
pre -indi vidual and impersona l neutral field within which it unfold s. It is
therefore in a differ ent way that [the virtual) is pr oduced by bodies. Th e
questio n is now about bodies taken in their undifferenti ated depth and in
their measureless pulsation . Thi s depth acts in an original way, by means
1!.f its power to orBanize suifaces and to envelop itsclf with in sUlfaces. (Deleuze,
LOBic 1!.f Sense, p. 124; emphasis in th e or iginal)
I have replaced refer ences to 'se nse' in this ext ract by ' the vir tual'. (The
term 'se nse' is closely related to 'v irtual mul tipli city' , but refers to th e
re lation between virtuality and language, a relat ion I do not ex plore at all in
this book.) The capacity of matt er to form sur faces, eve n surfaces at
equilibrium, constitu tes the most primitive form of self-organizat ion. The
surfaces of liqu id or solid bodies are , indeed , specia l or singu lar zones of
those bodies, very different fro m the ord inary bulk mat erial that they
enve lop . The bulk of a liqu id body, a lake or ocean, for instance , consists of
a populatio n of mo lecules on . which forces of attraction are exer ted in all
directions. At the surface of this body, on the othe r hand , there ex ists a
changing sub-population on which forces are exe rted inward but not
ou twa rd. This gives those surface molecules special prop ert ies not displayed
b thu bulk. In parti cular , they will possess a certa in amount of free energy
(·ncrgy available for do ing work) which acco unts for the surface's spon
t,lIWOUS tendency to contract or minimize its ex te nsion (a 's ur face tension '
NOTES
which explains why dropl et s of water spon taneo usly acquire a round shape).
See Neil Kensington Adam, The Physics and Chemistry '?! Suifaces (Do ver, New
York, 1968), pp . 1-7.Even at equilibr ium, the surfaces of ind ividuated bodies are capable of
spo ntaneo usly giving r ise to asymmetr ical distributions of events, a distribu
tio n which is the signature of the quasi-causal operator. This is part icularly
dear in the case of electrical phenomena occ ur ring at the surface of contact
between different phases of matt er .
When two conducti ng phases are in contact, a difference of electrica l
potential is generally established between them . The establishme nt of this
'phase boundary pot ential ' is int imately associate d with the formation of
an 'e lectrical doubl e layer', at the surface, i.e . an unsymmetrical distribution
f!I electrically charBed particles near the phase boundary, with an excess of
posit ive charges tow ards the phase which assum es a positive pot enti al and
of negative charges toward s the phase assuming negat ive pot enti al.
(p. 300; my emp hasis)
Here is Dele uze 's versio n of the same ideas,
Everything happ ens at the surface in a crystal which develops only on the
edges. Undoubtedly, an organism is not developed in the same manner
. . . But membranes arc no less important, for they carry potent ials and
reBcncrate polarities . Th ey place int ernal and ex te rna l spaces into contact
without rega rd to distance . Th e intern al and the exte rna l, depth and
height, have biological significance only through th is topoloqical surface '?!contact . Thus, even biologically it is necessary to und erstand that ' the
deepest is the skin'. The skin has at its disposal a vita l and properl y
superficial potential ene rgy . And just as [virtual) eve nts do not occ upy
the surface hut rather frequent it, superficia l ener8Y is not local ized at the
.<U~face but rather bound to its formation and riformation . (Deleuze, LOBie '?!Sense, p. J03; my emphasis)
77. The term 'line of flight ' is used in two ways, one to refer to relati ve , the
other to absol ute movements towards the virtual. A relative line of flight
refers to actual assemb lages, like those I described above when discussing
emhryogenesis and ecosys tems, defined by affects and relations of speed and
510 \\'n ss.
'omparntive rates of flow in these lines produce phe nomena of re lative
slowness or viscosity, or on the contrary, of acceleration and ru pture .
t\1I this, lines and measurable speeds, consti tute an assem blage. (I cleuzc
.uul Guattari , " Thousand Plateaus, p. 4)
NOTE S
I said that in these assemblages relative accelerations (neoteny, symbiosis)
allow an escape from rigid morphologies, the term 'relative line of flight '
refe rring to th ese ph enomena, among others. An absolute line of flight is a
further acce leration or boosting of th ese relative escapes whi ch allows th em
to leave th e extensive and intensive altogeth er.
Th ese relative movements should not be confused with th e possibility of
. .. an absolute line of flight ... The former are stratic or int erstratic [that
is, conce rned with exte nsities or intensities], wh ereas th e latter conce rn
the plane of consistency .. . There is no doubt that mad particles leave
minimal trace of their passage through th e strata as they accelerate, escaping
spatio-te m poral and even exi stential coo rdinate s as they tend towards ..
th e state of unformed matter of th e plane of consistency . (pp . 55-66)
And it is th ese absolute lines that creat e the heterogeneous virtual
continuum . 'Moreover, th e plan e of consistency does not preexist . . . the
lines of flight that draw it and cause it rise to the surface , th e becomings
that compose it' (p . 270) .
n . Philosophy is a const ructivism , but constructivism has two qualitatively
different complementary aspects : th e cre ation of concepts and th e laying
out of a plan e ... Concepts are absolute surfaces or volumes, formless
and fragm entary, whereas th e plan e is the formless, unlimited absolute,
neither sur face nor volume but alwa ys fractal ... Concepts are events
hut the plane is the horizon of events , the reservoir or reserve of purely
conceptual events .. . (Deleuze and Guattari, What is Philosophy ?, p . 36)
Here the term 'concept' does not refer to 'concepts of the understanding',
that is, to semantic or representational entitie s, but to virtual multiplicities :
'Every co nce pt . . . is a multiplicity although not every multiplicity is
conce ptual' (p. 15). Without this definition referen ce to conc epts as surfaces
or volumes (tha t is, as manifolds) would be meaningless. That virtual
mult ipliciti es cannot be conceived as int elle ctual con cepts is clear from the
following ex tract, wh ere th e term ' Idea ' gives a better rendering of what
'concept' mean s:
If the Idea elim inates variability, thi s is in favour of what mu st be called
var iety [a synonym of manifold I or multiplicity . The Idea as concre te
universal stands opposed to conce pts of the und erstanding. (Deleuze ,
[)!JJcrence and Repetiti on, p. 17 3)
7lJ. Dclcuzc and Guatta ri, What is Philosophy] , p. 126 .
HO . Ikl cuzl·. Lon ic r1 Sense, p. 148.
NOTES
4 VIRTUALITY AND THE LAWS OF PHYSICS
1. T he rejecti on of totalities and th e definition of social ontology as composed
en tirely of individuals op erating at differ ent scales needs to be defended in
detail. I am aware that th e way I present it here is rough and hardly
compelling. Moreover, a convincing case for thi s point of view needs of
lic e ssity to have a historical dim ension, that is, it need s to give the details
of .pccific individuation processes , for institutions, cities and nation states. I
have applied this ontology in th e conte xt of a historical analysis of W estern
history in Manuel DeLanda, A Thousand Years r1 Nonlinear History (Zone
Books, Ne w York 1997) .
lhcrc are man y approaches to th e question of the disunity of science. Some
p.rrticularly useful ar e John Dupree , The Disorder r1 Thinps. iHetaphysical
l-outulutions c1' the Disuni ty r1 Science (Harvard University Press , Cambridge,
199 5); Jerry Fodor, 'Special Scien ces, or Th e Disunity of Science as a
\ orking Hypothesis' , in The Philosophy c1'Science, eds. Richard Boyd, Philip
(;. Ispl·r and J. D. Trout (MIT Press, Cambridge , 199 3); Peter Galison,
' In t ro duct ion : Th e Context of Disunity ', in The Disunity ofScience, eds. Pet er
( ;.llisun and David J . Stump (Stanford Univ ersity Press, Stanford, 1996);
Vndrcw Pickering, The Alanale ifPractice. Time, AaenC)', and Science (University
ot Chicago Press, Chicago , 1995) .
lronical ly, some conte m porary socio logists of scien ce who are highly critical
01 the philosophers' s approach mak e the mistake thinking that a nov el
Pl'ro,ich to the study of science demands th e elimination of causal relations.
'I' I I. M. Co llins , Chanai na Order (University of Chicago Press, Chicago,
I'}')l), pp . 6-8 .
It is hard to tell wh ether Collins trunks causes do not exist , thus siding
\ IIh l lumc, or whether he thinks we should susp end belief in them as a
I II ' lhodological man oeu ver to highlight th e 'social ' aspects of scientific fields.
I II lat ter interpretation would avoid my criticism (that he is siding with the
old" ' 1 and most conse rvative philosophy of science) but it would sti ll be
" Jl" 11 to crit icism in a differ ent way: bringing 'society' as a totality into the
nalvsis.
I \II 11.ll'king , Represeminq and Interven inq (Cambridge University Press, Cam
11/1.1 'I ' , 1992) , P: 46. (My emphasis) In conte mporary philosophy th e re vival
II I 1 ,111. •Ility as a productiv e or gene tic relationship, one to be studied
II1Jlll'ic,d ly not mer ely conce ptually, was foreshadow ed by the philosopher
1.11 III Hungl' in 1959 , although the degr ee to which he has influenced
l U I n -ut aut ho rs is hard to evaluate . His key hook in this r 'spe t is Causality
"hi lIoJefTl S k nce (Dover, Ne w York. 1979). Her I adopt man y of Bunge 's
II \\ 1111 product ivity and depart onl y in the terminology. Il l' uses the term
NOTES
'd et ermination' for the general relation (including linear, nonlinear and
statistical causality) reserving th e term 'causality' for linear causality, so as
not to depart from tradition. I myself pr efer to speak of causal relations in
gene ral, taking the linear case as an untypical case , since th e point of my
discussion is to break with tradition in th ese matters.
5. Th e entire group of new philosophers that have tak en the 'causal turn ' are
unanimous in their rej ection of th e deductive-nomological model of expla
nation (as w ell as related models which replace deduction by induction, and
exceptionless laws by statistical laws) for its emphasis on logico-linguistic
form at the expense of causal -productive processes. See Bunge, Causality and
Modern Science, PI" 290-1; Nan cy Cartwright , How the Laws rf Physics Lie
(C larendon Press, Oxford, 1983), PI'. 132- 3; W esley C. Salmon, Scientific
Explanation and the Causal Structure i!f the World (Prince ton University Pre ss,
Prin ceton , 1984), PI" 26-32; Dupree, The Disorder eif Thinas, Pl' 178- 9.Deleuze som etimes echoes th e philosophical mischaracterization repres
ented by th e nomological -deductive model wh en he asserts that the object
of science is ' funct ions that are pr esented as propositions in discursive
syste ms' (Deleuze and Guattari, What is Philosophyi , p. 118).Alth ough in his early work Deleuze is very careful to differentiate
between mathematical functions which are close to lingui sti c statem ents
(such as algebraic functions) from those that ar e not (differe ntial functions),
in his last work where the differences between science and philosophy are
most dramatically stated, he lapses into a less car eful state me nt of the
question . Elsewhere (I" 128) he adds that '[TJhe fact that scien ce is
discursive in no way means that it is deductive ', but gives as an example of
non -deductiv e activity th e use of co mpute rs in the study of nonlinear
functions. I believe th e non -deductive aspe ct needs to be stressed mu ch
mo re and exte nde d to modelling pra ctices much older than com pute r -based
ex pe rime ntation . 1 have alread y arg ued that Delcuze' s main point, the
in.H!fficiency '?I J im ctions to capture the virtual , can be mad e without subordinat
ing mathematical models to propositions, that is, by showing that function s
define indi vidu ation processes in such a way as to st res s th e direction
tow ard s the actual.
6 . Ron ald N . Giere, Explainino Science. A Coq nitive Approach (University of
'hicago Press, Chicago , 1988), 1'.82. (My em phasis)
7. ~omment i ng on a particular case of deri vation, that of th e mod el of th e
simple pendulum in one dim ension from the two-dim ensional case, Giere
says
Th e move from the mass-on -a-spring exam ple to the Simple pendulum
see ms lo me a clear case of what Kuhn called 'direct mod eling '. T he
NOTE S
two examples are not just special cases of a gene ral relationship . One
manages to reduce th e pendulum, a two-dimensional system , to the on e
dim ensional case only by means of a judicious approximation that restricts
the pendulum to small angl es of swing . In particular, the ste p from th e
original application of Newton 's laws to the two-dimensi onal pendulum
to the one- dimensional version is not a matter of purely mathematical,
or logical, deduction . •Approximation' is a valid rul e of deduction only in
physicists ' jokes about mathematicians. (ibid ., 1'.71; sec also PI" 76-80)
" Ily.1 Prigogine , From Beino to Becoming (W. H. Freeman . New York, 1980),
p. 19." ( ',lTlwright, How the Laws rf Pby sics Lie, PI" 54-5 .
II I 'This fits better with my picture of a nature best described by a vast array
,.1 phenomeno logical [or causal] laws tailored to spe cific situations, than with
"'"' govern ed in an orderl y way from first principles, ' (ibid., p. 66).On Gie re's view see Giere , Explain ing Science, p . 85, and PI" 90-1 on
I", d ews on Car twright's work.
( ,,'twright, How the Laws eif Physics Lie, p. 107.I ),borah G. Mayo, Error and th e Growth eif Experimental Knowledge (University
," Chicago Press, Chicago, 1996), p. 128.
(" twright, How the Laws eif Physics Lie, PI" 96-7."'JI,is Kline, Mathematics and the Physical World (Do ver , New York , 1981),
I' 1-1 0 . (My emphasis)
\I, ,,ri s Kline, Ala thematical Thought from Ancient to Modern Times. Vol. 2
,I I li,rd l.lniversity Press, New York, 1972), p . 580. More generall y, on
II" lrist orv of variational techniques see Chapters 24 and 30.
l ;jvell appro priate variational principles each with an associated multiple
IIlk >ral and scalar int egrand, we can produce all the important partial
old ('rent ial equations in physics: the wave equation, th e diffu sion
l' I".ll ion, Po isson's equat ion , Shrodingcr 's equation, and each of Max
\\'(, 11'S equat ions . . . Such thinking bears fruit. General relativity and
' 1",lI1l UI11 mechan ics both originate d from variational principles . (Don . S.
1'1II01lS , Peifecl Form. Variat ional Principles, Meth ods and Applications in
I I /IIl'lIIary Php ics [Princeton Uni ver sity Press, Princeton , 1997J, p. 11 I)
I ,I, PI" 17 27. In a passage where Deleuze contrasts the propositional
II" ...HI. 10 the probl em atic one (o r what amo unts to the same thin g, an
1'1" ".J( h to though t in terms of its conditions as opposed to its producti ve
,". ,,) . II(' co mpares lhe Kanti an co nce ption of the con cpt of •shortes t
It 11I1l' ' (.IS a representational schema) to the conce ption made possihle hyIII I ,til II Ills of vari.u ious. Th e term 's hortes t ', as he S.l)'S,
N O T E S
may be understood in tw o ways: from the point of view of conditioning,
as a schema of the imagination which det ermines space in acco rdance
with the concept (the straight line defined as that which in all parts may
be supe rimpose d upon itsel f) - in this case the difference rem ains
ex ternal, incarn ated in a rul e of construction . . . Alte rna tively, fro m the
genetic point of view, the shortest may be understood as an Idea
[multiplicity] which . . . interi ori zes the difference between straight and
curved, and expresses this internal difference in the form of a reciprocal
determination [differ ential relati ons] and in the minimal condi tions C!f an
inteqral , (De leuze , D!fference and Repet it ion , p. 174)
J 8 . Leo nard Euler, quot ed in Stephen P. Tim oshenko, History c1 St renqtb C!fMaterials (Dover , New York , 1983), p . 31. (My emphasis)
19. Far from being conce rne d with so lutions, truth and falseho od primarily
alTect problems. A solution alwa ys has the truth it deserv es according to
the probl em to which it is a response , and a problem always has the
solution it deserves in pr oportion to its own truth and falsity - in other
word s, in proportion to its sense. (Deleuze, D!fference and Repetiti on,
p. 159)
In what follow s I will not speak of ' true problem s' but of 'correct ' or
'well-pose d problem s' but this constitutes, 1 believe , only a harml ess
terminological departure from Deleuze.
20. Kline , Mat hematics and th e Physical World , p. 441 . Within this traditi on , the
unifying power of Hamilton 's principle was almost inevitabl y inte rpre ted as
consisting in the gene rali ty of its truth , and axiomatic versions of classical
mechanics were produ ced in the nineteenth century (by Heinrich Hertz, for
example) to marry the unifying pow er of variati onal principles with the
concept of general truth . See Rob ert B. Lindsay and Henry Margenau ,
Foundations tifPhy sics (O x Bow Press, Woodbridge , 198 I), pp. 118- 20.
In a Deleuzian onto logy eliminating essentialism from physics involves
replacing clear and distinct truths (axioms and theorems) by problems, that
is, replacing dedu ctively connec te d linguistic propositions in the Euclidea n
geo metry mould by problem s defined by singu larities (events) and affects.
Greek geo metry has a general tend ency on the one hand to limit problem s
to the benefit of theorem s, on the othe r to subor dinate problem s to
theor ems themselves. The reason is that theorem s see m to ex press and
d rv .lop the prop rt ies of simple essences whereas problem s co n ern only
el'ents and Cf./ fections ... As a result , however , the Beneti c point of view is
fordbly rdegatcd to an inferio r rank : proof is given that so mething cannot
NOTES
be rather than that it is and why it is (hence the fre quency in Eucl id of
negative, indirect and [redu ctio ad absurdum I arguments .. .). Nor do
the essen tia l aspects of the situatio n change with the shift to an algebraic
and analytic point of view. Problem s are now traced fro m algebraic
equations . . . How ever just as in geo metry we imagine the probl em
solved, so in algeb ra we operat e upon unknown quanti ties as if they wer e
known: th is is how we pursue the hard work of reducing problem s to the
form of propositions capable of sen-ing as cases of solution. W e see this
clearly in Descartes. The Cartesian meth od (the search for the clear and
distinct) is a meth od for so lving suppose dly given probl em s, not a meth od
of inventi on appropriate to the const itution of problems or the und erstand
ing of questions. (Deleuze , Difference and Repetuion, p. 160.)
Delcuzc, D!fference and Repetition, p. 189.' For Probl em s-Ideas are by nature unconscious: they arc extra-proposit ional
aru! sub-representati ve , and do not resembl e the propositions which represent
the affirmations to which they give rise' (I" 267; my emphasis).
I,HI I lacking, Bepresent lnq and ln terveninp , p. 41. (Emphasis in the original) In
.ldcl ition to ignoring causes and downplaying explanations, positi vist philo -
ophy holds a ' verificationist ' theory of meaning (if the truth of a stateme nt
,.II H1ot he tes ted the state me nt is meaningless) , a belief that verification
111\ olves comparison with raw data (data fro m the senses) and a disbelief in
theore tical (or uno bserva ble) ent ities. Hacking later on also expresses some
do ubts about the ro le of ex planations (PI" 52- 5) but this is, I believe ,
limited to thei r ro le as argume nts for realism . Hacking is well known for his
, h,lI11pioning of causal inte rventions in expe rimental reality as cri te ria for
I ••rlism , or for belief in unobservable entities .
\ II I(H'US on Whv questions is not meant to link these matter s to a specific
'.\ ut.ut ic: form, and is Simply a matter of case of ex position . Clearly, such
' 1" 1"\1 iOlls may be paraphrased in other ways : the requ est for a causal
I ' 1'1,11 1,11 ion ex pressed by the question ' Why did event X occ ur?' may be
' 1'I"l"\sed hy ' How was event X produced ?' or something like that. Though
1 1,, 1" 11"11' doe s not refer to Why questions he does differentiate between
'I" ' ,l illllS with simple propositions as answers (which subordinate the
' I" ' ,I jllil to a search for essences) from those mor e properly problematic .
IC,tioll,dism want ed to tie the fate of Ideas [multiplicities] to abst ract and
.1",111 ,'ssenc:es; and to the ex tent that the probl ematic form of Ideas was
1< ·' ·Il~lIi"led. it eve n wanted that form tied to the question of essences
11\ orlu-r words , to the ' W hat is X?' ... It should be noticed how few
I'flllm0l'lH"rs have placed their trust in the quest ion ' \ hat is X?' in orde r
til },.I\"I ' 1<i" ,ls. (' ..rtainly not risto t lc, O nce ti ll" diah-ct ir Itl ll' .irt of
NOTES
posi ng problem s] brew s up its matter instead of being applied to
propaedeutic ends , the qu est ion s ' How mu ch ' , 'How' , ' In what cases '
and 'Who ' abo und . . . These qu estion s are th ose of th e acciden t , the
event , the mult iplicit y. (Dc leuze, D!fference and Repetit ion , p . 188)
A more important omiss ion in my discussio n is that it does not include
Dclcuzc's distinction between problem s and questions. Problem s are th e
episte mological cou nterpar t to vir tual multiplicit ies, while questions (which
invo lve an imperative, a request or dem and for an exp lanatio n, for example)
are the sources of probl em s or the counte rpart of the quasi-causal operator.
Th er e are also episte mo logica l co unter parts to the inte nsive and th e act ual,
W e dist ingu ished four instances; imperati ve or onto logica l questions;
dia lectica l pr ob lem s or th e th em es that eme rge fro m them; symbo lic
fields of so lvability in which these problem s are 'scientifically ' ex pressed
in acco rdance to th eir conditions; th e so lutions given in th ese field s when
the probl em s are incarna ted in the actuality of cases. (I' . 200)
24. Alan Garfinkel , Forms if Explanation (Yale Uni versity Press, New Haven,
198 1), p . 21. O ther phil osoph ers have develope d similar approaches to Wh y
qu estions and the ir relat ion to the distributions of the relevant and the
irrelevan t . See , for ex ample, Salmon, Scientific Explanation and the Causal
. tru ctute '!.( the World, PI'. 1-6. See also Salmon's discussion of Van Frassen 's
approach to Wh y questions and contrast spaces (PI" 102- 6) which, unli keGarfinkel's, is complete ly lingui stic.
) 5. Alan Garfinkel, Forms ifExplanation, p. 40.
/6 . lbid.; p. 64 . Garfinke l takes this characterization of state space fro m Rene
Thorn, cre ator of catastro phe theory and of the conce pt of st ructural
stability . Here th e term 'crit ical point' may refer to both th e unstable
scparatrix tha t defines (as a repeller) th e border of a basin of attraction, or
to a bifurcation which defines the poin t of structural instability at which one
distribution of attractors changes into another.
27 . Deleuzc, DyJerence and Repetiti on , p. 159.
)H . Alan Garfin kel, Forms ifExplanation, PI" 53-8.)9. Ihid., Pl" 58-62.
lO. 11M, p. 168.
I I. Robert M. May, ' Chaos and the Dynamics of Biological Pop ulations ' , in
Dynami cal Chaos, ed. M. V. Berry (Lo ndon Royal Society, (987), PI" 31-2.
May's focus in this essay is chaotic at tractors, bu t he does m en tion periodic
at tractors , (T he latter are less controversial in populat ion st udies than the
Iormer .) I avoid discuss ion of 'chaos ' in the main t ext du e to thc excessive
hypc surrounding the subject, bu t more importantly, because on to logically
NO TES
the key notion is that of 'a ttractor ' no t the particular chao tic case . T hat is,
th e key is quasi-causality itself not anyone of its particular forms .
.~ 2 . Deleuze , Difference and Repetition , p . 212.
n . lbid. , p. 211.\4 . Dele uze views the so lving of a virtua l pr oblem by individuation processes as
an 'explanation ' or rather , an 'explication" , Th is term is used to refer to the
cancelling out of int en sive differences during a pr ocess of indivi duation, the
hid ing of int ensit y under the extensities and qua lities it gives rise to .
It is not surprising th at , strictly speaking, difference sho uld be ' inexplic
able'. Difference is explicated, but in syste ms in which it tends to be
canceled ; this means on ly th at difference is essentially implicated, that its
being is impli cation .. . Intensity is developed and ex plicated by means
of an ex tension whi ch relates it to th e ex te nsity in which it appears
outside itse lf and hidden ben eath quality . (De leuze , D!fJerence and Repe
tition, p. 228)
Some scien tists to day (C hr is Langto n, for instance) are begi nn ing to view
some processes of morphogen esis as involving th e solutio n to computational
problems.
A material near its critical transit ion point bet ween th e liqui d and th e gas
states , mu st, in effec t, come to a global decision abou t whether it must
settle down to a liqu id or to a gas . T his sounds almost anth ro po mo rphic,
hut th e results rep orted here sugges t tha t we must think abo ut such
syste ms as effectively computing thei r way to a minimum energy state.
( .hristopher G. Langton , ' Life at th e Edge of Chaos ', in Artificial Life II,·ds. Christopher G. Langton, Charles Tay lor, Doyne Farmer and Steen
Rasmussen (Addison- W esley, Redwood City, 1992), P: 82.
lh is, in fact, occurs in a differen t context. Deleuze never makes this point
n -l.u ivc to theoretical and ex pe rimental phYSiCS, but I believe his idea can be
I xtcnded in that direction . Th e act ual extract reads,
ot on ly do linguistic variables of expression enter into relations of
formal opposition or distinction favorab le for th e extraction of constants;
non .lin guisti c variables of content do also . As Hjelmslev notes, an
ex 1'1' .ssion is divided, for example , into phon ic units in the same way a
co nten t is divided into social, zoologica l, or phys ical uni ts . .. Th e
nvt work of hinaritics, or arborescences, is applicable to both sides. There
/I . however, no analytic resemblan ce, correspondence or co'!(ormity betll'ccn the
I II' '' planes . Rue their independence does not preclude isomorplusm . . . (DelCUZI'
,11 1< 1 Cuauarl, /1 Thousand Plateaus , p. 108; my emphasis)
NOT E S
\6 . Bunge, Causality and Modern Science, p . 175 . (My emphasis)
17. In a linear system the ultimate effect of the combined action of two
differ ent causes is merely th e superposition [e .g . addition] of th e effects
of each cause taken individually. But in a nonlinear syste m add ing a small
cause to one that is alre ady present can induce dramatic effects that have
no com mon m easure with th e amplitude of th e cause. (Gregoire Nicoli s
and lIya Prigogine, Exploring Complexity [W. H. Freeman, New York
1989J, p. 59)
\8 . Bunge , Causality and Modern Science, P: 127.
\9. lbid., p. 49 .
40 . Th is is W esley Salmon's character ization of statistical causality, meant to
replace pr evious versions state d in terms of high probabili ty. Th ese old er
vers ions, due to th c absoluten ess of th e probability value (near = 1), ar e
simply weak enings of nccessity (the case with probability = I) whereas
enhance d probability is not. Th e latt er demands that we know the prior
probabilities (th e probability of occurrence of an event without th e pr esen ce
of the cause) as well as th e posterior probabilities. Whether or not the value
uf the enhanced probability is near = 1 is not an issue in Salmon's version ,
hence it really br eaks with necessit y not just weak ens it. See Salmon,
Scient!fic Explanation and the Causal Structure rif the World, pp. 30-4 .41. lbid ., p. 203.
42. Bunge, Causality and Modern Science, Chapter 6 .
4\ . Ibid., Chapte r 8.
44 . Deleu ze and Guattari, A Thousand Plateaus, p. 408. (Emphasis in the original)
45 . Ian Hackin g , Representing and Interveninq, P: 158. (My em phasis) Hacking
explicitly compares expe r ime ntalists and artisans, both suffering a relatively
low er soc ial status due to th eir involvement with an active materiality, on e
that does not ob ey Sim ple theoretical laws or allow exte rn al forms to be
imposed on it as a command (p. 151). In classical m echanics perhaps th e
bcst examples of these tw o scient ific caste s are th e theorists Isaac Newton
or Rob ert Boyle, on one hand, and th e expe rime ntalist Rob ert Hooke, on
the othe r . As one scientist puts it, ' unlike Newton, Ho oke was inten sely
interested in what went on in kit chens, do ckyards, and buildings - the
mundane mechanical arenas of life .. . Nor did Hooke despise craftsme n,
and hc pr obabl y got the inspiration for at least som e of his ideas from his
friend the gre at Lond on clockmaker Th omas Tompion . .. ' Gam es Edward
Go rdon , The Science rif Structures and Mat ertals [Scientifi c Ameri can Library.
19881,p·18) .
46 . 'Phenomena accumulate . For example, Willis Lamb is trying to do optics
without photons. Lamb may kill off the photons [i.c. create a new theory or
NOTE S
a new paradigm for optics] but the photoelectric effect will sti ll be there '
(Hacking, Represenring and Intervening, P: 56. Also see pp. 155-62).,17. Ibid. , pp. 83-4.
IX. lbid., p . 265. (Emphasis in the original)
I (J. Pickering , The Mangle rif Practice, p . 70 .
r, O. Dclcuze, in fact, do es not refer to learning in a laboratory conte xt , but his
idea of lcarning as involving an intensiv e assemblage or a problematic.field is
d earl y applicable to th e case of expe rime ntal physics. Here's how Del euz e
expresses this idea,
For learning evolves entirely in th e comprehension of problem s as such
. . . Learning to swim or learning a foreign language means composing
the singular points of on e's own bod y or of on e's own language with
those of another shap e or cleme nt which tears us apart but also propels
us into a hitherto unknown or unh eard-of world of problems. (Diffe rence
and Repetiti on, p. 192)
And he adds that this composition of on e's singularities and affects with
tho se of water (in the case of swimming) or with thos e characterizing the
ounds and patterns of a language, forms a problematic field (p. 165). A
' problematic field' refers to a het erogeneous assemblagc since , as he says,
' le,lI'lling is the . . . structure which unites difference to differen ce, dissimi
I.II·ity to dissimilarity, without mediating between them' (p . J66).
11>,.1., p. 164., Il.ll.:king, Representing and Intervening, P: 209.
IIII IIg c , Causality and Modern Science, p. 71. (Emphasis in th e original)
Ih leuz e , D!flerence and Repetition, p . 25.1/.,,1.• p. 177 .
t; " neralizing , we can say that a dynamical theory is approximately true
just if the modelin g geome tric structure approximates (in suitable
I "spects) to the structure to be mod eled: a basic case is where trajectories
III the model closely track traj ectories encoding physically rea l behaviors
(or. at least , track them for long enough) . (Pete r mith, Explaining Chaos
1C.llnhri dge Un iversit y Press, Cambr idge, 1998], p. 72)
r rhur " lberall , Towards a General Science I?!' Viable Systems (Mc Graw-Hill ,
• \\ York, 1972 ), p . 7 . (My emp hasis)
I I 1111 ( ;oodman, 'Seven tri ctures on Similarity' , in Problem and Projects
dIu),)" Merrill , Indianapolis, 1972 ), p . +45. Goodman 's att ack on the noti on
" uui l.rritv was as caustic as it was influential. Similarity, he said 'ever
.. I til so l\'\' philosophi cal pr obl em s and oven-orne obsta les, is .1 pn··
NOTES
tender, an impostor, a quack . It has, ind eed, its place and its uses, but is
more often found wh er e it do es not belong, professing powers it does not
possess' (p,437) . Tod ay' s generat ion of realist philosopher s who have
resuscitated this notion have learned Goodman' s lesson that any two things
are similar in some respel.t or another , and that therefore when ever valid
judgments of similarity are made the relevom respects in which things may be
said to he alike mu st be speci fied (p . 444) . But thi s, of course, simply
changes the task to on e of speci fying distributions of th e relevant and the
Irrelevant, and that is just wha t a problematic approach is supposed to do.
At this point the usual reply by defender s of similarity is to fall back on
subjectivism and say that questions of rel evan ce and irrelevance are int erest
relative.
But far from sett ling the issue , to re latlvize r elevance to subjective
interests is fatal to realism . If there ' s one lesson to be learned from recent
sociology of scie nce it 's that , as a matter of empirical fact , the inter ests of
scientist s canno t be viewed as being purely episte mo logical, born from some
essent ial rational ity or a driving curiosity . If we are to relativ ize relevance
to inter ests th en we should bring th e full repertoire of interests here,
including not only selfish profession al and institutional interest s but also
those that may be deri ved from a sci entist' s membership in class or gende r
hierarchies, for example . Th e rampant relativi sm that thi s manoeuver has
sometimes given rise to should be a cautio nary lesson for any defender of
realism. Alan Garfinkel sometimes expresses him self as if the choice of
contrast space, that is, the choice of how to pose a problem, is relativ e to
human inter ests and values, as in th e different values held hy the pr iest and
the thief in his example, But qu esti ons of explanatory stability seem to point
to an object ivity of the distribution s of the rel evant and the irrelevant.
Whateve r relativity there may be in explanations it is an objecti ve one ,
depending on the existence of indi vidual s with their own eme rge nt causa l
capacities at many levels of scale. Human values would enter the picture in
thechoice of on e or another of these level s of scale as the level of interest,
but a correct ex p lanation, as Garfinkel says , ' will see k its own level'
(Garfi nkel , Forms l!I Explanat ion , P: 59) .
Gilles Dcleuze, D!fftrence and Repetition, p. 16 3.
There is nothing in the ordinary meaning of the words ' universal' and
' ~ i ngu lar ' that marks the philosophical distin cti on Deleuze is attempting to
draw here , In fact, ana lytical philosoph er s use the wo rds 'gen eral' and
'universal' alm ost inte rchangeably, and the terms ' part icular' and 's ingular '
JS rloscly relat ed . In D1ference and Repetition uni versality and singularity an'
(loth properties of object ive probl em s, the former defining their o nto logie.l l
-utus ,1S vir tual entit ies (capable or di vl.' rgt'n t act ualizat ion) the lat ter till'
NOTE S
status of that which defines th eir condition s (d istribut ions of the rel evant and
the irrelevant ). The very first page of thi s book state s 'Ge nerality, as
gt'nl'rality of the particular, thu s stands opposed to re peti tio n as unive rsality
of the singular ' (p. 1). Yet, Deleu ze is not consistent in his usage , and
else where he says th at the ' splendid ste rility or neutrality [of multipliciti es]
. . . is indifferent to the uni ve rsal and the singular , to the general and th e
particular, to th e personal and the co llec t ive' (Gilles Deleu ze , LOBic if Sense
[Col umbia University Press, Ne w York , 1990], P' 35) .
Ml. Dialect ic is the art of problems and qu estions ... Howev er , dialectic
loses its peculiar power when it remains conte nt to trace problems from
propo sitions : thus begin s th e history of th e long perver sion 'vhich places
it under the pO\\'er of the negative , Ari stotle writes: Th e differen ce
between a problem and a proposition is a differen ce in the turn of phrase .
{Dcleuze , Difference and Repetition, P' 158)
I . 1.1lI Stewart and Martin Golubitsky, Feaiful Symmetry (Blackwell, Oxford,
1')9 2), P: 4 2. (Emphasis in the ori ginal)
II } I Jcle uzc , D~fJerence and Repetit ion , P: 162.h' llu- impact of gro up th eory on physics is r evealed not onl y by the fact that
1111' ('hange fro m classical to relativistic physics can be described in group
tlu-un-t lc terms (Einste in replaced the old Ga lilean group of transformations
Ii ) ' another one , the Poin car e gro up) but also by th e fact that the switch to
n-Ia t lvi:..t ic mechanics involv ed a change of cognit ive st rate gy in which
mva rinnccs und er transformation s becam e more imp ortant than the physical
I.I\\"\ the mselves. As the physicist Eugen e Wi gner puts it ,
[Einste in's ] papers on speci al relativity .. . mark the reversal of a trend :
until then the prin cipl es of invariance were derived from the law s of
mo tion . . . It is natural for us now to deri ve the laws of natu re and to
tes t their validi ty hy means of the law s of invariancc, rath er than to
derive the laws of invarian ce fro m what we believe to be the laws of
o.uun-. The ge ne ral theory of relativity is the next mil eston e in th e
!.btury of invariance .. . It is the first attempt to derive a law of nature
Ii )' M'lt'ding the sim plest invari ant equat ion . .. (Euge ne P. Wi gner,
' Invartancc in Physical Theory ', in S/mmetries and Rtfteetiom , eds. W alter
Moon' and Michael Scr iven [O x Bow Press, W oodbridge , 1979], P' 7)
11111'b Kline, ,lIathemat ical Thouahr from Ancient to Modern Times, Vol. 2
I I ) lor d Llnivcrsity Pr ess, New York , 1972), P: 7 59. Thi s idea can he
• I'l. li lll' d hy .1Ilalogy with the use o f tra nsfo rmat ion groups to d assil)'
I t 1I1 1H·lrk.,1 figu n's. W hen un c s,lys thai a cube remains Invariant unde r a
NOTES
group of rotations (e .g. the set containing 0, 90, 180 and 270 degree
rotations) one means that, after performing one such transformation the
cube's appearance remains und1anged: an observer who did not witness the
transformation would not be able to tell that a change has in fact occ urred.
In a similar way, when Galoi s found a group of permutations that left
algebraic relations invariant he found a measure eif our isnorance eif the solutions,
since we cannot distinguish them from one another after they have been so
transformed.
65. Deleuze, D!iference and Repetiti on, PI" 180-1.
66 . Ibid . , PI" 179-80. That Deleuze views the progressive specification of a
problem as a kind of symmetry- breaking cascade (a term he never uses,
preferring Galois's idea of an 'adjunction of fields ') is clear from this ext ract:
On the contrary, 'solvability ' must depend upon an int ernal character
istic : it must be determined by the conditions of the problem, engendered
in and by the problem along with the real solutions. Without this
reversal, the famous Copernican Revolution amounts to nothing . More
over , ther e is no revolution so long as we remain tied to Euclidian
geome try : we must move to . . . a Riemannian-like differential geometry
which tends to S hoe rise to discontinuity on the basis ifcont inuity , or to ground
solutions on the conditions of the problem. (I" 162; my emphasis)
67. Ian Ste wart , Does God PIa)' Dice? The Math emat ics if Chaos (Basil Blackw ell,
O xford , 1989), 1'1'.38- 9.
68 . onlinear equations , due to factors like the occurrence of higher pow ers of
the dependent variable, do not obey supe rp osition. On the differ ences
between the linear and the nonlinear , and on the (rare) conditio ns for the
exact solvability of nonlinear equations (auto nomy and separability) , see
David Acheson, From Calculus to Chaos: An Introduct ion to Dynamics (Oxford
University Press, New York 1997), Chapter 3.
On the superposit ion principle as crite rion to distinguish these tw o types
sec David K. Campbell, ' Nonlinear Science. From Paradigms to Practical
ities ' , in From Cardinals to Chaos, ed. Necia Grant Cooper (Cambridge
University Press, New York, 1989), p. 219.
69 . Stewart , Does God Pia)' Dice?, p. 83. (Emphasis in the original)
70 . June Barrow-Green , Poincare and the Three Body Problem (American Math
ematica l Society, 1997), PI" 32-8 .
O n the history of this appro ach prior to the work by Poincare see Kline ,
Mathematica l Thou,qhtIrom Ancient to Modern Tim es, Pl" 72 1-5.
71 . We alwa ys find the tw o aspects of the illusion : the natural illusion wh ich
involves tracing pr oblem s from suppose dly preexistent propositions,
N OTES
logical OpiniOnS, geo metrical theorems, algebraic equatio ns , physical
hypoth eses or transcend ental judgments; and the philosophi cal illusion
which involves evaluating problem s according to their 'solvability' - in
other words, according to the extr insic and variable form of th e possibility
of their finding a solutio n. (Deleuze, D!ffirence and Repetit ion, p. 161)
7 1 . Bunge, Causality and Modern Science, PI" 203 - 4. (Emphasis in the original)
The fact that nonlinear theori es are rare is not so much a peculiarity of
natu re as a sign of the infancy of our science. Nonlinearity involves large
mathe matical diffi culti es; beside being math ematically clumsy, it affects
the very symbolic representation of physical entities. Thus forces that add
nonlin earl y (as gravitational forces do) cannot be exactly represented by
vecto rs since the addition of the latt er conforms to the superposition
' principle ' . From the moment it was realized that the laws of ferromag
netism are nonlinear, it has been more or less clearly suspected that all
physical phenomena may turn out to be at least weakly nonlinear,
linearity being onl y an approximation which is exce llent in som e cases
but only rough in othe rs. (I' . 168; emphasis in the original)
l Iclcuzc, Difference and Repetition, p. 189.
Index
.IITects 62-5,69-70,75, 141,167,199 n.35, 218 n,46
assemblage 56-8, 62-4, 93, 136,142-3,236 n.50
.u tractor 15, 20, 31- 2, 36- 7, 50, 55,
79,90,109-1 1, 134, 148,214
n.19axiomatic 121, 154, 179
I" 'coming 84, 101-2, 107bifurca tion 19-20,32,50,79,86,
109-10, 134, 207 n.61
, .Iusality 75 , 119- 20, 126, 129,137-40, 142, 144-5, 228 nA
, mtinu urn 22-3,27,69,74-6, 107,
158, 161, onvcrgcnce
and communication betwe en virtualser ies 76 -7, 104, 160- 1,206
n.59, 208 n.67, 220 n.53
.uu l subjectivity 162- 3, 171
divergence.1IId ramification of virtual ser ies
74- 5, 104, 160, 169, 176, 205
n.58
.incl actualization 22, 28, 64 , 118
,. "'nc<' 3- 4 , 9, 16, 28, 39- 40 , 78,106, 119, 121, 128, 183 nn. IO &
11 , 231 n.20, 232 n.23t ' u-nsiv« 26 -7, 46, 51 3, 58, 64, 85,
144, 163
identity 4, 9, 22, 40, 42, 74, 86, 107,
118, 193 nn.59 & 60immanence 3,10,13,28,41,75,80,
103, 110-11 , 146, 177,2 18 n,49individuation 29,40,43,45-6,51 ,
71,84,97-8 , 101, 117, 142,145,1 6 1,1 64,1 71,1 95 nA, 202
n.5 1, 234 n. 34
intensive 4, 26-7,45-6, 50, 55, 58 ,
64,85,92-4,98, 135, 143-4,159,1 6 1- 2, 165, 167, 169,1 71,
173- 5, 188 n.28, 199 n.30invariant 18, 24, 69, 75, 83, 86- 7 ,
150, 210 nA, 238 n.63
law 83, 106, 118-19, 121, 123- 4 ,
141, 146, 150,180,210 n.4, 221
n.57
metric
space 24-6,51,56-7,69,73, 172,179,186 n.22, 187 n.26
time 84,88, 106,108-9,213 n. 17
multiplicities 9, 13, 22, 28, 32,40- 1,69,72-5, 105-8, 111- 13, 129,
135,148,1 59,1 66,1 70,1 74,
181 n. 1, 182 n.6
natural kind 9, 13, 43, 46
necessity 37-8, 192 n.54, 235 nA O
non-equilibrium 66-8nonlinear 37,5 2-3,66-7,87, 119,
123,131,136,140 1,144,153 ,
155, 235 n. 17, 239 n.6
I N 0 E X
ordinal series 70,73 -7,104,113 ,
159, 170,204 n.56
phase transition 19- 20, 27, 61, 78-80,103- 4, 107, 197 n.23, 209 n.76
plane of consistency 69, 77, 112-13,
115, 158, 166, 170,226 n.77
population 47-8, 122
possibility 10,13,29,33-5,37,40,
206 n.59
problems 5, I I, 102, 115, 125, 127,
129- 33, 135- 7,1 40,1 44 - 5,
146- 7, 149- 52, 154, 168,
176- 7, 23 1 nn.19 & 20, 239n. 34, 240 n.72
progressive differentiation 17, 25, 28,
54,69, 102, 105, 151, 164, 177,
185 n.20
quasi-causal
opera tor 75- 8, 80, 103, 108,
I l l-IS, 136, 160, 165-6, 168,
170, 174- 6, 219 n 52,220 n.53 ,
223 n.71, 224 nn.74 & 75
re lations 127,129,133-4,140,
146, 207 n.62
rates of change I I , 49, 53, 95-6,
99- 100
relevance 5, 13, 90, 130 - 2, 144 - 5,
148, 151
resemblance 4,9, 10,2 1,28,40-2,
60, 68, 136, 147- 8, 193 nn .59 &
60, 237 n. 58
singularity 5,15-16,31,36,64-5,72,77,92,94, 108, 125-7, 131,
134,137,141,146-8,154,167,
203 n.53state space 14-15,30-2,62, 130,
134, 136, 145-6, 154, 175
symmetry-breaking 18-19, 21, 24-6,
74,86-7,105,107,135, lSI,
185 n.20, 201 n.46, 239 n.66
time scale 87,90- 1, 108, 214 n. 19
topologica lfeature (forms, constraints) 15- 16,
29,31,72, 110-11, 147-8 , 183
n.S, 197 n. 19space 23, 52-3, 56, 69, 74, 179,
226 n.76
time 105-9,222 nn.58 & 59transcendence 3, 10, 13, 4 1, 80, 107,
113, 177
truth 5, 121,123, 134, 147, 177
typology 41-2,47-8,68, 177, 193
nn.59 & 60
unity 13, 113universal
as concrete entity 22, 28, 70, 112as mechanism-independence 16, 55,
75,79,92-3,127, 129, 133,207
n.6 1
as opposed to the general 148- 50,
237 n.59
Virtuality 33-8,44,65-8,78, 102,105- 9 , 127, 135, 147, 154, 165,
174- 5, 219 n.52, 225 n.76