Wiener Cynernetique

7/26/2019 Wiener Cynernetique

1/194


2/194

TITUTEOFTECHNOLOGY

eofmustnot

withoutthe

ublisher.

Y,1949

sof America


3/194


4/194


5/194


6/194


7/194


8/194

d hadinfactbeena participantin

minaronthesubjectin 1911-1913.

wasessentialtohavesomeone

emathematicalquestionscritically.

emberofthegroupuntilDr. Rosen-

944andthegeneralconfusionofthe

eetings.

bluethandIhad sharedtheconvic

areasforthegrowthof thesciences

n neglectedasano-man'slandbe

hedfields.SinceLeibnizthere has

has hadafullcommandofall the

ay.Since thattime,sciencehas

ofspecialists,in fieldswhichshowa

sivelynarrower.Acenturyago there

butthere wasaGauss,a Faraday,

are fewscholarswhocancall

ansorphysicistsorbiologistswithout

atopologistoranacousticianor a

d withthejargonof hisfield,and

andallits ramifications,but,more

regard thenextsubjectassomething

threedoorsdown thecorridor,and

itonhis ownpartasan unwarran

econtinuallygrowingandinvading

likewhatoccurredwhenthe Ore

adedsimultaneouslybytheUnited

theMexicans,andthe Russiansan

oration,nomenclature,andlaws.

c work,asweshallsee inthebodyof

nexploredfromthe differentsidesof

cs,electricalengineering,andneuro-

singlenotionreceivesa separate

d inwhichimportantworkhas been

ed;whilestill otherimportantwork

bilityin onefieldofresultsthat may

sicalinthenext field.

nsofsciencewhichofferthe richest


9/194

edinvestigator.Theyareatthe

torytothe acceptedtechniquesof

onoflabor.If thedifficultyofa

athematicalinessence,tenphysio

maticswillgetpreciselyasfar asone

athematics,andnofurther.Ifaphy

athematics,workstogetherwithama

ophysiology,theonewillbeunable

msthattheothercanmanipulate,and

o puttheanswersinany formthat

r.Rosenbluethhasalwaysinsisted

ftheseblank spacesonthemapof

ebyateam ofscientists,eachaspecia

chpossessingathoroughlysoundand

hthefieldsofhisneighbors;all inthe

ofknowingoneanother'sintellectual

ngthesignificanceofacolleague'snew

kenona fullformalexpression.The

avetheskilltoconductaphysiological

avetheskill tounderstandone,to

stone.Thephysiologistneednot be

hematicaltheorem,buthemustbe

gicalsignificanceandtotellthemathe

dlook.Wehad dreamedforyears

dentscientists,workingtogetherin

fscience,notassubordinatesofsome

utjoinedbythedesire,indeedby the

erstandtheregionasa whole,andto

gthof thatunderstanding.

atterslong beforewehadchosen

igationsandourrespectivepartsin

nthisnewstep wasthewar.1had

methatif anationalemergency

initwouldbe determinedlargely

ontactwiththeprogramofcompu

byDr.VannevarBush,andmyown

g Leeonthedesignof electricnet

important.Inthesummerof

of myattentiontothedevelopmentof


10/194

hesolutionofpartialdifferentialequa-

restedinthese,andhadconvinced

blem,ascontrastedwiththeordinary

elltreatedbyDr.Bush onhisdifferen

e representationoffunctionsofmore

so becomeconvincedthatthepro

oyedintelevision,gavetheanswer

ctthattelevisionwasdestinedto

ingbythe introductionofsuchnew

pendentindustry.

ningprocessmustvastlyincrease

th ascomparedwiththenumber

inarydifferentialequations.Toac

tsinareasonabletime,itthus became

edofthe elementaryprocessesto

dinterruptingthestreamof these

sentiallyslowernature.It also

ormtheindividualprocesseswithso

thattheenormousrepetitionof theele

dnotbringaboutacumulativeerror

ccuracy.Thusthefollowingrequire

and multiplyingapparatusofthe

dbenumerical,asinanordinaryad

onabasis ofmeasurement,asinthe

;

s,whichareessentiallyswitchingde

ectronictubesratherthanongears

dertosecurequickeraction;

ththepolicyadoptedinsome

ellTelephoneLaboratories,itwould

icalinapparatustoadoptthescale

tiplication,ratherthanthe scaleoften;

eof operationsbelaidouton

ere shouldbenohumaninterven

awereentereduntilthefinal results

hatalllogicaldecisionsnecessaryfor

emachineitself;


11/194

ainanapparatusfor thestorageof

hemquickly,holdthemfirmlyuntil

y,erasethemquickly,andthenbe

thestorageofnewmaterial.

ogetherwithtentativesuggestions

hem,were sentintoDr. Vannevar

inawar.Atthatstageof theprepara

seemtohavesufficientlyhighpriority

onthemworthwhile.Nevertheless,

ichhavebeenincorporatedintothe

utingmachine.Thesenotionswere

ofthethoughtof thetime,and1do

claimanythinglikethesolerespon

n.Nevertheless,theyhaveproved

hatmymemorandumhadsomeeffect

gengineers.Atanyrate,as weshall

k,theyareallideas whichareofin

hestudyof thenervoussystem.

thetable,andalthoughithas

ledtonoimmediateprojectby

elf.Ouractualcollaborationresul

hichwaslikewiseundertakenforthe

thebeginningofthewar,the Ger

dthe defensivepositionofEngland

nyscientiststothe improvement

enbeforethewarit hadbecome

airplanehadrenderedobsoleteall

rectionoffire,andthat itwasneces

rolapparatusallthecomputations

nderedmuchmoredifficultbythefact

ncounteredtargets,anairplanehas

appreciablepartofthevelocityof

tdown.Accordingly,itisexceeding

missile,notatthetarget,but insuch

getmaycometogetherinspaceat

e musthencefindsomemethodof

onoftheplane.

xtrapolatethe presentcourseofthe


12/194

This hasmuchtorecommendit.

nd curvesinflight,thelessis its

stimeithasto accomplishamission,

nadangerousregion.Otherthings

asstraightacourseas possible.

rstshell hasburst,otherthingsare

lprobablyzig-zag,stunt,orin some

tion.

elyatthe disposalofthepilot,and

sortofintelligentuseof hischances

d pokerplayer,forexample,hehas

odifyhisexpectedpositionbeforethe

ouldnotreckonthechancesofhitting

ptperhapsin thecaseofavery waste

er hand,thepilotdoesnot havea

omaneuverathiswill.Foronething,

exceedinglyhighspeed,andanytoo

coursewillproduceanacceleration

scious,andmaydisintegratethe

ycontrolthe planebymovinghis

ewregimenofflowthatis established

evelop.Evenwhenit isfullydeve

heaccelerationoftheplane,andthis

ustbeconverted,firstintochangeof

ngeofposition,beforeiti sfinally

viatorunderthestrainofcombat

moodto engageinanyverycompli

oluntarybehavior,andisquitelikelyto

tivityin whichhehasbeentrained.

onof theproblemofthecurvilinear

hile,whethertheresultsshould prove

ortheactualuseofa controlapparatus

prediction.Topredictthefuture

certainoperationon itspast.The

annotberealizedbyanyconslructible

ertainoperatorswhichbearit acertain

ctrealizablebyapparatuswhichwe

rofessorSamuelCaldwelloftheMass

nologythattheseoperatorsseemed


13/194

diatelysuggestedthatwe trythem

tialanalyzer,usingthisasaready-

fire-controlapparatus.Wedid so,

discussedinthebody ofthisbook.

fengagedinawar project,inwhich

myselfwerepartnersinthe investi

iction andoftheconstructionof

etheories.

econdtime 1hadbecomeengaged

co-electricalsystemwhichwasdesigned

anfunction inthefirstcase, the

dpatternofcomputation;and inthe

thefuture.In thissecondcase,we

sionof theperformanceofcertain

fire-controlapparatus,itis true,the

mesin directlybyradar,butin

s ahumangun-pointeroragun-

thefire-controlsystem,andactingas

ssentialto knowtheircharacteristics,

mmathematicallyintothe machines

eirtarget,theplane,isalso humanly

e toknowitsperformancecharacter

otheconclusionthatan extremely

ryactivityiswhat thecontrolengineers

ussthis inconsiderabledetailinthe

noughto sayherethatwhenwe

agivenpattern,thedifferencebetween

yperformedmotionis usedasanew

ulatedtomoveinsuch awayasto

hatgivenbythepattern.For example,

eof ashipcarriesthe readingofthe

tillerwhichsoregulatesthe valvesof

movethetillerinsuch awayasto turn

ller turnssoasto bringtheotherend

setamidships,andinthatway registers

wheel astheangularpositionof

onor otherdelayingforcewhich

tiller willincreasetheadmissionof


14/194

side,andwilldecreaseiton theother,

sethetorquetendingtobring thetiller

sthe feedbacksystemtendstomake

eringenginerelativelyindependent

ertainconditionsofdelay etc.,a

uewillmakethe rudderovershoot,

eed-backintheotherdirection

vershootstillmore,untilthe steering-

doscillationorhunting,and breaks

ksuchasthat byMcColl',we

onoffeed-back,theconditionsunder

andtheconditionsunderwhichit

omenonwhichweunderstandvery

ativepointofview.

p alead-pencil.TodothisI have

However,forallofus butafew

notknowwhatthesemusclesare;

omists,therearefewif anywhocan

ouswillingin successionofthecon

ncerned.Onthecontrary,whatwe

.Oncewehavedeterminedonthis,

haway thatwemaysayroughly

hepencil isnotyetpickedupi s

Thispartofthe actionisnotin full

ha manner,theremustbea

m,consciousorunconscious,ofthe

failedtopickthe pencilupateach

onthepencil,thisreportmay be

itis moregenerallykinaesthetic,

gue,proprioceptive.Iftheproprio-

nting,andwedonotreplacethemby

e,weareunabletoperformtheact

dfind ourselvesinastateof what

iaof thistypeisfamiliarin the

tralnervoussystemknownastabes

anixms.VanNostrand,1945.


15/194

theticsenseconveyedbythespinal

troyed.

ed-backislikelytobeas seriousa

vityasa defectivefeed-back.Inview

owand myselfapproachedDr.

ecificquestion.Is thereany

hichthepatient,in tryingtoper

kepickingup apencil,overshoots

uncontrollableoscillation?Dr.

yansweredustliatthereissuch awell-

calledpurposetremor,andthatit is

ytothe cerebellum.

nificantconfirmationofourhypo

reofat leastsomevoluntaryacti

urpointof viewconsiderablytrans

gneurophysiologists.Ihecentral

appearsasa self-containedorgan,

ensesanddischarging intothemus

eof itsmostcharacteristicactivities

ar processes,emergingfromthe

uscles,andre-enteringthenervous

organs,whethertheybe propriocep-

alsenses.Thisseemedtous tomark

hatpart ofneurophysiologywhich

mentaryprocessesofnervesand

anceofthenervoussystemas aninte

snewpoint ofviewmeriteda

andpublishedx.Dr.Rosenblueth

percouldonlybe astatementotpro

xperimentalwork,andwedecided

ourplanfor aninterscientificinsti

wouldfurnishanalmostidealcenter

gineeringplane,ithadalready

wand myselfthattheproblemsof

communicationengineeringwere

gelow, Behaviour.Purpose&Teleology,Philo-

p.18-24(1943).


16/194

centerednotaroundthe technique

utaroundthemuch morefundamen

whetherthis shouldbetransmitted

ornervousmeans.Themessageis

quenceof measurableeventsdistri

whatiscalleda time-seriesbythe

nofthefutureof amessageisdone

itspast,whetherthis operatoris

athematicalcomputation,orbya

paratus.Inthis connection,wefound

echanismswhichwe hadatfirst

bytwotypesof error,ofaroughly

thepredictionapparatuswhichwe

madetoanticipateanextremely

ddegreeof approximation,this

salwaysattainedatthe costofan

bettertheapparatuswasfor smooth

besetintooscillationby smalldepar

ndthelongerit wouldbebeforesuch

Thusthe goodpredictionofasmooth

more delicateandsensitiveapparatus

ictionof aroughcurve; andthe

aratusto beusedina specificcase

sticalnatureof thephenomenonto

ngpairof typesoferrorseemedto

nwiththecontrastingproblemsof the

momentumtobe foundinthe

hanics,asdescribedaccordingtohis

pedthatthesolutionofthe problem

sonlyto beobtainedbyanappealto

eriestobepredicted,it wasnotdifficult

yseemedto beadifficultyin thethe

wasactuallyanefficienttoolfor sol

ion.Assumingthe statisticsofatime

oderiveanexplicit expressionforthe

ctionby agiventechniqueandfora

his,wecouldtranslatethe problem

edeterminationofa specificopera-


17/194

toa minimumaspecificpositivequan

ator.Minimizationproblemsofthis

dbranchofmathematics,thecalcu

anchhas arecognizedtechnique.

que,wewereabletoobtainan expli

blemofpredictingthefutureofa time

nature;andevenfurther,to achieve

ssolutionby aconstructibleappa

east oneproblemofengineering

ynewaspect.Ingeneral,enginee

tobeanart ratherthanascience.By

sort toaminimizationprinciple,we

tona farmorescientificbasis.Itoc

notanisolatedcase,butthat therewas

ngworkin whichsimilardesignpro

he methodsofthecalculusofvaria

hersimilarproblemsbythe same

astheproblemofthedesignof wave

ssagecontaminatedbyextraneous

lbackgroundnoise.Wethenfacethe

ginalmessage,orthe messageunder

agemodifiedbyagivenlag,byan ope

edmessage.Theoptimumdesignof

pparatusbywhichitisrealized depends

he messageandthenoise,singlyand

cedinthe designofwavefilters

merlyofanempiricalandratherhapha

witha thoroughscientificjustification.

deofcommunicationengineering

e,abranchofstatisticalmechanics.The

nicshasindeedbeenencroachingon

rmorethana century.Weshallsee

isticalmechanicsinmodernphysics

eforthe interpretationofthenature

mmunicationengineering,however,

sticalelementisimmediatelyapparent.

ationisimpossiblesaveasa trans.


18/194

nlyonecontingencyis tobetrans

mostefficientlyandwiththeleast

sageatall. Thetelegraphandthe

irfunctiononlyif themessagesthey

riedina mannernotcompletely

ndcanonly bedesignedeffectivelyif

sagesconformstosomesort ofstatis

mmunicationengineering,wehadto

yofthe amountofinformation,inwhich

tionwasthat transmittedasasinglede

bablealternatives.Thisideaoccurred

severalwriters,amongthemthestatisti

nonoftheBellTelephoneLabora

er'smotivein studyingthissubjectis

tisticaltheory;thatofShannonin the

tion;andthatofthe authorinthe

ageinelectrical filters.Letitbe

thatsomeofmyspeculationsinthis

estotheearlierwork ofKolmogo-

aconsiderablepartofmywork was

wascalledto theworkofthe Russian

f informationattachesitselfvery

on instatisticalmechanics:thatof

tof informationinasystemisa

ganization,sothe entropyofasystem

ofdisorganization;andtheone is

other.This pointofviewleadsus to

sconcerningthesecondlaw ofther

yofthe possibilityoftheso-called

estionsariseindependentlyinthe

rcatalysts,andtheir studyisessential

ngofsuch fundamentalphenomena

ismandreproduction.Thethird

oflife,that ofirritability,belongsto

terpolationuudExtrapolationvonstationarenZufal-

Sci. U.S.S.R.Sec. Math.,5.3-14(1941).


19/194

tiontheory,andfallsunderthegroup

ndiscussing1.

earsago,thegroupof scientists

dmyselfhadalreadybecomeaware

esetofproblemscenteringabout

ndstatisticalmechanics,whetherin

ssue.Ontheotherhand,wewere

elackofunityof theliteratureconcer

bytheabsenceof anycommontermi

nameforthefield.Aftermuchcon

totheconclusionthatallthe existing

yabiasto onesideoranotherto serve

thefieldas wellasitshould ;andas

sts,wehavebeenforcedto coinat

ekexpressiontofillthe gap.Wehave

eld ofcontrolandcommunication

chineorin theanimal,bythename

mfromtheGreekxugepv/lTY^orsteers

,wewishto recognizethatthefirst

backmechanismsisanarticleongover

byClerkMaxwellin1868 2,and

oma LatincorruptionofxugspwiT/is.

efactthatthe steeringenginesofa

earliestandbestdevelopedforms

s.

eticsdoesnotdatefurtherbackthan

all finditconvenienttousein refer

edevelopmentofthefield.From1942

opmentofthesubjectwentaheadonin

asof thejointpaperofBigelow,

weredisseminatedbyDr.Rosenblueth

Yorkin1942underthe auspicesof

n,anddevotedtoproblemsofcentral

stem.Amongthosepresentat that

cCulloch,oftheMedicalSchoolof

hohadalreadybeen intouchwith

tisLife"! CambridgeU.Press.Cambridge,Eng.,

.Soc.(London).March5,1868.


20/194

lf,andwhowasinterestedin the

fthe cortexofthebrain.

nelementwhichoccurs repeatedly

cstheinfluenceofmathematical

patron saintforcyberneticsoutof

ouldhavetochooseLeibniz.The

ersabouttwoclosely relatedconcepts

mbolismandthatofacalculus of

descendedthemathematicalnotation

hepresentday.Now,justas the

sitselfto amechanizationprogres

dthe deskcomputingmachinetothe

chinesofthepresentday,sothe

bnizcontainsthe germsofthemachina

gmachine.Indeed,Leibnizhimself,

,wasinterestedin theconstruction

hemetal.It isthereforenotinthe

meintellectuali mpulsewhichhasled

hematicallogichasat thesametime

echanizationofprocessesofthought.

chwecan followisonewhich can

berofsymbols.Thesesymbols,in

tothe notionofinfinity,butthis

nsumup inafinitenumberof stages;

aticalinduction,whereweprovea

arameternforn =0,andalso prove

fromthe case/7,thusestablishingthe

uesofn. Moreover,therulesof

mechanismmustbefinitein number,

peartobe otherwise,througharefe

finitywhichcanitselfbe statedin

sbecomequiteevident,bothtothe

dtothe intuitionistslikeWeyl,that

hematico-logicaltheoryissubjectto

s asthosethatlimitthe performanceof

weshall seelater,itis evenpos

ytheparadoxesofCantorand of

entofRussell,andowemuchto his


21/194

kforhisdoctor'sthesis attheMassa

ologytheapplicationofthe techniques

ebraof classestothestudyof

icalengineering.Turing,whois

whohavestudiedthe logicalpossibi

intellectualexperiment,servedthe

thewaras aworkerinelectronics,

programwhichthe NationalPhysi

onhasundertakenforthedevelop

nesofthemoderntype.

mthefieldof mathematicallogic

s.He hadbeenastudentof Car-

so beenincontactwith Professor

ofbiophysicists.Letit beremarked

hascontributedmuchtodirecting

maticallymindedtothepossibilities

lthoughit mayseemtosomeofus

dbyproblemsof energyandpoten

assicalphysicstodothe bestpossible

mslikethenervoussystem,which

ed energetically.

unetofallunderMcCulloch'sinflu

workquiteearlyonproblemscon

ibres bysynapsesintosystems

es.IndependentlyofShannon,

eof mathematicallogicforthedis

allswitchingproblems.Theyadded

rominentin Shannon'searlierwork,

ysuggestedbytheideas ofTuring:

rameter,theconsiderationofnets

ynapticandother delays1.

metDr.J.Lettvinof theBoston

ymuchinterestedin matterscon

sms.Hewasaclosefriend ofMr.Pitts2,

withhis work.HeinducedMr.Pitts

dtomaketheacquaintanceofDr. Ro-

ndon MathematicalSociety,II,2.42,230-265(1936).

tts,W., Alogicalcalculus oftheideasimmanent

Math.Biophys.,5 115-133(1943).


22/194

welcomedhimintoourgroup.

chusettsInstituteofTechnologyin

er toworkwithme,andto strengthen

oundforthestudyofthenewscience

bythattimebeenfairlyborn, but

alreadythoroughlyacquaintedwith

europhysiology,buthadnot hadthe

yengineeringcontacts.Inparticular,

hDr.Shannon'swork,andhe hadnot

epossibilitiesofelectronics.Hewas

nIshowedhimexamplesofmodern

edtohimthatthesewere idealmeans

heequivalentsofhisneuroniccircuits

me,itbecameclearto usthatthe

chine,dependingasitdoesonconse

mustrepresentalmostanidealmodel

thenervoussystem.Theall-or-none

eoftheneuronsis preciselyanalogous

n determiningadigitonthe binary

eofus hadalreadycontemplatedas

sofcomputingmachinedesign.The

mechanismfordeterminingwhethera

tputfromotherselectedelements

equatestimulusfor thedischarge

usthaveits preciseanalogue

.Theproblemofinterpretingthe

moryinthe animalhasitsparallelin

gartificialmemoriesforthemachine.

onofcomputingmachineshad

alfor thewareffortthanthefirst

aveindicated,andwas progressing

esnot toodifferentfromthosewhich

ated.Harvard,AberdeenProving

yofPennsylvaniawerealreadycon

heInstituteofAdvancedStudiesat

husettsInstituteofTechnologywere

eld.Inthisprogramtherewas a

mechanicalassemblytotheelectrical


23/194

oftento thescaleoftwo,fromthe

ectricalrelay,fromhumanlydirected

ydirectedoperation;andinshort

hanthelastwasin conformitywith

ntDr.Bush.There wasacontinual

nterestedin thesefields.Wehadan

ateourideastoourcolleagues,inpar

ard,Dr. vonNeumannoftheInsti

,andDr.Goldstineofthe Eniacand

versityofPennsylvania.Everywhere

chearing,andthevocabularyofthe

ontaminatedwiththetermsofthe

psychologist.

edingsDr.vonNeumannandmyself

ntmeetingof allthoseinterested

etics,andthismeetingtook placeat

of1943-1944.Engineers,physiolo

swereallrepresented.Itwasimpossi

hamongus,as hehadjustacceptedan

thelaboratoriesofphysiologyofthe

ologiainMexico,butDr.McCulloch

heRockefellerInstituterepresented

nwasunabletobe present;how

eofa groupofseveralcomputing-

articipatedinthemeeting,whileDr.von

yselfwerethemathematicians.The

resentationofcyberneticproblems

milarlythe computing-machinedesign

dsandobjectives.Attheend of

ecleartoall thattherewasasub

deasbetweentheworkersinthe

eineachgroupcouldalreadyuse

tterdevelopedbytheothers,and

bemadeto achieveacommon

orethis,thewarresearchgroup

Weaverhadpublishedadocument,

ted,coveringtheworkof Mr.Bige-

orsandwavefillers.It wasfound


24/194

aircraftfiredid notjustifythedesign

rvilinearprediction,buttheprinciples

ctical,andhavebeen usedbythe

gpurposes,andinseveralfieldsofrela

typeof integralequationtowhich

roblemreducesitselfhasbeenshown

roblems,andinmanyotherproblems

alinterest.Thusinonewayor ano

wtheideasof predictiontheoryandof

communicationengineeringalready

hestatisticiansandcommunication

tesandGreat Britain.Italsosawmy

owoutofprint,andaconsiderable

ersbyLevinson,Wallmann,Daniell,

nto fillthegap.I myselfhave

xpositorypaperunderwayforseve

avedoneon permanentrecord,but

etelyundermycontrolhavepreven

Finally,afterajoint meetingatthe

cietyandtheInstituteof Mathemati

orkin thespringof1947,and devo

sticprocessesfromapointofview

cs,IhavepassedonwhatI haveal

scripttoProfessorDoobof theUniver

opedinhisnotationandaccording

eMathematicalSurveysseriesof

calSociety.Ihadalreadydeveloped

eoflecturesinthe mathematics

hesummerof1945.Sincethen,my

or2,Dr.Y. W.Lee,hasreturnedfrom

seonthenewmethodslorthe design

pparatusin theM.I.T.electrical

nthefallof1947,and hasplansto

ecturesup intoabook.Atthesame

ernmentdocumentistobereprinted.

bluethreturnedtoMexicoabout

thespringof1945,I receivedaninvi-

hysics(M.I.T.), 1947.

hysics(M. I.T.),1932.


25/194

athematicalSocietytoparticipatein

adalajarathatJune.Thisinvitation

sionInstigadorayCoordinadorade

undertheleadershipofDr.Manuel

mIhavealreadyspoken.Dr.Rosen-

some scientificresearchwithhim,

eCardiologia,underits director

dedmeitshospitality.

Mexicoatthattime.Dr.Rosen-

ntinuealineofwork whichwehad

Walter13. Cannon,whowasalso

visitwhichunfortunatelyprovedto

odow iththerelationbetween,on

nic, andphasiccontractionsinepi

d, thetonicspasm,beat,andfibril

thatheartmusclerepresentedan

theinvestigationofconductionmecha

furthermore,thatthe anastomoses

artmusclefibres presenteduswith

antheproblemofthe nervoussynapse.

fulto Dr.Chavezforhisunquestion

hasneverbeenthepolicyof the

senbluethtotheinvestigationofthe

haveanopportunitytocontribute

directions: thestudyofphenomena

yinuniformconductingmediaof two

hestatisticalstudyof theconduc

etsof conductingfibres.Thefirst

atheoryofheart flutter,thelatter

standingof fibrillation.Rolhlines

a paper1,publishedbyus,and

earlierresultshaveshowntheneed

frevisionand ofsupplementation,

grevisedbyMr.OliverG. Selfridge

uteofTechnology,whilethestatis

studyofheart-musclenetshasbeen

h,A.,ConductionofImpulsesin CardineMuscle.

205:265


26/194

tofneuronalnetsby Mr.WallerPitts,

monGuggenheimFoundation.The

gcarriedonbyDr. Rosenbluethwith

mosoftheInstitutoNacionaldeCar-

ArmyMedicalSchool.

goftheMexicanMathematicalSo

dIpresentedsomeofour results.We

onclusionthatourearlier plansofcol

mselvestobepracticable.Wewerefor

ancetopresent ourresultstoa larger

946,Dr. McCullochhadmadearrange

yfoundationforthefirstof aseriesof

wYorkandtobe devotedtothepro

meetingshavebeenconductedinthe

kedoutmostefficientlybyDr.Frank

nizedthemonbehalfoftheFounda

gettogetheragroupof modestsize,

ntyinnumber,ofworkersinvarious

hem togetherfortwosuccessivedays

papers,discussions,andmeals

dthe opportunitytothreshouttheir

rogressinthinkingalongthe same

eetingshas beenthegroupthathad

1944,butDrs. McCullochandFre

eenthepsychologicalandsociological

,andhaveco-optedintothegroupa

ogists,sociologists,andanthropolo

gpsychologistshadindeedbeen

g.Hewhostudiesthe nervoussystem

dhewhostudies themindcannottor-

uchofthepsychologyofthe pasthas

morethan thephysiologyoftheorgans

holeweightof thebodyofideaswhich

intopsychologyconcernsthephy

ehighlyspecializedcorticalareascon

ofspecialsense.Fromthe beginning,

heproblemofthe perceptionofGes-

ormationofuniversals,wouldprove

sthemechanismbywhichwerecog-


27/194

irrespectiveofitsposition,its size,

sistusinsuchmatters,and toinform

htbemadeof ourconceptsfortheir

gussuchpsychologistsasProfessor

hicago,thelate Dr.KurtLewinof

teofTechnology,andDr.M.Ericsson

pology,itismanifestthat theim

ndcommunicationasmechanismsof

yondtheindividualintothecommunity.

pletelyimpossibletounderstandsocial

eofantswithouta thoroughinvestiga

mmunication,andwewerefortunate

Dr.Schierdainthismatter.Forthe

organization,wesoughthelpfrom

atesonandMargaretMead;while

tuteof AdvancedStudieswasour

eldof socialorganizationbelonging

eryimportantjointbookongames

theway,representsa mostinteres

ationfromthe pointofviewof

o,althoughdistinctfrom,thesubject

Lewiriandothersrepresentedthe

ofopinion samplingandtheprac

dDr.F.C. S.Norlhrupwasinte

osophicalsignificanceofourwork.

eacompletelistof ourgroup.

pto containmoreengineersandma

lowandSavage;moreneuroanatomists

chasvonBoninandLloyd,andso

n thespringof1946,waslargely

sbythoseofus whohadbeenpresent

ndtoa generalassessmentoftheim

present.Itwasthesenseof the

indcyberneticsweresufficiently

tothosepresentto warra'ntaconti

ntervalsof sixmonths;andthat

g,weshouldhavea smallmeeting

mathematicallytrainedtoexplainto


28/194

eas possiblethenatureofthe mathe

d.

eturnedtoMexicowiththesupport

tionandthehospitalityoftheInstituto

continuethecollaborationbetween

elf.Thistimewedecidedtotakea

romthetopicoffeed-back,andtosee

xperimentally.Wechcsethecatas

andthequadricepsextensorfemoris

cuttheattachmentofthemuscle,

owntension,andrecordeditscontrac

nically.Wealsousedanoscillograph

selectricalchangesinthemuscle

ithcats,firstdecerebratedunder

madespinalby athoracictransaction

strychninewasused toincreasethe

clewasloadedtothepoint wherea

odicpatternofcontraction,whichis

geofthe physiologist.Weobserved

,payingattentiontothephysiological

don themuscle,thefrequencyof

oftheoscillation,anditsamplitude.

s weshouldanalyzeamechanical

tingthesamepatternofhunting.We

hemethodsofMcColl'sbookonservo-

eplaceto discussthefullsignificance

e nowrepeatingandpreparingto

owever,thefollowingstatementsare

probable:that thefrequencyofclonic

sitivetochangesof theloadingcon

ted;and thatitis muchmorenearly

ntsoftheclosedarcefferent-nerve-

body-afferent-nerve-central-synapse-

hingelse.This circuitisnoteven

inearoperatorsifwetake asour

erof impulsestransmittedbythe

,butseemstobecomemuchmore

numberofimpulsesbyitslogarithm.

tthatthe formoftheenvelopeof


29/194

nerveisnot nearlysinusoidal,but

curveismuchmorenearlysinusoidal;

gsystemwithconstantenergylevel,

imulationmustbesinusoidalinall

eroprobability.Again,thenotionsof

remuchmorenearlymultiplicative

rexample,acompleteinhibition

zero,anda partialinhibitionmeans

quantity.It isthesenotionsof

whichhavebeenused1in thedis

urthermore,thesynapseisa coin

outgoingfibreis onlystimulatedif

pulsesina smallsummation-time

d.Ifthisthresholdis lowenoughin

mberofincomingsynapses,the

estomultiplyprobabilities,andthatit

telylinearlinkis onlypossibleina

pproximatelogarithmicityofthe

rtainlyalliedtotheapproximate

er-Fechnerlawofsensationintensity,

lyafirstapproximation.

thatonthislogarithmicbasis,and

econductionofsinglepulsesthrough

eneuromusculararc,wewereableto

tionstotheactualperiodsof clonic

quealreadydevelopedbytheservo-

ationofthefrequenciesofhunting

ystemswhichhavebrokendown.We

ationsofabout13.9persecond,in

oscillationsvariedbetweenfrequen

rallyremainedwithinarangevarying

nd17.Underthecircumstances,this

notthe onlyimportantphenomenon

ereis alsoarelativelyslowchangein

slowerchangein amplitude.These

bynomeanslinear.However,suffi-

ClonusfromtheInstituto Nacional


30/194

constantsofalinearoscillatingsys

rstapproximationasthoughthey

s thoughovereachpartofthe

avedasitwould ifitsparameters

atthetime.Thisis themethod

fphysics asthatofsecular perturba

udytheproblemsofbase-leveland

thisworkhas notyetbeencom

bothpossibleandpromising.There

thoughthetimingofthe mainarcin

o-neuronarc,theamplificationof

ableinoneand perhapsinmore

of thisamplificationmaybeaffected

cesseswhichrunmuch higherinthe

anthespinalchain primarilyrespon

us.Thisvariableamplificationmay

evelof centralactivity,bytheuse

tics,bydecerebration,andby many

tspresentedby Dr.Rosenblueth

etingheld intheautumnof1946,and in

kAcademyofSciencesheldat thesame

fusingthenotionsofcyberneticsover

werepleasedwithourresults,

eneralpracticability ofworkinthis

lessthatthetime ofourcollaboration

t ourworkhadbeendone undertoo

esirable topublishwithoutfurther

n.Thisconfirmationwhichnatu

futationwearenowseekingin the

47.

onhadalreadygivenDr.Rosenblueth

tofanewlaboratorybuildingat the

ologia.Wefeltthatthe timewas

ytothem thatis,to Dr.Warren

partmentofphysical sciences,and

arge ofthedepartmentofmedical

e basisofalong-timescientificcollabo

ourprogramata moreleisurelyand


31/194

reenthusiasticallybackedbyour

.GeorgeHarrison,DeanofSciences,

veoftheMassachusettsInstituteof

egotiations,whileDr.IgnacioChdvez

eInstituto NacionaldeCardiologfa.

becameclearthatthelaboratory

houldbe attheInstituto,bothin

onoflaboratoryequipmentandto

stthe RockefellerFoundationhas

tofscientificcentersinLatinAmerica.

asforfive years,duringwhichIshould

otheryear attheInstituto,while

endsixmonthsoftheinterveningyears

attheInstitutois tobedevotedtothe

ofexperimentaldatapertainingto

ermediateyearsaretobedevotedto

,andaboveall,totheverydifficult

oplewishingtogo intothisnew

whichwillsecurefor themboththe

physical,andengineeringbackground,

cewithbiological,psychological,

McCullochandMr.Pittsdida pieceof

erneticimportance.Dr.McCulloch

mof designinganapparatustoenable

edpagebyear.The productionof

ughtheagencyof aphotocellisan

tedbyany numberofmethods;the

e patternofthesoundsubstantially

of thelettersisgiven,whateverthe

ogueoftheproblemofthepercep

challowsus torecognizeasquare

enumberofchangesofsizeand

ch'sdeviceinvolvedaselective

tfora setofdifferentmagnifica

dingcanbeperfomedautomatically

sscanning,toallowa compari

givenstandardfigureoffixedbut

ewhichIhad alreadysuggestedatone


32/194

gramof theapparatusbywhich

onecameto theattentionofDr.von

sked,Isthis adiagramofthe

rtex ofthebrain? Actingonthis

,withtheassistanceofMr. Pitts,

the anatomyandthephysiologyof

histheorytheoperationofscanning

onsplaysanimportantpart.Thiswas

1947bothattheMacy meetingand

orkAcademyofSciences.Finally,

vesacertain periodictime,which

thetimeof sweepinordinary

usanatomiccluestothis timein

consecutivesynapsesnecessaryto

erformance.Theseyieldatimeof

condfora completeperformanceof

dthis istheapproximateperiodof

hm ofthebrain.Finally,thealpha

dence,hasalreadybeenconjectured

obeimportantin theprocessofform

eivedaninvitationto participate

nceinNancyon problemsarisingfrom

ted,andonmyvoyagethereand

weeksinEngland,chieflyas aguest

B.S.Haldane.Ihad anexcellent

se doingworkonultra-rapidcom

lyatManchesterandatthe National

ddinglon,andabovealltotalk over

yberneticswithMr.Turing atTedding-

hologicalLaboratoryatCambridge,

cetodiscussthework thatProfessor

weredoingonthe humanelementin

gsuchanelement.I foundtheinte

s greatandwellinformedin England

dtheengineeringworkexcellent,

the smallerfundsavailable.Ifound

andingofitspossibility inmany

aldane,H.LevyandBernalcertainly


33/194

mosturgentproblemsontheagenda

hilosophy.Ididnotfind, however,

beenmade inunifyingthesubjectand

ds ofresearchtogetheraswehad

es.

ancyonharmonicanalysiscon

uniting statisticalideasandideas

neeringina mannerwholelyinconfor

wofcybernetics.Here1mustmention

Blanc-LapierreandM.Loeve.I

nterestinthe subjectonthepartof

gists,andphysicalchemists,particu

rmodynamicaspectsinso farasthey

oblemofthenature oflifeitself.

tsubjectin Bostonbeforemydepar

Gyorgi,theHungarianbiochemist,and

dantwithmyown.

chvisitis particularlyworth

ague,ProfessorG.deSantillanaof

M.Freymann,ofthefirmof Her

stedof methepresentbook.1am

ehisinvitationasM Freymannisa

fthepresentbook,as wellasagood

gup toit,havebeendonein Mexico.

neof thedirectionsofworkwhich

acymeetingshassuggestedconcerns

onandthe techniqueofcommunica

iscertainlytruethatthe social

kethe individual;thatitisbound

mmunication;andthatit hasadyna

cessesofafeedbacknatureplay an

both inthegeneralfieldsof anthro

ndinthe morespecificfieldofeco

rtantwork,whichwehavealready

nnandMorgensternonthetheoryof

geof ideas.Onthisbasis,Drs.Gre

etMeadhaveurgedme,in viewofthe

ofthesociologicalandeconomicpro

confusion,todevotealargepart


34/194

ussionofthissideofcybernetics.

their senseoftheurgencyofthe

pe thattheyandothercompetent

msofthis sortwhichIshall dis

hisbook,1canshare neithertheir

hefirstclaimonmy attention,nor

cientprogresscanberegisteredin this

ciabletherapeuticeffectinthepresent

inwith,themainquantitiesaffecting

cal,butthe runsofstatisticsonwhich

velyshort.Thereis nogreatusein

eeconomicsofsteelindustrybefore

oftheBessemerprocess,norin com

berproductionbeforeandafter the

bileindustryandthecultivationof

sthereany importantpointinrun

ntsof venerealdiseaseinasingletable

odbeforeandthat aftertheintroduc

rthe specificpurposeofstudyingthe

Foragoodstatisticof society,we

ntiallyconstantconditions,justasfor

weneedalenswith alargeaperture.

lensis notappreciablyincreasedby

perture,unlessthelensis madeofa

sthatthedelayoflight indifferentparts

properdesignedamountbylessthana

h.Similarly,theadvantageoflongruns

aryingconditionsisspeciousandspu

ncesareverypoortesting-grounds

chnique:aspooras thestatistical

beto abeingofthe orderofsizeofa

ctuationswhichweignorefroma

epreciselythemattersofgreatest

bsenceofreasonablysaferoutine

eelementofthejudgmentoftheexpert

estobe madeofsociological,anthro

uantities,issogreatthatitis nofield

otyet hadthebulkofexperience

expert.I mayremarkparenthetically


35/194

softhetheory ofsmallsamples,once

nationofits ownspeciallydefined

amethodforpositivestatisticalinfe

otinspire mewithanyconfidence,

tisticianbywhomthemain elements

ationare eitherexplicitlyknownor

dinwhichmy expectationsof

temperedbyanunderstandingofthe

hwemayhope toobtain.There

1 ultimatelyhopetoaccomplish

heaidofcyberneticideas,butin

on furtherdevelopments.Oneof

sthesesforlostorparalyzedlimbs.

cussionofGestalt,theideasof com

avealreadybeenappliedbyMcCulloch

acementoflostsenses,inthe construc

nabletheblindtoread printbyhea

uggestedbyMcCullochlakesover

efunctionsnotonlyof theeyebutof

manifestpossibilityofdoing

seof artificiallimbs.Thelossofa

tonlythe lossofthepurelypassive

mentor itsvalueasmechanicalexten

elossofthecontractilepowerofits/

lltheloss ofallcutaneousandkinars-

nginit.Thefirst Iwolossesarewhat

owtriestoreplace.Thethirdhasso

Inthe caseofasimplepeg-legthis

hatreplacesthemissinglimbhas no

wn,and thekinaestheticmechanismof

tetoreportitsownpositionand velo

iththearticulatedlimbwithamo

naheadbythe patientwiththeaid

ure.Hehasnoadequatereportof

andthis interfereswithhissureness

ain.Theredoesnotseemto beany

uippingtheartificialjointsandthe

h strainorpressuregauges,which


36/194

r otherwise,saythroughvibrators,

epresentartificiallimbremoves

edby theamputation,butleavesthe

per receptors,muchofthisataxia

ndthe patientshouldbeableto

ewe alluseindriving acar,which

ut withamuchsurer gait.What

gshouldapplywithevenmore force

reofthemanikinfamiliarto all

ogyshowsthatthe sensorylossin

baloneis considerablygreaterthan

hip-jointamputation.

reporttheseconsiderationstothe

onow Ihavenotbeenable toaccom

whetherthesameideas havealready

ces,norwhethertheyhavebeentried

mpracticable.Incase theyhavenot

acticalconsideration,theyshould

tefuture.

herpointwhichlbelievetomerit at

eartomethatthe modernultra-rapid

principlean idealcentralnervous

automaticcontrol;andthatits inputand

formofnumbersordiagrams,butmight

thereadingsofartificial sense-organs

or thermometers,andtheperformance

hthe aidofstrain-gaugesorsimilar

manceofthesemotor organsandto

,tothecentralcontrol systemasanarti

earealreadyin apositiontoconstruct

stanydegreeofelaboratenessof

Nagasakiandthepublicawareness

doccurredtomethat wewerehere

social potentialityofunheard-ofim

vil. Theautomaticfactory,theas

nagents,areonlysofarahead ofus

nesstoput suchadegreeofeffort

as spent,forexample,inthedeve-


37/194


38/194

entsorlesshas nothingtosellthat

ytobuy.

haveasocietybasedonhuman

rselling.To arriveatthissociety

anningandagood dealofstruggle--

othebest,maybe ontheplaneof

hoknows ?Ithus feltitmydutyto

dunderstandingofthe positionto

nterest intheconditionsandthe future

aborunions.I didmanagetomake

ersonshighupin theC.I.().,and

ryintelligentandsympathetichearing.

ualsneitherInor anyofthemwasable

asithad beenmypreviousobser

thintheUnitedStales andinEngland,

thelabormovementareinthehands

el,thoroughlywell trainedinthe

op-stewardshipanddisputesconcer

ofwork; andtotallyunpreparedto

al, technical,sociological,and

concerntheveryexistenceof labor.

asyenoughtosee:the laborunion

omtheexactinglifeof aworkmaninto

ministratorwithoutanyopportunity

dforthosewhohave thistraining,

rallyinviting;nor,quite naturally,

such people.

tributedtothenewscienceof

moralpositionwhichis,to saythe

e.Wehavecontributedtothei nitia

h,asI havesaid,embracestech

reatpossibilitiesforgoodand for

verintotheworldthat existsabout

Belsenand Hiroshima.Wedonot

ppressingthese newtechnicaldeve

heage,andthe mostanyofuscan do

edevelopmentofthesubjectintothe

sibleandmost venalofourengineers.

e thatalargepublicunderstands


39/194

ofthepresentwork,andto confine

efields,such asphysiologyand

fromwarandexploitation.Aswe

whohopethat thegoodofa better

societywhichisofferedbythis new

eandoutweightheincidental

gtotheconcentrationofpower

ated,byitsveryconditionsofexistence,

nscrupulous).1writein1947,and I

isaveryslight hope.

esshis gratitudetoMr.WalterPitts,

eorgesDubeandMr.FredericWebster

nuscriptand preparingthematerial

ologia.


40/194

ONIANTIME

ngfamiliartoeveryGermanchild.

estehen

t?

ehen

hlet

fehlet

hl .

owestthouhow manystarsstand

?Knowestthouhow manyclouds

orld?TheLord Godhathcounted

wholegreatnumberbelacking.

stingthemeforthephilosopher

,in thatitputsside bysidetwo

e similarityofdealingwiththeheav

almosteveryotherrespectofferan

myistheoldestofthe sciences,

ngtheyoungesttobeginto deserve

arastronomicalphenomenacanbe

es,whilea precisepredictionof

nerallynot easyandinmanyplaces

heanswertothe firstquestionis

knowhowmanystarsthereare.In

minor uncertaintiesconcerningsome

stars,a starisa definiteobject,emi

gandcataloguing; andifahuman

rsas wecallthesecatalogues


41/194

ONIANTIME41

ntensethanacertainmagnitude,

antto usinthe ideaofadivineDurch-

ther.

weretoaskthe meteorologistto

slerungofthe clouds,hemightlaugh

atientlyexplainthatinall thelan

ereisnosuch thingasacloud,defined

permanentidentity;andthatif there

sthefacilitiesto countthem,norishe

ngthem.Atopologicallyinclinedme

defineacloudas aconnectedregion

ityofthepart ofthewatercontent

xceedsacertainamount,hut this

theslightestvaluetoanyone,and

extremelytransitorystate.What

rologistissomesuch statisticalstate

17,1950.Sky38%overcast.Cirro-

hofastronomywhichdealswith

meteorology:thestudyofgalaxies

ersandtheirstatistics, aspursuedfor

hta,butthisisaveryyoung branch

anmeteorologyitself,andissomething

sicalastronomy.Thistradition,

icatoiy,Durchmusterungaspects,

atherwiththe solarsystemthanwith

s.Itis theastronomyofthesolar

associatedwiththenamesof

o,andNewton,andwhichwasthe

cs.

plescience.Evenbeforetheexis

amicaltheory,evenasfarback asthe

dthateclipsesoccurred inregularpre

backwardsandforwardsovertime.It

f couldbetterbemeasuredbythe

rcoursesthaninany otherway.The

esolarsystemwasthe revolutionofa

s,whetherinthe formofthePtolemaic

Copernicantheoryoforbits,andinany


42/194

era fashionrepeatsIhepast.The

alindrome,andthebookofastro

kwardsasforwards.Thereis nodiffer

sand directionsbetweenthemotion

sand oneruninreverse.Finally,

byNewtontoa formalsetofpostu

cs,the fundamentallawsofthis

bythetransformationofthetime-

e.

otion-pictureofthe planets,speed

epictureofactivity,and weretorun

stillbeapossible pictureofpla

wtonianmechanics.Onthe other

motion-picturephotographofthe

athunderhead,andreverseit,it

ng.Weshouldseedown-dralts

,turbulencegrowingcoarserin tex

steadoffollowingthe changesof

eit, andsoonindefinitely.

weenthe astronomicalsmdthe

whichbringsaboutallthesedifferences,

rencebetweentheapparentreversibi

dthe apparentirreversibilityof

efirst place,themeteorological

ast numberofapproximatelyequal

rycloselycoupledto oneanother,

temofIhesolar universecontains

berofparticles,greatly diversein

eanother inasufficientlylooseway

ingeffects donotchangethegeneral

bserve,andtheveryhighordercoup

negligible.Theplanetsmoveunder

totheisolationofa certainlimited

anyphysicalexperimentwecanset

aredwiththe distancesbetween

nthesun,are verynearlypoints.

andplastic deformationstheysuffer,

nearly rigidbodies;or wherethey

sare atanyrateofa relativelyslight


43/194

ONIANTIME43

tivemotionoftheir centresiscon

theymoveisalmostperfectlyfree

din theirmutualattraction,their

yconsideredtolie atIheir centresand

ureofthe lawofgravityfromthe

minute.Thepositions,velocities,

f thesolarsystemareextremely

ndthecomputationoftheirfuture

oteasy indetail,iseasyand pre

herhand,inmeteorology,thenum

s soenormousthatanaccurate

onsandvelocitiesisutterlyimpos

reactuallymade,andtheir future

mputed,weshouldhavenothingbut

gureswliichwould needaradical

ouldbeof anyservicetous.The

mperature,turbulence,etc.,areall

singlephysicalsituation,buttoa

ationsofwhich onlyoneactualcase

gsofallthe meteorologicalstations

slylaken,theywouldnotgive abil

essarytocharacterizetheactualstate

ewtonianpointofview.Theywould

sconsistentwithan infinityofdiffe

most,togetherwithcertain apriori

vingasaprobabilitydistribution,a

ssibleatmospheres.UsingtheNew

systemofcausallawswhatever,all

futuretimeis aprobabilitydistribu

esystem,andeventhispredictability

oftime.

ystem,in whichtimeisperfectly

obabilityandpredictionleadto

betweenpastandfuture,becausethe

answersareasymmetrical.IfIset

bring thesystemIamconsidering

sentinsucha waythatIfix certain

sonablerightto assumethatcertain

wnstatisticaldistributions.1then


44/194

ributionofresultsaftera giventime.

hIcanreverse.Inorderto doso,

ckout afairdistributionofsystems

ononourpart,wouldendupwithin

ndfindoutwhatthe antecedent

meago.However,forasystem

positiontoendup inanytightly

sorare anoccurrencethatwemay

dwecannotbaseourexperimental

countingmiracles.Inshort,we are

ation tothefutureis differentfrom

lourquestionsareconditionedby

ranswersto thesequestionsare

micalquestionconcerningthe

inconnectionwithIhe timeof

areobservingremoteheavenlybodies

din whichthereseemstobeno

atureofourexperiment.Whythen

ermodynamicswhichisbasedon

bservationsstandusinsuch'good

answeris interesting,andnottoo

softhestarsarethroughthe agency

semergingfromtheobservedobject,

anperceiveincominglight,butcan

t;or atleasttheperceptionof

edbyan experimentassimpleand

ght.In theperceptionofincoming

yeora photographicplate.We

eptionofimagesbyputting themin

metimepast:wedark-condition

ges,andwewrapour platesinblack

Itis clearthatonlysuchaneyeand

e tous:ifwe weregiventopre-images

andifwehad toputourplates in

emand developthembeforeusing,

erydifficultartindeed.This being

stars radiatingtousandto thewbole

nystarswhoseevolutionis inthe


45/194

ONIANTIME45

attractradiationfrom thewhole

ractionfromuswill notbeperceptible

actthat wealreadyknowourown

usthepartofthe universewhich

t-futurerelations,asfarastheemission

oncordantwithour own.Thevery

ansthatits thermodynamicsislikeour

tingintellectualexperimenttomake

tbeing whosetimeshouldrunthe

suchabeing allcommunication

ble.Anysignalhemightsendwould

eamofconsequentsfromhispoint of

urs.Theseantecedentswouldalready

ould haveservedtousas the

ignal,withoutpresupposingan

ntit. Ifhedrew usasquare,we

is figureasitsprecursors,and it

ouscrystallizationalwaysperfectly

mains.Itsmeaningwouldseem to

eswereadinto mountainsandcliffs.

wouldappeartousas acatastrophe

plainablebynaturallaws bywhich

exist.Ourcounterpartwouldhave

rningus.Withinanyworld wthwhich

directionoftime isuniform.

etweenNewtonianastronomyand

esliein anintermediateposition,but

meteorologythantoastronomy.

veseen,containsacosmic meteoro

atextremelyinterestingfieldstudied

knownasthetheoryoftidal evolution.

treattherelativemovementsofthe

movementsofrigidbodies,butthis is

rth,forexample,isnearlysurrounded

erthe moonthanthecenterofthe

ctedto themoonthanthesolid part

rontheotherside islessstrongly

ighteffectpullsthe waterintotwo


46/194

nd oneoppositelothemoon.In

esehills couldfollowthemoon

reatdispersalofenergy,andconse

ostpreciselyunderthemoonand

ywouldconsequentlyhaveapull on

greatlyinfluencetheangularposition

ns.However,thetidalwavethey

tangledupanddelayedoncoastsand

eBehringSeaand theIrishSea.It

thepositionofthe moon,andtheforces

urbulent,dissipativeforcesof a

cesmet inmeteorology,andneeda

ed,oceanographymaybecalled

drosphereratherthanofthe atmos

themoonback initscourseabout

herotationof theearthforward.They

fthemonthandof thedayevercloser

edayof themoonisthemonth,

entsnearlythe samefacetotheearth.

thisisIheresult ofanancienttidal

containedsomeliquidorgas or

dgive undertheearth'sattraction,

patelargeamountsofenergy.This

utionisnotconfinedto theearthand

rvedto somedegreethroughoutall

espastithas seriouslymoditiedthe

oughin anythinglikehistorictimes,

mparedwith therigidbody

solarsystem,

tronomyinvolvesfrictionalproces

snotasingle sciencewhichconforms

tonianpattern.Thebiologicalsciences

reof one-wayphenomena.Birthis

ath,nor isanabolism,thebuildingup

eof catabolism,theirbreakingdown.

notfollowapatternsymmetricalin

thegerm-cellstoformthefertilized

arrow pointedthroughtimeinone


47/194

ONIANTIME47

ydirectedfrom thepastintothe

yindicatesadefinitelong-timetrend,

tedthoughitmightbe,fromthesimple

dleofthe lastcenturythistrendhad

entistswithanhonestlyopenmind,

heproblemofdiscoveringitsmecha

hroughthesamegreatstep bytwo

ametime :CharlesDarwinand

astherealizationthata merefor

vidualsofa speciesmightbecarved

essone-directionalorfew-directional

e varyingdegreesofviabililyofthe

romthepointofviewof theindivi

antdogwithoutlegswill certainly

rd thathasdevelopedthemechanism

yhaveabetterchanceforsurvivali*

edfromtheimpedingprojectionsof

whetherfish,lizard,ormammal,will

mshape,powerfulbodymuscles,and

chwillcatchthewater;and ifitis

e pursuitotswiftprey,its chances

ts assumingthisform.

samechanismbywhicha moreor

combinedintoa ratherdefinitepat

lholdstoday,thoughwehave a

fthemechanismonwhichitdepends.

enus afarmorepreciseanddis

tythanthatheldby Darwin,whilethe

etimeof deVrieson,hascom

tionofthestatisticalbasisof mutation.

atomyof thechromosome,andhave

e listofmoderngeneticistsislong

lofthese,suchasHaldane,havemade

delianismaneffectivetoolforthe

fthetidal evolutionofSirGeorge

son.Neithertheconnectionofthe

fthefather,nor thechoiceofthe


48/194

ortuitous.In tidalevolutionaswellas

haveamechanismbymeansof

ty,thatof therandommotionsofthe

fthemoleculesof thewater,iscon

cessintoa patternofdevelopment

on.Thetheoryoftidalevolu

stronomicalapplicationoftheelder

fDarwins,SirCharles,isone ofthe

ntummechanics.Thisfactmaybe

essrepresentsanevenfurtherinva

deas ofstatistics.Thesuccession

ann-Gibbsrepresentsaprogressive

micstostatisticalmechanics:lhatis,

enaconcerningheatand temperature

Newtonianmechanicsisappliedtoa

,notwitha singledynamicalsystem,

tionofdynamicalsystems;andin

ernnot allsuchsystemsbutan

them.Abouttheyear1900it became

methingseriouslywrongwith ther

whereitconcernedradiation.The

werto absorbradiationsofhighfre

helawofPlanck thananyexisting

ntheoryhadallowed.Planckgavea

iationthequantumtheorywhich

noughTorthesephenomena,butwhich

eremainderofphysics; andNils

asimilarlyadhoc theoryoftheatom.

ohrformedrespectivelythethesis

nantinomy.Thesynthesisis the

edbyHeisenbergin1925,in whichthe

micsof(iibbsisreplaced byastatis

thatofNewtonandGibbsfor large-

hichthe completecollectionofdata

stisnotsufficientto predictthefuture

thusnottoomuchto saythatnot

omy,buteventheNewtonianphy

oftheaverageresultsofa statistical


49/194

ONIANTIME49

countofanevolutionaryprocess.

onian,reversibletimetoa Gibbs-

adits philosophicalechoes.Berg-

encebetweenthereversibletimeof

ewhappens,andtheirreversibletime

whichthereis alwayssomething

heNewtonianphysicswasnotthe

asperhapsthecentralpoint inthe

vitalismandmechanism;althoughthis

esiretoconserveinsomeform or

of thesoulandof Godagainstthe

heend,as wehaveseen,thevita-

adofbuilding awallbetweenthe

physics,thewallhasbeenerectedto

ssthatboth matterandlifefindthem

atthe matterofthenewerphy

ewton,butitis sometingquiteas

morphisingdesiresofthevitalists.

m-theoreticianisnottheethicalfree

dTycheisasrelentlessa mistressas

reflectedinitstechnique.The

ayswereland-surveyors,astronomers,

e seventeenthandearlyeighteenth

ersandgrindersoflenses.Asin

enmadetheirtoolsinthe imageof

thingbuta pocketorrery,moving

stialspheres;and iffrictionandthe

arole init,theyare effectstobeover

motionof thehandsmaybeasperio

The chieftechnicalresultofthis

elofHuyghensandNewtonwasthe

forthe firsttimeitwas possibleto

respectableprecision,andtoconvert

toceansfromathing ofchanceand

erstoodbusiness.Itis theenginee

edthemanufacturer,andtothe

engine.FromtheNewcomenengine


50/194

the centralfieldofengineeringhas

vers.Heathas beenconvertedinfo

ndtranslation,andthe physicsof

entedbythatofRumford,Carnot,

csmakesitsappearance,ascience

yirreversible;andalthoughtheear

seemtorepresentaregion ofthought

hthe Newtoniandynamics,Ihetheory

gyandthe laterstatisticalexplana

r secondlawofthermodynamics

ationofenergythatprinciplewhich

iencyobtainablebyasteam-engine

mperaturesoftheboiler andthecon

used thermodynamicsandtheNewto

isticaland thenon-statisticalaspects

yeighteenthcenturiesare theage

hteenthandthenineteenthcenturies

m-engines,thepresenttimeis theage

trol.Thereisin electricalengineering

ermanyasthe splitbetweenthetech

d thetechniqueofweakcurrents,and

tinctionbetweenpowerandcommu

his splitwhichseparatestheage just

earenowliving.A ctually,commu

ealwithcurrents ofanysizewhatever,

nespowerfulenoughtoswing massive

shesit frompowerengineeringisthat

nomyofenergybuttheaccurate

issignal maybethetapof akey,

pofatelegraphreceiverat theother

ransmittedandreceived through

ne; oritmaybe theturnofa ship's

ularpositionof therudder.Thus

ngbeganwithGauss,Wheatstone,and

eivedits firstreasonablyscienti

f LordKelvin,afterthefailureof the

he middleofthelastcentury ;and

sperhapsHeavisidewhodidthe


51/194

ONIANTIME51

ernshape.Thediscoveryofradar

worldwar,togetherwiththeexigencies

fttire,havebroughtto thefielda

edmathematiciansandphysicists.

aticcomputingmachinebelongtothe

hwascertainlyneversoactivelypur

thepresentday.

esinceDaedalusorHero ofAlexan

icertoproduceaworkingsimulacrum

waysintriguedpeople.Thisdesire

tomatahasalwaysbeen expressed

niqueoftheage.Inthe daysofmagic,

ister conceptoftheGolem,that

eRabbi ofPraguebreathedinlife

neffableNameofGod. Inthe

atonbecomestheclockworkmusic

pirouettingstifflyon top.Inthe

utomatonisaglorifiedheatengine,

efuelinsteadoftheglycogenofthe

hepresentautomatonopensdoors

pointsgunsto theplaceatwhicha

plane,or computesthesolutionofa

magicalautomatonliesalongthe

of developmentofthemodern

mtohavehad muchofaninfluence

ught.Itis fardifferentwiththe

sideahasplayeda verygenuine

earlyhistoryofmodernphilosophy,

neto ignoreit.

onsiderstheloweranimalsasau

d questioningtheorthodoxChris

avenosouls tobesavedordamned.

matafunctionissomethingthat Des

everdiscusses.However,the

ofthemodeof couplingofthehuman

n will,withitsmaterialenviron

tesdoesdiscuss,althoughina very

eplacesthiscouplingintheone


52/194

nowntohim,thepinealgland.As

ngwhetherornot itrepresentsa

atterandofmatteronmind heis

ydoesregard itasa directaction

butesthevalidityof humanexperience

worldtothe goodnessandhonesty

inthismatteris unstable.Either

whichcaseitis hardtoseehow

allyexplainsanything,orHeisanactive

ishardtosee howtheguarantee

e anythingbutanactiveparticipation

usthecausalchainofmaterialpheno

usalchainstartingwiththe actof

s inustheexperiencescorresponding

n.Once thisisassumed,itis entirely

espondencebetweenourwill and

duceintheexternalworld toasimilar

thepathfollowed bytheOcca-

alebranche.InSpinoza,whoisin

rofthis school,thedoctrineofOcca

rereasonableformofassertingthat

eenmindandmatteristhat oftwo

fGod;butSpinozais notdynamically

o attentiontothemechanismof

whichLeibnizstarts,butL eibniz

sSpinozais geometricallyminded

ofcorrespondingelements,mind

mofcorrespondingelements: the

onceivedafterthepatternof the

stanceswhichdo notrisetothe degree

souls,andwhich formpartofthat

uldhaveattributed tomatter.Each

seduniverse,withaperfect causal

fromminusinfinityintimeto the

;butclosedthoughthey are,they

rthrough thepre-establishedharmony

hemtoclocks, whichhavesobeen


53/194

ONIANTIME53

togetherfromthecreationforall

madeclocks,theydonotdrift

sis duetothemiraculouslyperfect

or.

orldofautomatawhich,asis natu

ns,heconstructsafterthemodelof

onadsreflectoneanother,thisreflec

ransferofthecausalchainfrom one

llyasself-containedas,orrather

thepassivelydancingfigureson

avenoreal influenceontheout

ectivelyinfluencedbyit.As he

ws.Theapparentorganizationof

thingbetweenafigmentandamiracle.

solarsystemwrit small.

theautomatawhicharehumanly

hernaturalautomata,theanimalsand

estudiedfrom averydifferentaspect.

degradationofenergyaretheruling

vingorganismis aboveallaheat

rglycogenorstarch,fats,andproteins,

,andurea.Itis themetabolicbalance

tion;and ifthelowworkingtempe

ttractattentionasopposedtothe

esofaheatengineof similarefficiency,

ornerandgliblyexplainedbya contrast

ergyofthelivingorganismandthe

engine.All thefundamentalnotions

energy,andthechiefof theseisthatof

gofthebodyis abranchofpower

hisisthepredominatingpointof

yminded,conservativephysiologists;

ght ofsuchbiophysicistsasRashevsky

ssto itscontinuedpotency.

ealizethatthe bodyisveryfarfrom

dthatits-componentpartsworkinan

vailablepowerismuchlesslimited

e.Theelectronictubehasshownusthat

ourceof energy,almostallofwhich


54/194

ffectiveagencyforperformingdesired

sworked atalowenergylevel. We

suchimportantelementsastheneurons,

omplexof ourbody,dotheirwork

itionsas vacuumtubes,withtheir

pliedfromoutsidebythe circulation;

whichis mostessentialtodescribe

energy.Inshort,the newerstudy

emetalor intheflesh,is abranchof

ng,anditscardinalnotionsarethose

turbanceornoise aterm taken

ngineerquantityofinformation,

n.

withautomataeffectivelycoupled

merelybytheirenergyflow,their

ow ofimpressions,ofincoming

nsofoutgoingmessages.Theorgans

eceivedare theequivalentsofthe

organs.Theycomprisephoto-electric

r light;radarsystems,receivingtheir

hydrogen-ion-potentialrecorders,

e;thermometers,pressure-gaugesof

sandsoon.Theeffectorsmaybe

dsorheating-coilsor otherinstru

s.Betweenthereceptoror sense-organ

ntermediateset ofelements,whose

ei ncomingimpressionsintosuch

redtypeofresponseinthe effectors.

scentral controlsystemwillvery

concerningthefunctioningofthe

secorrespondamongotherthingsto

dotherproprioceptorsofthe human

ganswhich recordthepositonofa

tionofamuscle,etc. Moreover,the

eautomatonneednotbeusedat

or storedsoastobecomeavailableat

e analogueofmemory.Finally,

isrunning,itsveryrulesof operation

angeon thebasisofthedata which


55/194

ONIANTIME55

ceptorsin thepast,andthisis not

ng.

arenowspeakingarenotthe

t,northehopeof somefuturetime.

mostats,automaticgyro-compassship-

pelledmissilesespeciallysuchasseek

ftfire-controlsystems,automatically

s,ultra-rapidcomputingmachines,

unto beusedlongbeforethe war

am-enginegovernorbelongsamong

echanizationofthesecondworldwar

n,and theneedofhandlingthe

ergyottheatomwillprobablybring

tofdevelopment.Scarcelyamonth

pearsontheseso-calledcontrol

chanisms,andthepresentageisas

mechanismsasthenineteenthcenturywas

neortheeighteenthcenturytheage of

omataofthepresentageare coupled

forthereceptionofimpressionsand

ons.Theycontainsense-organs,

entofanervoussystemto integrate

fromtheone totheother.Theylend

scriptionin physiologicalterms.Itis

y canbesubsumedunderonetheory

ysiology.

anismstotime demandscareful

hattherelation input-outputisa

ndinvolvesa deQnitepast-future

so clearisthatthe theoryofthe

tisticalone.We arescarcelyever

nceofa communication-engineering

To functionadequatelyitmustgive

eforawholeclass ofinputs,andthis

actoryperformancefortheclassof

yexpectedtoreceive.Thusits

sianstatisticalmechanicsrather than

mechanics.Weshall studythisin


56/194

hapterdevotedtothetheoryofcom

onexistsinthesame sortofBergso-

anism;andhencethereisno reason

swhytheessentialmodeoffunctioning

ldnot bethesameasthat ofthe

alismhaswon totheextentthateven

othetime-structureofvitalism;but

yisa completedefeat,forfromevery

eslightestrelationto moralityor

csisfully asmechanisticastheold.

newpointofviewmaterialisticis

:theascendancyofmattercharac

th-centuryphysicsfarmorethan the

alismhas cometobebut littlemore

mechanism.Infact,thewhole

versyhasbeenrelegatedtothelimbo


57/194


58/194

termineinacompletelyNewtonian

momentaandpositionsforany

possibletomakestatementsabout

meofthesewill havethecharacterof

ystemwill havecertaincharacteristics

tainother characteristicswithpro

arenotionswhich includecomplete

possibility,butincludemuchmore

twith abulletofthe dimensions

tIhit anyspecificpointonthe target

ughit isnotimpossiblethatI hitit;

ccase Imustactuallyhitsome

eventofprobabilityzero.Thusan

atofmy hittingsomepoint,may

blageofinstancesofprobability

rocesseswhichis usedinthetech

ticalmechanics,althoughit isused

hereclearly awareofit,is thereso

encyintoan infinitesequenceof

safirst,a second,athird,andso

knownprobability;andthe expression

rgercontingencyasthesumof the

pecialcontingencies,whichform an

cannotsumprobabilitiesin all

probabilityof thetotaleventfor

zerosiszero whilewecansum

cond,a thirdmember,andsoon,

tingenciesinwhich everytermhasa

positiveinteger.

esetwocases involvesrathersubtle

gthenatureofsetsof instances,and

werfulmathematician,wasnevera

bleforaclassto beinfinite,andyet

tiplicityfromanotherinfiniteclass,

eintegers?Thisproblemwassolved

tcenturybyGeorgCantor,andthe

consider allthedistinctdecimal


59/194

LMECHANICS59

on-terminating,lyingbetweenzero

heycannotbearrangedin one,two,

trangelyenough,alltheterminating

arranged.Thusthedistinction

atisticalmechanicsis notontheface

serviceofLebesguetothe Gibbs

mplicitrequirementsofstatistical

tingenciesofprobabilityzeroand

itiesof contingenciescanactually

siantheorydoesnotinvolvecon

r,wasnotdirectlybasedon the

nics,butonwhat lookslikeavery

yoftrigonometricseries.Thisgoes

nturyphysicsofwavesandvibrations,

stionofthegeneralityofthesets

emwhichcanbesynthesisedoutof

system outofthosevibrations,

thepassingoftimesimplymultiplies

mfromequilibriumby aquantity,

ndentonthetimealone,andnot on

tionis expressedasthesumofa

coefficientsareexpressedasaverages

tiontoberepresented,multipliedby

.Thewholetheorydependson the

fa series,intermsofthe averageofan

ttheaverageofa quantitywhich

mzerotoA,and zerofromAto 1,

astheprobabilitythatthe random

rvalfrom0to A,ifit isknowntolie

words,thetheoryneededforthe

closetothetheorjr neededforanade

bilitiescompoundedfromaninfinite

he reasonwhyLebesgue,insolving

solved thatofGibbs.

sdiscussedbyGibbshavethemselves

n.Ifweconsidera certainvery

edynamicalsystem,withNdegrees

positionandvelocitycoordinates


60/194

alsetof2 Ncoordinates,Nofwhich

positioncoordinatesandN thegene

eterminea2N-dimensionalspace

volume;andifwe takeanyregion

ointsflowwiththe courseoftime,

2N coordinatesintoanewset

time,thecontinualchangeof the

snotc hangeits2N-dimensional

snotso simplydefinedastheseregions,

ratesasystemof measureofthetype

of measure,andintheconservative

aretransformedinsucha wayasto

nt,thereisone othernumerically

mainsconstant: theenergy.Ifall

ctonlyononeanother,andthere are

positionsand fixedorientationsin

expressionswhichalsoremain

vectors :themomentum,andthe

thesystemasawhole. Theyarenot

tthe systemisreplacedbya system

om.

ms,theremaybeother quantities,

rgy,themomentum,andthemomentof

changedasthesystemdevelops.

systemsinwhichanotherinvariant

tontheinitialcoordinatesandmomenta

dregularenoughto besubjecttothe

donLebesguemeasure,areveryrare

ensel.Insystemswithout otherinva

thecoordinatescorrespondingto

totalmomentofmomentum,andinthe

rdinates,themeasuredeterminedby

mcoordinateswillitselfdeterminea

smeasurein spacewilldetermine

surfaceoutof afamilyoftwo-dimen

pleifourfamilyis thatofconcentric

betweentwoconcentricspheresclose

Measure-PreservingHomeomorphismsandMetri

Math.(2),42,874-920(1941).


61/194

LMECHANICS61

dbytakingasonethe totalvolumeof

spheres,willgiveinthe limitamea

eofasphere.

easureona regioninphasespacefor

ntum,andtotalmomentofmomentum

supposethat therearenoother

ntitiesinthesystem.Letthetotal

egionbe constant,oraswecan make

e.Asour measurehasbeenobtained

ntime,in awayinvariantintime,

eshallcall thismeasurephase

kenwithrespecttoit phaseaverages.

ryingintimemayalsohavea time

"(/)dependson/,itstime averagefor

future:

nics,bothtime averagesandspace

rilliantideaofGibbsto trytoshow

eragewere,insomesense,thesame.

otypesofaveragewererelated,Gibbs

nthemethodbywhich hetriedto

utterlyandhopelesslywrong.For

me.Evenatthetime ofhisdeath,

ntegralhad justbeguntopenetrateto

enyears,itwas amuseumcuriosity,

ngmathematicianstheneedsand

hematicianasdistinguishedasW.F.

ingto dowithittill hisdyingday.

whena groupofmathematicians

Birkhoff2,finallyestablishedthe

sgood'searlyworkrepresentedanimportantstep

esgueIntegral.

orie.Erg.d. Math.(1936).


62/194

Gibbsstatisticalmechanics.Weshall

ationswere,in thestudyofergodic

ina systemfromwhichallthe

vedasextracoordinates,almostall

pacepassedthroughallcoordinates

thesishecalledtheergodic hypothesis,

v,work,and 6'coe,path.Now,

herel,andothershaveshown,there

rethathypothesisistrue.No diffe-

area intheplane,evenif itisof

rsofGibbs,includingat theend

wthisin avagueway,andreplaced

si-ergodichypothesis,whichmerely

oftime,asystemgenerallypassesinde

n theregionofphase-spacedeter

ants.Thereis nologicaldifficulty

merelyquiteinadequatefor the

seson it.Itsaysnothingaboutthe

temspendsin theneighborhoodof

ageandof measuretheaverage

ononeoverasetto bemeasuredand

were mosturgentlyneededtomake

n ordertoappreciatetherealsignifi

eneed amorepreciseanalysisofthe

as thenotionoftransformationgroup.

lyfamiliartoGibbs, ashisstudyof

vertheless,itispossibleto maintain

mattheirfull philosophicalvalue.

viside,Gibbsisoneof thescientists

ticalacumenoftenoutstripshislogic,

whileheisoften unabletoexplain

ence,it isnecessarythatthere

notstandisolated.In aworldruled

performedbyanirrationalGod

we shouldbeforcedtoawait each

ofperplexedpassiveness.Wehave


63/194

LMECHANICS63

thecroquetgameinAlice inWon-

sareflamingos;theballs,hedgehogs,

abouttheirownbusiness;thehoops,

wisesubjecttolocomotorinitiative

sarethedecreesof thetesty,unpre

erule foragameora usefullawof

bleinadvance,andthatit applyto

y,it shouldrepresentapropertyof

hremainsthesame underthefluxof

nthesimplestcase,it isaproperty

oftransformationstowhich thesystem

tothenotionsof transformation,

nvariant.

mis somealterationinwhicheach

.Themodificationofthesolar system

tionbetweentime/Aandtimetz isa

ofcoordinatesofthe planets.The

dinateswhen wemovetheirorigin,

esto arotation,isatransformation.

occurswhenweexamineapreparation

onofa microscopeislikewiseatrans

ansformationAbyatransforma

mation,knownastheproductorresul

ralitdependsonthe orderofA

sformationwhichtakesthecoordi

.andyintoo;,whilezisunchanged;

o x,andyis unchanged;

, y intozandzintox ;whileAB

x , andzintoy . IfABandBA

ythatAand Barepermutable.

s,thetransformationAwill not

fthesysteminto anelement,butwill

ryelementisthe resultoftransfor

ase,thereisaunique transformation

ndA'*Aare thatveryspecialtransfor

eidentitytransformation,whichtrans

tself.Inthis casewecall A"1the


64/194

tAisthe inverseo fA" 1, tha tIis its

verse ofABis B^A1.

transformationswhereeverytrans

sethasan inverse,likewisebelonging

esultantofany twotransformations

belongstotheset.Thesesets are

groups.Thesetofalltranslations

orin athree-dimensionalspace,is

ndevenmore,itis atransformation

nownasAbelian,whereanytwotrans

epermutable.Thesetof rotations

otionsofarigid bodyinspace,are

vesomequantityattachedtoallthe

atransformationgroup.Ifthisquan

h elementischangedbythesame

p,whateverthattransformationmay

ofthegroup.There aremanysorts

fwhich twoareespeciallyimpor

inearinvariants.Lettheelements

groupbethe termswhichwerepre

complex-valuedfunctionofthese

ropriatepropertiesofcontinuityor

ndsforthe elementresultingfrom

nT,andif f(x)isafunctionof abso

ofabsolutevalue1dependingonlyon

characterofthegroup.It isan

slightlygeneralizedsense.Iff(x)

ers,clearlyf(x) g(x)isonealso,as

esentanyfunctionh(x)definedover

binationofthecharactersofthe group,


65/194

LMECHANICS65

rofthegroup,and a/r(T)bearsthesame

oesto f(x)in(2.03),then

x)intermsofa setofgroupcharac

for allTin termsofthecharacters.

acters ofagroupgenerateother

ationandinversion,anditmaysimi

stant1isa character.Multiplication

generatesatransformationgroupof

selves,whichisknownas thecha

group.

translationgroupontheinfinite

Tchangesxintox +T,(2.03)be

r),

''*,a(T)=eiVr.Thecharacterswillbe

charactergroup willbethe groupof

ox+ T,thushavingthesamestructure

willnotbethe casewhentheoriginal

onsabout acircle.Inthis case,the

anumberbetween0and2*, differing

multipleof2*, andwhile(2.06)will

raconditionthat

before,weshallobtain:

bearealinteger,positive,negative,or

huscorrespondsto thetranslations

heotherhandthe originalgroupis

theintegers,xandT in(2.05)arecon

de'Xx involvesonlythenumberbet

rsfromxby anintegralmultipleof2*.

essentiallythegroupofrotations

a givencharacter/",thevaluesof


66/194

h awaythatthedistribution isnot

multipliedby


67/194

MECHANICS67

ofitsmetricalinvariants.Theseare

measurewhichdo notundergoany

ansformedbythe grouparepermu

egroup.Inthis connection,weshould

ofgroup measure,duetoHaar1.

oupitselfisa collectionofobjects

ngmultipliedby theoperationsof

mayhavean invariantmeasure.

rtainratherwideclassof groupsdoes

inedinvariantmeasure,definablein

hegroupitself.

ationofthetheoryof themetrical

ansformationsistoshowthejustification

ofphaseaveragesandtimeaverages

yseen,Gibbstriedin vaintoestablish.

sbeenaccomplishedisknown asthe

emsstartwithan ensembleE,

measure1, transformedintoitself

ransformationT,orbyagroup of

sformationsT*,whereoo


68/194

)rfx,

ergesinthemeantoa limitf*(x)as

spectively,in thesensethat:

0.

ereergodictheoremofBirkhoff,

;whichmeansthat

x)aredeflnedasin(2.16)and(2.17).

at,exceptfor asetofvaluesof x

theso-calledergodicormetrically

transformationT,orthesetof

sinvariantnosetofpointsx which

or0.In suchacase,the setofvalues

m)forwhich f*(x)takesona certain

waj's either1or0. Thisisimpossible

sconstant. Thevaluewhichf*(x)

ys,is

eorem,wehave


69/194

MECHANICS69

m,wehave

ofa:ofzero measureorprobability0.

continuouscase.Thisisan adequate

changeofphaseaveragesand time

TorthetransformationgroupT> is

hasshownunder verygeneralcondi

ucedtoergodiccomponents.Thatis,

ofa;ofzero measure,Ecanbeseparated

blesetofclassesEnand acontinuum

measureisestablishedon eachEand

derTorT>. Thesetransformationsare

eintersectionof SwithE(y)and Snwith

asure(S(y))dy +Vmeasure(Sn).

heoryofmeasure-preservingtransfor

the theoryofergodictransformations.

y,let usremarkinpassing, may

nsformationsmoregeneralthan

ranslationgroupon theline.In

dtothe translationgroupinvjdimen

mensionsisphysicallyimportant.

mporalequilibriumisspatial homoge

thatofthehomogeneousgas,liquid,or

tionof three-dimensionalergodic

ergodicgroupoftranslationtransfor

nsappearsasthe setoftranslationsofa

uch thatoneoranotherexistsat agiven

h.


70/194

sofstatisticalmechanics,which

onintheclassicalthermodynamics,is

rilyapropertyofregionsin phase-

ogarithmoftheirprobabilitymeasure.

erthedynamicsofn particlesina

ts,Aand B.Ifmparticlesare inA,

racterizedaregioninphase-space,

robabilitymeasure.Thelogarithm

bution:mparticlesinA, n-minB.

tof itstimeina statenearthatof

nsethatfor mostofthetime,nearlyint

yn-n^inB,wheretheprobabilityof

-min Bisa maximum.Forsystems

rticles,andstateswithinthelimitsof

hismeansthatifwetakeastate of

opy,andobservewhathappenstoit,

sincreases.

amicproblemsoftheheatengine,

onsinwhichwe havearoughthermal

slikean enginecylinder.Thestates

opyare statesinvolvingmaximum

atureandvolume,fora smallnumber

umesandatthegiventemperature

efineddiscussionsofthermalengines,

inesliketheturbine,in whichagasis

licatedmannerthanina cylinder,do

nstooradically.Wemaystilltalk of

veryfairapproximationeventhough

ydeterminedexceptinastateof equi

volvingthisequilibrium.However,in

hofeventhis roughhomogeneity.The

eas shownbytheelectronmicroscope

essandfinenessoftexture,andits

correspondingfinenessoftexture.

thanthatofthespace-and-timescale

er,andsothetemperaturesreadby

ivingtissuesaregrossaverages,and

ofthermodynamics.Gibbsianstatis

beafairlyadequatemodelof what


71/194

ALMECHANICS71

picturesuggestedbytheordinary

t.Thethermalefficiencyofmuscle

ng,andcertainlydoesnotmean

tisticalmechanicsisthat ofthe

pposeagasin whichtheparticles

edistributionof velocitiesinstatisti

emperature.Fora perfectgas,thisis

etthisgasbe containedinarigid

sit,containinganopeningspanned

y agatekeeper,eitherananthropo

emechanism.Whenaparticleof

tyapproachesthegatefromcompart

sthanaveragevelocityapproachesthe

,thegatekeeperopensthegate,and

h;butwhena particleoflessthanave

fromcompartmentAoraparticleof

cityapproachesfromcompartmentB,

way,theconcentrationofparticles

din compartmentBandisdecreased

oducesanapparentdecreasein

compartmentsarenowconnectedby

obtainaperpetual-motionmachine

estionposedbythe Maxwell

othingis easierthantodenythe

rstructures.Weshall actuallyfind

estrictestsensecannotexisti na

fwe acceptthisfromthebeginning,

rateit,weshallmissan admirable

thingaboutentropyandabout

al,andbiologicalsystems.

t,it mustreceiveinformationfrom

ncerningtheirvelocityandpointof

ertheseimpulsesinvolvea transfer

involveacouplingofthedemon

oftheincreaseofentropyapplies

stem,butdoesnotapply toanon-

tem.Accordingly,theonlyentropy


72/194

ofthesystemgas-demon,andnot

gasentropyismerelyoneterm

argersystem.Canwefind terms

lwhich contributetothistotal

demoncan onlyactoninforma

rmation,aswe shallseeinthe next

ativeentropy.Theinformationmust

lprocess,say someformofradiation.

sinformationiscarried ataverylow

ransferof energybetweenparticle

erabletimefarlesssignificantthan

.However,underthequantum

toobtain anyinformationgivingthe

ofaparticle, muchlessthetwo

eeffectonthe energyoftheparticle

nimumdependentonthefrequencyof

ation.Thusallcouplingisstrictlya

;andasysteminstatistical equilibrium

attersconcerningentropyandthose

ongrun, theMaxwelldemonisitself

ncorrespondingto thetemperatureof

bnizsaysof someofhismonads,it

small impressions,untilitfallsinto

ndisincapableofclear perceptions.

Maxwelldemon.

beaquiteappreciableintervaloftime

ditioned,andthistime maybeso

akofthe activephaseofthedemon

reasonto supposethatmetastable

st;indeed,itmaywellbe thatenzymes

emons,decreasingentropy,perhaps

eenfastand slowparticles,butby

cess.Wemaywellregardliving

mself,inthis light.Certainlythe

nismarealikemetastable:thestable

edeconditioned,andthestablestate

edead.Allcatalysts areultimately

esof reaction,butnottrueequili-


73/194

ALMECHANICS73

lystsandManalikehavesufficiently

bilitytodeservetherecognitionofthese

nentconditions.

hapterwithoutindicatingthat

rablywidersubjectthanwehave

ecertainmoderndevelopmentsof

emeasuretobe keptinvariantunder

efineddirectlyby thesetitself,

vance.Ireferespeciallyto thework

f,andtosomeofthe workofHure-

ool.

dtothe statisticalmechanicsof

field inwhichtheconditionsare

thestatisticalmechanicsofheatengines

suitedtoserveasa modelofwhat

ism.


74/194

ON,

enomenainwhichwhat isobserved

asequenceofnumericalquantities,

mperatureasrecordedby aconti

eter;ortheclosing quotationsofa

akenday byday;orthecompleteset of

shedfromdaytodaybythe Weather

s,continuousordiscrete,simpleor

sarerelativelyslowlychanging,and

entemployinghandcomputationor

uchasslide-rulesand computing

ongstothemoreconventionalparts

edis thattherapidlychanging

telephoneline oratelevisioncircuit

tusbelongjustastruly tothefieldof

lthoughtheapparatusby meansof

ndmodifiedmustin generalbevery

ctmustbe abletoputout resultspan

alterationsofinput.Thesepiecesof

eceivers,wavefilters,automaticsound

oderof theBellTelephoneLabora

tingnetworksandtheircorresponding

sencequick-actingarithmeticaldevices,

eapparatusofcomputingmachines

ffof computers,ofthestatistical

neededintheirusehas beenbuilt

s ithasintothe automaticrange-finders

-aircraftfirecontrol system;andfor


75/194

N,ANDCOMMUNICATION75

nof operationhastoworktoo

nlinks.

dtheapparatustodeal withthem

aboratoryorin thetelephonecircuit,

rding,preservation,transmission,and

this information,andhowisit

plest,mustunitary formsofinforma

oicebetweentwo equallyprobable

theother ofwhichisbound to

example,betweenheadsandtailsin

hallcall asinglechoiceof thissort

or theamountofinformationinthe

mentofaquantityknownto liebetween

iforma prioriprobabilitylieany

a l lsee tha t i fweputA=0andB=1,

inthe binaryscalebytheinfinite

... .whereal, a2,....,eachhas the

berofchoicesmadeand the

ormationisinfinite.Here

+a2++an+...

ntwhichweactuallymakeis per

on.If themeasurementhasauni

ngovera rangeoflengthblb2... bn....

ot equalto0,we shallseethatall the

andpossiblytoa*,are significant,

sarenot.Thenumberof decisions

rom:

"

ntityastheprecise formulaforthe

ditsdefinition.

hefollowingway:weknow opriori

n0 and1,anda posteriorithatitlies

e(0,1).Thenthe amountofinforma

sterioriknowledgeis


76/194

era casewhereouraprioriknowledge

acertainquantityshouldlie between

the aposterioriprobabilityisf2(x)dx.

ndoesoura posterioriprobability

thatof attachingawidthtothe

=fl(x)andy=/2(*).It willbenoted

gthevariabletohaveafundamental

resultswillnotin generalbethe

,oranyother functionofx.Since

,we shallhave:

hmofthe breadthoftheregionunder

somesort ofaverageoftheheightof

calof fl(x).Thusareasonablemea

ormationassociatedwiththecurve

)dx.

eas amountofinformationisthe

uallydefinedasentropyin similar

eregivenisnot theonegivenby

problems,althoughitisa statistical

to replaceFisher'sdefinitioninthe

stantover(a, b),andiszero

-^-^log, ^-=log,^^

informationthatapointisin the

he measureofthedifference,

=log

se ofapersonalcommunicationofJ.von

Documents

Wiener Cynernetique