WHITROW - On the Impossibility of an Infinite Past

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Brit.

J. Phil. Sd.

29

1978),

39 45

Printed m Great Britain

39

O n the

Impossibility

of an

Infinite Past

by

G. J.

W ITROW

" T h e i d e a of inf ini ty proves to be an aux i l ia ry in t roduc ed by t h o u g h t to

facilitate

its

ope ra t ions

. . . its

app l i ca t ion

to the real world is a

m i s u s e . "

( Ha n s Va i h i n g e r [ 1 9 2 4 ] ,

p. 62.)

1

History of the Problem.

2 Disproof of the Possibility of an Infinite

Sequence

of

Discrete

Past

Events.

3 Unrelated Sequences of Past Events.

4 Transformations of Time-Scale.

5 Conclusion.

The problem of the age of the universe and whether, in principle, it must

be finite or could be infinite has a long history and views have been divided.

It is my contention, contrary to the opinion now widely held, that the basic

philosophical problem has not been resolved by Cantor s theory of infinite

sets.

I

HISTORY OF THE PROBLEM

My point of view is a modern version of one that can be traced back at

least as far as the Christian philosopher Joannes Philoponos of Alexandria,

in the first half of the sixth century A . D . In a treatise criticising the neo-

Platonist Proclos, believed to have been written not long after 529, Philo-

ponos argued that if the world were eternal it would follow that an infinity

of successive acts could have been enumerated, but this is impossible.

1

This philosophical objection to the possibility of an infinite past, i.e. an

infinite chain of past events, has not, however, been accepted by all Chris-

tian believers in the creation of the world. A contrary opinion was expressed

by St Thomas Aquinas (1225-1274), who, influenced by Aristotle, main-

tained, in his Sumrna

Theologica

(ia, xlvi, 2) That the world has not always

existed is to be held by faith alone, and cannot be demonstratively proved.

In his opusculum De Aeternitate Mundi contra Murmwantes, in which he

1

See Philoponos, J.: Contra Proclum de Aeternitate Mundi, ed.) H. Rabe. Leipzig:

Teubner, 1899 pp. 8 et seq

Received June

1977

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40 G. Whitrow

crit ic ised those whom he des ignated as "m urm urers agains t A ris tot le" ,

he argued at some length that i t was logically possible for the universe to

have been created by God out of nothing and yet to have existed from all

eternity as A ristot le believed. T h e m edieval followers of St A ug ustine,

himself a neo-Platonist , rejected this view. Prominent among those who

argued agains t i t was St Thomas 's contemporary St Bonaventure

( 1 2 2 1 -

74). Of the various arguments that he formulated agains t the idea of the

eternity of the world

1

the most compell ing is that the infini te cannot be

bridged: if the universe had no beginning an infini te number of celest ial

revolutions must have taken place, and therefore the present day could

not have been reached (Gilson [1938], p . 192). Ho wever, he was not aware

of all the subtleties of infinite sets that were revealed last century by

Bolzano, Cantor and others , and he was definitely m istaken when he argu ed

that 'If no anterior day precedes the present by an infini te durat ion, then

all the anterior days precede i t by a fini te durat ion and therefore the dura-

t ion of the world had a beginning. ' {ibid.

The possibility of an infinite past was also rejected by the redoubtable

Richard Bentley (1662-1742) in the sixth of his Boyle Sermons in 1692.

He argued that the world cannot be eternal and there cannot have been

an infini te number of past revolutions of a planet around the Sun. 'For,

consider the present revolution of the Earth . . . God Almighty, i f he so

pleaseth, may continue this motion to perpetuity in infini te revolutions

to come; because futuri ty is inexhaustible, and can never be spent or run

out by past and present m om ents. But the n, if we look backwards from

this present revolution, we may apprehend the impossibil i ty of infini te

revolutions on that s ide; because al l are a l ready past , a nd so were once

actually

present,

and consequent ly are f inite . . . . For surely we cannot

conceive a preteriteness (if I may say so) still backwards in infinitum, tha t

never was present , as we can endless futuri ty that never wil l be present .

So that one is potentially infinite, yet nevertheless the other is actually

finite.' (Bentley [1838], p. 134.)

T h e same p oint of view was taken by Im m anuel Kan t when, in discuss ing

the firs t of his four antimonies of pure reason, he sought to establish the

impo ssibil ity of the hyp othesis that up to a given m om ent such as the present

there has passed away an infinite sequence of successive states of the

1

In par t icular , in h is commentary on the Sentences of Peter Lombard, St Bonaventure

{Opera Ornna, Quarrachi, 1882, II, 21) pointed out that, if the world were infinitely old,

there would have occurred an inf in i te number of annual revolut ions of the Sun around

the ecliptic, but since in each such period there are (roughly) twelve revolutions of the

M o o n , i.e. lunar m on ths, one infinity would th us be twelve t imes another—a conclusion

that he rejected. Despite his rejection of this conclusion, his argument anticipates the

m od ern idea tha t an infinite set, unlike a finite set, can be pu t in one—one corre lation

with a sub-set of itself.

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On the Impossibility o f an Infinite P ast 41

universe. 'Now', argued Kant, ' the infini ty of a series consists in the fact

that i t can never be com pleted by successive synthesis . It th us follows th at

it is impossible for an infinite world-series to have passed away, and that

a beginning of the world is therefore a necessary condit ion of the world's

exis tence. ' (Kant : Critique of Pure

Reason,

Kem p-Sm ith t rans lat ion, p .

217).

This argument as s tated by Kant i s imprecise by modern s tandards .

Kant did not clearly dist inguish between the question of whether there

was, or was not , a firs t event in the world's history and the question of

whether th e total dura t ion of past t im e is finite or infinite. Kant's discussion

of the hypothesis that the world had no beginning in t ime is formulated

as an argu m ent against the idea tha t the pa st sequence of successive 's tates

of things ' is , in the language of modern mathematics, an open set with no

first member. Whether or not an infini te measure is to be associated with

i t depends on the choice of temporal uni t .

Kant 's idea of successive 's tates of things ' presupposes that there is a

unique t ime-sequence for the whole universe. However, as I have argued

elsewhere ([1966], p. 568), the objection to the possibility of an infinite

past applies to any set of discrete events forming a sequence in t ime, e.g.

the swings of a pendulum, or the successive cl icks of a Geiger counter

produced by the emission of part icles from a radioactive source.

As I have al ready indicated, in my view Kant 's argument cannot be

automatical ly disposed of by appealing to the modern theory of infini te

sets and sequences. Indeed, al l reference to temporal concepts has been

purged from the modern theory of infini te sets and sequences, which are

thought of as aggregates and not as being produced in t ime. Of course,

any temporal sequence of events can, i f we wish, be thought of as merely

an aggregate, but only by omitt ing al l reference to i ts production in t ime.

Ka nt 's argu m ent, however, concerns successive act ions or events occ urring

in t im e. It does not conflict with the idea tha t infini te sequences an d series

in general are legit imate objects of thought, but i t rejects the occurrence

of an infinite sequence of past events as an actual possibility.

2 DISPROOF OF THE PO SSIB ILI TY OF AN IN FI NI TE SEQUENCE

OF DISCRETE PAST EVENTS

Bentley's point was tha t an infinite seque nce of past eve nts is not analogous

to an infini te sequence of future events . All past events are empirical

certaint ies , in the sense tha t we can, at least in p rinciple, be presen ted with

evidence ( 't races ') of their oc currence . On the o ther ha nd, al l future even ts

are no more than empirical possibil i t ies , however s trongly we may

happen to believe that they will in fact occur. Moreover, the reason why


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42 G. WUtrow

we tend, i f we are 'rat ional ' , to place more fai th, on the whole, in the

occurrence of a part icular pas t event rather than in that of a part icular

future event is not because retrodict ion is more rel iable than predict ion

(after all retrodict ion is a form of 'pred ict ion '), but because a ny genuine

trace that has been left to us of the occurrence of a part icular past event

was m ade when

that

event occurred, whereas there can be no corresponding

' t races ' avai lable now of a part icular future event m ade sim ultaneously

with the occurrence of that event . That is why, on the whole, i t is safer

to pu t mo re trus t in our m em ories than in ou r forecasts , fall ible thoug h the

former often prove to be.

Let us now consider a temporal sequence of discrete events , or acts ,

E

n

of any kind, where

n

takes integral v alues—negative, zero and posit ive.

Since, at this s tage of the argument, the question of durat ion need not be

considered, i t is unnecessary for these events to be 'evenly spac ed' in t im e.

Start ing from E

o

, which I des ignate ' the present ' (i.e. a present event) ,

consider the sequence of events E

n

for which n > o. Th es e will be labelled

'future events ' . If the set E

n

(n > o) is infinite, we can d esc ribe it, following

Bentley's terminology, as 'potential ly infini te ' . This s imply means that

we assume that each E

n

will have a successor E

n+ 1

, for each positive (or

zero) integer

n.

It cannot mean that there wil l occur, in the ful lness of

t ime, events E

x

, where a denotes the Cantorian cardinal aleph-zero. The

objection to this h app ening is i l lustrated by what B ertrand Russell ([1937],

p . 358) has cal led the 'Tris t ram Shandy paradox' . Tris t ram Shandy, in

Sterne's famous novel , on finding that i t took him two years to write an

account of the firs t two days of his l i fe lamented that material for his

biography would thus accumulate faster than he could deal with i t , so that

he could never come to an end. 'Now I maintain' , argued Russell , ' that ,

if he had lived forever, and not wearied of his task, then even if his life

had continued as eventfully as i t began, no part of his biography would

have remained unwri t ten. ' Since Tris t ram Shandy wri tes in a year the

events of a day, the events of the nth day will be writ ten about in the nth

year, and since any assigned day will be the nth day, i t wil l eventually be

writ ten about. If Tristram Shandy actually succeeded in l iving for an

infini te number of

years ,

then the t ime would eventually arrive, as Russell

said, when al l the days of his l i fe would have been writ ten about. Such a

si tuation, however, can never in fact occur. For, as t ime goes on, not only

does he get no nearer his goal but , on the contrary, the goal continually

recedes .

Now let us consider the sequence of events E

n

for which n < o. If we

can describe it as infinite, then each

E_?

(writing n = —r) was preceded

by an event E_

T

_

1

. For m any contem porary phi losophers this appears to


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On the Impossibility of an Infinite Past

43

be al l that is meant by saying that a past sequence of events is infini te.

My objection to this conclusion is that s ince a ll past events have—and

not m erely

a ny

past event has—occurred, the situation envisaged is strictly

analogous to the si tuation that confronts us in the Tristram Shandy

paradox when

we

consider the possibility of

events E

x

eventually oc curring.

If n events occurred in sequence before E

o

, then there must have occurred

an event designated

E_^,

in m y n otat ion. Sim ilarly if aleph-zero events

occurred before

E

o

,

the n there m ust actually have occurred ( in t ime p as t)

events

E_^.

But, s tart ing from any one such event , i t would have been

impossible to at tain the event E

o

, jus t as from the event E

o

it will never

be possible to attain events

E

x

in the future. Consequently, the concept

of an infinite sequence of past events is incapable of fulminating in the

present event .

1

Hence, just as every future sequence of discrete events

from

E

o

onw ards will always be finite, in th e sense tha t nev er will an eve nt

E

n

be at tained where n is not fini te, so in every past sequence the total

number of events , however large, can never be infini te.

3 UNRELATED SEQUENCES OF PAST EVENTS

T he re rem ains the poss ibil ity that the world m ight be such that there are

an infini te number of

unrelated

past sequences

E_r

such th at , al thou gh

each sequence can necessari ly only contain a fini te number of events ,

there is no definite finite integer

N

such that none of the sequences con-

cerned contains more than

N

members . In that sense, and in that sense

only, could there be a past situation analogous to a potentially infinite

sequence of future even ts . Nevertheless, there s t i ll could no t be the p ossi-

bility of an infinite number of past 'states of the universe'. For, if there is

a temporal sequence of past 's tates of the universe'

(cf.

the technical con-

cept of 'cosmic t ime' in modern theoretical cosmology with—for the

purposes of this discussion—the usual continuous variable t replaced by

a sequence of discrete instants . . . t_

T

_

1

, £_

n

t_

r+ ]

. . . t

0

,

where

t

0

denotes

the prese nt), i t m ust itself c onsti tute a tem pora l chain of the type tha t

I have been considering, and hence th e nu m ber of events in the chain,

whatever this number may be, must necessari ly be fini te.

Hence, the only possibil i ty that the universe does not have a fini te

past arises if no temporal sequence of successive states can be associated

with i t . Consequently, a s teady-state s tat ic universe, for example an

Einstein universe, is not open to objection if i t is assumed that nothing

1

Th ere is symme try, in this respect, of future and past: an infinite sequence actual

infinity, or aleph-ze ro) of past events corresp onds to an actually infinite aleph -zero)

seque nce of future events. Th e latter concept has to be rejected and l ikewise, in my v iew ,

the former.


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44 G. WUtrow

ever happ ens in i t , so that s trict ly speaking i t is timeless. O n the other ha nd ,

in the steady-state expanding world-models devised, in 1948, by Hoyle

and by Bondi and Gold, respectively, al though the past history of every

observable galaxy is finite, there exists a unidirectional cosmic time t

which is assum ed to be wi thout beginning or end. These mo dels are ther e-

fore open to the objection discussed above, even if overall they do not

change with t ime.

1

4 T R A N S FO R M A T I O N S O F T I M E - S C A L E

On passing from the sequence of discrete epochs . . .

t_

T

_

lt

f_

r

,

t_

r 1

,

. . . t

0

, . . . tn_

v

tn , / „+ , . . . to a co ntinuous t im e variable t, we can consider,

as Milne and I did forty years ago in connection with the world-models

based on the techn ique of kinem atic relativi ty, different m onotonically

increasing mathematical t ransformations of t. For example, i f t > o in

a world-mod el tha t expands un iform ly from an init ial point-l ike sing ulari ty

{e.g.

the ini t ial instant t = o corresponding to an epoch of the order of

io

1 0

years ago), we can introduce another variable

T

such that

r = t

0

log(t/t

0

)+ t

0

,

where t

0

is the 'present ' value of t, and

T

= t

0

, 6T = dt, at t = t

0

. A l-

thou gh formally T is negatively infinite w hen t = o, the r-scale applies

only to events for which t > o an d no t to the ini tial point-like singu larity,

s ince according to the r-scale the model is s tat ic. If the firs t event after

the initial singularity occurs when, say, t =

io "

2 4

seconds, it will be as-

signed a fini te epoch on the r-scale and be the firs t event to wh ich tha t

scale can be applied.

5 C O N C L U S I O N

Kant assumed that , i f the concept of t ime could be associated with the

universe as a whole, i t would be quasi-Newtonian, but he argued that the

universe could not have an infini te past nor have existed for only a

fini te part of an infini te past (other ' leg' of the First Antimony of Pure

Reason). He therefore concluded that t ime does not pertain to the

universe i tself but merely to our way of regarding i t . Instead, I conclude

t ha t : either the u niverse is such that there is no 'cosmic t im e' (in th e

modern technical sense) applicable to i t , o r else—as I prefer to believe—

the universe is such th at cosmic t im e exists but i ts total past rang e is fini te,

in the sense explained above.

1

Th ese m odels have now been rejected by m ost astronom ers because they are believed to be

incompatible with observat ions, e.g. the cosm ic background 3K black-body radiation

discovered in 1965 by Penzias and Wilson.


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On the Impossibility of an Infinite Past

45

According to current views,

1

theoretical wo rld-mo dels (i.e. m athem atical

models of the large-scale basic framework of the universe) based

on

general

relativity m ust contain singularit ies ,

e.g.

an initial singularity corresponding

to an epoch of 'world-creation'

8

at a fini te past epoch. This conclusion is

consistent with cu rren t interp retat ions of the observational results (optical

and radio) tha t defini tely rule ou t , inter alia, all steady-state world-models,

both those without and those with mutual recessional motion of the

galaxies.

Imperial College of Science and Technology

REFERENCES

BENTLEY,

R. [1838]:

Se rmo ni Prea ched at Boyle t Lecture, etc.

(ed i ted by A. Dyce . )

L o n d o n .

GILSON, E. [1938]:

The Philosophy of St. Bonaventure

( t rans la ted by I . T re thowan and

F. J . Sheed) . London: Sheed and Ward .

HAWKING, S. W. and ELLIS, G. F. R. [1973]: The Large Scale Structure of Space-Time.

Cambridge Universi ty Press.

RUSSELL, B. A . W. [19 37] : Principles o f Ma thematics. Cambridge Universi ty Press.

VAIHINCER,

H . [ 1 9 2 4 ] : The

Philosophy

of At If ( trans la ted by C. K. Ogd en) . Lo nd on :

Rout ledge and Kegan Paul .

WHITROW, G. J. [1966]: 'Time and the Universe ' in J. T. Fraser (cd.): The

Voices

of Time.

New Y ork: Brazil ler.

1

See Hawking and Ell is [1973].

1 r

World-creat ion ' s ignif ies a f ini te past . To claim more than that takes us beyond science

into theology.

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