Aristotel Infinity

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Aristotelian Infinity

Author(s): Jonathan LearReviewed work(s):Source: Proceedings of the Aristotelian Society, New Series, Vol. 80 (1979 - 1980), pp. 187-210Published by: Wiley on behalf of The Aristotelian SocietyStable URL: http://www.jstor.org/stable/4544958 .

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XII*-ARISTOTELIAN INFINITYby JonathanLear

Philosophershave traditionallyconcernedthemselveswith twoquite disparatetasks: they have, on the one hand, tried to givean account of the originand structureof the world and, on theother hand, they have tried to provide a critique of thought.With the conceptof the infinite,both tasksare united. Since thetime of Anaximanderthe apeiron has been invoked as a basiccosmologicalprinciple.' And the conceptual change that occursas the apeiron of the Presocratics s refined and criticized byPlato and Aristotle, to the developmentof Cantor'stheory ofthe transfiniteand its critique by Brouwer,is one of the greathistories of a critique of pure reason. For whether or not theworld is infinitelyextended in space or time would, for all weknow, make no differenceto the qualityof our local experienceof the world. Nor, for all we know, are we able to distinguishon the basis of how our movement seems to us whetherwe aremoving througha continuousor a discreteworld. "The infinite",says Aristotle, "first manifests itself in the continuous",2forinstance, in motion, time and magnitude.But he does not relyon our experienceof a continuousworld to establishthe world'scontinuity.In PhysicsVI. 1, 2, he offers a series of theoreticalargumentsto show that a magnitude must be infinitelydivisibleand that if a magnitude is continuous so too must be motionand time. To providea critique of the infinite, Aristotlehad toprovide a critique of thought concerning the infinite; yet theresult of such a critique was supposed to provide insight intothe nature and structureof the world in which we live.The infinite needs to be invoked, according to Aristotle, toexplain three distinct phenomena: the infinite divisibility ofmagnitudes, the infinity of numbers and the infinity of time(Physics206ag-I 2). Yet an understandingof Aristotle'saccountof magnitude,number and time is hinderedby the fact that his* Meeting of the AristotelianSociety held at 5/7 Tavistock Place, W.C,r

Qn Monday, May 19, I980 at 6.3o p.m,

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I88 JONATHAN LEARtheory of the infinite poses a seriousproblem of interpretation.It is often said that, for Aristotle,the infinite could exist onlypotentiallyand not actually.But it is not at all clear what sucha doctrine amounts to. Normally when we say something ispotentially b we imply that it is possible that it should beactually(D;3yet it is commonlythoughtthat Aristotledeniesthisin the case of the infinite.A similarproblem about the relationbetween possibilityandactualityconfronts ntuitionistmathematiciansn their treatmentof the infinite. Intuitionistscriticizeclassicalmathematicians orsupposingthat they understandquantificationover infinite do-mains.4They arguethat we do not have a conceptionof what itis for a mathematicalstatementto be true that transcendsourunderstandingof how it would be possible to determinethat itis true. Thus they do not accept the unrestrictedvalidity of theLaw of Excluded Middle '(Ex)F(x) v 7 (Ex)F(x)',even withF a decidable predicate, when the quantifiersrange over e.g.the infinite totality of natural numbers,since for some F theremay be no way we can even in principledecide which, if any,disjunctis true. In such cases,they argue,we cannot claim thatwe know that one of the disjunctsmust be true even thoughwe do not know which. The intuitionistswill, however, allowthe validity of '(Ex)F(x) v 7 (Ex)F(x)' when the quantifiersrange over any finite subdomain of natural numbers, forthey argue it is possiblethat we shoulddecide which disjunctistrue.Yet consider the sentence, 'There is some even number lessthan io100 that is not the sum of two prime numbers'. Intui-tionists considerthe disjunctionof this sentence and its negationvalid becausethey assertthat the sentenceis in principledecid-able. But in what sense is such a sentence decidable? In thephilosophyof mathematics his questionhas been askedby strictfinitists,5but it could equallywell be askedby a militantrealist;and Dummett is correct to stress how importantit is that theintuitionistshould be able to answer this question.' The firstanswer the intuitionistoffers-that there is a mechanical pro-cedureby which we could in principleclheckeach even numberless than io100 and determine whether or not it was the sumof two primes-is inadequate.Fortheremay well not be enoughink, paper,computertime etc. to decide the questionbeforethe



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ARISTOTELIAN INFINITY I89heat death of the universe.But then (to use a familiar phrase)what can the intuitionists'assertion hat the sentenceis in prin-ciple decidableconsistin? If the intuitionistscannot show thattheir claimsof possibilityare well founded, then they are facedwith one of two equally grave consequences.Either they mustcontent themselveswith a smallfragment of finite mathematics;i.e. their position collapsesinto some variant of strict finitism.Or they must concede to the realists that they too have anepistemically ranscendentnotion of truth and that the disagree-ment with the realists s one of degree,not of kind.

Aristotlewas able to offer at least a partial solution to his'problemof potentiality'but, unfortunately or the intuitionists,his solution is not one of which they can take advantage.To seethis we must firstrecognizethat a commonlyaccepted interpre-tation of Aristotle is unacceptable.I

Hintikka has argued that, for Aristotle, every genuine possi-bility is at sometime actualized.7Aristotle's heoryof the infiniteseems to provide a counterexample,since it appears that theinfinite has merely potential existence and is never actualized.However, according to Hintikka, the alleged counterexampleis only apparent; he resolvesthe tension as follows.8Thoughthe claimthat 'theinfiniteis potential'may appearodd, Aristotledenies that 'potential' s being used in a deviant way: he says,rather, that 'to be' has many senses (2o6ai6-27). A lump ofbronzemay at one time potentially be a statue and at a latertime actually be a statue, becausethe processof sculpting it isone that can be completed. The resultof the sculpting processis an individualstatue, a tode ti. By contrast,the infinite is inthe same sense in which a day is or a contest is (2o6a21-25).The point is that there is no moment at which the day existsas an individual entity (tode ti). Rather, one moment of theday occurs after another. Similarlywith the infinite: if onebegins dividinga line, there is no moment at which the infiniteexists as a completed entity. One can continue the processofdivisionwithout end. But Aristotleallowedthat when a processis occurring,whether it be the passing of a day, the OlympicGames or the division of a line, we may say that it is actually



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I90 JONATHAN LEARcoming to be (2o6a21-25, 2o6b13-14). Aristotle is aware, saysHintikka,that "the distinctionbetweenactualityand potentialityapplies also to the kind of existence that is enjoyed inter aliaby the infinite, by a day, and by the Olympic Games . . .Although there is perhaps a rather loose sense in which theinfinite may be said to exist only potentially,in the exact andproper sense in which it exists potentially, it also exists actu-ally."9 (My emphasis).Thus, in Hintikka'sinterpretation,theinfinite does not constitute a counterexample o the principlethat every possibility s at some time actualized.

Though some variant of Hintikka's interpretationis com-monly accepted, I do not think that it can be correct.Aristotleis trying to account for three distinct phenomena-the infinitedivisibilityof magnitudes, he infinityof numbersand of time--and an adequateinterpretationmust see him as providingthreedistinct, if related, solutions. For instance, there are at leasttwo important asymmetriesbetween the case of a day orcontest on the one hand and the case of an infinite divisionon the other. First,the successivedivisionsof a given magnitudeare discrete acts that occur one after another.The flow of timeduring the courseof a day is, by contrast,continuous.Second,though there is no moment at which either the day or thedivision will result in a completedentity, there at least comesa time in the passage of the day when we can say that theday is over. By contrast, there is no processwhich could cor-rectly be considered the actualization of an infinite divisionof a line. For any such process will terminate after finitelymany divisions. To see this more clearly, consider what sortof processmight be consideredthe actualizationof an infinitedivision. It could not be a physicalprocessof actually cuttinga finite physical magnitude, for, obviously, any physical cutwe make in such a magnitude will have finite size and thusthe magnitude will be completelydestroyedafter only finitelymany cuts. Nor could it be a processof theoreticaldivision:`i.e. a mental operationwhich distinguishespartsof the magni-tude. For no mortalcould carryout more than a finitenumberof theoreticaldivisions."And even if, like Aristotle,we believedin the permanenceof the species, there is no way in which atheoretical divider of the presentgenerationcould pass on hiswork to a divider of the next generation.



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ARISTOTELIAN INFINITY I9IHowever, this is a problem for Hintikka, not for Aristotle.Having compared the way a day or contest is with the way

the infinite is, Aristotlecontinues:For generally the infinite is as follows: there is alwaysanother and another to be taken. And the thing takenwill aways be finite, but always different (2o6a27-29).12

In general,the infinite is in the sensethat there is alwaysanotherto be taken. (2o6a27-28). The essenceof Aristotle's nterpreta-tion of the way in which the infinite is said to exist is thatthere will alwaysbe possibilities hat remainunactualized.Hin-tikkasaysthat Aristotle'sclaim that the infiniteexistspotentiallyand not actually (2o6aI6-ig) "is a very misleading way ofspeaking . . . because it muddles an important distinction."13Aristotle s not muddled: he is makingan importantdistinctionto which Hintikka is insensitive.A magnitude is infinite bydivision (apeirondiairesei)and, in virtue of this, it is possibleto begin actuallydividing it. Of course,any such actual processwill terminate after only finitelymany divisions.But that doesnot mean that the actual processdoes not bear witness to themagnitude being infinite by division. Whether it does or notdependson whether,after the terminationof that or any otherprocess, there remain other divisions which could have beenmade. For whatever reason the actual division terminates, asterminate it must, the reason will not be that all possibledivisionshave been exhausted.Hintikkahas failed to distinguishbetween an actual processbearing witness to the existence ofthe potential infinite and an actual process being a witnessto the existence of the actual infinite. No actual process ofdivision could bear witness to a length being actually infiniteby division. However, an actual processof division which ter-minates after finitely many divisions, having failed to carryout all possibledivisions,is all that a witness to the existenceof the potential infinite could consist in. While such a processis occurring one can say that the infinite is actually comingto be, one division occurring after another. (The contrast iswith a process that might be occurring but is not). (2o6a23-25,2o6bI3-14). But even as he says this, Aristotle can insist thatthe infinite by division is potential and not actual (2o6ai6-I8,



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192 JONATHAN LEAR206bI 2-I6) because the actual process can only reveal thelength to be potentiallyinfinite.

Hintikkacites a passagefromMetaphysicsVIII. 6 which hethinkssupportshis interpretation.The infinite does not exist potentiallyin the sense that itwill ever exist actually and separately; it exists only inthinking. The potential existence of this activity ensuresthat the process of division never comes to an end, butnot that the infinite exists separately. (Io48bi4-17, Hin-tikka'stranslation)

The central claim of this passage is that the infinite does notexist as a separate,individualentity. The problemfor Hintikkais to explain how the potential existence of a mental activitycould guaranteethat the processof division never comes to anend. Hintikka'sresponseis to invoke the so-called principleofplenitude: that, for Aristotle,no potentialitygoes forever un-actualized.Thus, Hintikka concludes, not only does Aristotle'stheory of the infinite not constitute a counterexampleto theprinciple of plenitude, the principle is in fact requiredin theexpositionof the theory.This conclusionis, I think, unjustified.Whetheror not Aristotlesubscribed o the principleof plenitudeis extremelycontentious.That does not mean one should neverinvokethe principle.Indeed its invocationmight lend the prin-ciple support if it helped significantlyto harmonize the data,preservethe appearances.But Hintikka invokes the principlein an instance where it is incredible that it should hold. Arewe really supposedto believe that the potential existence of amental activity ensuresthat the processof divisionnever comesto an end because, n the fullnessof time, therewill be an actualmental activity that will ensure that the activity never comesto an end? Hintikka acknowledgesthat the translationof thesecond sentence quoted is disputed. The Greek sentence is:to gar mg hypoleipeinten diairesinapodidosi to einai dunameitautentenenergeian, o de chorizesthaiu. (I 048b 14-17) Hintikkahas had to taketo einai dunameitautenten energeianas the sub-ject,while I would agreewith Rossthat it is morenatural to taketo gar me hypoleipein en diairesinas the subject."4The sentencewould then be translated:



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ARISTOTELIAN INFINITY I93The division not being exhausted ensuresthat this activity(of dividing) is potentially, but not that the infinite existsseparately.

On this reading, it is precisely because there are possible divi-sions that will remain unactualized that the line is potentiallyinfinite.II

Now it is easy to be misled into thinking that, for Aristotle,alength is said to be potentially infinite because there could bea process of division that continued without end. Then it isnatural to be confused as to why such a process would notalso show the line to be actuallyinfinite by division.However,it would be more accurate to say that, for Aristotle, t is becausethe length is potentially infinite that there could be such aprocess. More accurate, but still not true, strictly speaking.Strictly speaking there could not be such a process, but thereason why there could not be is independent of the structureof the magnitude: however earnest a divider I may be, I amalso mortal. But even at that sad moment when the processof division does terminate, there will remain divisions whichcould have been made. The length is potentially infinite notbecause of the existence of any process, but because of thestructure of the magnitude.This interpretation s borne out by Aristotle's first positiveremarkson the nature of the infinite:

To be is said on the one hand potentially and, on theother hand, actually; and the infinite is on the one handby addition and on the other hand by division. On theone hand it is not possiblethat the magnitudebe actuallyinfinite, as has been said, but, on the other hand it ispossible that the magnitude be infinite by division, forit is not difficultto refute [those who argue in favour of]the atomic 'lines'. Therefore what remains is the infinitethat existspotentially.(2o6ai4-I7)

This is a curious passage. There are three men . . . de .clauses,and they establishan odd set of contrasts.To be is saidon the one hand potentiallyand on the other hand actually;N



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194 JONATHAN LEARthe infinite exists on the one hand by addition and on theother hand by division.The third contrastis:

On the one hand it is not possible that a magnitude beactually infinite, as has been said, but on the other handit is possiblethat the magnitude be infinite by division.One would expect Aristotle to contrast the impossibilityof amagnitude being actually infinite with the possibility of itsbeing potentially infinite. This, I submit, is what is beingcontrasted.For what is meant by the claim that a magnitudecannot be actually infinite? It is merely the claim that therecannot be an infinitelyextended magnitude.Aristotlesays that'it has been said' that there cannot be an actually infinitelength-and what has been said, in Physics III. 5, is thatthere cannot be an actual magnitude (or body) of infiniteextension.So it seemsthat Aristotle s, on the one hand,equatinga magnitude'sbeing actually infinite with its being infinitelyextended and, on the other hand, equating a magnitude'sbeing potentially infinite with its being infinite by division.Evidence for this interpretation s providedby Aristotle'scon-clusion:

Therefore what remains is the infinite that exists poten-tially. (2o6aI7)Having eliminated the actual infinite and allowed only theinfinite by division, Aristotle concludes that what remains isthe potential infinite. The point he is making is that the struc-ture of the magnitudeis such that any division will have to beonly a partial realizationof its infinite divisibility: there willhave to be possibledivisions that remain unactualized.Aristotle'streatmentof the infinite by addition furthercon-firms this interpretation (2o6b3-33, 207a33-bI5). One mightreasonably hink that the infinite by additionwould be describedby a processwhich took some given unit length and repeatedlyadded it. However, Aristotle categoricallydenies that such aprocesscouldbe a witness to the infinite by addition.The reasonAristotlegives is that there is no actual infinitelyextendedbody(2o6b20-27). Thus we cannot think of such a processas show-ing that the infinite by addition exists even potentially. Forsuch a processcould not (even if we were immortal)go on with-



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ARISTOTELIAN INFINITY 195out end. And given any such actual process of repeatedlyadding a given unit length, when it terminates after finitelymany additionswe cannot be guaranteedthat we can say thatmore additionscould have been made. Thus no actual processof repeatedlyadding a uniform finite length can bear witnesseven to the potential existenceof the infinite by addition. Theonly way such a finite process could be considereda witnessto the potential existence of the infinite by addition is if therewere an actually existing infinite body.With characteristicunderstatementAristotleshows how theproblem of the infinite by, addition can be reduced to theproblemof the infinite by division. "The infinite by addition"he says "is in one way (pis) the same as the infiniteby division"(2o6b3-4). The point is that given a successivedivision of afinite length AB, as prescribedin Zeno's Dichotomy i/2AB,I /4AB, i /8 AB . . . one can use this division to form aprocessof addition I/2AB + I /4AB + i /8AB + . . . Sucha process could be a witness to the infinite by addition,though the addition of the lengths will never exceed the finitelength AB. In no other way can the infinite by addition besaid to exist, even potentially(2o6bU2-I3). But in so far asthe infinite by addition does exist potentially,we say it existsin the same fashion as the infiniteby divisionexists: i.e. thereis alwayssomethingoutsideof what has been taken (2o6bi6-18).Again we have evidence for equating the potentialinfinitewiththe infinite by division and seeing both as existing in virtueof the continuousstructureof a magnitude and not in virtueof the existenceof any process.One of the puzzles Aristotlehopes to solve is how numberscould be infinite (2o6awo-12). Numbers, for Aristotle, are notamong the basic constituentsof reality, but are the result ofabstraction rom physicalobjects (Met. M3). If we are to speakof an infinity of numbers, their existence must somehow beguaranteed.His solution is a direct consequenceof his analysisof the infinite by addition (207bI-I5). Physics III. 7 openswith an admissionthat, on his account, the most natural con-ception of the infiniteby addition-that it is possibleto exceedany given length-is impossible(207a33-35). Thus one cannotassigna numericalunit (i) to a standardunit length and hopethat an infinityof numbers will be guaranteedby the fact that



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I96 JONATHANLEARthis length can be repeatedlyadded without end. Nevertheless,Aristotle concludes, it is always possible to think of a largernumber becausethe divisionsof a length are infinite (2o7bio-ii). What he has in mind is that the infinity of numbers isguaranteed n much the same way as is the infinite by addition.Given a finite length AB we can assignnumbersto the succes-sive divisions:

I/2AB, I/4AB, i /8AB, i / i6ABI 2 3 4

Thus, thoughAristotlethinks that number is not separable romthings it numbers (2o7b1 3) he nevertheless ees the numberingas dependent upon the concept under which the things num-bered fall.1"In this case the number cannot be thought of asthe numberof unit lengths,but ratheras the numberof divisions.The motivationfor Aristotle'sanalysisof the infinityof num-bers is clearlystated:It is always possible to think of a greaternumber, for thedivisions of a magnitude are infinite. So it is potentiallyinfinite,but not actuallyinfinite;but the thing taken alwaysexceedsany definitenumber(2o6aIo-13).

By reducing the problemof the infinityof numbers to that ofthe infinite divisibility of a magnitude, Aristotle is able toreaffirmthe potentialityof the infinite and deny its actuality.And the key to the potentiality of the infinite is that givenany number a greaternumber could be found; for given anydivision of the line, another could be made.But we must be clear about what possibility Aristotle isclaimingif we are to understandhow his treatmentdiffersfromthe intuitionists' reatment.For the intuitionistsdo, in a way thatAristotledoes not, make a fundamentalappeal to actual humanabilitiesto justifytheir claims about the infinite. Both Aristotleand the intuitionistsagreethat a magnitude s infinitelydivisibleand both will interpret his as a claim that no matter how manydivisions of a magnitude have been made another could bemade. But in fact each is claiming a very different possibility.Aristotle is claiming that the magnitude is such that if therewere a divider who could continue to divide the length (i.e. a



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ARISTOTEIAN INFINITY 197Creative Mathematician,unhamperedby human physical andmental limitations) then no matter how many divisions hemade, he could always make another. Only the last 'could' isfundamentalto Aristotle'sclaim that the magnitudeis infinitelydivisible,for it is a claim about the structureof the magnitude,not the existenceof a process.On the questionof whethertherecould be someonecapableof carryingout the divisions,Aristotleis silent.'8 (Similarlywith numbers: Aristotle is only claimingthat if there were someone capable of continuallyabstractingnumbers,then he could always abstractanother number.)Theintuitionists'possibility-claim s, by contrast, inextricably tiedto an ability to carryout a certainprocedure.But if we takethis as a genuineclaim that a certainprocedure s possible, heneither the claim is too weak to generate the potentiallyinfinitetotalitythat the intuitionistwantsor the claim is false.If theclaimis that a man may actuallystartby makinga theoreticaldivisionand given any numberof divisionshe has made, he could makeanother,then such a claim will not providethe intuitionistwithmore than a vaguely determined totality. For example, eventhough it is absurd to suppose that there was a last heartbeatof my childhood,that only showsthat 'the numberof heartbeatsin my childhood' determines a vague, not an infinite, totality.For there are finite numberse.g. io100 larger than the numberof heartbeats n my childhood. Similarly,'the number of theo-retical divisionsI could actuallycarryout' specifiesonly a vaguetotality: it is absurd to suppose that some particulardivisionis the last division I could make, yet there are finite numbers,e.g. io010, largerthan the number of divisions I could actuallycarryout. If, however,the claim is strengthened, t also becomesfalse: there is no process of physical or theoretical divisionwhich could make io010 divisions. Unfortunately, the intui-tionists cannot follow Aristotle'sexample and reinterprettheirpossibility-claim n such a way as to avoid recourse to ourability to divide. For the core of the intuitionistcritiqueof therealist is that the realist'sconception of truth for a sentencetranscendshis ability to determine whether or not a sentenceis true. Thus the intuitionistmust maintainthat it is in principlepossible to decide e.g. whether there are two prime numbersx and y such that io010 divisionsof a line are equal to x + ydivisions.Nor can the intuitionist egitimately nvokea Creative



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198 JONATHANLEARMathematicianfor, accordingto him, our conceptionof truthis derivedfrom our own abilityto determinewhethera sentenceis true. Yet to decide such a question, the intuitionist mustembark on a processof construction that it does not appearpossible to complete. The intuitionist is thus in a radicallyunstableposition,and he cannot adopt an Aristotelianapproachto the potential infinite without sacrificinghis own critiqueofthe classicalmathematician.

IIISo far I have argued that for Aristotle it is because a lengthis infinite by division that certain processesare possible andnot vice versa. A problem for this interpretationmight bethought to arise with Aristotle'sconsidered response to Zeno(PhysicsVIII. 8, 263a4-bg). While we may think that thereexist infinitelymany pointson a line and that such existenceisnot dependent upon any process,Aristotleseems to deny this.He distinguishesbetween the potential and actual existence ofa point: a point does not actually exist until it has beenactualized (cf. 262aI2-263a3). A point on a line may beactualized if one stops at it, or reversesone's direction at it,or divides the line at it. Accordingto Aristotlea runner (callhim Achilles) would indeed be unable to traverse the finitelength AB if in the course of his journey he had to traverseinfinitely many actually existing points. However, continuousmotion along a length is not sufficientto actualize any pointalong the length. Thus Aristotle would be among those whothink that while it may be possible for Achilles to traverseAB in one minute by moving continuously across it, it wouldbe impossiblefor him to traverseit in two minutes if in thefirst thirty secondshe went to the midpoint (1/2AB) and thenrested thirty seconds, in the next fifteen seconds to the threequarterpoint (3/4AB) then rested I5 seconds and so on. Forsuch a 'staccatorun' to be successful,Achilles would have had(by the end of two minutes)to actualizeinfinitelymany pointson the length AB and this Aristotleheld to be impossible.I donot here wish to assess the adequacy of Aristotle's responseto Zeno :17 what is importantfor the presentdiscussion s thatAristotle appears to make the existence of infinitely many



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ARISTOTELIAN INFINITY I99points on a line dependentupon the existence of a process.Aristotle does deny the actual existence of infinitely manypointson a line and he does this becausehe takes certainkindsof processes-e.g. a staccato run-to be impossible.But he doesnot deny the potential existence of infinitelymany points andhe does not make this potential existence dependent upon theexistenceof any type of process.It is preciselybecauseinfinitelymany points do exist potentially that an actual staccato runwill be a witness to the infinite divisibilityof the length AB,even though Achilles' nimblenesswill give out after he hasactualizedonly finitely many points.Although Aristotle denies the actual existence of infinitelymany points on a line, this is not as odd as it might firstappear. For he denies that a line is made up out of points.A line is continuousand nothing continuouscan be made upof unextended points or of indivisible magnitudes (cf. e.g.23ia2o-b6, 227aiO-12). So since we need not, in fact shouldnot, think of a length as composedof points, we need not, infact should not, think of the points as actually existing.Michael Dummett has tried to envisagea 'realitynot alreadyin existence but as it were coming into being as we probe.Our investigationsbring into existence what was not therebefore but what they bring into existence is not of our ownmaking."8According to Aristotle, points do not actually existindependentlyof our 'probing'for them, yet they are not ofour own making.A point, for Aristotle,existsonly in a deriva-tive sense: it is, so to speak,a permanentpossibilityof division.'9But these are possibilitieswhich cannot all be actualized. InDe Generatione et Corruptione, Aristotle considers the problemsthat arise from supposingthat an infinitelydivisiblemagnitudehad actually been divided "through and through" (3i6ai5-3'7ai8). The situationAristotle s envisaging is that all possibledivisionsof a magnitude had actually been made. What thenwill remain? No magnitudes can remain, for magnitudes aredivisible and this would contradict the assumption that thedivisionhad been carriedout throughand through.20Nor couldthere be pointswithoutmagnituderemaining.A divisiondividesa whole into its constituents,yet one cannot without absurditythink of pointswithoutmagnitudeas constituentsof a length.Aristotle offers a paradigmaticallyAristoteliansolution. He



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200 JONATHAN LEARdistinguishestwo senses in which a line may be said to bedivisible 'through and through' (3i7a3ff). A length is divisiblethrough and through in the sense that it could be dividedanywherealong the length. But it is not divisiblethrough andthrough in the sense that it could (even potentially)be dividedeverywherealong the length. One can thus actualizeany pointbut one cannot actualize every point; for any processof divi-sion, there mustbe divisionswhich could have been made whichin fact were not made.

IVAristotleis attemptinga revolution n philosophicalperspective.This is a revolutionwhich cannot be appreciated f one thinksthat, for him, every possibilitymust be actualized. Aristotlewants to remove the infinite from its position of majesty.Theinfinite traditionallyderived its dignity from being thought ofas a whole in which everything is contained (207aI5-2i). Butthe view that the infinite contains everything arises, Aristotleargues,from a conceptualconfusion.

The infinite turns out to be the oppositeof what they say.The infinite is not that of which nothing is outside, butthat of which there is always somethingoutside . . . Thatof which nothing is outside is complete and whole, forwe define the whole as that of which nothing is absent,for example, a whole man or a wooden box. . . . By con-trast, that of which something, whatever it might be, isabsent is not everlasting. Whole and complete are eitheraltogether the same or of a similar nature. Nothing iscomplete (teleion)which has no end (telos), and the end isa limit (peras)"(2o6b33-207aI5).

Aristotle here presentsan argument to show that the infiniteis imperfectand incomplete: I. The infinite is that from whichit is always possibleto take somethingfrom outside (2o7ai-2).2. That of which nothing is outside is said to be completeandwhole (207a8-9). The examples Aristotlegives are paradigmsof finite self-containedobjects; they are individualentities,sub-stance or artifact. 3. The whole = the complete (aI3-14).But 4. Every completething has an end (a14). And 5. An end



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ARISTOTELLANNFINITY 201is a limit (aI4-I5). The reader is left to draw the conclusionfrom (3)-(5) that 6. The whole has a limit. Thus it would beabsurd to equate the whole and the infinite, for that wouldbe to say that the apeironhad a peras. Of course, we need notbe persuadedby this argument to recognizeits genius. This isan excellent example of how one can formulate an a prioriargument which nevertheless fails to establish its conclusionbecause the concepts it employs-in this case the concept of alimit-are not sufficientlyrefined.2' The claim that the apeironmust lack a peras would appear to Aristotleto be an analytictruth.

Having dethroned the infinite, Aristotlecan argue that:[The infinite] qua infinite does not contain, but is con-tained. Therefore[the infinite] qua infinite is unknowable,for matter does not have form (207a24-25).Aristotle often draws an analogy between the infinite andmatter. (Cf. e.g. 2o6b14-I5, 207a2I-25, 207a26, 207a35-bI,207b34-208a3.) It is this assimilationof the infinite (apeiron)to matter which lies at the heart of the conceptual revolutionhe is trying to achieve.22For Anaximander,the apeiron servedat least four functions. First it was a great unlimited massenveloping the world. Second, it was the temporal arche ofthe world. Third, it provided a permanent,unchanging prin-ciple which governedthe change and transitionobserved n thenaturalworld. Finallyit is that out of which things are formed.It is only this last function for which Aristotle had any use.For Aristotle,the world is finite and unenveloped,eternal andungenerated;the natural processescould be explained withoutthe governanceof a transcendentprinciple.What Aristotledidneed, however,was an underlyingstuff from which things areformed: thus the apeironwas, if with rough justice, impressedinto serviceas the materialprinciple.23

The infinite,for Aristotle, s immanentin nature, not a trans-cendent principle; thus he can say that we first encounter theinfinite in the continuous (2oobl7ff). The infinite, like matter,does not contain the world, but is contained; that which con-tains is form (207a35-bI). Most importantly,matter as suchis merelya potentiality: the only way it can exist actually is asinformed matter. (Cf. Met. 1050a15, 1049bI3, deAn 43oaio.



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202 JONATHAN LEAR

Cf. also Met. M3 I078a30-I). The infinite exists only poten-tially as matter does (2o6bl4ff). Due at least in part to itspotentiality, the infinite, like matter, is unknowable. (Cf.207a24-26, a3o-32). Matter qua matter is unknowablebecause it lacks a form and it is form that is knowable. Theinfiniteis unknowableboth because that which is indeterminateis unknowableand becausethat which the mind cannot traverseis unknowable.Here we encountera threadthat runsto the core of Aristotle'sphilosophy.Were the chain of causes of a given thing infinitewe would not be able to know the explanationof that thing,because the mind cannot traversean infinite series,i.e. a seriesthat has no peras. But we can know the causes of a thing;therefore they must be finite (Met.A.2). If those propertieswhich make a substancewhat it is were infinitein number,thenthe substancewould be unknowable.But we can know whata substanceis, thereforethere are only finitely many propertiesin its definition(An.Pst. 1.22). ThroughoutAristotle'swork thistheme recurs: the possibilityof philosophy-of man's abilitvto comprehend the world-depends on the fact that the worldis a finite place containing objects that are themselves finite.And the possibilityof philosophy s one possibility hat Aristotlespent his life actualizing.

VTime, for Aristotle, is also supposedto be potentiallyinfinite.But the claim that time is potentiallyinfinite is fundamentallydifferentfrom the claim that magnitude is potentiallyinfinite,for although a stretch of time is continuous and in virtue ofthis infinitely divisible, and although the continuity of timedepends ultimatelyon the continuityof magnitudes(232a23ff),it is not to the infinite divisibilityof time that Aristotlewishesto draw our attention when he says that time is infinite. Rather,in the case of time, he does for once seem to emphasize theidea of a process: time flows on and on.However, when one tries to combine Aristotle'sbelief in theinfinityof time with his belief that magnitudes can be of onlyfinite size, his natural philosophy seems to verge towardsinco-herence. First, time is supposed to be a measure of change



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ARISTOTELIANNFINITY 203(kinesis) (22ob32-22Ia7, 2I9bI-2).24 One measures a givenchange by picking out a motion (phora)25which is uniformand lettingthat be a standardagainstwhich the time of a givenchange is measured. The paradigmatic measure of change isregularcircularmotion and he took the motion of the heavenlybodies to be eternal (223b12-2I, cf Physics VIII.8,9). But,secondly,a body in motiontraversesa spatialmagnitude(cf. e.g.23IbI8ff). So if time is infinite (2o6bg-I2) and time is measuredby motion, why does not the infinityof time bear witnessto theexistence of an infinitely extended magnitude?

The obviousresponse s that the only motion which Aristotlethought could be regular,continuous and eternal was circularmotion (Physics VIII. 8-9). And circular motion is not trulyinfinitary (206b33-207a8). While traversingthe circumferenceof a circle, one can always continue one's motion, but onecannot properlycall the circle "infinite". For the circle to beinfinite, it would be necessary that each part traversed bedifferent from any part that had been traversedor could betraversedagain. And this necessarycondition is not fulfilled.So although the heavens have always moved and always willmove in a regularcircularmotion and thus provide a measureagainst which the time of other changescan be measured,theydo not themselvestraverse an infinitelyextended magnitude.This response,as it stands, is inadequate, for although thesphereof the heavensmay be finite, the path the heavensdescribethrough all time must be infinite. Aristotlehimself admits thatif time is infinitely extended then length must be infinitelyextended (233aI 7-2o). To avoid the problem one must somehowsever the tie between the passageof time and the path describedby a moving body. And this is preciselywhat Aristotledid.But he did not do it by as simple a sleight of tense asSimpliciusascribesto him.26Simpliciusdraws our attention to206a25-b3 at which Aristotle contrasts the passage of timewith the successive divisions of a length: while the parts of aline that are divided remain in existence, in the case of timethe moments cease to exist; past time has perished. Thoughthe contrast is genuine, it does not of itself embody a solutionto the problem. For Aristotleclearly says that a moving bodymoves acrossa spatialmagnitude,and thus there is a magnitudethat is traversedby a moving body, even though at any one



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204 JONATHAN LEARmoment the moving body is in only one place. That the timeat which a body moved has 'perished' does not imply thatthe magnitude or path it traversed has perished.Thus one isstill faced with the problem of why the path traversedby theheavensthroughall past time is not infinitelyextended.I would like briefly to suggest that Aristotle developed ananti-realistaccount of time, which enabled him both to showhow time could be potentially and not actually infinite andto avoid 'spatializing'time.To claim that Aristotle was an anti-realist with respect totime is not say that he succumbed to the sceptical aporiai ofPhysics IV.Io and concluded that time did not exist. It is onlyto saythat, in companywith McTaggartand Dummett,Aristotledid not believe one could give an observer-independentescrip-tion of time.27Time seemed to Aristotleessentiallyto involvechange (2i8b2ff), and it is to this aspect of time that a static,observer-independent escriptionof the sequenceof events can-not do justice. The claim that Aristotle thought time unrealamountsto no more than that he thought an adequatedescrip-tion of time must include a consciousnessthat is existingthrough time aware of the distinction between present, pastand future. In a famous passage which, because it comes atthe conclusion of his extended discussionof time, I take tocontain his mature thoughts on the nature of time, Aristotlesays:

It is worth investigatinghow time is related to the souland why time seems to be in everythingboth in earth, insea and in heaven . . . Someone might well ask whethertime would exist or not if there were no soul. If it isimpossible for there to be someone who counts, then itis impossiblethat somethingbe countable,so it is evidentthat neither is there number, for number is either thething counted or the countable.If there is nothing capableof counting, either soul or the mind of soul, it is impossiblethat time should exist, with the soul not existing,but onlythe substratum of time [sc. change], if it is possible forchange to exist without soul. (Physics IV.I4, 223aI 6-28,my emphasis)Time is above all a measureand, as such, could not exist were



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ARISTOTELIAN INFIMTY 205there no soul or mind of soul which could measure. This doesnot mean that the measurement s subjectiveor that any mea-surementa soul makesis correct: it only means that we cannotgive an adequate account of time without including in theaccount a soul which is measuring he change.Time exists, but in a derivative sense, dependent for itsexistence on a soul that is measuringchanges. For an event tooccur at some time it is necessarythat that event stand in adeterminaterelationto the present (222a24-29). From, say, thefall of Troy until now the heavens have revolvedcontinuouslya finite number of times. A soul can measure the time that haselapsed and the path describedby the heavens from the fall ofTroy until now will only be finitely long. In so far as we aretempted to conceive of there being a stretch of time from thatevent to the present,that stretchwill be finite.But what if one should considerthe entire previoushistoryof the world? For Aristotlethe world is uncreatedand eternal.So if one were able to measurea stretchof time encompassingall events in the historyof the world, that stretch would haveto be infinitely extended. But Aristotle denies that such ameasurements possible.His theory is, so to speak,omega-incon-sistent: each event is in time (en chrono), but all events arenot in time (22Ial-222ag). An event is in time only if it isencompassedby time; i.e. if there were events that occurredbefore and after it. So although each event in the history ofthe world is in time-and one can thusmeasurethe time elapsedfrom that event until now-one cannot treat all events in theprevious history of the world as being in time: one cannotmeasure the time elapsed in the entire previoushistoryof theworld.This solutionmay appearunsatisfying.In orderto appreciateit, one must confront anotherproblem.How can Aristotleevensay that the world is eternal? For the world to be eternal, itmust have existed at all times, always.What, for Aristotle,cansuch a claim consist in? To say that the world must alwayshave existed is to say that there is no time at which the worlddid not exist. But since time is the measureof change, and themotionof the heavensprovides he standardmeasure,were thereno world and thus no change, neitherwould there be any time.Thus the claim that the world is eternal seems in danger of



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2o6 JONATHAN LEARcollapsing from its vaunted position as a metaphysicalclaimabout the nature of the world into a trivial analytic truth. Ofcourse there was no time at which the world was not, for ifthe world were not there would be no time and a fortiori notime at which it was not.'It does not help to claim that "therealways was change and always will be change throughoutalltime" (266a6ff). For to claim that there alwayswas change isto claim that there is no previoustime at which there was riochange; but that is trivially true. For were there no changethere would be no time, since time is the measureof change.Similarly, the claim that there always will be change seemsno longer to be a metaphysicaldiscoveryabout the future, butan analytic triviality.Nor will it help to claim that time is ameasureof rest as well as of change (22ib7-14), for all rest isin time (b8-9) and thus time must extend in both directionsbeyond any given rest. One can say, for example, that a givenanimal rests because there is a finite interval in which theanimal has temporarily ceased its motion, but during thisinterval, as during the entire period of the animal's life, theheavenscontinueto revolve,providinga measureagainstwhichthe animal'schange and restcan be measured.One cannot thusthink of a stationaryheaven as resting and so as being in time.Nor will it help to claim that the world is ungenerated.(Cf. deCaelo L.IO-12.) For to claim that the world is ungenerated isto claim that there is no previoustime at which it came intobeing and this may be triviallytrue.However, Aristotledoes provideus with an argumentwhichenables us to break out of this circle of triviality. In PhysicsVIII.i he argues that the suppositionthat there was a firstchange leads to absurdity.

We say that change [kin0sis] s the actualityof the change-able thing in so far as it is changeable.It is necessary here-fore that for each change there are things capable of beingchanged . . . Further these things necessarily either cometo be (at some time they do not exist)or they are eternal.If therefore each of the changeable things came to be,before that change another change [metabolen kaiIcinesin]must have come to be, according to which thething capable of being changed or changing came to be.



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AMSTOTELIAN INFIMTY 207The suppositionthat these things existed always but un-changedappearsunreasonablemmediately,but even moreunreasonablef one goes on to investigate he consequences.For if, among the things that are changeableand capableof producingchange, there will at some time be somethingfirst producingchange and something changing, while atanother time there is nothing but somethingresting, thenthis thing must have previouslybeen changing. For therewas some cause of rest: rest being a privationof change.Thereforebefore this first change there will be a previouschange. (25 Iag-28)

Aristotle is arguingthat given any purportedfirstchange, theremust have been a change which existedbefore it. Thus we canunderstandhis claim that there has always been change asbeingmorethan an analytictruth, if we interprethim as claimingthat it is absurdfor there to have been a firstchange. Similarly,Aristotle'sclaim that the world is eternal should not be inter-preted in terms of an infinitely extended length of time, butonly as a claim that no moment could be the first (or last)moment of the world's existence.However, this claim does not establish that the temporalhistoryof the world consists n more than a vaguelydeterninedtotality. (It is, for example, absurdto suppose that any givenheartbeatwas the first (or the last) of my adolescence,yet thatonly shows that the heartbeatsof my adolescenceconstitute avaguely determinedtotality). I am not sure how disappointedAristotlewould be with this result. Time, he admits, dependsfor its very existence on the existence of a soul or mind thatis measuringchange; the 'infinity'of time consistsin the factthat no measurementcould be a measurementof a firstchange.We seem to have been led from the introduction nto a theoryof time of an observerof change,via considerations f vagueness,to the conclusionthat the 'observer' s not a mere observerofa phenomenon totally independent of him: for if the soulmeasuringchange had an analogous relationto time as, say,a lepidopteristhas to his butterflies, t is difficult to envisagehow we could conceive of time as constitutinga vaguely deter-mined totality.If Aristotle'stheory of time is coherent-and the answer to



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208 JONATHANLEARthat question waits upon a detailed study of the now-it slipsthrough the net of Kant's so called First Antinomy.28Kantthought that reason could construct two equally valid argu-ments, one to the conclusionthat the world had no beginningin time, the other to the conclusion that the world had abeginning in time. The proof that the world had no beginningis similar in structure to Aristotle's. To prove that the worlddid have a beginning, Kant supposes that the world had nobeginning and then infers that an [actually] infinite extensionof time must have passed before the present moment, whichhe takes to be impossible.Aristotle would accept both that theworld had no beginning and that it is impossible hat an actualinfinity of time should have elapsed, but he would reject thleinferenceand thus the argumentas invalid. From the fact thatthe world had no beginning, all that follows is that there canbe no measurementof a firstchange; and it is to the importanceof so understandingour temporal claims that Aristotleis draw-ing our attentionwhen he says that time is potentiallyinfinite.

NOTES* I would like to thank: M. F. Burnyeat,C. Farrar,T. J. Smiley, andR. R. K. Sorabji for offering valuable criticisms of an earlier draft; thePresident, Deans and Librarian of The Rockefeller University for theirhospitality during the summer, 1979, when this paper was written; IanSpence, who at age five asked me a question to which this paper is apartial response.1 Cf. C. H. Kahn, Anaximanderand the Origins of Greek Cosmology,New York, Columbia University Press, I960; P. Seligman, The Apeiron ofAnaximander,London, The Athelone Press, I962.2 Physics, III.1, 200bI7.3 D. Bostock, 'Aristotle, Zeno and the Potential Infinite', Proceedingsof the Aristotelian Society, 73, 1972-3. Cf. W. D. Hart, 'The PotentialInfinite', Proceedings of the AristotelianSociety, 76, 1975-6; J. Thomson,'Infinity in Mathematics and Logic', Encyclopedia of Philosophy, NewYork, The Macmillan Company, I967.4 Cf. e.g. Michael Dummett,Elementsof Intuitionism,Oxford, ClarendonPress, 1977; 'The Philosophical Basis of Intuitionist Logic', 'Platonism','Realism', in Truth and Other Enigmas, London, Duckworth, 1978.6 Cf. A. S. Yesinin-Volpin, 'The Ultra-intuitionistic Criticism and theAntitraditional Program for the Foundations of Mathematics',in A. Kino,J. Myhill, R. Vesley (eds.), Intuitionism and Proof Theory, Amsterdam,North Holland, I970.6 Michael Dummett, 'Wang's Paradox', Truth and Other Enigmas, op.cit. Cf. Crispin Wright, 'Language Mastery and the Sorites Paradox,'



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ARISTOTELIANNFINITY 209in G. Evans and J. McDowell, Truth and Meaning, Oxford, ClarendonPress, 1976.

7 Jaakko Hintikka, Time and Necessity, Oxford, Clarendon Press, 1973.8 Cf. 'AristotelianInfinity', in Time and Necessity, op. cit.9 Ibid. ii6.10 David Furley, Two Studies in the GreekAtomists,Study I. 'IndivisibleMagnitudes', Princeton, Princeton University Press, I967.11 I deliberately ignore those who are able to perform Benacerrafiansupertasks.See P. Benacerraf,'Tasks, Supertasksand the Modern Eleatics',Journal of Philosophy, 59, I962; J. Thomson, 'Comments on ProfessorBenacerraf'sPaper', W. C. Salmon (ed.), Zeno's Paradoxes, Indianapolis,Bobbs Merrill, 1970.12 I have relied on the Oxford Classical Text of Aristotle'sPhysics (W.D. Ross, ed., 1973), the translations unless otherwise noted are my own.13 Hintikka (1973) Op. cit., iI6. Cf. G. E. L. Owen, 'Aristotle on theSnaresof Ontology', New Essays on Plato and Aristotle,R. Bambrough(ed.),London, Routledge and Kegan Paul, I965.14 W. D. Ross, Aristotle'sMetaphysics,v. II, Oxford, Clarendon Press,

1975; 252-3. Cf. Physics 203b23-25.15 Cf. e.g. Physics IV.14, 224a2ff.lo This I disagree with the emphasis Weiland places on the relationof infinite divisibility and motion. Weiland correctly relates infinite divisi-bility to the possibility of certain processes occurring, but I do not thinkhe has accurately seen what this possibility consists in. Cf. W. Weiland,

Die aristotelische Physik, Gottingen, Vandenhoeck & Ruprecht, 1962,293-3I6. Compare his interpretationof Met. VIII.6, 1o48bl4ff (294) withthe one I have offered above. And cf. Physics208aI4ff.17 Cf. e.g. D. Bostock (1972-3) op. cit.; G. E. L. Owen, 'Zeno and theMathematicians', Proceedings of the Aristotelian Society, 58, 1957-8; J.Barnes, The Presocratic Philosophers, v.I, London, Routledge and KeganPaul, 248ff.18 Michael Dummett, Truth and ,OtherEnigmas, op. cit., i8.19 Ross has said, 'and when it does not rest but moves continuously, thepre-existence of the points on its course is equally presupposed by itspassage through them'. W. D. Ross, Aristotle'sPhysics, Oxford, ClarendonPress, 1979, 75. Aristotle would entirely disagree.20 I ignore the atomists' critique of Aristotle's arguments. See Furley(i967) op. cit.; J. Mau, 'Zum Problemdes Infinitesimalen bei den antikenAtomisten', Deutsche Akad. d. Wiss, z. Berlin, Institut fiur hellenistisch-romisch Philos. VerofJentlichung,Berlin, I954.21 Cf. e.g. C. B. Boyer, The History of the Calculus and Its ConceptualDevelopment, New York, Dover, 1959; chapter II.22 For the use of 'apeiron' in Anaximander,Homer and other early Greekwriters, see the appendix on the apeiron in C. H. Kahn (I960) op. cit,23 It is well known that Aristotle tried to present his philosophy as

embodying that which was correct in the philosophies of earlier thinkers,also that he was often unfair in his portrayal of others' thoughts. Cf. H.Cherniss,Aristotle'sCriticismof PresocraticPhilosophy, New York,Octagon,1976; G. E. L. Owen, 'Tithenai ta phainomena', S. Mansion (ed.) Aristoteet les Problemes de Mithode, Louvain, I96I.24 Aristotle says both that time is a number of change (2igbi-2) anda measure of change (22ob32-22iai). I have had to suppress a discussionof the distinction between numbering and measuring, but see J. Annas,

0



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210 JONATHAN LEAR'Aristotle, Number and Time', Philosophical Quarterly, 1975, and D.Bostock 'Aristotle'sAccount of Time', unpublishedmss. Cf. Met. X,j-2.25 Motion (phora) is a particular type of change, a change of place.Cf. 226a32ff.26 Simplicii, In Aristotelis Physicorum, Commentaria in AristotelemGraeca, v. IX, H. Diels (ed.), Berlin, I882, 494-5.

27 J. E. McTaggart, 'The Unreality of Time, Mind, I7, 1908; M.Dummett, 'A Defense of McTaggart's Proof of the Unreality of Time',Truth and Other Enigmas, op. cit. See also, J. Perry, 'The Problem of theEssential Indexical', Nod2s,XIII, 1979. I certainly am not claiming thatAristotle had made the distinctions involved in A-series and B-series. SeeG. E. L. Owen, 'Aristotle on Time', in J. Barnes et. al. Articles on Aristotle,V.3, London, Duckworth, 1979.

28 I. Kant, Critique of Pure Reason (N. K. Smith, translator), NewYork, St. Martin's Press, I965, 396-7. Cf. D. H. Sanford, 'Infinity andVagueness',PhilosophicalReview, 84, 1975

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Aristotel Infinity