aristotle dynamics and proportionality

8/8/2019 Aristotle Dynamics and Proportionality

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Aristotle, Dynamics and Proportionality

Author(s): Andrew GregorySource: Early Science and Medicine, Vol. 6, No. 1 (2001), pp. 1-21Published by: BRILLStable URL: http://www.jstor.org/stable/4130277

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ARISTOTLE, DYNAMICS AND PROPORTIONALITYANDREWGREGORY

University College, London

Aristotle's comments on the motion of free fall have received agood deal of criticism. Rightly so, some would say, as Aristotle waswrong in considering the speed of fall to be directly proportional toweight and clearly did not do the requisite experiments. We haveperhaps been too ready to assume that Aristotle's intention was togive a dynamics, in the modern sense of that part of mechanics whichdeals with force, mass and motion, when his main aim may well havebeen elsewhere.' Debate used to centre on how close Aristotle's sup-posed dynamics were to those of Newton, and what prevented Aris-totle from getting any further.2 Recent work by Wardy and Lloydthough has emphasised the context of Aristotle's comments onmotion, and has argued that given their disparate, tentative and dia-lectical nature we should take great care in how we evaluate them.3Attempts to see these comments as general laws are for Lloyd "insome respects clearly, and in others very probably, an overinterpreta-tion", while Wardy urges a "minimalist" interpretation and attacksthe "Whiggism" associated with the older view.4 In general I havesympathy for this attack on the older view, though I believe it haspushed matters too far in the other direction. I argue that Aristotleis concerned with a higher level project than dynamics, and that isthe establishment of a coherent theory of change in general. Hisapparently disparate comments on motion can be seen as argumentsrequired to establish this general theory. The latter relies in a funda-mental manner on a broad application of proportionality in order

' See e.g. I.E. Drabkin, "Notes on the Laws of Motion in Aristotle", TheAmericanJournal ofPhilology 59 (1938), 60-84.

2 See e.g. Drabkin, "Notes on the Laws";G.E.L Owen, "Aristotelian Mechanics",in Logic, Science and Dialectic. CollectedPapers in GreekPhilosophy,ed. M. Nussbaum(Cornell, 1985), 315-334.s See R.B.B. Wardy, TheChain of Change:A Study ofAristotlePhysicsVII(Cambridge,1990); G.E.R. Lloyd, The Revolutions of Wisdom.Studies in the Claims and Practise ofAncient GreekScience(Berkeley, 1987). Cf. H. Carteron, "Does Aristotle have a Mecha-nics?"in Articleson Aristotle,vol. 1: Science,eds.J. Barnes, M. Schofield and R. Sorabji,(London, 1975), 161-174; and F. de Gandt, "Force et science des machines", in Scienceand Speculation.Studies in Hellenistic Theoryand Practise, eds.J. Barnes, J. Brunschwig,M. Burnyeat, M. Schofield, and I.E. Drabkin (Cambridge, 1982), 96-127.4 Lloyd, Revolutionsof Wisdom,219. Wardy, TheChain of Change, 303.

? Koninklijke Brill NV, Leiden, 2001 Early Science and Medicine 6,'1


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2 A. GREGORYto eliminate infinities and paradoxes Aristotle believes to be inher-ent in Eleatic, Heraclitean and atomist accounts. Part of his strategyin preserving proportionality is the denial of a void, but I argue healso denies the possibility of instantaneous change and can accom-modate threshold changes (where no change occurs until a thresh-old is reached). Moreover, Aristotle extends this general theory tocover all types of change, and not just cases of what we would labeldynamics. As this line of argument will re-open the question of theextent to which Aristotle was committed to a direct proportionalitybetween speed of fall and weight, we will have to deal with the issueof experiment. It is sometimes said that there were simple, obviousand conclusive tests available to Aristotle, which would have shownhis views to be wrong. The fact that Aristotle did not conduct thesetests has led some commentators to draw adverse conclusions abouthis methodology.5 I argue that given the nature of Aristotle's project,the tests we find so seductive may not have been so simple, obviousor conclusive. Finally, there is a question of the sort of physicist Aris-totle was. Some have argued that Aristotle gives us metaphysics ratherthan physics, while others argue that Aristotle was in fact a "workingphysicist" with a methodology closer to the more empirical biologi-cal works.' I argue that Aristotle is engaged in an important pro-gramme that constitutes an ancient analogue to theoretical physics,and that we need to make space for Aristotle, the theoretical physi-cist, between these two extremes.

IDid Aristotle have an integrated theory of force, mass and motionthat could be considered to be a dynamics? At no point does hepresent his relevant views as a whole. His comments are scatteredthrough the Physicsand De Caelo, and later works do not developthese or put them into a more systematic form.' Nor do his com-

5 E.g. S. Sambursky,ThePhysical Worldof the Greeks London, 1956), 89: "Aristo-tle's approach...failed to observe the right balance between induction and deduc-tion and came to be dominated largely by the latter";or G.E.R.Lloyd,EarlyGreekScience:Thales oAristotleNewYork,1970), 115: "Aristotle ailed to carryout certainsimple teststhat would have indicated the inaccuracyof some of his propositions".6 E. Hussey, "Aristotle's Mathemati'cal Physics," in Aristotle'sPhysics.A CollectionofEssays, d. L.Judson (Oxford, 1991), 213- 242, findsAristotlea "meticulousempiri-cist"(p. 240).

7 I take it as uncontroversial hat the Mechanicss bya later Aristotelian.


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ARISTOTLE, DYNAMICS AND PROPORTIONALITY 3ments form the sort of coherent system we might expect. He is quiteaware that bodies in free fall move more quickly as they approachthe ground, and so accelerate:

There is proof that motion is not unlimited as earth moves more quickly thenearer it is to the centre, and fire more quickly the higher it is. If the move-ment were unlimited, so would be the speed unlimited, and if the speed,then the heaviness or lightness too." (De Caelo I/8, 277a27-30)8Yet Aristotle makes no attempt to integrate this insight with his otherviews or to develop a theory of acceleration. He is also distinctlyslack in his usage of key terms such as ioGxUgischus), 86vcaLg(dunamis), and 'onq (rhope),' quite in contrast, for example, to thedefinitions of key philosophical terms in MetaphysicsA.'o The com-ments on motion do not appear as conclusions that Aristotle arguesfor, but as premises useful for the generation of anti-infinity andanti-void conclusions." They often appear to be pitched at the wronglevel, appearing to be concerned with change in a more generalsense. Why then does Aristotle make these comments? Was he sin-gularly incompetent at constructing a dynamics? Or did he have amore fundamental purpose in mind?

IISome commentators have pointed out that prior to Aristotle, therewas nothing that could be described as dynamics.'2 Whilst true, thisunderestimates the problem. Aristotle sees the Eleatics as denyingthe possibility of change, and the Heracliteans as holding that every-thing is changing all the time. So perhaps the primary issue for Aris-totle is not a quantitative description of motion, but a coherent gen-eral theory of change. Such a theory for Aristotle has to be free from

8 Cf. Physics230b23-25, 265b12-16 and De Caelo277b5-9; see E. Hussey, Aristotle'sPhysicsBooksIII and IV (Oxford 1983), 199-200.9 See Drabkin, "Notes on the Laws," 72-73, G.E.L. Owen, "Method, Physics andCosmology", in Logic, Science and Dialectic. CollectedPapers in GreekPhilosophy,ed. M.Nussbaum (Cornell, 1985), 151-164, esp. p. 156. I will leave these terms untranslatedas the English renditions of ischus as "strength", dunamis as "force" or "power" andrhopeas "impulse" or "momentum" are apt to be misleading if taken in any technicalsense.0oMetaphysicsA12 and 01 do discuss dunamis, but in the broader sense of capac-

ity or power." See e.g. PhysicsIV/8, 215a14-18.12 Lloyd, Early GreekScience, 112, comments that "It is hardly an exaggeration tosay that before Aristotle there is nothing that can be called dynamics at all in Greekscience." Cf. Lloyd, Revolutionsof Wisdom,217.


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4 A. GREGORYinfinities and paradoxes, and in what follows I will try to show thathe pays particular attention to what might be called boundary con-ditions. One example of this is that unrestricted rest or change can-not be permitted, lest paradox result.'3Two points are worth noting,as in general form they will be thematic in what follows. Firstly, Aris-totle does not, and does not need to, specify the exact position ofthese boundaries. All he requires is that they exist. Secondly, theactual quantification of things that always change and always restdoes not arise as an issue. No direct empirical investigation will helphere, though empirical evidence may be relevant for cases wherecommonplace knowledge conflicts with an implication of a position,resulting in a paradox. Thus Aristotle's theories do not fly in theface of empirical evidence. His search is for what needs to be as-sumed in order to account for the phenomena.Within this context Aristotle's targets become well defined. Clearly,for example, some of Zeno's paradoxes are obstacles to be over-come.'" And so is the problem of the void in relation to motion.Without there being resistance provided by a medium, all bodies,whatever their weight, will move at the same speed, and that, accord-ing to Aristotle, cannot be so.'5He furthermore needs to argue againstcertain conditions that might generate infinities, as can be seen quiteclearly in the Physicsand De Caelo.'6 He is concerned to argue thatnothing finite can generate anything infinite, and that there can beno interaction between things finite and infinite. Furthermore, therecan exist no infinite or weightless bodies; there cannot be a void;nor can any agent of change or change itself be infinite, either inspeed or extent. Again, Aristotle does only what is required to avoidinfinities or paradoxes. He is not concerned, for instance, with es-tablishing what the greatest attainable speed might be, but only withestablishing that there cannot be infinite speed. All of the majorcomments on motion occur as premises in arguments that aim torefute theories of infinity or of the void." If this higher-level project

1S See e.g. PhysicsVIII/3, 253a32 ff., and cf. MetaphysicsIV/8.14E.g. the arrow,the Achilles and the stadiumparadoxes,on which see PhysicsVI.5 See PhysicsIV/ 6-9.16 See e.g. PhysicsI/2 184b15 ff. and De CaeloI/V 271b5 ff.

17 One might look at the following passages and what they argue for Physics215a24-216a21, no void; 266a10-b25, no finite magnitude can produce an infinite one; DeCaelo 273b30-274a18, no infinite weight; 274b33-275b4, no action between finiteand infinite, infinite and finite, or twoinfinites;277a13-blO,no infinite movement,speed or weight;301a21-b33,no weightlessbody. There are some passageswhere


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ARISTOTLE, DYNAMICS AND PROPORTIONALITY 5is Aristotle's main concern, this may help to explain why the com-ments on motion are so scattered and disparate.Underneath all of this lies an Aristotelian key principle. His con-clusions ultimately rely on a relationship of proportionality (thoughnot necessarily one of direct proportionality) between the factorsdetermining change:

In conclusion then, the basic principle is clear. There is alwaysa ratio be-tween changes, for they are in time, and between two determinate timeperiods there is alwaysa ratio, but there is no ratio between fullnessand thevoid. (Physics V/8, 216a8-11)18This principle is also critical in the struggle against Zeno in PhysicsVI, where a key argument is that it makes no sense to talk of achange at any instant of time, only of change between two instants,however small the separation.IIIThe generality of Aristotle's comments is significant. He uses lettersto represent the various factors determining change, and when hevaries these factors, he employs phrases such as "twice factor A leadsto half factor B", etc. It appears that such comments were meant tobe taken quite generally for all sorts of change ("heating, sweeten-ing, throwing or any sort of change")," not just for motion. So per-haps what Aristotle is primarily concerned with is that proportional-ity applies to all changes, rather than with the specific values in anyparticular case. This is by no means trivial in the context of the de-velopment of a fourth century theory of change designed to combatEleatics, Heracliteans and atomists, especially in relation to the wayin which Aristotle treats the factors involved in change. He believesthat key comparatives would lose their meaning if there were notsome relation of proportionality between them. Consider the wayinwhich pairs of comparatives such as heavier/lighter, quicker/slower,longer/shorter (distance) and shorter/longer (time) are related to

comments on motion are used as premises to establish other conclusions, notablytowards he end of DeCaelo see e.g. 294a12-23, 304bl11-23, 08b18, 309b15, 311a19-25). That is what one wouldexpect though; the primary unction in theseargumentsis to avoid infinities,but they are then used for other incidental purposes."8Cf. De Caelo 74a7 and Physics 38b29 ff.19De Caelo 75al ff., cf. Physics 50a28 ff., 266a26 ff.


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6 A. GREGORYeach other for Aristotle.20PhysicsVI/2 defines "quicker" to meanthat a greater distance is covered in the same time, or the samedistance in a shorter time. Take any two speeds, such that b is quickerthan a. They will map onto a pair of distances such that b' is furtherthan a'. That must be so for all pairs of speeds and their related pairsof distances. If this were not so, then there would be at least oneinstance where being quicker does not entail going further in thesame time. So the relation between speed and distance must be onethat is order preserving, and that means that there must be somerelationship of proportionality between speed and distance. Thatrelationship does not necessarily have to be one of direct pro-portionality. As long as we can write f (speed) a g (distance) [a func-tion of speed is proportional to a function of distance], then orderwill be preserved and the meaning of quicker/ slower, etc. will bepreserved.2' The idea that we can apply this analysis to all forms ofchange I shall call the "proportionality thesis". Aristotle's criticismof other theories of change is often that key comparatives wouldlose their meaning (e.g., with motion in a void). Aristotle has noneed to determine which relation of proportionality applies in eachcase; he merely needs to argue that in all cases there exists such arelation.

The proportionality thesis will not generate paradoxes or infini-ties unless zeros or infinities are fed into a proportionality relation.For any positive integer or fraction, we will get sensible results. Aris-totle has been careful to deny that there can be infinite mass, veloc-ity or force, and so too that there can be a medium with zero resist-ance or a body with zero mass. As I shall argue a little later, he deniesthe possibility of instantaneous change (change in zero time) as well.Thus by use of the proportionality thesis and careful considerationof boundary conditions Aristotle eliminates infinities from his gen-eral theory of change.

20 PhysicsVI/8 is also interestinghere, as quickerand slowerare here used onlyof that which occupies a period of time (238b29-31).21 Also note thatif c > b > a then c' >b' > a', and thatAristotlebelieves time andspace to be continua.


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ARISTOTLE, DYNAMICS AND PROPORTIONALITY 71VOne problem for the proportionality thesis might be the followingpassage from PhysicsVII/5:

If A is the mover, B is that which is moved, C is the distance moved, and Dis the time in which this occurs, then in the same time a UticqOtLsgdunamis)equal to A will move half B through twice C in half of D, and through C inhalf D; thus there will be proportion... but if A moves B the whole of C in D,half of A (call this E) will not move B, in time D or in some part of it, a partof C bearing the same proportion to the whole as A does to E. In fact, itmay be that it will not move it at all. For it is not the case that if the wholeioyg (ischus) produces a certain amount of movement, that half could moveit any specific distance in any time at all. Otherwise one man would be ableto move a ship, since both the LoXy6gischus) of the shiphaulers and the dis-tance through which they all move the ship can be divided into the samenumber. (Physics VII/5, 249b30-250al9.)We are faced with a possible breakdown of proportionality if a mancannot move a ship at all rather than move it very gradually. How-ever, while there may eventually be a breakdown as the task be-comes progressively harder, equally there may not. Nothing heredemands that a breakdown occur in all such cases. Lloyd com-ments that "This exception strongly suggests that the propor-tionalities adumbrated in this chapter are not intended to applystrictlyas general rules."22Wardy, in turn, comments on this passagethat we cannot:

Accommodate these cases within a purported systematic dynamics by con-struing them as exceptions to law-like generalisations. The central objectionis based on the fact that the exceptions are, in a crucial sense, unspecifiable.That is, these situations are not merely such that they invalidate an infer-ence: they further appear unsusceptible to any analysis which might isolatethose factors responsible for the withdrawal of the inferential licence.2'Certainly in the PhysicsVII/5 passage Aristotle does not specify un-der what conditions the proportionalities break down. However, ifI am right about Aristotle's general strategy, the reasons for thatare clear enough. In order to circumvent the difficulties faced inthis passage, specifically Zeno's millet seed paradox-if the drop-ping of one grain of millet makes no sound, why does dropping abushel of millet (i.e. many single grains together) make anysound?24-Aristotle does not need to make any such specification.

22 Lloyd, Revolutionsof Wisdom,221 (Lloyd's emphases).23 Wardy, Chain of Change, 317; and Lloyd, Revolution of Wisdom, 221, note 22.24 See Physics250a19-28.


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8 A. GREGORYNevertheless, this does not mean he believes that no such specifi-cations can be made.

Is the failure here due to something inherent in the moved, themover, or to the relation between the moved and the medium inwhich it moves?25In the absence of a medium, all velocities would beequal. There are no exceptions made here for small or smallerweights or lightnesses, nor for small or smaller forces applied to largeor larger masses.26So we must look to the relation between mediumand moved. Is that relation unspecifiable? Aristotle is aware that thereare certain apparent anomalies with his theory concerning the be-haviour of flat objects and very small objects. He says:

The shapes of bodies are not simply responsible for their motion down orup, but for their swiftness or slowness. It is not hard to see the reason forthis. The question here is why flat objects of iron or lead float on the sur-face of water, but others smaller and of less weight, be they round or elon-gated, like a needle, move downwards,and why certain things float in theair because of their smallness,such as gold dust and other earthyand dustythings. (De Caelo V/6, 313a14-21)

Here would seem to be a situation where Aristotle could simply saythat proportionality breaks down, particularly in relation to dust par-ticles in air, where he could simply repeat the PhysicsVII/5 formula.However, he decides for a rather more complex explanation:Since some continuous entities are more easily divided than others, and inthe same manner some things are better dividers than others, it is here wemay expect to find the reason. It is the easily bounded which is easily di-vided, and the more easilybounded the more easily divided, air more thanwater, and water more than earth. Further,a smaller amount of each kind ismore easilydivisible and separable.Thus bodies which have breadth remainwhere they are because they encompass a great amount, as the greater is noteasily parted, but bodies with the contraryshape move downwardsbecausethey cover little, and this is easier to divide. In air this happens much morereadily than water, in as much as it is more easilydivisible. Since the weighthas a certain toidg (ischus)to produce a downwardmotion, and the continu-ous body a certain LogXgischus)not to be parted, it is necessarythat theseoppose one another. For if the aoxu3gischus)to tear asunder and divide ofthe weight exceeds that of the continuous body, it will be forced down morequickly, but if it is weaker, then it will float. (De Caelo V/6, 313b7-22)

This explanation makes quite a good job of specifying the condi-tions under which an object will float or sink, at least theoretically.Floating would constitute a breach of the proportionalities set outat Physics 215a24 ff. Certainly this does not give us actual figures,

25See Physics V/8 215b1 ff. on the relation of mover, moved and medium.26 See Physics V/8, cf. De Caelo 01a ff.


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ARISTOTLE, DYNAMICS AND PROPORTIONALITY 9but then we should not expect that. As always, Aristotle merelywishes to remove an apparent anomaly, and does only what hebelieves is required for that. In doing so, he does set up a specifi-cation of the factors involved in principle, and it is clear enoughhow we might go on to be more precise. We could measure thebreadth of various bodies and their jXUo'gischus) for downwardmotion, and at least indirectly, we could measure the 'toxig (ischus)of various continua to hold together.VSo far we have mainly considered the case of bodies in motion, butthere is also the question of change of quality, &khowoLgalloiosis).One might try to distinguish sharply between the two, attempting topreserve a strict proportionality for (what we would call) dynamicscases while considering proportionality of any sort inappropriate fora&kk0otmgalloiosis). However, evidence given below and Aristotle'sdrive for a general theory of change militates against this. If we treatall these cases similarly, it may be that the difficulties of the treat-ment of akkomootg(alloiosis) spill over into the treatment of dynam-ics wrecking the possibility of proper proportionality.27Alternatively,it may be that the rigours of the analysis of bodies in motion giveproportionality a rather firmer grip on &kxhoLoag alloiosis) thanmight usually be allowed. While De CaeloIV/6 may give us a specifi-cation of where and why the proportionalities break down, there areother cases where such an analysis may be more problematic. This isbecause Aristotle applies his principles very broadly. PhysicsVII/5 isno exception here, where he asks:

Is this the same with qualitative change a&XoL6oe~ogalloioseos) and growth?Yes, for there is an increasor and a thing increased, in a certain amount oftime and to a certain extent the increasor affects the increased. So too thereis an alterer and an altered, and a certain amount of change, according tothe greater and the lesser, taking place in a certain amount of time. So intwice the time twice the change occurs, and twice as much change occupiestwice the time.28 (Physics VII/5, 250a28-bl)

27 See Wardy, Chain of Change.28 Cf. De Caelo274b34 ff. Aristotle goes on to point out that if we have twice theagent of change we can expect twice the change to take place in the same time, butif we have only half it may be that no change will occur, as with the previous exam-ples, thus drawing a tight parallel between cases we might catalogue as dynamics and

dXkkoLootgalloiosis), respectively.


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This might be thought to raise two sorts of difficulties, one aboutquantification and an allied one about specification of the break-down of proportionality. As Lloyd points out, Aristotle had "an es-sentially qualitative conception of the hot/ cold spectrum."29 Also,as Owen has commented, the Greeks had no thermometers and verylittle in the way of devices for "translating qualitative differences intoquantitative ones"."On the question of quantification, and the relation of to6Toov(to poion, quality) and ~i6oov (to poson, quantity), we might tryto take posos in rather a broader sense in this passage, to includenot only change that is quantifiable in a numerical manner, butchange that can be specified by comparatives too. Note that thisdoes not require us to say that a certain amount of one quality isequal to, greater than, or less than a certain amount of anotherquality. All that is required is that we can use these comparativesof the same quality. If we have a change involving two qualities, ifwe employ a greater amount of one quality, a greater amount ofthe other quality results. There is a reasonable way in which we cansay that there is a relationship of proportionality here, at least ofthe sort that Aristotle is interested in. So relating qualities togetherin the same way as we did for quantities earlier, for quality 1, whereb is qualitatively greater than a (e.g. b is hotter than a), then forquality 2 there is a corresponding b' which is qualitatively greaterthan a'. The key here again is order preservation such thatcomparatives retain their meaning.The English word "proportion" has very strong mathematical con-notations, although here, too, it is possible to speak of "a punish-ment in due proportion to the crime". The words that Aristotle typi-cally uses (vdahoyog, analogos; X6yog, logos;6oog, hosos) perhaps donot have such strictly mathematical connotations, and he may infact have something broader than a strict mathematical proportion-ality in mind. What Aristotle requires is that there be a k6yog (logos),in the sense of account or explanation, for all sorts of change. Insome specific situations, where quantification is indeed possible,k6yog (logos) can be a strict mathematical proportionality. Wherequalitative judgements are required, the proportionality thesis stillapplies in order to generate a k6yog (logos) where the meaning ofcomparatives can be preserved.

29Lloyd,Revolutionsf Wisdom,22.so Owen, "AristotelianMechanics,"157.


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ARISTOTLE, DYNAMICS AND PROPORTIONALITY 11

There are several examples where Aristotle is quite happy to talkof AXoLWotLg (alloiosis) usingjust "greater than", "equal to" and "lessthan". Elsewhere in the Physics,we read: "Let the greater dunamisalwaysbe the one which accomplishes the same act in a shorter time,whether that be heating, sweetening, throwing or any sort of change"(PhysicsVIII/10 266a26-28)." Rather more specific about what wemay be able to say about qualitative change is the following passagefrom De Caelo:

Suppose thatA is heated, pushed, affected in anywayor changed byB in a timeC; et Dbe less than B,and let us assume that the lesser createsa smallerchangein an equal given time. Suppose E to be that which is altered (dhkkotW~'evov,alloiomenon)y D. As D is to B, so will E be to some limited amount. We mayassume that in the same time an equal force will effect an equal change ('CoovXkkotoov,

sonalloioun),a lessera lesserin anequal time,and a greatera greaterand so in proportion of the greaterto the lesser. (DeCaelo /7, 275a1-10)A further reason for quoting these passages is that I do not think wecan single out PhysicsVII/5 as being exceptional in the way that itdraws qualitative change into the general scheme of proportionalityfor Aristotle.32 Physics 250a28 ff. might also be thought to raise aproblem about specification, in that if we cannot specify quantita-tively where the breakdown in proportionality occurs, then we can-not treat the proportionalities as general rules with specifiable ex-ceptions. However, perhaps we can extract a general principle fromDe CaeloIV/6, where we have the opposition of the LoxUIgischus) fordownward motion and that of the continuum to avoid being split.When the koXi'g ischus) of the changer to create change fails to ex-ceed the oLoYXgischus) of the changed not to be changed, then nochange occurs. Now that may be somewhat unhelpful for the deter-mination of precise figures, but the general principle is clearenough.33Here I would stress that I am not attributing any understandingof inertia or friction to Aristotle. His comments are quite generaland are operating on a level above that of dynamics. They imply,among other things, that for some sorts of changes there is athreshold to be overcome. And of course there were situationswhere a practical determination of thresholds would have beenalmost insuperably difficult, and certainly so for the ancient

s' Cf. Physics250a28 ff. and De Caelo275 a32 ff.32 Contra Wardy, Chain of Change,328.3 Cf. De Motu Animalium, 699b6-8; see Hussey, "Aristotle's Mathematical Phys-ics," 219-220, and Wardy, Chain of Change, 303, note 5.


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ARISTOTLE, DYNAMICS AND PROPORTIONALITY 13can in fact be read in terms of simultaneous change. Take De sensu446b28 ff., where Aristotle draws a distinction between motion, wherethe change first has to pass through half of the total distance, andahokoLotg (alloiosis), where change may take place all at once in theway in which water may freeze all at once. However, he adds that thisis not necessarily so in the case of a large body being heated or fro-zen, as one part may change first and then change others. One mightread this passage as dealing with instantaneous change, but if all theparts are affected in the same way at the same time, then there maybe simultaneous but not instantaneous change. If not then the changewill not be simultaneous but will spread from the most affected area.As it is clearly easier for a puddle to be affected in a uniform mannerthan a lake, we see the spreading effect more readily in large bodies.Nor does anything in PhysicsVIII/3 compel us to accept the ideaof instantaneous change. Here the stated intention (253a22-253b6)is to show that it cannot be the case that everything is always at rest,or that everything is always changing. What Aristotle wishes to denyis that all things apparently at rest are in fact changing, a thesis hemay have attributed to Heraclitus. This intention stands behind boththe example of the stone being split by roots and the reference tothe shiphaulers, where, Aristotle argues, there really is rest prior tothe beginning of change. We merely need to recognise that in somecases there is a threshold to be overcome before change begins tooccur.

In sum, then, the evidence for instantaneous change in Aristo-tle is far from compelling. There seems little reason to saddleAristotle with this view, especially as it would contradict the sophis-ticated analysis of PhysicsVI and would reintroduce problems withcomparatives.VIIWhereas we might accept that Aristotle is concerned with a generaltheory of change, we might still wish to insist that he is quite specificabout the motion of free fall. After all, he comments that:

If a certain weight moves a certain distance in a certain time, a greaterweight will move (that distance) in less time, and the proportion which theweights have, the times will have conversely, thus if half the weight will covera certain distance in a certain time, the whole weight will do so in half thistime. (De Caelo 1/6, 273b30-274a2.)


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14 A. GREGORY

Why does Aristotle commit himself here when he need not do so? Itmay simply be that he believes that this proportionality accountsreasonably well for the observed phenomena.36 But is this a goodanswer, considering that there might have been simple, evident andconclusive experiments to show this view to be false? So why wouldAristotle not have done these experiments? Here I want to set asidetwo sorts of reply, the first being psychological-it is rare to findanyone doing experiments to show their own views to be false-andthe second methodological-this is an example of Aristotle's moregeneral failure to appreciate the importance of experiment. In or-der to investigate how matters may have looked to Aristotle, let usreverse the usual question. Instead of asking why he did not do theexperiments we find so compelling, let us ask why he should havedone them.

Does he have any pressing need of an experimental investigationof free fall motion? It is evident to Aristotle that there is some rela-tion of proportionality between weight and the speed of fall, andthat is all he needs. He also has several commonplace observationsthat he believes support his view. In air, heavy objects do drop morequickly than light ones, and larger raindrops fall more swiftly thansmall ones. There is no perceived theoretical crisis or difficulty inAristotle's position that demands an empirical investigation.Nor is there anything counter-intuitive about Aristotle's dynam-ics, which might have led him to carry out further investigations. Ifanything, the opposite is true. In this context, consider the standarddemonstration experiment in elementary dynamics. In order to showthat all bodies fall at the same rate in a vacuum, a feather and apenny are placed in a vacuum and dropped together. Why is itthat this experiment has such didactic force? It is precisely becausethe result is counter-intuitive and needs to be seen to be fully be-lieved.

VIIIHowever, it might be argued that doing such tests is obvious-that,as it were, the motivation comes from the tests themselves. But werethese tests obvious to Aristotle? Which experiments should he have

36 For a surveyof the context and natureof Aristotle'svarioususesof proportion-ality,see Hussey,Aristotle'sPhysicsII andIV,186-188.


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designed, and which criteria and parameters should he have used?When we design experiments, we know which factors to maximise(weight difference) and which to minimise (air resistance) in orderto show Aristotle to be incorrect; and hence we end up with thefeather and penny in a vacuum. As all of Aristotle's comments in-volve double and half, the simplest and most evident test is to taketwo objects, one double the weight of the other but roughly the samesize, and drop the heavier from twice the height of the lighter. Inter-estingly, there is so little difference between the times that the ob-jects hit the ground that one might well think this confirms ratherthan refutes Aristotle's views. It might be suggested though that thisis not the right experiment to do. Rather, we should be droppingweights in a ratio of ten to one. But then, why that experiment ratherthan 'Aristotle's'? Just because we know the critical factors and howto exploit them?While we may accept that Aristotle had no means of producingany vacuum suitable for experimentation and hence had no way oftesting his own claim that this phenomenon was indeed impossible,what about experimental possibilities of diminishing or nullifyingair resistance? Assuming that there were no satisfactory experimentswith air alone, could he not have employed less viscous or moreviscous media? Given Aristotle's situation, I see no reason why heshould have opted for any less viscous medium, given the difficultiesof managing an experiment in air. More satisfactory tests could beconducted in a more viscous medium. And indeed, atDe Caelo294b4ff., Aristotle mentions the commonplace observation that earth sinksin water, and the greater the piece the quicker it sinks."Aristotle's definition of weight is also important here. Considerthe passage in De Caelo 273b30 ff., where an object with twice theweight is said to fall twice as quickly, or consider the following pas-sage:

We call absolutely light then that which moves up to the extremity, andabsolutely heavy that which moves down to the centre. By relatively (nQo6gkkho; ros allo) light and lighter we mean which of two things having weight

7 Another option for producing manageable speeds would have been for Aristo-tle to think up something like Galileo's inclined planes experiment. However, Aris-totle would have been suspicious of such a combination of natural and enforcedmotion, and there is indeed a questionable leap in Galileo's reasoning from steepplanes to free fall. On a plane the motion is rolling or sliding, and there is friction,while there are none of these in free fall. This makes the conceptual move from thesteep plane to free fall suspect.


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16 A. GREGORYand the same size moves naturallydownwardsmore quickly. (DeCaelo V/2,308a28-32)"3

For Aristotle, to be heavier is to fall more quickly. If something doesnot fall more quickly, it is not heavier. If so, one can see why Aristo-tle would not conceive of the ten-to-one experiment. Quite simply,if a body does not fall at ten times the rate, then it is not ten times asheavy.Aristotle recognises that bodies in free fall accelerate. Does thatmean that he also believes that they get heavier? If he does, then hemight quite reasonably believe that the simple proportionality be-tween weight and speed in free fall holds, but the speed and weightincrease pro rata. There is even a commonplace observation thatmight suggest this. If you catch something falling from a consider-able height, it feels heavier at the moment that you catch it, com-pared to the weight it seems to have once you have caught it. Now itmay well be the case that Aristotle has not thought through all theimplications here, as he is interested in a general theory of changeand not dynamical minutiae. However, there is an important pointemerging here: Aristotle's recognition of acceleration in free fallmeans that experiments needed to become considerably more so-phisticated in order to show him to be conclusively incorrect.39The sophistication of equipment and experimental techniquesavailable to Aristotle must therefore be our final consideration. Physi-cal experiments could not but remain very rough in the context ofancient Greece. There were no clocks to measure accurately the timeof fall, and no means of guaranteeing simultaneous release. If onealters the experiment to get simultaneous release, by dropping theobjects from the same height, then one has the problem ofjudging,in the absence of clocks, whether the difference in drop time is sig-nificant. While the simple and evident tests are by no means conclu-sive, for the tests to be conclusive, they must stop being simple orevident. But again, we must ask here what motivation there wouldhave been for a programme of complex and precise experimentsgiven that observation and simple experiments both supportedAristotle.

38Cf. De Caelo 77b3.9 As long as acceleration is proportional to something, whether proximity tonaturalplace (DeCaelo 77a30, 277b5-9?),or distancefrom the startingpoint (Phys-ics 265b12-16?), then a new proportionality relation could be formulated. SeeSambursky 1956) p. 76 ff.


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ARISTOTLE, DYNAMICS AND PROPORTIONALITY 17IXAnother point of criticism might be that Aristotle's programme ap-pears to be largely one of definition and does not have an adequateempirical content. It is important to separate two claims here, viz.that Aristotle constructs his views a priori, and that his views are notempirically testable. To a certain extent the former is true, thoughAristotle does cite some commonplace observations. The generalproportionality thesis may indeed be in principle untestable. Thebest defence here is to point out that Aristotle is not alone in thismatter. Consider Newton's first law of motion: "Everybody preservesin its state of being at rest or of moving uniformly straight forward,except insofar as it is compelled to change its state by forces im-pressed upon it.'"40 s this something that is strictly empirically test-able? Hanson comments that:

Although we have no reason for supposing that particles free of unbalancedforces do exist, the law tells us what would happen if they did. This has thedoubly awkward consequence that (1) we cannot experimentally investigatethe properties of such bodies (2) the law, being a hypothetical claim, can-not be shown to be false.4'Let us also consider Newton's second law, the famous formula F=ma.One might argue that this too is not an empirical law, but is rather adefinition of the relation between the concepts of force, mass andacceleration in Newtonian mechanics.42 The rest of Newtonian me-chanics are then constructed around these basic conceptual foun-dations, the adequacy of which is judged by how well the systemdeals with the phenomena.In the same vein, we may also deal with one of the fundamentalprinciples of special relativity: "If, relative to K, K' is a uniformlymoving co-ordinate system devoid of rotation, then natural phenom-ena run their course with respect to K' according to exactly the same

40 I. Newton, Philosophiaenaturalis principia mathematica (London, 1726), lex 1.4~ N.R. Hanson, "Newton's First Law: A Philosopher's Door into Natural Philoso-phy," in BeyondtheEdge of Certainty,ed. R.G. Coldony (Pittsburgh, 1983), 6-28, p. 13(Hanson's emphases). Also see B. Ellis, "The Origin and Nature of Newton's Laws ofMotion," in ibid., 29-68, pp. 31-41 on the first law. A particle free from unbalancedforces is one which is not "compelled to change its state".42 See Ellis, "Origin and Nature", 52-62; and Hanson, "Newton's First Law". Thefoundational problem here is very similar to Aristotle's. Such definitions cannot betested directly (what measure of force is there other than as the product of mass andacceleration?), only in relation to other definitions, which can give other means ofmeasurement.


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18 A. GREGORY

general laws as with respect to K."43Is this principle strictly empiri-cally testable? The driving force of special relativity is effectively thatthis principle apparently contradicts its other fundamental princi-ple, that all observers will measure the same speed for light in avacuum irrespective of the motion of either the observer or the lightsource. Special relativity then requires us to rethink our concepts oftime, length and simultaneity, in order to make these two funda-mental principles compatible. There is no question here that theconceptual foundations of Newton's mechanics are superior to thoseof Aristotle, and Einstein's superior to Newton's. They sequentiallygenerate systems, which are simpler, more comprehensive and moreaccurate in their treatment of the phenomena. However, all havesomething in common. Aristotle's proportionality thesis, Newton'sfirst law and the relativity principle are all subtle uniformity princi-ples which, though they may not in themselves be empirically test-able, serve as a basis to build a theory of motion.It is well known that Einstein performed no experiments in gener-ating his theory of special relativity, though he was of course awareof the experimental background in mechanics and other fields. Nordid Newton present his laws of motion as the result of empiricalresearch, though he too would have been well aware of much ex-perimental work. Ellis comments that "[n]o detailed experimentalevidence existed that would have warranted the acceptance of New-ton's laws of motion in the seventeenth century... Newton's lawswereprimarily conceptual in origin, in that they represented a new way ofconceiving of dynamical problems.""To argue along these lines is of course not to deride the work ofeither Newton or Einstein, but merely to emphasise how the historyof the development of theories of motion indicates that there is strongevidence for the idea that the foundational principles of those theo-ries are neither empirically derived, nor in the strictest sense empiri-cally testable. It is possible to imagine that Aristotle, Newton andEinstein were all wrong in pursuing physics in their respective man-ners. But this implies that we can hardly single out Aristotle for criti-cism. Of course, one may engage in blaming his theory by compar-ing it to contemporary rival models. But I strongly favour the alter-native conception, which is to claim that the activityhe engaged in isfundamental to physics, or at least to the earlier stages of the devel-

4 A. Einstein,Relativity: heSpecial nd theGeneralTheoryLondon, 1920), 13.44 Ellis,"Originand Nature",30-31.


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opment of physics. But if this is true, we have no reason for beingcritical of the nature Aristotle's programme, even though we may becritical, in proper historical context, of the actual definitions or theextent of his programme. That has important implications for howwe view Aristotle as a physicist. If between experimental physics andmetaphysics there is a role for the theoretical physicist (and I takeNewton and Einstein to be theoretical physicists in this sense), thenwe must ask how Aristotle fitted into that role.

XIf it is the case that Aristotle is engaged in a higher level project thanthe construction of a dynamics, then this explains much about thenature of his comments on motion. How ought we to evaluate thisproject? There are a good number of positive things we can sayaboutit. First, the breadth of this project is certainly impressive. Whileseveral presocratics had limited theories of change, Aristotle at-tempted to show the limitations of those theories and to establish ageneral theory of his own. Second, he tackled this matter at a veryhigh level of abstraction and for the first time used symbols to repre-sent the various factors determining change, treating those factorsin an idealised manner and thereby generating an important frame-work for investigating change. The critical point is here that for everychange there is a rate of change, so that after Aristotle, the debatecould move to consider what rate of change it really was. Havingspoken of generality and abstraction, it is important for us to distin-guish between the contributions made by Aristotle and by Zeno.Whether Zeno's project was a destructive one in support of Parme-nides or instead a deliberate provocation is not a matter I can dealwith here. What may be said, however, is that Zeno presented diffi-culties in formulating a theory of motion in a highly general andabstract manner, with a significant move towards a mathematicaltreatment, which Aristotle then built upon or reacted to. The latter,however, dealt with these issues at another level of generality in at-tempting to create a general theory of change free of paradoxes andinfinities.

What sort of physicist was Aristotle then? One of the underlyingthemes of this paper has been the depiction of Aristotle, at leastwhen he is dealing with these sorts of questions, as someone en-gaged in an ancient analogue to theoretical physics, rather than as


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20 A. GREGORYan empirical physicist or a metaphysician. What drives theoreticalphysics? I would suggest that it is an interest in foundational princi-ples, a quest for ever more general theories and a drive to removeparadoxes and anomalies from theories. Clearly Aristotle was inter-ested in foundational principles, and one of the important aspectsof his work is that he may well have been the first person to formu-late such principles clearly and comprehensively. In fact, he had tospend considerable time and effort on those principles and theirimplications for the simple reason that this was virtually a green fieldsite. Those foundations, in the form of his proportionality thesis,run somewhat deeper than is generally recognised.The quest for more general theories has alwaysbeen driving phys-ics, but has become more evident with the unification of varioustheories in the nineteenth century, and with the conscious quest fora unification of general relativity and quantum mechanics in thetwentieth.45 One thing I hope to have brought out in this paper isjust how general Aristotle intends his theory of change to be, andthat he intends a unification of various sorts of quantitative and quali-tative change in one comprehensive theory.The drive for unification in twentieth century physics has also high-lighted another important aspect of theoretical physics, which is theassault on theoretical anomalies, and in particular what might becalled 'infinity busting'. The central task of the unification of relativ-ity and quantum mechanics has long been beset with difficulties in-volving infinities. These have to be arbitrarily ignored in order tomake sensible calculations, and physicists strive to produce theoriesthat avoid these difficulties.46 One of the central questions in theo-retical cosmology is whether there can be such a thing as a singular-ity, which because of its zero volume generates infinities.47 A signifi-cant part of Aristotle's programme isjust this sort of infinity busting.He is concerned to remove anything that would generate infinitiesfrom his theory, as such infinities would be ikoyog (alogos). One ofhis key criticisms of other theories is that they would allowjust such

45 See, for example, the unification of electromagnetism (itself a unification)with optics, and of both of these with kinematics to form special relativity.46 The processof 'renormalisation'.The firstidea of quantisationcan be seen as

a response to an infinity paradox, the ultra-violetcatastrophein black body radia-tion.47 I.e. if a singularityhasanymass it has infinite density,any energyit has infinitetemperature,etc. IwouldarguethatAristotlewouldcome out againstthe possibilityof a naked singularity.


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ARISTOTLE, DYNAMICS AND PROPORTIONALITY 21infinities. It is quite possible then to see Aristotle's quest for a gen-eral theory of change as an important ancient analogue to theoreti-cal physics.I have made no comment on the development of Aristotle'sthought in this paper. It may well be the case that his methodologyor interests changed during his career. It may even be the case thathis scientific work became more empirical as he grew older. How-ever, if Aristotle did do good work in theoretical physics, as I haveargued here, then we need to be suspicious of the assumption thatAristotle's most empirical scientific work was also his best.48

SUMMARYWhat ought we to make of Aristotle'sapparentlydisparate comments on bodiesin motion? I argue that Aristotle is concerned with a higher level project thandynamics and that is the establishment of a coherent theory of change in gen-eral. This theory is designed to avoid the paradoxes and infinities that Aristotlefinds in Eleatic, Heraclitean and atomist accounts, notably in relation tocomparativessuch as 'quicker' and 'slower'.This theory relies on a broad appli-cation of proportionality to all typesof change, not merely those we would label'dynamics'.To support this I argue that Aristotle denied the existence of the voidand the possibility of instantaneous change, and that he could accommodate'threshold' changes within his scheme. If this is so, then the aims of Aristotle'scomments on motion become more comprehensible, and it will be understand-able whyAristotle was more concerned with the application of proportionalityingeneral rather with the investigationof specific cases in dynamics.

48Mythanks to Prof. R.W.Sharpleswho commented on an earlier draft of thispaper, and to ESM's woanonymousreaders.

aristotle dynamics and proportionality

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