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Chapter 1
Plato versus Aristotle
A. Plato
1. The Socratic background^1
Plato’s impetus to philosophize came from his association with Socrates, and Socrates was preoccupied with questions of ethics, so this was where Plato began. A point which had impressed Socrates was that we all used the notions of goodness and beauty and virtue – and again of the particular virtues such as courage, wisdom, justice, piety, and so on – but could not explain them. Faced with a question such as ‘what really is goodness?’, or ‘what really is justice?’, we soon found ourselves unable to answer. Most ordin- ary people would begin by seeing no real problem, and would quite confidently offer a first-off response, but Socrates would then argue very convincingly that this response could not be right. So they would then try various other answers, but again Socrates would show that they too proved unsatisfactory when properly examined. So he was led to conclude that actually we did not know what we were talking about. As he said, accord- ing to the speech that Plato gives him in his Apology (21b–23b), if there is any way in which he is wiser than other people, it is just that others are not aware of their lack of knowledge. On the questions that concern him he is alone in knowing that he does not know what the answers are. It may be disputed whether and to what extent this is a fair portrait of the actual Socrates, but we need not enter that dispute. At any rate it is
(^1) This section involves several conjectures on my part. I have attempted to justify them in my (1986, pp. 94–101).
2 Philosophy of Mathematics
Plato’s view of Socrates, as is clear from Plato’s early writings, and that is what matters to us. Now Socrates was not the only one to have noticed that there were such problems with the traditional Greek ethics, and other thinkers at the time had gone on to offer their own solutions, which were often of a subjec- tivist – or one might even say nihilist – tendency. For example, it was claimed that justice is simply a matter of obeying the law, and since laws are different from one city to another, so too justice is different from one city to another. Moreover, laws are simply human inventions, so justice too is simply a human invention; it exists ‘by convention’ and not ‘by nature’. The same applies to all of morality. And the more cynical went on: because morality is merely a human convention, there is no reason to take it seriously, and the simple truth of the matter is just that ‘might is right’.^2 Socrates did not agree at all. He was firmly convinced that this expla- nation did not work, and that morality must be in some way ‘objective’. Plato concurred, and on this point he never changed his mind. He always held that when we talk about goodness and justice and so on then there is something that we are talking about, and it exists quite independently of any human conventions. So those who say that might is right are simply mistaken, for that is not the truth about what rightness is. But there is a truth, and our problem is to find it. However, this at once gives rise to a further question: why does it seem to be so very difficult to reach a satis- fying view on such questions? And I believe that Plato’s first step towards a solution was the thought that the difficulty arises because there are no clear examples available to us in this world. Normally the meaning of a word is given (at least partly) by examples, for surely you could not know what the word ‘red’ means if you had never seen any red objects. But Plato came to think that the meaning of these words cannot be explained in the same way. One problem is that what is the right thing to do in one context may also be the wrong thing to do in another. Another problem is that we dispute about alleged examples of rightness, in a way in which we never seriously dispute whether certain things are red or one foot long, or whatever. In such cases we have procedures which are generally accepted and which will settle any dis- puted questions, but there are no such generally accepted procedures for
(^2) For some examples of such opinions see, e.g., the views attributed to Callicles in Plato’s Gorgias (482c–486c), and to Thrasymachus in Plato’s Republic (336b–339a). See also the theory proposed by Antiphon in his fragment 44A (D/K).
4 Philosophy of Mathematics
that are equal to it, to give a larger square that is four times the given area. Then draw in the diagonals of the squares as shown:
It is now easy to argue that the central square, formed by the four diag- onals, is twice the area of the one that we began with. As Plato describes the lesson, at each step of the proof Socrates merely asks the boy some question, and the boy provides the correct answer, as if this at least is a point that he already knows. (For example: does the diagonal of a square bisect that square into two equal halves? – Yes. And what do we get if we add four of those halves together? And so on.) The moral that we are invited to draw is that the boy already possessed the knowledge of each individual step in the argument, and so Socrates’ questioning has merely brought back to his mind some result that he really knew all along. Here Plato is evidently exaggerating his case, for although our slave-boy might have known beforehand all the premises to the proof, there is obviously no reason to think that, before Socrates questioned him, he had ever put those premises together in the right way to see what conclusion followed from them. But this criticism is of no real importance to the example. We may press home the force of Plato’s illustration in several ways that Plato himself fails to emphasize. First, it is obvious to all readers that Socrates must know the answer to the problem before he starts, and that that is how he knows what figure to draw and what questions to ask about it. But this cannot be essential, for it is obvious that whoever first discovered this geometrical theorem did not have such help from someone else who already knew it. As Plato might have said, one can ask oneself the right questions, for mathematical discovery is certainly possible. Second, although Plato insists that the slave-boy was just ‘drawing knowledge out of himself ’, he does not insist as he might have done that this knowledge cannot be explained as due simply to his perception of the diagram that Socrates has drawn in
Plato versus Aristotle 5
the sand. At least three reasons might be given for this: (i) the diagram was no doubt somewhat inaccurate, as diagrams always are, but that does not prevent us from grasping the proof; (ii) since it is indeed a proof that we grasp, we can see on reflection that the result will hold also for all other squares, whatever their size, and not just for the one drawn here; (iii) more- over, we see that this is a necessary truth, and that there could not be any exceptions to it, but no one diagram could reveal that. So one moral that Plato certainly wishes us to draw is that mathem- atics uses proof , and that a proof is seen to be correct a priori , by using (as he might say) the ‘eye of the mind’, and not that of the body. Another moral that he probably wishes us to draw is that in this example, as in other cases of mathematical proof, the premises are also known a priori (e.g. one knows a priori that the diagonal of a square bisects it into two equal halves). If so, then mathematics is a wholly a priori study, nowhere relying on our perceptions. (But we shall see that if he did think this when he wrote the Meno – and I guess that he did – then later in the Republic he will change his mind.) A final moral, which he certainly draws explicitly to our atten- tion, is that since we ‘draw this knowledge out of ourselves’ (i.e. not rely- ing on perception), we must have been born with it. It is, in later language, ‘innate’. From this he infers further that the soul (or mind, i.e. psych Å) must have existed before we were born into this world, and so is immor- tal. But the Meno is itself rather evasive on how a previous existence might explain this supposedly innate knowledge, and for this I move on to my next passage.
(ii) Phaedo 72e–77d The theme throughout Plato’s dialogue Phaedo is the immortality of the soul. Several arguments for this are proposed, and one of them begins by referring back to the Meno ’s recollection theory. But it then goes on to offer a rather different argument for this theory. Whereas the Meno had invoked recollection to explain how we can come to see the truths of mathematics (i.e. by means of proof), in the Phaedo its role is to explain how we grasp the concepts involved. Although the example is taken from mathematics in each case, still the application that is ultimately desired is to ethics, i.e. to such concepts as goodness and justice and so on. The example chosen here is the concept of equality, and the overall structure of the argument is completely clear. The claim is that we do under- stand this concept, but that our understanding cannot be explained as due to the examples of equality perceived in this world, for there are no
Plato versus Aristotle 7
in this world are ‘defective’, because they are also examples of inequality, so they cannot explain what we do in fact understand. It is clear that we are expected to generalize: there are no satisfactory examples in this world of any of the things that mathematics is really about. For example, arithmetic is about pluralities of units, units which really are equal to one another in all ways, and are in no way divisible into parts, but there are no such things in this world ( Republic , 525d–526a; cf. Philebus 56d–e). Or again, geometry is about perfect squares, circles, and so on, which are bounded by lines of no thickness at all, and the things that we can perceive in this world are at best rough approximations ( Republic , 510d–511a). Mathematics, then, is not about this world at all, but about what can metaphorically be called ‘another world’. So our understanding of it can be explained only by positing something like an ‘experience’ of that ‘other world’, which on this theory will be something that happens before our birth into this world. I should add one more detail to this theory. The recollection that ordin- ary people are supposed to have of that ‘other world’ is in most cases only a dim recollection. That is why most of us cannot say what goodness is, or what justice is, or even what equality is. We do have some understanding of these concepts, for we can use them well enough in our ordinary thought and talk, but it is not the full understanding that would enable us to ‘give an account’, i.e. to frame explicit definitions of them. So the philosopher’s task is to turn his back on sense-perception, and to search within himself, trying to bring out clearly the knowledge that is in some sense latent within him. For that is the only way in which real understanding is to be gained. We know, from the case of mathematics, that this can be done. This gives us reason to hope that it can also be done for ethics too, for – as Plato sees them – the cases are essentially similar: there are no unam- biguous examples in this world, but we do have some (inarticulate) understanding, and only recollection could explain that.
3. Platonism in mathematics
Henceforth I set aside Plato’s views on ethics. What is nowadays regarded as ‘Platonism’ in the philosophy of mathematics has two main claims. The first is ontological: mathematics is about real objects, which must be regarded as genuinely existing, even though (in the metaphor) they do not exist ‘in this world’. This metaphor of ‘two worlds’ need not be taken too
8 Philosophy of Mathematics
seriously. An alternative way of drawing the distinction, which is also present in Plato’s own presentation, is that the objects of mathematics are not ‘perceptible objects’, but ‘intelligible objects’. This need not be taken to mean that they exist ‘in a different place’, but perhaps that they exist ‘in a different way’. The main claim is just that they do exist, but are not objects that we perceive by sight or by touch or by any other such sense. The second claim is epistemological: we do know quite a lot about these objects, for mathematical knowledge genuinely is knowledge , but this knowledge is not based upon perception. In our jargon, it is a priori know- ledge. This second claim about epistemology is quite naturally thought of as a consequence of the first claim about ontology, but – as I shall explain at the end of this chapter – there is no real entailment here. Similarly the first claim about ontology may quite naturally be thought of as a con- sequence of the second, but again there is no real entailment. However, what is traditionally called ‘Platonism’ embraces both of them. Platonism is still with us today, and its central problem is always to see how the two claims just stated can be reconciled with one another. For if mathematics concerns objects which exist not here but ‘in another world’, there is surely a difficulty in seeing how it can be that we know so much about them, and are continually discovering more. To say simply that this knowledge is ‘ a priori ’ is merely to give it a name, but not to explain how it can happen. As we have seen, in the Meno and the Phaedo Plato does have an account of how this knowledge arises: it is due to ‘recollection’ of what we once upon a time experienced, when we ourselves were in that ‘other world’. This was never a convincing theory, partly because it takes very literally the metaphor of ‘two worlds’, but also because the explana- tion proposed quickly evaporates. If we in our present embodied state cannot even conceive of what it would be like to ‘experience’ (say) the number 2 itself, or the number 200 itself, how can we credit the idea that we did once have such an ‘experience’, when in a previous disembodied state, and now recollect it? Other philosophers have held views which have some similarity to Plato’s theory of recollection, e.g. Descartes’ insistence upon some ideas being innate, but I do not think that anyone else has ever endorsed his theory. Indeed, it seems that quite soon after the Meno and the Phaedo Plato himself came to abandon the theory. At any rate, he does not mention it in the account of what genuine knowledge is – a discussion that occu- pies much of the central books of his Republic , which was written quite soon after. Nor does it recur in his later discussion of knowledge in the
10 Philosophy of Mathematics
this relation also holds within each realm, e.g. as some visible things are images (shadows or reflections) of others. To represent this we are to con- sider a line, divided into two unequal parts, with each of those parts then subdivided in the same ratio. The one part represents the visible, and the other the intelligible, and the subdivisions are apparently described like this:
The stipulation is that A is to B (in length) as C is to D, and also as A + B is to C + D. (It follows that B is the same length as C, though Plato does not draw attention to that point.) The main problem of interpretation is obvious from the labels that I have attached to this diagram. Plato certainly introduces A + B and C + D as representing objects that are either visible or intelligible. He also seems to describe the sections A and B as each representing objects (namely: in A there are shadows and reflections, e.g. in water or in polished metal; in B there are the material objects which cause such images, e.g. animals and plants and furniture and so on). But, when he comes to describe the relationship between C and D, his contrast seems to be between different methods of enquiry , i.e. the method used in mathematics and the method that he calls dialectic. The first line of inter- pretation supposes that what Plato really has in mind all along is distinct kinds of objects , so (despite initial appearances) we must supply different kinds of object for sections C and D. The second supposes that Plato is really thinking throughout of different methods , so (despite initial appear- ances) we must supply different methods for sections A and B. I begin with the first. There are different versions of this line of interpretation, but the best seems to me to be one that draws on information provided by Aristotle, though the point is not clearly stated anywhere in Plato’s own writings. 7 Aristotle tells us that Plato distinguished between the forms proper and the objects of mathematics. Both are taken to be intelligible objects rather than perceptible ones, but the difference is that there is only one of each
A Images
B Originals
C Mathematics 1 4 4 4 4 2 4 4 4 4 3
D Dialectic
Originals
Intelligible
1 4 4 4 2 4 4 4 3
1 4 4 4 2 4 4 4 3 1 4 4 4 4 2 4 4 4 4 3
Images
Visible
(^7) As recent and distinguished proponents of this interpretation I mention Wedberg (1955, appendix) and Burnyeat (2000).
Plato versus Aristotle 11
proper form (e.g. the form of circularity) whereas mathematics demands many perfect examples of each (e.g. many perfect circles). So the objects of mathematics are to be viewed as ‘intermediate’ between forms and per- ceptible things: they are like forms in being eternal, unchangeable, and objects of thought rather than perception; but they are like perceptible things insofar as there are many of each kind (Aristotle, Metaphysics A, 987 b14–18). If we grant this doctrine, it will be entirely reasonable to sup- pose that section C of the divided line represents these ‘intermediate’ objects of mathematics, while section D represents the genuine forms. But the problem with this interpretation is whether the doctrine should be granted. One must accept that there is good reason for holding such a view, and that Aristotle’s claim that Plato held it cannot seriously be questioned.^8 But one certainly doubts whether Plato had already reached this view at the time when he was writing the Republic , and I myself think that this is very unlikely. For, if he had done, why should he never state it, or even hint at it in any recognizable way?^9 Why should he tell us that what a geometer is really concerned with is ‘the square itself ’ or ‘the diagonal itself ’, when this is his standard vocabulary for speaking of forms (such as ‘the beautiful itself’, ‘the just itself ’)? If he did at this stage hold the theory of intermediates, would you not expect him here to use plural expressions, such as ‘squares themselves’? There is also a more general point in the background here. The theory which Aristotle reports surely shows the need to distinguish between a form, as a universal property, and any (perfect) instances of that property that there may be. But I do not believe that Plato had seen this need at the time when he was writing the Republic , for the confusion is clearly present at 597c–d of that work. 10 For these reasons I am sceptical of this first line of interpretation. Let us come to the second. This second approach is to see the simile as really concerned through- out with methods of enquiry, and it is easy to see how to apply this line of thought. The sections of the line should be taken to represent:
(^8) The claim is repeated in chapter 1 of his Metaphysics , book M, where he also describes how other members of Plato’s Academy reacted to this idea. He cannot just be making it up. (^9) Republic 525d–526a can certainly be seen as implying that in mathematics there are many ‘number ones’. But I doubt whether Plato had absorbed this implication. (^10) It seems probable that Plato later recognized this as a confusion, for the dialogue Parmenides constructs several arguments which rely upon it and which quite obviously have unacceptable conclusions. (The best known is the argument that, since Aristotle, has been called the argument of ‘The Third Man’. This is given at Parmenides 132a, cf. 132d–e.)
Plato versus Aristotle 13
100 years later, was so clearly an improvement on what had come before it that it eclipsed all earlier work. We do know that there had been earlier ‘Elements’, and presumably they were known to Plato, but they have been lost, and we cannot say how closely their style resembled what we now find in Euclid. (Euclid distinguishes the premises into definitions, common notions, and postulates. Both of the last two we would class as axioms.) I think myself that it is quite a possible conjecture that earlier ‘Elements’ did not admit to any starting points that we would call axioms, but only to what we would classify as definitions. So I think it is quite possible that the ‘hypotheses’ which Plato is thinking of were mainly, and perhaps entirely, what we would call definitions. 13 But in any case what he has now come to see is that mathematical proofs do have starting points, and that these are not justified within mathematics itself, but simply assumed. For that reason he now says that they do not count as known (in the proper sense of the word), and hence what is deduced from them is not known either. In the Meno mathematics had certainly been viewed as an example of knowledge, but now in the Republic it is denied that status. This is a notable change of view. It may not be quite such a clear-cut change as at first appears, for at 511d there is a strong hint that one could apply the dialectical method to the hypotheses of mathematics, thereby removing their merely hypothet- ical status. If so, then the implication is that mathematics could become proper knowledge, even though as presently pursued (i.e. at the time when Plato was writing the Republic ) it is not. In the other direction I remark that something like the method of mathematics, i.e. a method which (for the time being) simply accepts certain hypotheses without further justification, can evidently be employed in many areas, including an enquiry into the nature of the moral forms. In fact Meno 86e–87c attempts to do just that in its enquiry into what virtue is (though the attempt does not succeed ( Meno 86e–99c)); and Republic 437a invokes a ‘hypo- thesis’, which is left without further justification, and which plays an import- ant role in its analysis of what justice is. In broad terms, Plato thinks of the method of mathematics as one that starts by assuming some hypo- theses and then goes ‘downwards’ from them (i.e. by deduction), whereas the method of philosophy (i.e. dialectic) is to go ‘upwards’ from the initial
(^13) This would explain why the elucidation offered at 510c (previous note) simply men- tions certain concepts – i.e. concepts to be defined? – and does not give any propositions about them.
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hypotheses, finding reasons for them (when they are true), until eventu- ally they are shown to follow from an ‘unhypothetical first principle’. While the ‘downward’ method is something which we have no difficulty in understanding, it is not easy to say quite how the ‘upward’ method is supposed to work, but I cannot here pursue that problem.^14 At any rate, the upshot is this. Plato himself did not always remain quite the ‘Platonist’ about mathematics that I described in the previous section. This is because he came to think that mathematics (as presently pursued) begins from unjustified assumptions – or from assumptions that are ‘justified’ only in the wrong way, i.e. by appeal to visible diagrams – and that this means that it is not after all an example of the best kind of know- ledge. Perhaps he also thought that this defect could in principle be remedied, but at any rate he has certainly pointed to a problem with the usual ‘Platonic’ epistemology: proofs start from premises, and it is not clear how we know that those premises are true. As for the ontology, he remains always a ‘Platonist’ in that respect. Either in the Republic , or (as I think more probable) at a later date, he became clearer about just what the objects of mathematics are, namely not the forms themselves but perfect exam- ples thereof. But in any case they remain distinct from the ordinary perceptible objects of this world, accessible only to thought rather than perception, for they have a ‘perfection’ which is not to be found in this world. That is the main reason why, in the Republic , he lays down for the aspiring philosopher a lengthy and arduous preliminary training in mathematics: it is because this subject directs our mental gaze away from ordinary material things and towards what is ‘higher’. (It may be that he also thought that the philosopher’s ‘upward path’ towards an ‘unhypothetical first principle’ would start from reflection on the hypotheses of mathematics. But that is a mere speculation.^15 ) In brief, the Republic indicates at least a hesitation over the epistemo- logy, but no serious shift in the ontology. By way of contrast, let us now turn to Aristotle, whose views on both these issues were very different.
(^14) A classic discussion of Plato’s ‘method of hypothesis’, which assembles all the relevant evidence, is Robinson (1953, chapters 6–13). I have offered a few observations myself in my (1986, chapter 8). (^15) Book VII outlines five areas of mathematics, which are all to be studied, until their ‘kinship’ with one another is seen (531d). Is that perhaps because such a study will allow one to begin on the project of explaining the several initial hypotheses, by seeing how each may be viewed as an instance of some more general truth that explains them all?
16 Philosophy of Mathematics
that we can perceive. That is the broad outline: the truths of mathematics are truths about perfectly ordinary objects, but truths at a high level of generality. Compared with Plato, it seems like a breath of fresh air. On the topic of epistemology Aristotle similarly claims that our know- ledge of such truths is again perfectly ordinary empirical knowledge, based upon perception in much the same way as all other scientific knowledge is based upon perception. He quite naturally supposes that if mathem- atics need not be understood as concerned with a special and ‘separate’ kind of object, then equally our mathematical knowledge need not be credited to a special and rather peculiar faculty for ‘a priori’ knowledge. So both the Platonic ontology and the Platonic epistemology are to be rejected. That is the broad outline of his position. Unfortunately we do not find very much by way of argument for it. Certainly, Aristotle very frequently states objections to Plato’s general theory of forms, construed as objects which enjoy a separate existence of their own. But that is not the end of the argument. For, as we have already noted (p. XX), Plato eventually came to distinguish between the forms themselves and the objects studied in math- ematics, regarding these latter as ‘intermediate’ between forms proper and perceptible things. Granted this distinction, one might for the sake of argu- ment concede to Aristotle that he has good reason for rejecting the Platonic view of forms, but insist that the question of the separate existence of the objects of mathematics is not thereby settled. For these objects are not supposed to be forms, but to be perfect examples. And mathematics might still need perfect examples, which exist ‘separately’ from all the imperfect examples in this world, even if the same does not apply to the forms themselves. There is only one place where Aristotle seriously addresses this question, namely in chapter 2 of book M of the Metaphysics , and his arguments there are less than compelling. I here pass over all the details, noting only this one general point. The two main arguments that Aristotle gives, at 1076 b11–39 and 1076 b39–1077a14, aim to show that if we must assume the existence of those intermediates that Plato desires, then we must also assume the existence of many other intermediates too, which will lead to an incredible and quite needless duplication of entities. But he never tells us why Plato thought that these intermediates were in fact needed, nor how his reasons should be countered. As I have said, the usual explanation is this: Plato held that mathematics was about perfect examples, and so – since mathematics is true – there must be perfect examples. But no examples that we can perceive are perfect, so there must somewhere be impercept- ible examples, available to the intellect but not to perception. Let us
Plato versus Aristotle 17
assume that this is indeed Plato’s argument for supposing that his ‘inter- mediates’ are needed. Then the chief weakness in the counter-arguments that Aristotle presents in Metaphysics M.2 is that they have nothing to say about what is wrong with this Platonic argument. They simply do not address the opposition’s case. Nor is there anywhere else in his writings where this line of thought is explicitly considered. So I now turn to consider what we might say on Aristotle’s behalf.
6. Idealizations
It is a vexing feature of Aristotle’s discussion that we cannot even be sure of whether he himself did or did not accept the Platonic premise that there are no perfect examples in this world. His main discussions are quite silent on this point, and although there are a couple of asides else- where they are not to be trusted. 18 What he should have done is to accept the premise for geometry but deny it for arithmetic, so let us take each of these separately.
Arithmetic We, who have been taught by Frege, can clearly see that Plato was mistaken when he claimed that this subject introduces idealizations. The source of his error is that he takes it for granted that, when numbers are applied to ordinary perceptible objects, they are applied ‘directly’; i.e. that it is the object itself that is said to have this or that number. But Frege made it quite clear that this is not so. In his language, a ‘statement of number’ makes an assertion about a concept , not an object, i.e. it says how many objects fall under that concept. (An alternative view, which for present purposes we need not distinguish, is that numbers apply not to physical objects but to sets of those objects, and they tell us how many members the set has.) To illustrate, one may ask (say) how many cows there are
(^18) In the preliminary discussion of problems in Metaphysics B, we find the claim that perceptible lines are never perfectly straight or perfectly circular (997 b35–998a6). But in the context there is no reason to suppose that Aristotle is himself endorsing this claim, rather than mentioning it as a point that might appeal to his Platonist opponent. On the other side, a stray passage in De Anima I.1 apparently claims that a material straight edge really does touch a material sphere at just one point (403 a12–16). But I am very suspicious of this passage, for as it stands it makes no sensible contribution to its context. I have discussed the passage in an appendix to my (2009b).
Plato versus Aristotle 19
what geometry claims about perfect circles may very well be true even if there are no perfect circles at all, for the claims may be construed hypo- thetically: if there are any perfect circles, then such-and-such will be true of them (e.g. they can touch a perfectly straight line at just one point and no more). One might ask how geometry can be so useful in practice if there are no such entities as it speaks of, but (a) this is a question for the Platonist too (since ‘in practice’ means ‘in this perceptible world’), and anyway (b) the question is quite easy to answer. We are nowadays familiar with a wide range of scientific theories which may be said to ‘idealize’. Consider, for example, the theory of how an ‘ideal gas’ would behave – e.g. it would obey Boyle’s law precisely 21 – and this theory of ‘ideal’ gases is extremely helpful in understanding the behaviour of actual gases, even though no actual gas is an ideal gas. This is because the ideal theory simplifies the actual situation by ignoring certain features which make only a small difference in practice. (In this case the ideal theory ignores the actual size of the molecules of the gas, and any attractive or repulsive force that those molecules exert upon one another.) But no one nowadays could suppose that because this theory is helpful in practice there must really be ‘ideal gases’ somewhere, if not in this world then in another; that reaction would plainly be absurd. Something similar may be said of the idealizations in geometry. For example, a carpenter who wishes to make a square table will use the geometric theory of perfect squares in order to work out how to proceed. He will know that in practice he cannot actu- ally produce a perfectly straight edge, but he can produce one that is very nearly straight, and that is good enough. It obviously explains why the geometric theory of perfect squares is in practice a very effective guide for him. We may infer that geometry may perfectly well be viewed as a study of the spatial features – shape, size, relative position, and so on – of ordin- ary perceptible things. As ordinarily pursued, especially at an elementary level, it does no doubt involve some idealization of these features, but that is no good reason for saying that it is not really concerned with this kind of thing at all, but with some other ‘ideal objects’ that are not even in prin- ciple accessible to perception. All this, however, is on the assumption that geometry may be construed hypothetically: it tells us that if there are per- fect squares, perfect circles, and so on, then they must have such-and-such
(^21) For a body of gas maintained at the same temperature, where ‘P’ stands for the pres- sure that it exerts on its container, and ‘V’ for its volume, Boyle’s law states that PV = k, for some constant k.
20 Philosophy of Mathematics
properties. That is helpful, because it implies that the approximate squares and circles which we perceive will have those properties approximately. But, one may ask, does not geometry (as ordinarily pursued) assert outright that there are perfect circles? That would distinguish it from the theory of ideal gases, and is a question which I must come back to.
7. Complications
Within the overall sketch of his position that Aristotle gives us in Metaphysics M.3, there are two brief remarks which certainly introduce a complication. At 1078 a2–5 he says, somewhat unexpectedly, that mathem- atics is not a study of what is perceptible, even if what it studies does happen to be perceptible. More important is 1078 a17–25, where he says that the mathematician posits something as separate, though it is not really separate. He adds that this leads to no falsehood, apparently because the mathematician does not take the separateness as one of his premises. A similar theme is elaborated at greater length in the outline given in Physics II.2, at 193 b31–5. There again we hear that the mathematician, since he is not concerned with features accidental to his study, does separate what he is concerned with, for it can be separated ‘in thought’, even if not in fact. And again we are told that this leads to no falsehood. On the contrary, Aristotle seems to hold that such a fictional ‘separation’ is distinctly help- ful, both in mathematics and in other subjects too (1078 a21–31). Our texts do not tell us what kind of thing a mathematical object is conceived as being, when it is conceived as ‘separate’. I think myself that the most likely answer is that it is conceived as the Platonist would conceive it, i.e. as existing in its own ‘separate world’, intelligible and not perceptible. Further, if – as seems probable – Aristotle concedes that perceptible objects do not perfectly exemplify the properties treated in ele- mentary geometry, then it will presumably be this mental ‘separation’ that smoothes out the actual imperfections. But it must be admitted that this is pure speculation, and cannot be supported from anything in our texts. 22
(^22) I believe that Aristotle holds that when a geometrical figure is conceived as separate, it is also conceived as made of what he calls ‘intelligible matter’ ( Metaphysics Z.10, 1036a1–12; Z.11, 1036b32–1037a5; H.6, 1045a33–6). This is what allows us to think of a plurality of separate figures – e.g. circles – that are all exactly similar to one another. For what distin- guishes them is that each is made of a different ‘intelligible matter’.
