Princeton/Stanford Working Papers in Classics

Aristotle’s Metaphysics M3: realism and the philosophy of QUA

Version 1.0

December 2006

Reviel Netz

Stanford University

Abstract: The article provides a new translation and interpretation of Aristotle’s Metaphysics M3, arguing that Aristotle uses there the QUA as a perspective of intellectual action: an operator on actions rather than a filter on objects. Instead of Aristotle’s mathematics being a science of “Objects QUA mathematical”, we should consider it as a science whose manner of action is “QUA mathematical”. A discussion follows as to Aristotle’s view that his QUA account salvages a realist reading of mathematics without invoking special mathematical objects. This view depends on the deceptively compelling assumption that a statement which is true QUA X is also true simpliciter. If this assumption is false – as I believe the experience of modern science suggests – then Aristotle was wrong and we must indeed either deny the reality of mathematics, or invoke special mathematical objects.

© Reviel Netz. netz@stanford.edu

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The study of Metaphysics M31 has reached a paradoxical state, typical perhaps for such passages. It is not that one can no longer be original, to the contrary: it is too easy to be original. The literature2 has already divided the subject up into a definite, large number of interpretative issues, and on each of these issues a definite, large number of answers were made possible. Combinatorics now guarantees that no careful reader is ever satisfied with any interpretation. We all feel we might just as well have a go ourselves - to try our luck with our own, new combination of the wheels - hoping that this combination will, finally, break open the lock.

This is precisely the hope of this article3. Added to this, somewhat unrealistic goal, the article also aims to contribute to the question of realism. Aristotle is trying to set up in this chapter a realist metaphysics (so that science is seen to be true), without postulating Platonic entities. This is done through an interpretation of the scientific practice based on the QUA analysis: what may be called ‘the philosophy of QUA’ or ‘the QUA theory’. Aristotle’s claim, then, is that the philosophy of QUA makes realism without Platonic entities possible. My aim in this article is to describe the QUA theory and to show how it is supposed to sustain realism without Platonic entities. This finally allows us to raise the question whether Aristotle’s claim is true. To anticipate, I shall conclude by suggesting that Aristotle’s claim may be ultimately false, but that this can be seen only through an analysis alien to Aristotle’s own approach: on his own terms, Aristotle was successful. Furthermore, the QUA theory itself, while perhaps falling short of sustaining a tertium between Platonism and anti-realism, still succeeds as an analysis of the scientific practice.

The structure of the article is as follows: • Section 1 offers a list of some possible questions in the philosophy of mathematics,

setting up the stage in this way for a reading of M3. • Section 2 describes the function of the QUA theory in its discourse-context: what is the

QUA theory meant to show in M3? • Section 3 describes the content of the QUA theory. • Section 4 brings in the rest of M3, to enable us to see Aristotle’s QUA theory in action. • Section 5 is a brief philosophical assessment of the QUA theory.

1By ‘M3’ I shall mean the passage 1077b12-1078a31, i.e. beginning at Bekker’s chapter 2 (lines 1077b12-17 are transitional, and 14-17, especially, must be read as belonging more to the following than to the preceding text). I also disregard 1078a31-1078b6, the discussion of value in mathematics, which I consider independent from ‘M3’ as defined here, but this is a more difficult problem to which I shall briefly return below. 2Some of the important landmarks are Mueller (1970), Annas (1976), Lear (1982), Hussey (1992), Cleary (1995), all of them offering an enlightening commentary on the chapter. Readers will see that I have no interest in polemic. I shall not attempt to register all points where my views agree or, more rarely, disagree, with those of these authors and others. I try to give credit wherever my position follows directly that of another scholar, and I apologise for the many omissions I suspect there are in this regard. 3 Thus this article focuses on a single passage by Aristotle. Even more, it is largely an exegesis of this passage alone, bringing in other Aristotelian passage only sparingly. A brief argument in favour of such a restrictive methodology is offered in n. 44 below, but see Netz (2002) for the general claim that Aristotelian passages, as a matter of discourse structure, tend to be isolated at the individual passage level.

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1. Introduction 1.1 Some Possible Questions Here are some possible questions in the philosophy of mathematics: 1. How is mathematics true of the world4? 2. Given that mathematics is not true of the world (in that the objects mentioned by mathematics are not among the objects of the world) how can it be that mathematics is true without qualification? 3. What accounts for the privileged status of mathematics? Put differently, what is so special about mathematics? 4. What is mathematics true of? Each of these questions betrays different presuppositions: that mathematics is in a sense true of the world (1), and in a sense it isn't (2); that it is true without qualification (2); that indeed it is true in some special, privileged way (3), and that it is true of something (4). All these presuppositions might be challenged, but I suggest that all were basically accepted in Plato’s circle. Notice also that question 4 is especially important for Platonism, which approaches this set of questions in a very efficient way. The answer to question 4 - that, with some variations, mathematics is true of mathematicals5 - yields immediately an answer to question 3, about the privileged status of mathematics. It also yields an obvious answer to question 2 - mathematics is true without qualification, without reference to the objects of the world, simply because the world is not at all what it is about. Platonism is less clear on question 1, but this question simply falls under the general problem of the relation between Platonic entities and things. The specific issues of the philosophy of mathematics are all solved by a single ontological assumption. It is quite clear that the main claim of M3, to the extent that it touches upon mathematics, is an answer to question 2, not question 4: in this I differ from some past interpretations (that took this passage to be, simply, “Aristotle’s Philosophy of Mathematics”). Aristotle’s main point, it seems to most readers now, is not that he has a better answer than Plato’s to the question “so what is mathematics of?”, but that the question “what is mathematics of?” is misguided. Further, I suggest, it is not that Aristotle is much better than Plato on question 1, on the way in which mathematics is true of the world. Aristotle was thought, by some readers, to show that mathematics is, in fact, true of the world, so that he answers questions 4 and 1 simultaneously. According to one type of such interpretation of Aristotle, mathematics is true of the products of a certain abstraction from physical objects; which products however do not have a separate ontological status (the nature of this operation of abstraction may then be specified in several ways). This interpretation had already been criticised for positing an epistemological mechanism which is not to be found in the text, besides making Aristotle vulnerable to modern, Fregean criticisms6. I shall return to discuss this issue in section 3 below, and my main claim shall be that the interpretations of M3, according to which

4I avoid the expression ‘physical world’, which is meaningless outside the context of certain physical and ontological theories. 5What are those ‘mathematicals’, and how they relate to forms (whatever they may be), is a question which need not detain us - and neither need we assume the existence of any consistent ancient Platonist answer. 6The best formulation of the theory of abstraction is Mueller (1970); the best criticism of that formulation is Lear (1982).

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‘mathematics is true of abstractions from physical objects’ fail as translations of the relevant passage. I shall thus say little, philosophically, about ‘abstraction’. 1.2 A note on Lear (1982) Moving on to the interpretation offered by Lear (1982), we may now see this as another interpretation in which Aristotle is seen to answer question 1. Lear’s theory runs as follows. For mathematics to be true, there must be something of which the arguments of mathematics hold. Further, Lear claims, it is false to assume that this something is not out there in the physical world7. The prop of Aristotle's philosophy of mathematics is the existence of some physical objects which satisfy mathematical definitions. The most obvious case is the heavenly sphere, which is perfectly spherical. This is a coherent theory, and it is Aristotelian in that it does not project a separate ontology for mathematics - which abstraction theories tend to do. But there is something disturbing about the central position given by this theory to some special physical assertions, nowhere even hinted at in M3. Of course I do not deny that according to Aristotle the heavenly sphere is perfectly spherical. But how does Aristotle argue for this? Take De Caelo II.4: a list of arguments, all physical and teleological. For instance, a sphere is appropriate for the circular motion of the heavens, as its circular revolution does not lead to a change in its location. Aristotle does not prove this claim, but its proof would be obvious - for instance, the radii are all equal in a sphere, hence they will all coincide following a revolution around the centre. The proof is indeed obvious but, and this is the problem, it is, and it must be, mathematical. The mathematical nature of such arguments, finally, makes Lear’s account untenable. To have the possibility of mathematics based upon truths of physics is a misfortune, to have it based upon truths of mathematics is carelessness. In more precise terms: it seems plausible, given the nature of mathematical and physical arguments, that

(i) any argument, showing that a specific physical object has a specific mathematical property, must be based, at least in part, upon some truth of mathematics8. Hence,

(ii) it must be possible to know some truths of mathematics while bracketing all the questions as to whether any specific physical object has any specific mathematical property9.

This type of considerations - order and the avoidance of circularity - is not irrelevant for M3. While M3 and its context make no reference to, say, the sphericity of the heavenly sphere, M2, immediately preceding our passage, uses the notion of conceptual priority, and Aristotle returns to this issue in M3 itself (we shall look at the issue of prioricity in somewhat more detail in section 4 below). That mathematics is conceptually prior to physics is obvious and is implied by Aristotle in

7 175: ‘This interpretation of Aristotle’s philosophy of geometry rests on the assumption that Aristotle thought that physical objects really do instantiate geometrical properties’. 8The most plausible argument for the claim that X has the mathematical property F would be that X is independently known to have property G, and, in general, property G implies property F. But an argument showing that ‘in general, property G implies property F’, property F being mathematical, is most likely to be mathematical at least in part. 9The negation of claim (ii) is that no mathematical truth is knowable unless it is also known that some specific physical object has some specific mathematical property. But from (i) we assume that this in turn is unknowable, unless some mathematical truths are known. So, on the assumption that mathematics is knowable, claim (ii) follows from claim (i).

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the M3, as we shall see; indeed, that the sphericity of the heaven is a prima facie mathematical question is explicitly asserted by Aristotle, as an aside in the Physics B2, a passage closely related to the Metaphysics M3 (and which Lear discusses in detail - though without mentioning the reference to the heavenly sphere).

So, to generalise the argument above, there must be some coherent way to engage in mathematics without relying upon information generated through any other discipline. It is not of course that the circularity of Lear’s theory makes mathematics impossible on his account - mathematics could be true as a brute fact, even though knowing it involves some leap of faith. However, Lear’s theory does make it impossible for Aristotle to give a rational account of the possibility of mathematics, which is precisely the position Aristotle is trying to distance himself from in M3. Aristotle’s point is that he is able to set up mathematics (and science) with minimal ontological assumptions. I agree with Lear that Aristotle need not ultimately assent to the presupposition lying behind question 2. But he does offer an interesting reply to this question - whereas, on Lear’s account, he ought merely to have dissolved it, showing its false presupposition. Somehow, the question of mathematical ontology is simply irrelevant for Aristotle. To see how this can be the case, we need to investigate the QUA theory. 2. Function of the QUA Theory 2.1 Translation of unit (a) M3 consists of five main textual units, which I shall call units (a) to (e). Unit (a), 1077b12-34, is the longest and the most important, as it gives what I call the QUA theory. (The remaining units are (b) 1077b34-1078a9, (c) 1078a9-17, (d) 1078a17-23 and (e) 1078a23-31.) I shall translate unit (a) here and discuss it in sections 2-3 (while the remaining units (b)-(e) will be discussed in section 4).

The structure of my sections 2-3 is a two-pronged assault on unit (a), as follows. I first analyze the discourse and logical structures of unit (a) as a whole, thus extracting its function, in this section 2; while the first two subsections of unit 3 (subsection 3.1-3.2) analyze the syntactic and logical properties of the main specific tool used in this unit, the QUA. These two lines of analysis combine in an interpretation of the content of the QUA theory, in subsection 3.3. Subsection 3.4 goes on to provide a further, vital background to the QUA theory - that of the Greek mathematical practice - and subsection 3.5, finally, is a summary of my interpretation. Unit (a) '12 So it has been sufficiently said that they <=mathematicals and forms> are not:

• substances more than bodies are, nor • prior to the sensibles in being (but only in conceptual priority), nor • is it possible that they are in some way separated.

15 But since neither is it possible that they are in sensibles, it is obvious that: • either they do not exist at all, or • they exist in some way (and therefore do not unqualifiedly exist).

For 'being' we say in many ways. For: just as the general <arguments and proofs> in mathematics are not about some separate objects besides the geometrical magnitudes and the numbers, but are about these - but not QUA these as having a geometrical magnitude, or as being divisible -

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20 <so> it is clear that it is possible that there are also arguments and proofs about the sensible magnitudes, but not QUA sensible but QUA just that <i.e. magnitudes>. For: just as there are many arguments QUA <things being> moved alone, apart from their <specific> definitions and attributes; 25 and it does not necessarily follow from them that:

• there is something, moved, separate from the sensibles, or that • there is some distinct nature in them,

so, in the case of moved things as well, there will be arguments and sciences, not QUA moved but QUA bodies only, and again QUA planes only and QUA lengths only, and QUA divisible and QUA indivisible while possessing position, and QUA indivisible only, 30 so that: since it is true to say without qualification, not only that separates exist but also that non-separates (e.g. that ‘there are moved things’), <therefore> it is true without qualification to say also that mathematicals exist, indeed that they are as they <=mathematicians> say they are’. 2.2 The discourse-structure of unit (a)

The structural setting of the passage needs to be clarified, as it is obscured by the standard division into chapters. This structure is given by three factors: transition, framing, and the ‘for’ clause10.

Lines b12-17 are a transition: ‘(A) So it has been sufficiently said that they are not: substances more than bodies are, nor prior to the sensibles in being (but only in conceptual priority), nor is it possible that they are in some way separated. | (B) But since neither is it possible that they are inside sensibles, it is obvious that: either they do not exist at all, or they exist in some way (and therefore do not unqualifiedly exist)’. The concessive structure ‘(A)| but (B)’ is not meant as a statement of a certain tension between the statements (A) and (B). Rather, it is a typical Aristotelian discourse-marker, signaling a transition: (A) is understood to sum up the preceding argument, while (B) is understood to make a fresh start. This structure is further enhanced by framing. Lines 16-17 are a clear echo of lines 1076a32-3511: ‘If mathematical objects exist, they must either exist in perceptible objects as some say, or separate from perceptible objects (some say this too), or, if neither, then either they do not exist or they exist in some other way’. These lines 1076a32-35 served to launch chapter 2. However, chapter 2 concentrated merely on a part of the demonstrandum suggested by lines 32-35: chapter 2 concentrated (as implied by lines 12-15) on the negative claim, that mathematicals do not exist in some ways. By echoing 1076a32-35 in this way, then, the text of lines 16-17 signals, again, the closure of chapter 2, and at the same time it

10These structures are described in more detail in Netz (2002). 11 Following Annas’ (1976) translation.

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signals a fresh start: the part of the argument which chapter 2 left aside. Notice now the following important complication. The function of lines 16-17, of signaling a transition, creates a certain tension, since they need to do, simultaneously, two things: to echo a preceding passage, and to lead to a new passage. To some extent, the wording of lines 16-17 is determined not by what follows them, but also by what precedes them, and the result is a certain gap inside line 17. The text runs as: ‘since neither is it possible that they are inside sensibles, it is obvious that: either they do not exist at all, or they exist in some way (and therefore do not unqualifiedly exist). For ‘being’ we say in many ways. For...’ and what follows is the main ‘for’ clause of the passage (as I shall immediately return to show), offering the QUA theory. But what does this last ‘for’, at the end of the last quotation, stand for? In other words, what is the QUA theory for? Is it meant to show that the claim said to be obvious is indeed ‘obvious’? Or that being is said in many ways? Or that mathematicals do not exist? Or that they do exist, though not in an unqualified way? And if so, is the stress on their existence, or on the qualified manner of their existence? Lines 16-17 have many implicit claims, none of which is obviously supported by the QUA theory. It is thus tempting to dismiss the ‘for’ and to read the following passage on its own, as if it were not part of a ‘for’ clause. This, I guess, is the background to the decision of several editors, most importantly Bekker’s decision, to put a new chapter in line 17 - a decision which may have led some interpretations astray. For this gap is an optical illusion, created by the fact that lines 16-17 are constrained by their double function, to close one passage and to open another (thus some of the implicit claims of 16-17 are there not because they are necessary for what follows, but because they provide closure for the preceding chapter). The QUA theory is indeed governed by the lines 16-17, and this is shown by another framing - not the framing of chapter 2, now, but the framing of unit (a) itself. Unit (a) ends with lines 31-34: ‘so that: since it is true to say without qualification, not only that separates exist but also that non-separates (e.g. that ‘there are moved things’), it is true without qualification to say also that mathematicals exist, indeed that they are as they say they are’. Just as the QUA theory is introduced by a g£r, ‘for’, following lines 16-17, so it leads on to an éste, ‘so that’, preceding lines 31-34. ‘For’ is a mark of a start of an argument supporting its preceding claim; ‘so that’ is a mark of a conclusion supported by the entire preceding argument, summed up. The two logical markers mirror each other - as do lines 16-17, 31-34 themselves, with their obvious verbal echoes. Thus lines 31-34 make it finally possible to focus on the implicit claim of lines 16-17 to which the QUA theory is relevant. We may do this by concentrating on the common core of 16-17 and of 31-34, which may be paraphrased as: Mathematicals exist (though in a qualified way), and mathematics is true. The stress is on existence and truth; the qualification of the existence is secondary in importance.

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This thesis, then, is what the QUA theory should be relevant for: most briefly, for the claim that mathematics is true. The framing makes such an understanding necessary - yet it is still problematic. the main remaining obstacle for our interpretation of the unit is that, at first glance the QUA theory does not seem to support any such statement as ‘mathematics is true’. I move on to try to remove this obstacle. 2.2 The claim of unit (a) Having described the discourse-structure of unit (a), I wish to point out another, logical, feature of this unit, namely the modals it uses. The argument has two kinds of claims. First, there are observations stated in the present indicative without any special modal operators which are supposed to be accepted facts about the scientific practice: the mathematical practice, in lines 17-20, and the physical practice, in lines 22-24. Second, there are conclusions which Aristotle draws from these practices, and which are all very cautious in their modals. In lines 20-22, Aristotle asserts that it is possible that there might be arguments about sensibles, QUA geometrical magnitudes. In lines 25-27 he asserts that it is not necessary that there be separate moved objects. In lines 27-30, finally, Aristotle uses the future tense, ‘there will be’ (arguments and sciences of a certain QUA description) which must have here not a tense force, but the modal force of possibility (which the future tense may always have in Greek): ‘there would be’ might perhaps be a better English translation. Thus Aristotle’s positive claim in this passage is extremely weak. He does not state positively that anything is the case. All he argues for is the possibility of some states of affairs and the non-necessity of others - which comes to the same thing. Aristotle’s main claim is in fact not difficult to describe, since all we need to do is to generalise from the three examples he gives: for instance that it is possible to have ‘general arguments’ in mathematics without there being mathematical objects over and above geometrical magnitudes and numbers, etc.. Generalised, this may be paraphrased as:

It is possible to have a separate science without having a separate set of objects dealt with by this science or, in other words: The existence of a separate science does not entail the existence of a separate set of objects. Notice that this is not about mathematics. The theory is much more general. If we apply it to mathematics, it might offer an answer to question 2, an answer which touches, negatively, upon the Platonic answer to question 4. This amounts to nothing as a positive statement, but it is a powerful negative statement. It is an argument for a non-sequitur, that from the existence of a science, the existence of a set of separate objects does not follow. Correlated to this, from the non-existence of separate objects, one cannot deduce the non-existence of a science. Now I return to the ‘for’ structure. The argument is meant to support the claim that mathematicals exist in some sense and that mathematics is true. Of course the QUA theory does not support this directly, but seen as a theory arguing for a non-sequitur its function becomes clear. An obvious Platonic counter-argument to Aristotle would be that the existence of mathematicals is necessary to guarantee the truth of mathematics, the truth of mathematics being a shared assumption

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in the debate. The Platonist would argue that Aristotle implies that there is no such thing as mathematics since, according to Aristotle, there is no separate set of mathematicals. So the Platonist would claim that Aristotle is refuted by the existence of mathematics. Aristotle now claims that mathematics does exist, and that this is possible without Platonic ontology, once the Platonic non-sequitur is uncovered, since the non-existence of a separate set of objects does not imply the non-existence of the science. Realism without Platonic entities is shown to be at least a possibility worth entertaining.

To make an interim summary, then, the QUA theory shows that the validity of a science, and the existence of special objects to which that science refers, are logically distinct questions. You cannot pass directly from the science to its ontology. This supports Aristotle, in allowing him to accept the validity of mathematics, without accepting the existence of mathematicals12. Briefly, my claim so far was that the function of the QUA theory is to argue that:

The existence of a separate science does not entail the existence of a separate set of objects.

3. Content of the QUA theory 3.1 A note on the syntax of QUA So far I have described the immediate function of the QUA theory in its context, and now I should discuss its content, and so, first, discuss the content of QUA itself: what is the meaning of the QUA expression? But once again it is doubtful whether content may be separated from function. QUA, the dative feminine singular of the relative pronoun, with a special idiomatic significance, is a unit of Greek syntax as much as of the Greek lexicon. Asking 'what is the meaning of QUA'? is almost like asking 'what is the meaning of the definite article?'. Such grammatical units have function, primarily, and meaning only derivatively from that function. So the first question to ask is what is the grammatical function of Å. This question - what is the grammatical function of Å - is in fact a very difficult question, and a full answer to it will demand at least a separate article. In this article I shall state, somewhat dogmatically, a position which seems to me implicit in most translations, and, at the purely grammatical level, it seems to me that this position was not challenged by the scholars of ancient philosophy with whom I had discussed this. So I hope the following, at the very least ‘captures the intuitions’ of most practitioners, and I try to make such intuitions somewhat more explicit. The first thing to notice about the grammatical function of Å is that it is not the same as the

12At this point mention may be made of 1078a31-1078b6, the discussion of value. One possible way to approach the function of this brief passage is as another, parallel NON-SEQUITUR argument, this time not to the effect that mathematics may be true without separate mathematicals, but to the effect that mathematics may be noble without separate mathematicals. This is certainly one thing this passage does, and in this way its position (and the abruptness of its beginning) may be accounted for. However Aristotle does not stress this role of the passage, focusing instead on what seems like a separate polemic (probably with Aristippus: cf. Cleary (1995) 339-343); and the structural features may be accounted for by noticing that the end of chapter 3 (on Bekker’s division) is also the end of a relatively long textual unit (Bekker’s chapters 1-3), following which Aristotle moves on to a separate discussion; the ending of chapter 3, then, is the natural position for an appendix on value in mathematics.

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grammatical function of QUA as this word is used in modern English13. Greek Å never lost the grammar of a relative pronoun and, as such, it governed the type SENTENCE. What follows a Greek Å is a sentence. Often, especially in Aristotle, this is a very elliptic sentence: typically, a sentence in which both subject and copula are dropped: 'Å man', for 'Å <he is> a man'. But still there is no reason to doubt that this is used as a form of the relative pronoun, since quite often you do get fuller sentences, and since this usage accounts for several syntactic features of the construction, the most important being:

• The avoidance of the definite article following the Å. (While the noun in the main clause tends to have the definite article: ‘study of man [definite marker] Å man [bare noun]’, instead of ‘study of man [definite marker] Å man [definite marker]’).

• The use of the nominative following the Å (instead of using the case of the repeated noun from the main clause: ‘study of man [gen.] Å man [nom.]’, and not ‘study of man [gen.] Å man [gen.]).

The significance of the use of the bare noun following the Å is clear, since in Greek the use or avoidance of the definite marker is often significant as a way of distinguishing subject and predicate in simple predicative sentences of the form ‘X is Y’ (the subject gets a definite marker while the predicate is a bare noun). In the expression ‘Å man’, subject and copula are elided, but the predicate status of ‘man’ is signaled by the absence of the definite marker. The use of the nominative is meaningful, in that it shows that the expression ‘man Å man’ does not form a cohesive unit, with an attraction of the case. This should be compared to the expression ‘X æj Y’, ‘X as Y’, where the case of Y is often attracted, in Greek, to the case of X. This is interesting, since etymologically Å and æj are similar (both are adverbial forms of the relative pronoun). The difference is that æj, much more commonly used than Å, has had its grammar considerably diversified. No longer limited to a use as a form of the relative pronoun, it could be used also as something akin to a preposition, combining two nouns, X and Y, into a larger noun-phrase, ‘X æj Y’, ‘X as Y’. Because this larger noun-phrase may be interpreted as a single cohesive unit, it is natural that the case of Y may be identical to that of X.

It should thus be clear that the Aristotelian Å phrase is an elliptic relative clause. This is connected (as I shall immediately explain) to the following point.

In Greek prior to Aristotle, Å has a strictly adverbial function. Further, I have made a TLG survey of all the uses of Å in Aristotle (there are about 900 tokens of the word, though often two or more tokens occur in a single sentence, and many occurrences do not display the special idiomatic usage; altogether, there are about 200 passages in Aristotle where idiomatic Å is used). In this survey, I have not come across any passage where there is a need to doubt that Å is used adverbially. In Aristotle, just as in the preceding literature, it is always possible to read the Å as governing a sentence which specifies the manner of some verb, or of some noun of activity, preceding the Å. The Å phrase is adverbial, in the sense that it forms part of the Verb-Phrase, not the Noun-Phrase (I

13QUA merits an entry in the OED, where it has two main meanings: ‘in so far as’, and ‘in the capacity of’. The first meaning governs the type SENTENCE, while the second meaning governs the type NOUN-PHRASE. It seems to me that the standard Aristotelian ellipsis (which I describe in the main text) has made some readers mistake the Greek Å as if it ought to be translated by the second OED meaning, and I argue that this second OED meaning already departs from Greek grammar.

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immediately give an example). Our modern English QUA, on the other hand, may be used adverbially but is often used

adjectivally, as a component in a noun-phrase - just because it has become akin to a mere preposition, no longer strictly a relative pronoun. Thus, in our expression 'MAN-QUA-MAN', the 'QUA-MAN' phrase is used by us adjectivally, rather like the use of 'TALL' in 'THE TALL MAN', creating a more complex noun-phrase which may be, for instance, a subject in a sentence, e.g. 'MAN-QUA-MAN IS RATIONAL', with the intended constituent-structure of a Noun-Phrase followed by a Verb-Phrase: [MAN-QUA-MAN] [IS RATIONAL].

I have heard such expressions used by philosophers. I suppose we feel, even when uttering such expressions, that these are not properly Aristotelian (their implied ontology, to be sure, seems to be more Platonist than Aristotelian) but my claim would be even stronger: that this is not only un-Aristotelian but possibly also un-Greek, that is, not a grammatical expression in the Greek we find used in Aristotle’s writing. In Aristotle, Å is not yet akin to a preposition and is instead simply a form of the relative pronoun; it is correspondingly used adverbially; and thus, in the expression 'MAN Å MAN IS RATIONAL' the phrase 'Å MAN' should be taken not with the noun 'MAN' but with the verb-phrase 'IS RATIONAL'. Man is rational. In what respect? In that he is a man. The structure is Noun-Phrase MAN, followed by a Verb-Phrase IS RATIONAL, with the added adverbial qualification, forming part of the Verb-Phrase, QUA MAN.

In the literature, it is common to use the metaphor of ‘operator’ to describe QUA. QUA is understood to take one object and to transform it into another: say, take MAN and transform it into MAN-QUA-MAN. Wieland (1961), for instance, sees the essence of QUA in its identifying a part of an object, which however is not a spatial part: so once again, identifying an object-within-an-object, an operator transforming one object into another. This can easily lead to misleading interpretations, in which chimerical objects are conjured out of the Aristotelian text. Thus, to mention a case from outside the philosophy of mathematics, it seems that some readers attribute to Aristotle an interest in the object ‘being QUA being’: but there is no such object, because Å does not attach to objects to create other objects (thus all Aristotle ever speaks about, in this case, is of the study Å being, of being - and the only object he ever studies is being). Returning to the philosophy of mathematics, some such chimerical object must lie behind the various ontological interpretations of M3, whether abstractionist or physicalist. Those interpretations seem to read Aristotle as if he was describing the object of mathematics as ‘physical objects QUA mathematical’. But clearly he does not do this: the ‘Å mathematical’ phrase specifies the manner of some activity, not an object. 3.2 A notation for the QUA theory Generally speaking, then, Å must be seen not as an operator transforming one object into another, but (to the extent that such metaphors are legitimate) as an index transforming one operation into another: it is attached not to objects (Noun-Phrases) but to operations-upon-objects (Verb-Phrases). Thus, instead of analyzing ‘Man is rational QUA man’ by:

*Is rational (QUA man (man))

(i.e., the predicate ‘is rational’ holding of the object, resulting from applying the operation ‘QUA man’ to the object ‘man’), I offer to analyze this by

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Is Rationalman (man)

(i.e. the predicate ‘is rational’, indexed by ‘man’, holds of ‘man’).

I now move on to analyze to the QUA theory in M3 (and I also return to use the Latin word, now referring again to Aristotle's Å by the modern QUA). Aristotle uses QUA in two main contexts relevant to us. One is the 'belong' context, where the QUA phrase supplies the manner of a predicate belonging to a subject:

Aristotle’s formulation Suggested notation X belongs to Y, QUA X X belongsx to Y and the other is a cluster of uses with 'argument' as the core, the QUA phrase supplying the manner of the argument: Aristotle’s formulation Suggested notation An argument about X, QUA X An argumentx about X

This latter usage is the one operative in our passage: the QUA phrase supplies the manner for arguments, proofs and sciences: lÒgoi and ¢pode…xeij in line 22 and, implicitly, line 19; lÒgoi in line 23; lÒgoi and ™pistÁmai in line 28.

In a wider view still, we may say that Aristotle can have, first, the simple expression: Aristotle’s formulation Suggested notation An argument about A Argument (A) Or he may index this by the QUA phrase, e.g., to take the common case: Aristotle’s formulation Suggested notation

An argument about A, QUA A ArgumentA (A) And, in the more generalised (though less common) case, we may have: Aristotle’s formulation Suggested notation

An argument about A, QUA G ArgumentG (A)

My task is to interpret this expression. But the first thing to notice is that we should not over-interpret this expression, i.e. we should

not insist upon a specific interpretation, which might also be over-specific. Aristotle uses the QUA phrase in what may seem to be a loose way, and he does not stop to define it himself14. It seems that

14In another context (APo I.4), he tries to characterise - not, it seems, to define - the expression ‘kaq’ aØtÒ’, among others, and he asserts this expression be equivalent, in some sense, to the QUA phrase (73b28-29).

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the sense of the expression - and not only its function - is already given by explaining its adverbial nature. When we say that ‘an argument about man, QUA man’ is an argument about man, whose manner of argument is defined by ‘man’ - an argument which approaches ‘man’ through the framework defined by the concept ‘man’ - we say very little beyond an unpacking of the syntax. But this already supplies us with an interpretation of the expression. A loose interpretation, no doubt, but perhaps the only one which was present to Aristotle’s mind. It may be correct to say that all Aristotle meant by

ArgumentG (A) was ‘an argument about A, inside a G-ish framework’ - the sense of ‘framework’ left open.

It is possible to suggest a somewhat more specific analysis, at least as a way of sharpening our intuitions about the QUA theory. For a modern reader, vague ‘frameworks’ may be more difficult to follow than specific logical analyses. I thus move on to offer such a specific logical analysis, which I believe is a way of capturing Aristotle’s intuitions. One advantage of such a precise analysis is that it allows showing in precise terms the compatibility of my approach and Lear’s. However I insist that, ultimately, Aristotle’s theory was less specific, so that, having used the logical analyses for this purpose, we should return to be using a more vague sense of QUA.

So I move on to the logical analysis. We have already moved, in the paragraph before last, from ‘an argument whose manner of argument is defined by ‘man’’, to ‘an argument which approaches ‘man’ through the framework defined by the concept ‘man’’. To derive a more precise analysis, remember that the QUA phrase is an elliptic relative sentence: an argument about man QUA man is an argument about ‘man insofar <as he is a> man’ and a very simple further extension of this is:

'An argument whose only special presupposition is that this object under consideration

(namely man) is a man'. Or in more general, an argument about A, QUA G is ‘an argument whose only special presupposition is that A is G’. In the notation offered above:

Def. (1): ArgumentG (A) ≡ Argument whose only special presupposition is G(A).

Of course, the same definition may hold, with the necessary substitutions, for the other members of the same family of expressions, such as ‘Science QUA’, ‘Proof QUA’, etc. This allows the following definition of the usage of QUA with the relation ‘belongs’: Def. (2): 'F belongG to A ≡ F belong to A, shown by an argumentG(A)'.

This is the closest Aristotle ever comes to defining this ubiquitous expression.

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(I ignore the complication, which need not concern us at this moment, of Aristotle's requirement that this argument must be in a sense the primary argument showing this result). Once again, this is not a statement Aristotle ever made, but it is one natural way of producing a coherent description comprising both main usages of QUA.

Note that an immediate advantage of this set of definitions is that they allow us to derive the definition of QUA offered in Lear (1982) which, transcribed into the same notation, becomes: Def. (3): 'F belongsG to A ≡ F belongs to A, and it follows from G(A) that F(A)'.

Lear’s definition may, alternatively, be taken as more basic than definition (2): given definition (1), definitions (2) and (3) are interchangeable. Definition (3) is a more minimal interpretation of the Greek, while definition (2) has the advantage of combining the two relevant Aristotelian usages of QUA. My main interest in producing the two definitions together is in showing that my analysis is compatible with Lear’s. Nothing in the following, however, relies upon my logical unpacking of QUA, and the rest of my argument uses a looser sense of ‘framework’ or ‘manner’ as the meaning of QUA. 3.3 An interpretation of the QUA theory Now, going finally from the meaning of QUA to the meaning of the QUA-theory, we can interpret the theory as follows. Suppose that a science has the form: Sciencex(y) that is, the object of the science is y, the manner is QUA-x. Usually Aristotle uses the QUA-phrase with x and y identical, e.g. Scienceman(man), sciencebeing(being), but the point of the QUA theory is that x and y may be different: the object and the manner need not be identical. What Aristotle is saying is that the existence of Sciencemathematical magnitudes(x), does not entail the existence of Science (mathematical magnitudes). This is indeed precisely what is required for the function of the QUA-theory, to argue that (following the argument of section 2 above):

The existence of a separate science does not entail the existence of a separate set of objects. Or, put now in the notation above:

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The existence of sciencex does not entail the existence of science (x): Sciencex ≠ Science(x), Argumentx ≠ Argument (x).

This is the content of the QUA theory. 3.4 The QUA theory and the Greek mathematical practice I repeat again the major caveat: the logical analysis offered in subsection 3.3, even if coherent, is not Aristotle’s. All I may claim is to have captured Aristotle’s intuitions with my alien symbolism. Partly, this rests upon the syntactic analysis above, following which the index notation may be read as an open-ended typographic symbol, capturing the equally open-ended syntactic symbol used by Aristotle. More importantly, however, I need to look at the material which might have informed Aristotle’s intuitions, and to show that its behavior was consonant with the analysis above. I have argued that, according to Aristotle, what constitutes the identity of a scientific argument is not its object, but its manner, which may be spelled out as its set of presuppositions. I now move on to show that this is, in a very transparent way, true of the mathematics Aristotle would most likely have been familiar with. I quote from Euclid’s Elements II; I shall argue that what we see there is an example of argument-QUA, and that the logical structure of such arguments might have informed Aristotle’s understanding of science15.

This is the General Enunciation of Elements II.5 (fig. 1):

'If a straight line is cut into equal and unequal <segments>, the rectangle contained by the unequal segments of the whole, with the square on the <line> between the cuts, is equal to the square on the half'. Immediately following that, the Particular Setting-out and Definition of Goal: 'For let some line, <namely> the <line> AB, be cut into equal <segments> at the <point> G, and into unequal <segments> at the <point> D. I say that the rectangle contained by the <lines> AD, DB, with the square on the <line> GD, is equal to the square on the <line> GB'. And following that comes an argument with some special constructions and logical moves, from which I quote a typical argument:

‘But the <area> GM is equal to the <area> AL, since the <line> AG, too, is equal to the <line> GB’; Following a sequence of such moves, comes the Conclusion:

15An obvious presupposition of this argument, then, is that at least some mathematical arguments known to Aristotle had a structure similar to that of Elements II.5; this is usually accepted.

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'Therefore if a straight line is cut into equal and unequal <segments>, the rectangle contained by the unequal segments of the whole, with the square on the <line> between the cuts, is equal to the square on the half'. The first thing to notice is that there is no special moment in which Euclid does the QUA: nothing like ‘a moment of QUA’ could have informed Aristotle’s intuitions. Euclid never says anything like 'and let AB be considered QUA line'. The QUA is not a special moment inside the proof - there is no need for an act of abstraction. QUA, instead, is inside the manner of the proof. For let us look closely at the argument quoted above: 'But the <area> GM is equal to the <area> AL, since the <line> AG, too, is equal to the <line> GB'; The object of this argument are some particular lines, with some particular properties, e.g. their particular lettering, or if you wish, their particular size, colour, location upon the page. So the object of the argument is not the particular line QUA the general line, or anything like this. The object of the argument is simply the particular line. However, the manner of the argument is different. It is an argument QUA equal lines, an argument resting on the lines being equal - if you wish, an argument whose only presupposition is the equality between the relevant lines. It is something about equal lines, that if you construct parallelograms on them as you did here, the parallelograms will be equal. No further presupposition is required for the argument to hold. So the argument is proved, not QUA AB being called this way, or QUA AB being located or coloured in this way, but QUA AG being equal to BG. In such ways, the argument is not QUA a line being called AB, but QUA a line being cut into equal segments. In other ways, the argument is QUA a line being cut into unequal segments. And thus the argument may safely conclude as it does that

'if a straight line is cut into equal and unequal <segments>, the rectangle contained by the unequal segments' etc. Stated in this general language. Because the argument is QUA a line being cut into equal and unequal segments, it will hold of lines being cut into equal and unequal segments, whatever their names, colours, positions etc.16 But it is unnecessary to believe in the existence of lines devoid of specific colour and position, lines which are only cut into equal and unequal segments and that's all the properties they have. You can have a valid argument about lines being cut into equal and unequal segments, without having to project an ontology containing the thing which is just a line being cut into equal and unequal segments.

16The ease with which we are able to identify in this case the ‘QUA’, the specific presuppositions of the argument, is misleading - and is only approximately the case even in this simple example. Generally speaking, a deep mathematical understanding is required in order to analyse the precise presuppositions involved in an argument, and this process of analysis may go on indefinitely, as shown in Lakatos (1976). This is important, since the fact that an argument may not give away immediately its specific QUA makes it possible to have an argument seemingly about one thing, and in reality about another. The prose you speak - the presuppositions of your very speaking - is always something you gradually discover.

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We may restate our interpretation of Aristotle’s QUA theory as:

Argumentx ≠ Argument (x) (i.e., an argument may seem to be about a certain object, but its manner of argument may be different).

This basic principle is used positively by Greek mathematicians. Even though they study a particular object in the diagram, the manner of their arguments has a wider applicability, and the intension of the argument is correspondingly wider. Is there any evidence for such an understanding, from the Greek mathematical writings themselves? Greek mathematicians usually do not speak on such theoretical issues, but sometimes they happened to be more philosophically oriented. Eratosthenes, for instance, had dedicated to king Ptolemy a mathematical instrument, the mesolabion, with a brief explanation: something akin to popular science, then. Following the proof, Eratosthenes makes several second-order comments, and among other things he notes that (Heiberg (1913) 94.2-4): '... and the sizes and the proportions may be as anyone wishes them; for the arguments of the proof will yield the conclusion in the same way'. Eratosthenes concedes that he gave a proof implying some specific sizes and magnitudes. However, he also insists that his conclusion is invariant to such specific measurements. The arguments of the proof yield the conclusion, without reference to precise measurements, hence the proof has an invariance relative to such changes in sizes and proportions. Once again, what makes the proof valid for its domain of application is its manner, the fact that the arguments yield their conclusion in a certain way; not the object of the proof, but the manner of the proof. Exactly the same principle is explained by Proclus, in the course of the commentary to the first proposition of the Elements (In Eucl. 209.24-):17

'And then comes the general conclusion: 'An equilateral triangle has therefore been constructed upon the given straight line'. For even if you make the line double that set forth in the setting-out, or triple, or of any other length greater or less than it, the same construction and proof would fit it'. We may change the object and keep precisely the same proof, the same text. In this way the text is indeed always about some particular object, but it is QUA something more general than this particular object. Notice that I am not going beyond the text in ascribing the QUA locution to Proclus. What we have just read is his example of the general theory of the mathematical conclusion, from a bit earlier in the same commentary:18

'They pass therefore to the universal conclusion in order that we may not suppose that the result is confined to the particular instance. This procedure is justified, since for the demonstration

17 Friedlein (1873) 209.24-210.5. 18 Friedlein (1873) 207.12-25.

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they use the objects set out in the diagram not QUA these particular figures, but QUA figures resembling others of the same sort. It is not QUA having such and such a size that the angle before me is bisected, but QUA being rectilinear and nothing more. Its particular size is a character of the given angle, but its having rectilinear sides is a common feature of all rectilinear angles. Suppose the given angle is a right angle. If I used its rightness for my demonstration, I should not be able to infer anything about the whole class of rectilinear angles; but if I make no use of its rightness and consider only its rectilinear character, the proposition will apply equally to all angles with rectilinear sides'. I quote this extensively, because this is, with a few qualifications, my interpretation of Aristotle's QUA theory. To repeat, I have described the function of this theory in the M3 as showing that: The existence of a separate science does not entail the existence of a separate set of objects. And this is because in general, the existence of a valid argument does not directly determine the identity of the objects for which the argument is true. What is an argument true of? Of anything, QUA which the argument was conducted - which may even fail to include the original object used or intended by the scientist. Proclus does differ from Aristotle, in the following respect. Although they both rely upon the same logical principle, they use it to make different, almost contradictory points. Proclus uses this principle to upgrade mathematics, to show that it is true not of the particular diagram but of the general set of mathematical objects; Aristotle in M3 uses this principle to downgrade mathematics, to show that it is true not of mathematicals but simply of any set of objects satisfying mathematical arguments (whose identity Aristotle can leave open)19. This is a difference in the use of the QUA-theory, not in the content of the QUA-theory itself. In both the M3 and the Proclus text, the QUA-theory has the same significance: that the apparent, immediate objects of an argument, and its manner of argument, are distinct and so one may seem to argue about one thing while in reality arguing about another. If you take as ‘the apparent objects’ something like mathematicals, the QUA theory shows that mathematics need not assume them; if you take as ‘the apparent objects’ something like the diagram, the QUA theory shows that mathematics need not be tied down to the diagram. The results are different, moving in opposite directions20; but the theory is the same.

It is important to see how Aristotle did use the QUA theory in the Proclean direction, in De Mem. 449b31-450a721: [Immediate context: ‘... It is not possible to think without an image’] ‘For the same effect occurs in thinking as in a geometrical demonstration; for in the latter case, though we do not make any use of the fact that the size of the triangle is determinate, we none the less do draw <a triangle which is> determinate in its size; and similarly the thinker: even if he is not thinking of something with a size, places something with a size before his eyes, but thinks of it not QUA having a size; and

19 I shall return to discuss the nature of the set of objects satisfying mathematics, according to Aristotle, in subsections 4.4-4.5 below. 20I return to discuss those ‘opposite directions’, of Platonism and Aristotelianism, in subsection 4.3 below. 21I keep close to the translation in Sorabji (1972).

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if its nature is that of things: which have a size, but not a determinate one - he places before his eyes something with a determinate size, but thinks of it simply as having size’. This passage mentions three cases, the first and third perhaps identical. The first is demonstrating about triangles. The diagram22 has a triangle with a determinate size, but the proof is for triangles in general. Why the generality? Because we did not make any use of the fact that the size of the triangle is determinate. That we are allowed to gloss this as the QUA effect can be seen from the continuation of the argument, where two cases are given: thinking with an object having a (determinate23) size, but not QUA an object having a size; and thinking with an object having a determinate size, QUA having a size24.

I stress that Aristotle’s use of this example was not to make a special claim about mathematics, but to use a fact which was understood to his audience (or to himself) about mathematics: that one can use particulars for thoughts (such as images), without thereby tying thought down to that level of particularity: the domain of application would be given not by the original choice of object, but by the QUA-index of the argument itself. Aristotle is clear that this effect is not specific to mathematics: mathematics is chosen simply as the field in which this effect is clearest. Given the many arguments in Greek mathematics such as Elements II.5, in which a seemingly particular argument shows a general claim, this is not surprising. This then is an important result: mathematics was the discipline in which the QUA methodology could be seen at work.

I argue that arguments similar to Elements II.5 could have formed an important background to Aristotle’s notion of QUA, and that we are therefore allowed do as I did: to use such arguments to unpack the intuition behind the QUA theory.

Before moving on, it is worth noticing the following interesting result: because the QUA theory is a theory about methodology and not about ontology, a Platonist such as Proclus can easily accommodate it as part of his own understanding of mathematics, using it as a justification of the mathematical method without taking on board its intended ontological implications. This is possible, because the ontological implications of the QUA theory, in itself, are very weak: the theory does not show that mathematicals do not exist, it merely shows that it is possible to have an argument seemingly about one thing, and in reality about something else. 3.5 The QUA theory: a summary I sum up, using the notation introduced in this section.

• The QUA analysis is the observation that arguments and ‘belong’ statements may be indexed as Argumentx, Belongx. (By which we do not need to intend anything more specific than ‘arguments inside a certain X-ish framework’ ‘belongs as seen through an X-ish framework’).

• On the basis of the QUA analysis, the claim of the QUA theory is an almost immediate result: Argumentx ≠ Argument (x), that is, to have an argument in an X-ish framework is

22Or the particular image - we do not need to go into the detail of the psychology of De Mem. 23I suppose that all particular objects (of the kind which can be placed in front of one’s eyes) have not only size, but a determinate size. 24Note that there is no need to spell out the negative corollary, ‘but not QUA having a determinate size’. An argument whose index is fully given by a certain QUA is thereby unindexed by all other relevant QUAs.

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not the same as to have an argument about X. • As regards mathematics, Aristotle makes a further, specific, suggestion, based on the

general QUA theory. Mathematical arguments are characterized as mathematical, not through their object (argument (mathematical)), but through their manner: a mathematical argument is argumentmathematical.

• This last claim is especially plausible, since a central logical feature of Greek mathematical arguments - their general import - is based upon their manner, and not their object.

4. Beyond ‘unit (a)’: the QUA Theory at Work In terms of the possible questions from section 1 above, we can see that the QUA theory offers a way of dealing with question 2 (to rephrase: assuming that the objects mentioned by mathematics are not among the objects of the world, how can mathematics be true without qualification?). Mathematical arguments are not arguments (mathematicals) but argumentsmathematical: so their validity is independent from the identity of their objects. This is of course also a powerful indirect argument against the Platonic answer to question 4 (what is mathematics true of?). Since the validity of mathematics can be sustained without invoking a special ontology, this ontology loses its most powerful attraction.

Platonism has other attractions, and the rest of M3 does two things simultaneously: to show several possible developments of the QUA theory; and to show how they help to dissolve other specious attractions of Platonism. In this way, the QUA theory is enhanced, while a richer account of mathematics-without-mathematicals emerges, on the basis of this QUA theory. 4.1 Unit (b): 1077b34 ‘And just as it is true to say without qualification that the other sciences 35 are ‘of’ this:

• not of the incidental <property> (e.g. as <would have been the case if we were to say> that <medicine> was of white, if: white was healthy, but QUA it is healthy)25

• but of that, whichever, QUA <that> it is; of each of the things26: 1078a if <it is> QUA healthy, <it is> of healthy, if <it is> QUA man, <it is> of man: so also with geometry.

It is not as if, had that, of which27 it <=geometry> is, happened to be perceptibles, while it <=geometry> is not QUA perceptibles -

it is not as if the mathematical sciences would <then> be of perceptibles (yet nor would they be of anything separate, different from them:

25I keep the manuscript reading, e„ ØgieinÕn tÕ leukÒn, Î d’ œstin ØgieinÒn, against Jaeger’s text e„ tÕ ØgieinÕn leukÒn, ¹ d’ œstin Øgieniou. 26I keep the manuscript reading, ™ke…nou Î ™stˆn ˜k£stou, against Jaeger’s text ™ke…nou oá ™stˆn ˜k£sth. 27This is a loose expression: clearly Aristotle means by ‘of which’, what he earlier meant by ‘about which’, so this ‘of’ is not the technical ‘of’ developed in this passage (we must assume this, or otherwise the sentence becomes self-contradictory). An Aristotelian failure to use a technical vocabulary consistently should not come as a surprise.

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5 and many <properties> are attributes of things, per se: QUA each of the attributes, which belong to them <=to the things>:

since there are proper attributes <to an animal>, both QUA <the animal being> male and QUA <the animal being> female, yet there is nothing ‘male’ nor ‘female’ separate from animals.

So that < many attributes belong> also QUA lengths alone or QUA planes)’.

This unit has attracted attention (e.g. Annas (1976) 149, Lear (1982) 168-169) because it seems to relativise the distinction between ‘essential’ and ‘incidental’. In fact, one has the sense that readers are often mystified by this passage, so that they are perhaps tempted to over-interpret its philosophical significance – failing, I would suggest, to see its function in context. Clearly, in terms of discourse structure, we expect at this point a footnote to unit (a). this is because, on the one hand, there was no clear discourse break between units (a) and (b) while, on the other hand, unit (a) seems to be very self-enclosed. Thus unit (b) depends upon unit (a), in logic and discourse structure, but not the other way around. We expect some subservient relation, unit (b), for instance, clarifying a minor point that might arise from unit (a).

This, I suggest, is precisely what unit (b) does. Unit (a) has shown that mathematics might be characterized as sciencemathematical, and not science (mathematical), so that we need not assume the separate existence of mathematicals in order to have valid mathematical arguments. An obvious Platonist objection might be that while mathematical arguments, on Aristotle’s account, may be valid, they are no longer mathematical. If a typical mathematical statement, according to Aristotle, holds true of, say, a physical object:

Statementmathematical (physical)

then this statement - so the Platonist - is a statement of Physics. According to Aristotle, then, there is only physics (or whatever it may be, of which mathematical arguments hold). Aristotle now denies this counter-argument, and all he needs to show is that: In the situation sciencex(y), the science is properly characterized by x and not by y (it is of x, and not of y). In terms of the Greek, Aristotle used in unit (a) the distinction between per… (translated by ‘about’) and Î (translated by ‘QUA’): science could be per… x, Î y. Unit (b) introduces the bare genitive (science of object [genitive], translated by ‘science of object’), and the claim is that we should normally use the bare genitive in accordance with the ‘QUA’, not in accordance with the ‘about’. This plausible claim is clarified with the aid of the ‘incidental’ terminology: medicine may speak of some white body, but clearly it is not a science of whiteness, since the whiteness of the body is incidental, and what makes it medicine which speaks of the body, is the healthiness of the body. In other contexts, Aristotle is still free to say that some attributes are incidental simpliciter while others are essential, but this is not the issue here. Aristotle does not discuss here the notion of the incidental, but uses it, in order to make a point about the structure of sciences: the structure of the sciences is given not by the ultimate ontology of the world, but by the range of possible frameworks.

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4.2 Unit (c) 1078a9 'And so, to the extent that they are about things prior conceptually, and simpler, to that extent they will have more precision 10 (this is what simplicity is), so that they will:

• be more <precise> without magnitudes than with magnitudes • and most <precise> without motion. • and if <they have> motion, they will be most <precise when they have> primary motion,

for it is the simplest, • and <more precise> than this, uniform <primary motion>28.

And the same reasoning applies to harmonics and optics. For neither studies QUA sight, or QUA sound, 15 but as lines and numbers (indeed these <lines and numbers> are intrinsic properties of them <sight and sound>). And similarly mechanics'.

The ‘and so...’ at the beginning (kaˆ ... d¾) is a clear signal of a fresh start (note that this is the only d» in the chapter). Unit (c) uses the terminology of unit (a), but it already makes a separate claim. I shall return to the overall structure of the chapter at the end of this section. Unit (c) provides Aristotle's answer to question 3 from section 1.1: What accounts for the special status of mathematics? This was another thing which a Platonist philosophy explained with the help of mathematicals, and Aristotle needs to show that he can account for this, as well. Now, since the description of the methodology was about science in general, and not only about mathematics, it is not surprising to see that mathematics, according to Aristotle, is special in a quantitative, not a qualitative way. Its methodology is that of science in general, but its arguments presuppose very little and therefore are especially precise. In this interpretation, I follow the important analysis in Cleary (1985) according to which the ‘without’ description is independent from the ‘QUA’ description29. To put Cleary’s analysis in my terminology: to have an argument QUA x is simply to have argumentx, while to have an argument ‘without y’ is to have such an argumentx that x does not include y (in some sense of logical inclusion which requires further specification).

Since every argument must have some presuppositions, it is obvious that any argument is indexed in some way, is QUA-ed in some way: there is no such thing as a QUA-less science. Similarly, it seems plausible that no argument presupposes the totality of information about its subject-matter, and therefore every argument is ‘without’ something or other. The point about mathematics therefore is not some special qualitative logical feature it has: that it indexes its arguments in the QUA way. The special thing about mathematics is quantitative: its indexing is especially minimal, it is ‘without’ many things, hence it is ‘simple’.

Aristotle’s claim is very convincing (and thus he can afford to give many examples, as the

28My punctuation differs slightly from Jaeger’s, following a suggestion by David Sedly. 29Thus, ‘without’ is a quasi-technical term in this context, a fact which becomes important in the interpretation of unit (e) below.

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facts actually seem to support his theory!) but he offers no more than a sketch of a theory: the key notions of ‘simplicity’, ‘precision’ and ‘conceptual priority’ are left vague, and their interrelations are merely hinted at. This is comparable to the QUA concept itself. Aristotle introduces a key notion, but spells it out only to the extent which is necessary for the development of a convincing argument. This is more than just evidence for brevity, or for rhetorical cynicism. This technique, of leaving key notions only partly spelled out, is in fact a sound method: by omitting the spelling out of notions such as ‘QUA’ or ‘simplicity’ (beyond what is absolutely necessary for the argument to carry conviction) Aristotle does two things. Negatively, he avoids leaving hostages to logical fortune (why should he risk offering a definition of ‘simplicity’, which might be mistaken, as long as the basic argument about ‘simplicity’, however spelled out, is correct?). Positively, he has the chance, at least, of concentrating on the main logical issues: the issue about simplicity and mathematics is not what simplicity precisely is, but that simplicity, and nothing else, accounts for the special nature of mathematics. In other words, we see that Aristotle is following a certain ‘QUA methodology’ of his own: he uses concepts, QUA is necessary for his arguments to carry conviction and not more than that (similarly, I argued above that unit (b), as well, did not study the essential/incidental distinction QUA ontological, but approached it in a more limited manner, focused on the structure of sciences). I shall briefly return to the implications of this, Aristotelian ‘QUA methodology’ in section 5 below. 4.3 Unit (d) a17: 'So that if one, positing as separate from the accidents, makes a study of something concerning them, QUA just-such <i.e. whatever was separated>, he shall commit no invalid move through this, just as he would not <commit an invalid move> when he draws in the ground and he says that a <line> one-foot long is not one-foot long30. 20 For the false <step> is not among the premises. So everything would be studied best, if one would posit the inseparable, separating it - as the arithmetician does, and the geometer'.

In terms of polemical function, Aristotle confronts here a Platonic argument which is especially embarrassing for his theory: that mathematicians simply seem to behave platonically: they seem to refer to ideal objects. In other words, Aristotle imagines himself to be in Glaucon’s place in the Republic, and when Socrates tells him31

‘You know how [mathematicians] begin by postulating odd and even or the various figures

and the three kinds of angle... they make use of visible figures and discourse about them, though what they really have in mind is the originals of which these figures are images: they are not reasoning, for instance, about this particular square and diagonal which they have drawn, but about the Square and the Diagonal; and so in all cases ... [seeking] to behold those realities which only

30I keep the manuscript reading, t¾n podia…an fÍ m¾ podia…an, against Jaeger’s podia…an fÍ t¾n m¾ podia…an. This is not crucial for the logic of the argument, but I believe the Aristotelian choice may be meaningful; I discuss this in he main text. 31Republic 510c2-511a1, following Cornford’s (1941) translation.

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thought can apprehend’. Aristotle is incapable of answering ‘no, I do not know this’. Aristotle finds that he needs to concede Socrates’ description of the mathematical practice. And this is of course especially embarrassing, because the implication is that Platonism, and not Aristotelianism, gives an account of the phainomena. The Platonist would say that Aristotle’s ‘mathematics’ is a coherent science - yet it is the brainchild of a philosopher that no practicing mathematician would ever recognise as his discipline.

Remarkably, Aristotle is able to escape this embarrassment, with the aid of the analyses developed in units (a) and (c). Thus, his answer to Socrates is ‘yes, I concede that this is the practice: and I am able, unlike you, to account for the specific methodological advantages of this practice, in this way showing that this practice does not have the ontological implications which you naively attribute to it’. The main idea is obvious to us by now. Science, in a way, brackets ontology. The choice of objects to be discussed by the scientist does not commit the scientist ontologically: the domain of applicability of a valid argument is defined not by the choice of objects, but by the manner of argument concerning those objects. It is thus not only possible, but indeed obligatory to pick objects for scientific discourse not for their ontological merits, but for their heuristic convenience. Indeed, there is nothing wrong about a completely fictional object, as long as the manner of argument is valid. For heuristic purposes, the best fictional object would be the simplest object (following the argument of unit (c)) - which is the Platonic object. So there is nothing wrong in mathematicians talking as if they discussed Platonic objects.

It is easy to see that this Aristotelian passage is, indeed, an upside-down version of the divided line. This context, I believe, may also be relevant for the interesting textual complication of this passage. As noted in n. 29, I keep the manuscript reading, which is 'calling the one-foot-long not one-long-foot-long'. Editors have changed this into 'calling the not one-foot-long one-foot-long', and this is because this passage resembles three other very passages, one each from the two Analytics, and one each from the two books of the Metaphysics M (the one we saw above) and N32:

Prior Analytics I.41: ‘...like the geometer who calls this a foot-long line, this a straight line,

and says that they are breadthless, though they are not, but does not use these things as though he were deducing from them’.

Posterior Analytics I.10: ‘Geometers do not suppose falsehoods, as some have asserted. They say that you should not use falsehoods but that geometers speak falsely when they say that a line which is not a foot long is a foot long or that a drawn line which is not straight is straight. But geometers do not conclude anything from the fact that the lines which they have themselves described are thus and so; rather, they rely on what these lines show’. Metaphysics N2: ‘...like geometers when they assume a line to be a foot long when it is not a foot long. But ... geometers do not make any false assumptions (it is not a premise in their reasoning)’.

32 APr. 49b35-37, following Smith’s (1989) translation, APo.76b39-77a3, following Barnes’ (1975) translation, Metaphysics 1089a21-25, following Annas’ (1976) translation.

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In all of these passages, the same components recur: an hypothetical element in mathematics; the possibility of falsehood deriving from that hypothesis; an example of declaring a line to be of a certain length. It is a reasonable assumption, given the frequency and the relative opacity of those passages, that Aristotle recalls a discussion he or his readers were expected to be familiar with. It is also reasonable that this discussion was inspired, at least in part, by the divided line passage or some similar, unwritten discussions (as shown by the reference to ‘mathematical hypothesis’ and the question whether such hypotheses might not undermine the validity of mathematics)33.

The topic for this background discussion could have been this. Mathematicians make false hypotheses at the start of their proofs: 'This is one-foot-long'. This is false, because the line in the diagram is not one-foot-long. Does this mean that mathematics is false? Aristotle, probably not alone among Platonists, does not think so. To summarise briefly, the argument seems to be that assertions of the type ‘this is so and so’ can at least in some cases be logically neutral, as they are mere heuristic aids to the reasoning, picking an object which aids the understanding but which does not impinge upon the actual logic at play. So this is a special case, most probably recognized before Aristotle, of what we saw already: at least in some cases, the choice of an object is immaterial, because what counts is not the object mentioned by the argument, but the argument itself. And so you might as well - now this is Aristotle - take a quasi-Platonic object as your object of study.

What must be stressed is the following: discussions concerning those issues in the Academy must have dealt with assertions such as 'this line in the diagram is one-foot-long', not assertions such as 'this is an ideal mathematical line'. What Aristotle does here is to put those discussions upon their head. While the Platonist would patronizingly concede that the mathematician may have recourse to diagrams, since these can be construed as useful heuristic and no more, Aristotle concedes that the mathematician may have recourse to Platonic objects, since they, in turn, can be construed as useful heuristic and no more. Hence, I tentatively suggest, the turn-about in the example, from a one-foot not being one-foot, to a non-one-foot being one-foot - echoing, perhaps intentionally, the upside-down nature of this Platonic pastiche. The main feature of this unit, regardless of our interpretation of its intertextual context, is that here, indeed, Aristotle comes closest to offering a philosophy of mathematics. There seem to be, for him, three kinds of objects in a mathematical proof: 1. The token sign (A line in the diagram). 2. The token signified (A Platonic, ideal line).

3. The type of things for which the conclusion holds true. In this I offer, effectively, an analysis of the Aristotelian concept ‘about’ (an analysis Aristotle

33The reference to one-foot-lines may show, I think, that the discussion attempted to understand

something like the divided line passage in light of the Meno: Socrates does not give in the divided line a concrete example of mathematical practice, and when philosophers look for such examples they often look for them not in the works of mathematicians, but in the works of fellow-philosophers (not a silly thing to do, in fact, if you are trying to understand the philosopher’s view of mathematics). The geometrical lesson in the Meno is the most detailed piece of mathematics in the Platonic corpus, and it is one of very few places in Greek mathematics where lengths of lines in feet are at all mentioned: I am trying to sketch the type of heavily intertextual discussions, to which this passage, I believe, belonged.

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himself did not offer): a mathematical argument can be said to be ‘about’ each of the three above. And the funny thing, emerging from unit (d), is that neither of the tokens 1 and 2 belong to the type 3. The token sign does not belong to the type of things for which the conclusion holds, and this can be proved, I think, on the basis of a reconstruction of the nature of Greek diagrams, which were schematic and non-representational34. As for the ideal Platonic object, we just saw that this is not one of the things of which the conclusion holds, for the reason that it does not exist and is merely a virtual object set up as a heuristic tool. It has, for Aristotle, the same kind of function as the diagram. All of this, however, does not undermine the possibility of the existence of the type of things of which the conclusion is true. The many falsehoods of mathematics do not undermine its validity, not because they cancel each other out but because they are all, as it were, orthogonal to the real vector of mathematical validity. They are about this or that object, not about the manner of arguments, the logical shape of arguments. So we can just ignore ontology while considering the validity of mathematics.

It is true that, so far, nothing precludes the possibility of mathematics being a vacuous science. We saw Elements II.5: it was a valid general argument, because it showed that, given a certain property, another must follow as well, and this would be true for the domain of the validity of the derivation, regardless of the object chosen for the particular proof. It was an argumentx and therefore it could be directly transferred to whatever satisfied the antecedent ‘has argumentx valid for it’, ‘can be viewed inside the x framework’. However, if a mathematical (or scientific) statement is typically based on such arguments, showing that if something can be seen in a certain framework, then certain properties follow, then it is always possible that the antecedent would never materialise. Like a remnant of the cold war, Euclid's Elements II.5 might rust in its silo. Had some line been cut into equal and unequal segments, it would have been launched, the squares and the rectangles would have been equal. Yet no one cut a line into equal and unequal segments, the button was never pressed. The validity of the theorem remained vacuous, without any application. But notice that this is a kind of vacuity with which Aristotle could easily deal. In this case, Elements II.5 would be not vacuously true, but potentially true - i.e. true. It is a substantial feature of the universe, that were lines to be cut in a certain way, some equalities would result, even if no one has ever cut lines in that particular way: we need not worry (as far as the mathematics is concerned) about the question whether the antecedent ever materialises or not.

One may raise the following difficulty: could Elements II.5 be a dummy? Which could never be launched? Suppose that the argument of II.5 is valid, and yet suppose also that, for some reason, the antecedent of the conditional just cannot come to pass. For instance, lines cannot be cut precisely in the middle, or they cannot be cut at all, or simply one cannot produce lines. This worry does not seem to preoccupy Aristotle, as if it were just obvious to him that geometry may be actualized. I shall return to this in the next subsection. I explain: this worry does not preoccupy many modern philosophers of mathematics, because their view is that II.5 shows the derivation of one statement from another, and that's it; the truth of the P or the Q, taken separately from the P Q, is just immaterial. For Aristotle, however, the truth of science and of mathematics in particular was about reality, not just about logical

34Netz (1999), chapter 1. Indeed, this should be obvious once stereometry is considered: spheres must be represented in diagrams by circles and a circle, however accurately drawn, is not a sphere.

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derivations35. Thus he should have provided some argument why the antecedents of mathematics are of such a type that they are at least potentially true. But such a possible argument is no more than hinted at by unit (e), an extremely brief and tantalizing argument. I repeat a21-23, from which unit (e) takes of: 'So everything would be studied best, if one would posit the inseparable, separating it - as the arithmetician does, and the geometer'. This we have read already: we are allowed the fictions of the Platonic objects. The worry now is: how can we say that those fictions ultimately hold of anything? 4.4 Unit (e): a23: 'For man is, QUA man, one and indivisible. But the <arithmetician> first posits an indivisible one, and then studies whether anything

follows, QUA indivisible, for man. 25 While the geometer does not <study> QUA man or QUA indivisible but QUA a solid

<magnitude>. For these which would belong to him even if in some way he was not indivisible - it

is clear that they <=divisibility properties> may also belong to him without them <= without the presuppositions ‘man’, ‘indivisible’>36.

So that, because of this, the geometers speak correctly, and they speak about beings, which really are;

30 for being is double: • 'entelechy'; • and 'as matter'.

Let us try to follow this line of thought developed here. First of all, as in unit (a), ‘framing’ and the ‘for’ clause provide the structure of the passage. Lines 28-30 (‘so that... really are’) are a faint echo of lines 21-23 (‘so everything... the geometer’). The passage in between starts from a ‘for’ and leads to a ‘so that’. And so the main thrust of unit (e) is, once again, the main claim of unit (a): ‘mathematics is true’. (In this way unit (e) as a whole forms a certain echo of unit (a), and the very last words return us to the beginning of the passage, about ‘ways of being’: hence a sense of closure for the entire chapter.) Here however the sense of ‘mathematics is true’ clearly has to do not with methodology alone, but also with ontology: it is not only true, but is true of things ‘which really are’: the kind of things for which mathematics may be said to be true, has a valid claim to existence. This is crucial for the main project, of sustaining realism without Platonic entities. To put it anachronistically, Aristotle now needs to show that sciences are true (without Platonic entities) not only in some limited instrumental sense, but in the sense that they state the truth about external realities. For Aristotelianism to stand as an alternative to Platonism, this unit must succeed, and in

35Burnyeat (1982). 36I italicise the ‘without’ to stress that it is the same quasi-technical use as in unit (c) above (I owe this observation - the key to the understanding of this unit - to Alexander Sherman).

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the next, section 5, we shall therefore need to examine the success of the chapter in light of the success of this unit. The claim is that the objects of which mathematics speaks really exist, under Aristotle’s own account. The way Aristotle approaches this claim is this: he sets out what seems like a difficulty for this claim, and shows how this difficulty may be removed. We take a concrete example, with one object, and with two mathematical sciences: man, arithmetic, geometry. Aristotle wants to show that both disciplines actually apply to the concrete man. Our interpretation of this passage depends on what we see as the difficulty Aristotle sensed about this application. The most important thing to notice is the asymmetry Aristotle deliberately constructs between the two sciences. He starts with the general statement, that ‘man is, QUA man, one and indivisible’, and in this light an asymmetry is created between the different indexing of arithmetic and of geometry. Arithmetic indexes with ‘indivisible’: arithmeticindivisible(man), and this ‘indivisible’ index is obviously legitimate, since it is one of the essential features of ‘man’.

However geometry does not take any such index, as stressed by Aristotle. He does not only assert that geometry does not index with ‘indivisible’, but adds the further remark that it does not index with ‘man’. Of course it doesn’t - who ever thought it did? Aristotle mentions this not to teach us this particular point, that geometry does not index with ‘man’, but to stress the main feature of the way in which geometry indexes in relation to ‘man’: its index is such that does not belong to man, QUA man. The index is ‘solid’:

geometrysolid(man).

This is in contrast with arithmetic, whose index does belong to man, QUA man (I shall immediately explain why this does not belong to man, QUA man). This is the difficulty, then: while it is easy to see how an ‘arithmetic (man)’ type of application holds, it is difficult to see how a ‘geometry (man)’ type of application holds. A science whose index is appropriate to a given subject-matter will obviously hold of that subject-matter: what is an index if not that37? Arithmetic is clearly in a better position, then: without postulating any separate realm of ‘units’ clearly there are units around. But what about solids, planes and lines, which seem to be ‘without’ essential properties, to ‘abstract’ essential properties? How are we assured of (at least the potential) presence of objects which are appropriately indexed by such properties? This is the asymmetry of the difficulty, and that the difficulty is asymmetric is clear when we read the solution, which is asymmetric as well:

‘For these which would belong to him even if in some way he was not indivisible - it is clear that they <=divisibility properties> may also belong to him without them <= without the presuppositions ‘man’, ‘indivisible’>’.

37I am right now speaking on behalf of Aristotle, and I shall qualify this view, myself, in section 5 below.

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To paraphrase, we are given here two routes for finding properties. Route one, the red route, is to take the counterfactual case of the divisible man, and to spell

out his properties (presumably, with the aid of geometry). Since this man is not indivisible, it is no longer improperly indexed by geometry. Thus we may use geometry to derive a set of properties, which I call ‘counterfactual man properties’. It is obviously assumed that this is a subset of ‘(actual) man properties’ (presumably because counterfactual man differs from actual man by having one less defining property, and the assumption is that sets of properties are a direct projection from sets of defining properties: I shall return to this crucial point in section 5 below). Thus geometry, applied to counterfactual man, would reveal a subset of the properties of actual man.

Route two, the green route, now completes the argument. For let us reinstate man as he is, indivisible, and study it geometrically, i.e. ‘without’ the presupposition ‘indivisible’. Aristotle claims this:

The properties revealed by the green route are the same as those revealed by the red route.

In other words: there is no distinction between (i) studying a counterfactual divisible man, and (ii) studying man as he actually is, but without the assumption of divisibility. The two routes converge: there is no duty on counterfactuality. Hence, by studying man geometrically, we reveal a subset of the properties of man, in spite of the strange nature of the geometrical index.

To repeat, there is a systematic asymmetry between arithmetic and geometry as applied to man. One applies directly, to actual man; the other applies directly only to counterfactual man, and can apply to man only through the complex two-routes argument. The theme of this unit is this duality, and how it is solved.

Why is there such a duality to begin with? This is not the skeptical worry: are there straight lines at all? I.e., is anything straight? - not an important worry for Aristotle, because this can be easily solved by an argument from potentiality. The worry derives from the distinction between essential and incidental properties (so Aristotle is very far from relativising such a distinction in this chapter!). Even if something happened to be straight (say, my hand, stretched out), this may be an incidental property of it (I may bend my hand again) and thus not a property of it at all. ‘Fine’, replies the naive Aristotelian: ‘this is no problem, as long as it is a property of something: surely there is something for which the property of straightness is ultimately not incidental’. Not necessarily, cautions Aristotle: he has been careful to remove any ontological commitments. There is no separate realm for mathematics, nothing that simply answers for mathematics - nothing to guarantee that mathematical properties will essentially belong to it. Thus Aristotle’s realism puts him in a position where he is not required to project any ontological conclusions from his structure of sciences. The existence of a bona fide science shows strictly nothing about the ontic structure of the world, and does not demand, in particular, the existence of essences (the shadows of Platonic entities) appropriate to it. Realism, however - Aristotle explains - demands no such essences: properties shown by a science which does not deal with essences will still belong to things, even if they do not belong to things as they essentially are.

Consider in particular ‘man’: for instance, consider me. An arithmetician may tell me that my wife and I form a couple, and this will be true of me: it reveals an existential reality about me, the individual that I am. But now I encounter a geometer, and I am told that my volume is n cubic centimeters. And this, fully analysed, is not a truth about me at all, nor about any other complete

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entity. What the geometer has told me translates into: ‘if I compress your body into the shape of a cube, without losing any volume in the process, then I can produce of you a cube whose side is equal to the cubic root of n’. Such geometrical claims may be true, but they are no longer true about me: following this cubic reduction, I shall lose my identity and, in Aristotelian terms, I shall cease being an entity at all. The geometrical cubic reduction is true not of me - the complete Aristotelian entity that I am - but is true only of the matter of which I happen to be made. Both of which, Aristotle finally explains, have a title to existence:

30 ‘for being is double:

• 'entelechy'; • and 'as matter'’.

The most important thing to realise about this last, most brief and most tantalizing sentence, is that Aristotle still thinks in terms of the duality developed in the preceding argument, and with the specific example: man, arithmetic and geometry. Arithmetic is true of man the entelechy; geometry is true of man the matter. Both exist, and so both arithmetic and geometry are true of existent things38.

The significance of the example is easy to generalize: If a property P belongs to some object O (at least potentially), but there is no complete entity to which it essentially belongs, the statement ‘P belongs to O’ remains a true statement about existent things. This self-obvious principle allows the following, remarkable result: Even in the minimal ontological case where there is no complete entity to which any mathematical property essentially belongs, mathematics may be true, and true about existent things. That is, it is not only in the terms of a Platonist ontology that mathematics is ontologically neutral, but also in the terms of an Aristotelian ontology. In order to say that mathematics is true about existent things we need specify nothing about complete entities and essences.

This negative claim is all that Aristotle needs in this chapter, in order to show how minimal are the ontological presuppositions of science. It is clear that he would also wish to show that there are some things of which mathematical properties truly hold, essentially or otherwise. He does not show this in any detail here, this not being his concern, but he does sketch a possible analysis: the two main mathematical disciplines are arithmetic and geometry; arithmetic holds of entelechies, unproblematically (this is because it assumes indivisibility, which obviously belongs to them); while geometry holds of matter. Why is this last statement true? What is it about the geometrical set of assumptions - the geometrical framework - which makes it fit matter?

Aristotle might suggest in this passage something along the following lines: whatever is

38It is plausible that the preceding ‘geometers’ are not the common synecdoche ‘geometer’=’mathematician’, but actually means ‘geometers’. Then the claim is that geometers speak about existent things - because of the two-routes argument in lines 26-28 (that arithmeticians speak of existent things was never problematic).

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enmattered and therefore spatially extended is guaranteed to have geometry potentially true for it, because whatever is enmattered and spatially extended is potentially divisible, and geometrical antecedents are usually of the type 'let X be cut' (which is the case in Elements II.5, and has a great measure of truth in general, especially if you think of the main objects - planes, lines and points - as potential results of cuts). Thus, in a paradoxical but also a satisfying way, ‘divisibility’, the very property which guarantees the inapplicability of geometry to entelechies, also guarantees its applicability to matter. 4.5 Structure of M3 The main division of M3 is between units (a)-(b), on the one hand, and units (c)-(e), on the other hand. The first part is a diminuendo, unit (b) being no more than a footnote to unit (a). The second part is generally speaking a crescendo, unit (c) leading on to unit (d), while unit (e) is somewhat like a footnote to unit (d) - but such an important footnote that it is, if anything, at the same level of importance as unit (a). This cycle of diminuendo and crescendo finally leads on to a certain echo of the beginning of (a) at the end of (e), with a satisfying sense of closure.

Units (a)-(b) put forth the QUA theory and show how it dissolves the Platonist answer to question 2: ‘assuming that there are no separate mathematical objects, how can mathematics be true without qualification?’ - answered by unit (a); ‘and how can it be true as mathematics?’ - answered by unit (b). Units (c)-(e) do several things simultaneously. First, they display the fertility of the account offered in unit (a): each of the units is an application of the analysis of science in terms of the QUA theory. Second, these units account for a couple of phainomena: the place of mathematics in the structure of sciences (unit (c)), and the mathematical quasi-Platonist practice (unit (d)). Finally, they begin to indicate (very briefly) how Aristotle might have developed a philosophy of mathematics in another, less polemic context, and unit (e), especially, is an attempt to account for mathematics not just ‘outside Platonism’ but also ‘inside Aristotelianism’. Because M3 is so deeply embedded in a polemic context - as shown in Cleary (1995) - this remains as no more than a tantalizing suggestion, as described in section 4.4. The real purpose of M3 is polemical: to argue against Platonism, and to show that its entities are not necessary for sustaining realism. In the next section, I shall raise the question of the success of Aristotle’s alternative account to Platonism. To sum up the results of this section, however, it might be useful to approach the more positive issue, of Aristotle’s own philosophy of mathematics, on Aristotle’s behalf, or on the M3’s behalf. What is the positive philosophy of mathematics suggested by M3? I repeat the questions from section 1.1: 1. How is mathematics true of the world? The answers to this will differ, depending on the different mathematical disciplines. The difficult case is apparently that of geometry, and the answer is complex: partly, geometry is true of the world because the world is enmattered (why this follows, we are not told: perhaps because of the divisibility of matter; at any rate, this seems to be the suggestion, and it is in itself plausible); partly, geometry is true of the world potentially. (That is, even if no line is cut in the middle, lines could be cut in the middle; but in Aristotelian science the ‘potentially’ qualification is hardly a qualification at all and so there really is no need to stress this last point). 2. Given that mathematics is not true of the world (in that the objects mentioned by mathematics are

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not among the objects of the world) how can it be that mathematics is true without qualification? This is the only question in the philosophy of mathematics which is directly discussed in M3, and the main moral of the chapter is this: that, in general, questions of scientific validity are independent from questions of ontology: questions abut the framework are separate from questions about the object (unit (a)). Indeed, sciences are defined by the frameworks from which they set out, not by the objects which they set out to capture (unit (b)). 3. What accounts for the privileged status of mathematics? Put differently, what is so special about mathematics? This has been answered clearly (if not in detail) in unit (c): mathematics is marked by the relatively minimal presuppositions inherent in its arguments; the mathematical frameworks are especially minimal. 4. What is mathematics true of? Clearly Aristotle thinks that mathematics is true of the world. But this is not an argument developed in any detail in M3. The worry, that mathematics may be vacuous, is gradually removed, not by giving a set of objects of which mathematics holds (except for a very brief argument in unit (e)), but by the basic approach of the QUA theory. Once we are used to thinking in terms of the QUA theory, it seems much less surprising that mathematics will finally hold of things.

We may think of this in the following way. Suppose that a science is like an ad sent to the personals column. It projects an hypothetical set of the individuals who satisfy the properties of the ad or of the science (an ad which asks for a non-smoker is like a science whose framework involves the assumption of non-smoking). Obviously, such ads or sciences cannot guarantee in themselves that their sets would be non-empty. The requirements might be such that no individual would satisfy them all: this is contingent. However, what is not contingent is the following principle: the lesser our requirements, the larger the set is likely to be. A set defined by truly minimal requirements is most likely to be non-empty. According to Aristotle, mathematics is like this: it is the science with minimal requirements. Assume indivisibility and nothing else, and you get arithmetic; assume divisibility and nothing else, and (perhaps) you get geometry. Assume nothing, apparently (except, perhaps, ‘quantity’), and you get general mathematics. You just cannot assume any less. Thus, if anything is ever to be true, than this is at least mathematics and, for this reason at least, we should not be surprised that mathematics is true. 5. Realism and the Philosophy of QUA: an Assessment In this section, I briefly develop a philosophical claim independent from the exegesis of Aristotle, and then return to assess (using this general philosophical discussion) Aristotle’s claim that realism can be sustained without Platonic entities.

Let us assume, then, the principle that every scientific argument is characterized by a perspective, whose precise nature we need not specify here. Thus, any scientific argument will hold or fail to hold, in the first instance, relative to that perspective, while the question of the validity or the relevance of the perspective itself comes later. This may be because scientific statements, typically, are conditionals: they state logical entailments or causal relations, or they offer explanations: all of the form ‘P, because Q’. Perhaps we have no choice: to persuade, we must give an argument, and an argument is a finite passage from claim A to claim B and so, ultimately, the only thing we can ever persuade anyone about is the passage from A to B: a point understood by

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Aristotle39. At any rate, it is typically possible to separate the question of the validity of the scientific derivation, from the question of the truth of the antecedent. Does the earth move around the sun? Are organisms made of cells? Is there a significant unity underlying Mediterranean societies? All important questions - yet, independently of the answer to such questions, it is always possible to proceed as if they were already answered in the affirmative - producing in this way a scientific argument within a given perspective. So far we were given no cause for concern: given a true derivation and a true antecedent, the consequent will hold as well. So, on this analogy, if the argument holds inside the perspective, and if the perspective itself matches reality, then the argument finally matches reality as well. So long as our framework matches reality, or to the extent that our framework matches reality, so long, and to that extent our results would match reality as well: this seems to have been Aristotle’s view. This complacency, however, is misplaced, and we need to remind ourselves of the fact that a valid argument inside a valid framework may fail to hold of reality simpliciter.40

Let us first distinguish between two senses of ‘framework’. One may be called ‘approximation’, the second may be called ‘abstraction’41. A typical approximation is, for instance, to deal with a physical, imperfect sphere as if it were mathematically perfect. Then it can be shown that (on that assumption) the area of its surface is four times the area of a great circle inside the sphere. This will be approximately true of the physical sphere, that is the area of its surface would be very nearly four times the area of a great circle inside it: this is an approximation, and, typically, it makes a false assumption, that all the radii in the physical sphere are equal. The assumption is qualitatively false but, in some quantitatively meaningful way, it is very nearly right: almost all radii are very nearly equal. So this is a case where we adapt a framework, in the full knowledge that it fails to match reality. Such frameworks are a compromise between truth and expedience: for a certain price in truth, we gain considerably in computability. It is clear that such perspectives, strictly speaking, do not match reality, and therefore claims made through such perspectives, strictly speaking do not match reality either. Another type of perspective is ‘abstraction’. An instance of this would be to take the sphere (whether physical or mathematical) as a continuous set of points42. This is an ‘abstraction’, because the sphere, after all, is not just a continuous set of points: it is a very special way of organizing such continuous points in space. What is most typical of abstraction is that, unlike approximation, while it is not the whole truth, it is the truth and nothing but the truth. An approximation is strictly speaking false, but an abstraction is strictly speaking true: the sphere is a continuous set of points. And, on this perspective, we can show some interesting results: for instance, that the sphere contains an infinite number of points. We can then say that the sphere, strictly speaking, contains an infinite numbers of points. It is not just that the sphere contained them given the perspective: the perspective, while abstract, was also true, and therefore the result, of containing infinitely many points, is simply true as

39This is one of the main themes of the Posterior Analytics, argued especially in the first three chapters. 40 The following discussion is related to familiar arguments in current philosophy of science, mainly following Cartwright (1983). My aim is to show that, following such current recent analyses, Aristotle may be shown to endorse, tacitly, a false presupposition, critical to his project of a realism without Platonism. 41 Of course, the term ‘abstraction’ is used here without reference to the ‘abstractionist’ readings of M3. 42 Let us assume for the sake of the argument that the assumption of continuity holds of the physical world, and is not just a convenient approximation.

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well. This is because - so the assumption runs - we have assumed only true things (if a subset of the truth) and therefore our conclusions should contain a (subset of the) truth This assumption, that abstraction guarantees validity because it projects a subset of the true claims about the object, is extremely important. It seems to have been Aristotle’s underlying assumption in the key argument of unit (e) (1078a26-28): ‘For these which would belong to him <=to man> even if in some way he was not indivisible - it is clear that they <=divisibility properties> may also belong to him without them <= without the presuppositions ‘man’, ‘indivisible’>’. I have explained this argument in section 4.4 above as follows:

• We are given man as it is: indivisible. • We envisage another, counterfactual, not-indivisible-man. • We derive a set of ‘a (counterfactually) not-indivisible-man’s properties’. • We assume that this set is a subset of ‘actual man properties’:

• This is valid on these assumptions: that counterfactually not-indivisible-man differs from actual man by the lack of a defining property,

• and that sets of properties are a direct, monotonic projection from sets of defining properties: add defining properties and you add some extra, derived properties, remove defining properties and you remove some extra, derived properties.

• We now go back to actual man. We know that we have a subset of his properties, discovered through the counterfactual route: we may simply attach to him this subset, and equate this with the subset of properties which belong to man QUA not indivisible.

This is a description of abstraction at work, and it reveals a central chain of assumptions. First, it is assumed that the results revealed through abstraction hold of the object, simpliciter: If (F belongsx to A), then (F belongs to A), Which in turn relies upon the assumption that defining properties project extra-properties in a direct, monotonic way. To explain: in any system of dependencies (logical, causal, etc.), in which some properties, events, etc., can be said to be ‘consequents’ of others, it is possible to ask this question: Given A, B, and their two sets of consequents: consequents (A), consequents (B), what can we say of the further, third set, consequents (A and B)? We shall say that the system is compositional if it always satisfies:

Consequents (A and B) = consequents (A) and consequents (B).

Consider for instance ‘enmassed’ and ‘positively charged’. One of the consequents of ‘enmassed’ is ‘mutually attracted’: two pieces of enmassed matter attract each other. So:

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Mutual attraction is among consequents (enmassed). At the same time, ‘mutual attraction’ is not among the consequents of ‘positively charged’. In fact, two pieces of matter with positive electric charge repel each other. So:

Mutual attraction is not among consequents (positively charged). Clearly, then, Mutual attraction is not among consequents (enmassed and positively charged) Two pieces of matter that are both enmassed and positively charged might happen to attract each other, but we can not predict this would be the case just on the basis of these two properties: the scientific consequence, which followed for ‘enmassed’ alone, no longer follows when this predicate is composed with another such as ‘positively charged’.

In this case, then, the system is non-compositional. Another way of looking at it is that derived properties do not project monotonically from the defining properties: by adding a defining property (‘positively charged’), we have removed an extra, derived property (‘mutually attracted’). Suppose one were to say, then, ‘I shall abstract from electric charge, considering matter only QUA enmassed’: while obviously possible, and while this is clearly an act of abstraction (and not of approximation), the results of the abstraction would not necessarily hold for the domain from which we have started. What is an abstraction, after all? It is an analysis of a domain into its defining properties, and the application of the results of the abstraction to the domain are a synthesis, a re-composition of the domain (what Aristotle calls ‘subtraction’ and ‘addition’, as shown in Cleary (1985)). But such a re-composition would hold only in the special case of a compositional domain or, in other words, If (F belongsx to A), then (F belongs to A) is reliably true only under the special condition: The index x is compositional with all the remaining applicable indices. It might be that, with due logical care, compositionality can be guaranteed in some areas, perhaps in mathematics. On the other hand, some areas may be essentially non-compositional. The world, in particular, does not seem to operate with smooth logical summations, but, instead, with messier parallelograms of forces: the consequents of (A and B) in the world are, most often, neither among the consequents of A nor among the consequents of B, but are instead some compromise between the two. Seen from the horizontal perspective, the object moved from point H to point K; seen from the vertical perspective, it moved from H to L. Both statements are true - with the perspective-proviso attached. None is true simpliciter: the object has moved, simpliciter, from H to M. H K L M

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Here, of course, we have an over-arching perspective which gives us access to truth simpliciter, but we should concentrate on the case were we are not yet given such a privileged point of view. In this case - perhaps the normal position of the scientist - we have two approaches to studying a given problem, both legitimate; which taken together yield an inconsistency: the one predicts the point K, the other predicts the point L. This should cause us neither surprise not alarm, since this effect is a predictable result of the non-compositionality of the field. In fact, what we have is not an inconsistency, but a complementarity. The two approaches provide us with two, complementary ways of studying the same problem, and as long as we remember that the two start from different presuppositions - as long as we do not make a direct attempt to synthesize the two approaches - there is no contradiction. The principle is simple, then: in a non-compositional field, complementarity is the duty paid on abstraction. If you study things through different perspectives, you should expect that your claims would sometimes fail to hold for the object itself. Furthermore, you should expect that the results provided by different perspectives would sometimes fail to mesh together. The results obtained through one perspective should be incompatible with the results obtained by another.43 What is true of the object, QUA electrically charged, and what is true of the object, QUA possessing mass, may fail to mesh together, so that ultimately both theories - electrical and gravitational - may fail to hold, simpliciter, of anything; so to assume that they are true simpliciter we must postulate the separate existence of their theoretical referents. In general, if we wish to save our intuition that science is true simpliciter, we must postulate theoretical entities of which it is true. As mentioned above, the analysis offered here of the relationship between perspectives and reality is related to that of Cartwright (1983), ‘How the Laws of Physics Lie’. It is interesting that Cartwright herself saw her argument for (what I call) the non-compositionality of perspectives, as an argument against realism as regards scientific laws.44 For her, the fact that laws of nature fail to agree with reality, is an argument for those laws being false. But this is of course precisely what a Platonist denies. For a Platonist, the fact that laws of nature fail to agree with (ordinary) reality is precisely an argument why super-ordinary, Platonic reality should be assumed over and above ordinary reality. The Platonic argument works on the presupposition of the truth of science. As I have tried to show in this article, Aristotle does not wish to contest this presupposition, and his aim is to show only that Platonism does not follow: realism is sustainable without Platonic entities. In order to show this, Aristotle argues that sciences may be true, and true of existent things, even without projecting

43 An obvious consequence for hermeneutics is that an author who seriously who adopts different frameworks for different arguments, and who takes seriously those frameworks, should display inconsistencies. To this extent, the arguments produced by such an author are best understood at the individual level of the particular argument, relative to the particular framework. I assume Aristotle is a prime case of such an author; hence my tendency to understand M3 apart from Aristotle’s corpus taken as a whole. 44 So in Cartwright (1983). In Cartwright (1989), however, she shows the necessity of assuming the reality of capacities. In both these works, then, Cartwright shows the absence of the third way between anti-realism and Platonism, settling for anti-realism in 1983, as regards laws, and settling for Platonism in 1989, as regards capacities. Such a differential attitude to Platonic entities is of course plausible; all I insist upon is that the non-compositionality of perspectives shows that while one may settle for either anti-realism or Platonism, for different objects, one must settle for one of them.

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ontological domains peculiar to those sciences. Sciences do not carve up the domains of ontology, but the domains of methodology, and what characterizes a science is its methodology (or perspective). A science such as mathematics does not have any special ontological domain, and its (at least potential) truth for existent things is provided by the existence of things of which the mathematical perspective (at least potentially) holds. Thus, for Aristotle, some sciences such as mathematics are ontologically parasitical. They do not have their own domain, but rely upon other ontological domains, of which they just happen to be true. As seen in unit (e), Aristotle’s claim, which seems plausible, is that parasitical truth is no less a truth than any other truth: even if a science such as mathematics does not hold directly of anything, still it holds indirectly of entities which, among other things, have that science (for instance mathematics) true of them. However, Aristotle’s seemingly plausible claim is in fact wrong: if a science merely holds of things which have it true of them, among other things, then its truth becomes contingent upon the compositionality of that science with the other relevant sciences, and we can no longer assume in general that this science is true simpliciter. Parasitical truth is, therefore, less of a truth than direct truth. Aristotle’s argument, that the QUA theory makes it possible to sustain realism without Platonic entities, is therefore problematic. The Platonic-Aristotelian exchange may be paraphrased like this. Plato’s starting-point is that the world is so intricately dialectical, so deeply embedded in an irrational flux, that no rational theory can relate to it directly, without intermediaries. Rational sciences, organized around rational objects, must be postulated as intermediaries between the knower and the world, so as to make genuine knowledge possible.45 Aristotle, in fact, accepts this - at the epistemological plane. He accepts that intermediaries between the knower and the world are necessary, but takes them as purely epistemic entities, without ontic consequences. Aristotelian sciences, just like Platonic sciences, are not merely reports on the world of flux: they follow rational principles and so, like their Platonic counterparts, are at a certain remove from the world. Ontologically, however, this has no consequence, since what constitutes those sciences is not their objects, but their approaches: all the sciences are different perspectives on the same world, the only world there is. These perspectives are no more than intermediaries or interpreters for the world; when we wish to say, ultimately, what it is that we know, we can simply get rid of the intermediaries: the things we know, are truths about the world. The irony is that, in hindsight, on the basis of our modern experience of a large interacting body of sciences (not available to Plato or to Aristotle themselves), we can see that the very perspectival analysis of the structure of science sustains, finally, the Platonic intuition of the need for intermediaries of ontic standing. Just because sciences are different perspectives on the world (and do not merely pick up different parts of the world), sciences can not be simply assumed to be true of the world. What is seen to be true through a given perspective need not be true simpliciter so that, to make science true simpliciter, we must assume the existence of its theoretical entities.

To sum up, then: in Metaphysics M3, Aristotle attempted to offer a middle position between Platonism and anti-realism. Both are indeed counter-intuitive: Platonism poses a difficulty to the

45 There are of course many arguments in Plato’s writings for the existence of forms or of other transcendent entities, and there are many modern interpretations of all of these arguments. The brief paraphrase I offer follows, essentially, Jordan (1983), but it is meant as no more than an approximation to one strand of Platonic argument, seen at the background of M3.

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intuition that there is no transcendent order of reality beyond ordinary reality itself, while anti-realism poses a difficulty to the intuition that science is true. The enduring philosophical interest of M3 stems from its attempt to sustain, simultaneously, these two highly compelling intuitions. What is remarkable is not that Aristotle failed, but that he was so nearly right: there is no reason why he should have come up with the analysis of the non-compositionality of perspectives and, without this analysis, he was, I suggest, justified in his position. In historical terms, then, we can say that the position of the M3 is a valid response to Platonism. In philosophical terms, we can say that, while the non-compositionality of perspectives need not, in itself, make us decide between Platonism and anti-realism, it does show that an account such as Aristotle’s cannot sustain a middle position between the two. We are back where Plato left us: either there are transcendent entities, or science is, strictly speaking, false. References

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