¹ Tarán, Leonardo, Speusippus of Athens (Leiden 1981)CrossRefGoogle Scholar. I refer to this work simply by page number.

² For a substantially different reading of these passages see Merlan, Philip, From Platonism to Neoplatonism³ (The Hague 1968) 98–140 CrossRefGoogle Scholar.

³ For an example of a more flexible attitude on this question see Robin, Léon, La théorie platonicienne des idées et des nombres d'après Aristote (Paris 1908) 229–232 Google Scholar.

⁴ The central work in what might be called the counter tradition is Stenzel's, Julius Zahl und Gestalt bei Platon und Aristoteles³ (Bad Homburg vor der Höhe 1959)Google Scholar.

⁵ ‘On the Platonist doctrine of the ἀσύμβλητοι ἀριθμοί’, CR xviii (1904) 247–260 Google Scholar.

⁶ See Aristotle's Metaphysics ii (Oxford 1958) 427 Google Scholar.

⁷ See The riddle of the early Academy (Berkeley and Los Angeles 1945) 34–37 Google Scholar, or Aristotle's criticism of Plato and the Academy, i (Baltimore 1944) 513–517 Google Scholar. (In the sequel I refer to these works as Riddle and Criticism, respectively.)

⁸ The passages cited by T. (14), who also refers to Wilson, Ross, and C., are Phd. 96e–97b with 101b–c, R. 525c–526b, Cra. 432a–d, and Phlb. 56d–57a. For a clear statement of the relevant interpretation of the first three of these passages by C. see the first reference in the preceding note. There C. uses the first passage to show that Plato believed in a form corresponding to each number, a point which is not in dispute; the dispute is over the character of these forms. The second passage shows that λογιστική compels the soul to discuss numerical forms, but it also provides evidence that λογιστικοί study ‘congeries of units’, since Socrates imagines someone asking them, ‘What kind of numbers are you talking about … in which each unit is all equal to every other, not differing in itself and having in itself no part at all?’ C.'s attempt to construe ‘each unit’ to mean ‘the unity of each of the numbers’ strikes me as far-fetched, but, even if one accepts it, calling the passage evidence for the received view would, I think, be to confuse evidence for an interpretation with a reading of a passage based on the interpretation. The Cratylus passage, too, does not seem to provide any real evidence for the received view. C. uses it to describe how numbered groups might be said to fall short of numerical forms, namely by lacking their unity; but the Philebus passage mentioned by T. suggests another way: the units in numbered groups lack the absolute equality of the units in ideal numbers. In this respect the Philebus passage counts against the received view, but I am not confident that T. would really want to make use of it in this connection. C. (‘Some war-time publications concerning Plato’, AJP lxviii [1968] 189–191 n. 79Google Scholar) uses it as an argument against Plato's believing in ‘intermediates’, numbers sharing some properties with forms and some with numbered groups.

⁹ Defenders of the received view usually treat Platonic ideas as concepts or universals (T.: ‘For Plato the ideas are the hypostatization of all the universals.’ [13] Cf. Sir Ross, David, Plato's Theory of Ideas [Oxford 1951] 225 Google Scholar), and the objections raised against the theory at the beginning of the Parmenides as to one degree or another insignificant. (See, e.g., Ross 87 or C, ‘Parmenides and the Parmenides of Plato’, AJP liii [1932] 135–138.) Others, of course, interpret the ideas as paradigmatic instances and find serious difficulties in the objections of the Parmenides. (See, for example, the papers by Gregory Vlastos and Peter Geach in R. E. Allen [ed.], Studies in Plato's Metaphysics [London and New York 1965]Google Scholar) Paradigmatic instances of numbers as conceived by the Greeks would almost certainly be ‘congeries of units’, but numerical universals presumably would not be. The scope of this paper precludes further discussion of the character of ideas in general, and I will content myself with trying to show that even if one accepts the notion that ideas are hypostasized universals, the received view of ideal numbers is illconceived.

¹⁰ For expositions of the view T. evidently has in mind see, e.g., Russell, Bertrand, Introduction to mathematical philosophy (London 1919) 1–19 Google Scholar, or Frege, G., The foundations of arithmetic² , translated by Austin, J. L. (Oxford 1959)Google Scholar. The applicability of the cardinal-ordinal distinction to Greek notions of number is doubtful (See Klein, Jacob, Greek mathematical thought and the origin of algebra, translated by Brann, Eva [Cambridge, Mass. and London 1968], vii, 46–60 Google Scholar), but if one is going to apply it, the Greek conception of number as a ‘congeries of units’ is closer to a cardinal than an ordinal notion. (Cf. Mueller, Ian, Philosophy of mathematics and deductive structure in Euclid's Elements [Cambridge, Mass. and London 1981] 69 Google Scholar.)

¹¹ I have inserted the references to Aristotle from T.'s footnotes. I should perhaps mention that T. does not indicate the incidental references in the dialogues to which he refers.

¹² The passage says that Plato did not make there be an idea of numbers because he did not posit ideas in cases where one speaks of before and after. The passage has played a prominent role in the received view because, I suspect, Cook Wilson's reading of it was taken to solve a long-standing interpretive problem. (See C., Criticism 513.) For the passage had originally been read as asserting that Plato did not believe in numerical forms. The reading of the received view does eliminate a difficulty, but I know of no satisfactory reconciliation of the reading with the traditional interpretation of the theory of forms as universals, according to which ‘to each kind of thing to which we apply a common name there corresponds a single idea’ (13). C. (521) appears to argue that for Plato an idea of number in general would be redundant because for him ‘each idea of number is … just its unique position as a term in the ordered series of numbers’, so that an idea of number in general would be identical with this series. I find it quite unlikely that Plato would have thought of the idea of two as just the first position in the number series and not as, e.g., the property which all pairs share. But if he did, why should he have thought of the property of being a member of the series as identical with the series? Wouldn't the same considerations lead naturally to the view that a genus is identical with the collection of its species?

¹³ C. (Criticism 514) suggests that the absence of an order for arithmetic number depends on the fact that arithmetic numbers are [sometimes] related by inclusion. But there obviously are well-ordered series related in this way, the best known example being the von Neumann ordinals: φ, {φ}. {φ, {φ}}, … (Many textbooks in set theory include an account of these ordinals, usually with a heavy dose of mathematical symbols. Paul Bernays gives a more discursive presentation in ‘A system of axiomatic set theory’, in Müller, Gert H. (ed.), Sets and classes [Amsterdam, New York, and Oxford 1976)] 19–24 Google Scholar.)

¹⁴ C. does not really give any explanation at all. He writes:

These ideas of number are, as universals, ἀσύμβλητοι and, as ἀσύμβλητοι, entirely outside one another in the sense that none is part of another; thus (?) they form a series of different terms which have a definite order. (Criticism 514)

I have queried the ‘thus’ because for C. what precedes the semicolon is true of all ideas, but they do not constitute a series with a definite order.

¹⁵ I have discussed Aristotle's mathematical ontology in ‘Aristotle on geometric objects’, AGPh lii (1970) 156–171 Google Scholar; Jonathan Lear offers an alternative account in ‘Aristotle's philosophy of mathematics’, PhR xci (1982) 161–192 Google Scholar. Aristotle actually says very little that is specific about the ontological status of numbers. He standardly speaks in a general way about mathematical ontology, and illustrates his views by reference to geometry.

¹⁶ A version of the first position is adopted by Nelson Goodman; see his Problems and projects (Indianapolis and New York 1972) 149–200 Google Scholar. The second is a form of the principle that indiscernibles are identical associated first and foremost with Leibniz; for an exposition of Leibniz's view see, e.g., Broad, C. D., Leibniz, edited by Lewey, C. (Cambridge 1975), 39–43 Google Scholar.

¹⁷ For arguments to this effect see, e.g., Putnam, Hilary, Philosophy of Logic (New York, Evanston, San Francisco, and London 1971)Google Scholar. The issues involved here are clarified when one talks about congeries of congeries, a level of abstraction which Greek mathematics does not seem to have reached. Modern mathematics is unthinkable without the distinction between a set of numbers and a set of sets of numbers. For traces of the raising of the analogous issue for numbers in antiquity see Aristotle's suggestion at Metaph. 1044a 2–5 that a number should not just be a σωρός, and Socrates' argument at Tht. 204b–205a that there is no difference between τὸ ὅλον and τὸ πᾶν.

¹⁸ Modern discussions of the indiscernibility of identicals have focused on physical rather than mathematical objects. The notion of indiscernible nonidenticals had a substantive role to play in philosophy of mathematics only as long as integers were thought of as sets of units.

¹⁹ For a Neoplatonic example of this kind of reasoning see Syrian, in Metaph. 90.9–15.

²⁰ Metaph. 997b 12–24. For the reasons underlying my claim see my paper ‘Ascending to problems’ in Anton, John (ed.), Science and the sciences in Plato (Albany 1980) 103–121 Google Scholar.

²¹ In connection with his rejection of Speusippus' conception of number as untenable T. cites one Aristotelian argument:

It is strange that there should be some one which is a first of ones as they [ = Speusippus] say, and not a two which is a first of twos or a three of threes. For all are subject to the same argument. If this is the way things are in the case of number and one postulates that only mathematical number exists, the one is not [i.e., should not be] a principle (for such a one must differ from the other units, and, if so, there must be some two which is a first of twos and similarly for the other succeeding numbers). But if the one is a principle, the facts about number must be as Plato used to say, and there must be a first two and three, and the numbers must not be associable. (1083a 24–34)

T. analyzes this argument as follows:

[Aristotle] implies that the very notion of separately existing mathematical numbers is a contradiction in terms. If numbers have separate existence, there must be not only a first One, as Speusippus said, but also a first Two, a first Three, etc. For to exist apart any number would have to be a different entity from any other separately existing number, and this would mean that each number is ‘incomparable’ with every other number. Mathematical number, however, cannot be incomparable, since the component units are all comparable and undifferentiated. (24)

I find nothing in Aristotle's text corresponding to the sentence I have italicized. Nor do I see how the first half of this sentence could ‘mean’ what T. says it does, since the first half only says that separately existing things are different from one another. In fact Aristotle appears to be arguing simply that if Speusippus is going to postulate a first one as a principle of number, he ought, by parity of reasoning, to assert a first two, etc. But such a first two is incompatible with belief in mathematical numbers only since among them there is no first two; belief in a first two gets one back into Plato's idea numbers. What Aristotle says does not rule out the possibility of believing in separately existing mathematical and idea numbers.

²² Some thirty titles in incomplete list of Diogenes Laertius (iv. 4–5).

²³ T. does offer explanations of Aristotle's alleged mistakes and misunderstandings, but these rarely strengthen one's confidence that an error has been found. I do not intend to glorify Aristotle's skills as an expositor of the views of others. But even after finding him guilty on many counts of misrepresentation, I am reluctant to assume him generally guilty until proven innocent. However, without this assumption many particular charges of guilt seem hollow. On the other hand, the assumption has the value of consistency. Without it putting together an interpretation of Aristotle's accounts of the views of others with as few loose ends as, say, that of C., is very difficult, perhaps impossible. But to minimize the loose ends of an interpretation and maximize its consistency is not always to maximize its plausibility.

²⁴ Nothing turns on the choice of this term. Cf. T.'s remark on Metaph. 1085b27–34 (362). He, however, never gives more than conditional assent to Aristotle's ascription of ‘material’ principles to Speusippus.

²⁵ The various elements of the correlation crop up rather frequently in Aristotle; see, e.g. Metaph. 1084b26–27, 1090b21–23.

²⁶ In a paper called ‘Aristotle's approach to the problem of principles in Metaphysics M and N’ to be published in the proceedings of the 1984 Symposium Aristotelicum.

²⁷ It should perhaps be pointed out that Speusippus never explicitly calls one a number, but merely says things which imply it to be odd, prime, and a submultiple. T. takes for granted that ascribing these properties to one necessitates thinking of it as a number, a plausible but not absolutely compelling assumption.