¹ cf. Ross, W. D., Aristotle's Metaphysics (2 vols. Oxford, 1953), ii. 414.Google Scholar

² Annas, J., Aristotle's Metaphysics Books M and N (Oxford, 1977), pp. 145–6.Google Scholar

³ It might be considered helpful, in the light of this, to employ different terms for Plato's and Aristotle's ‘noetic’ numbers, but I think there are advantages in maintaining the same term for the two. The main advantage derives from the fact that Aristotle, in his discussion of numbers, is often explicitly criticizing Plato with the aim of challenging Plato's conception of the ontological status of non-sensible numbers, rather than drawing a different distinction which cuts across Plato's. Their respective views of sensible numbers are essentially the same, Plato characterizing them as those numbers which have ‘visible and tangible body’ (Rep. 525d).

⁴ cf. Klein, J., Greek Mathematical Thought and the Origin of Algebra (Cambridge, Mass., 1968).Google Scholar

⁵ Of some importance here is the use of the γνὼμων, a geometrical configuration which is ‘fitted’ to formations of monads showing similarity of kind between these formations. For a discussion of figurate numbers and the γνὼμων cf. Heath, T. L., A History of Greek Mathematics (2 vols. Oxford, 1921), i. 76–84,Google Scholar and his notes on Definition 16 of Book 7 of Euclid's Elements in his edition of The Thirteen Books of Euclid's Elements (3 vols. New York, 1956), ii. 287–90.Google Scholar

⁶ cf. Szab⋯, A., The Beginnings of Greek Mathematics (Dordrecht, 1978), pp. 257 f.CrossRefGoogle Scholar

⁷ Considerable problems arise when the doctrine of particular substances is extended to cover inanimate substances, and the topic has been the subject of considerable debate. Much of the literature on the question is cited and discussed in Hartner, E. D.,‘Aristotle on Primary ΟΥΙΑ’, Arckiv för Geschichte der Philosophic 57 (1975), 1–20.CrossRefGoogle Scholar

⁸ cf. Elders, L., Aristotle's theory of the One (Assen, 1961), pp. 65–7, for a detailed discussion of Aristotle's use of this expression.Google Scholar

⁹ Aristotle provides the same kind of account for infinity (Ph. Γ, 204 a 8–206 a 8), place (Ph Ph. Γ., 209b 1–210a 13) and time (Ph. A, 218b21–220a 26).

¹⁰ A good account of Aristotle's view of the role of noetic matter in geometry is given in Mueller, I., ‘Aristotle on Geometrical Objects’, Archiv für Geschichte der Philosophie, 52 (1970), 156–71.CrossRefGoogle Scholar

¹¹ I have defended this interpretation of the noetic matter of numbers in my ‘Aristotle on Intelligible Matter’, Phronesis, 25 (1980), 187–97; it would take us too far from our main topic to enter into the details of the justification here.CrossRefGoogle Scholar

¹² This is the way in which numbers are usually notated in Greek arithmetic (cf. Books 7 to 9 of Euclid's Elements) and it is upon these line lengths that arithmetical operations are performed. In multiplication, for example, the product of two lines is a rectangle having those lines at its sides. This procedure places severe conceptual and computational constraints on Greek arithmetic as compared to the arithmetic developed from the seventeenth century onwards: cf. Mahoney, M. S., ‘The Beginnings of Algebraic Thought in the Seventeenth Century’, in Gaukroger, S. (ed.), Descartes: Philosophy, Mathematics and Physics (New Jersey, 1980).Google Scholar

¹³ This point is different from Aristotle's more specific remark at Metaph. I, 1053 b 25–1054 a 10, where he argues that ‘one’ has different uses in different categories. What we are concerned with here is the equivocality of ‘one’ as applied to two things of the same category (substance).

¹⁴ cf. Lambros, C. H., ‘Are Numbers Properties of Objects?’, Philosophical Studies, 29 (1976), 381–9. Lambros presents what is, I think, a good prima facie case against Frege; whether it is conclusive is a different matter altogether, and one that I shall not take up here.CrossRefGoogle Scholar