David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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International Journal of Philosophical Studies 8 (1):23 – 45 (2000)
I argue here that a properly Platonic theory of the nature of number is still viable today. By properly Platonic, I mean one consistent with Plato's own theory, with appropriate extensions to take into account subsequent developments in mathematics. At Parmenides 143a-4a the existence of numbers is proven from our capacity to count, whereby I establish as Plato's the theory that numbers are originally ordinal, a sequence of forms differentiated by position. I defend and interpret Aristotle's report of a Platonic distinction between form and mathematical numbers, arguing that mathematical numbers alone are cardinals, by reference to certain non-technical features of a set-theoretical approach and other considerations in philosophy of mathematics. Finally I respond to the objections that such a conception of number was unavailable in antiquity and that this theory is contradicted by Aristotle's report in Metaph . XIII that Platonic numbers are collections of units. I argue that Aristotle reveals his own misinterpretation of the terms in which Plato's theory was expressed.
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References found in this work BETA
Penelope Maddy (1990). Realism in Mathematics. Oxford University Prress.
Paul Benacerraf (1965). What Numbers Could Not Be. Philosophical Review 74 (1):47-73.
Hilary Putnam (1967). Mathematics Without Foundations. Journal of Philosophy 64 (1):5-22.
W. D. Ross (1951/1976). Plato's Theory of Ideas. Greenwood Press.
John J. Cleary (1995). Aristotle and Mathematics: Aporetic Method in Cosmology and Metaphysics. E.J. Brill.
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