International Journal of Philosophical Studies 8 (1):23 – 45 (2000)
AbstractI 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
Realism in Mathematics.Penelope Maddy - 1990 - Oxford, England and New York, NY, USA: Oxford University Prress.
Philosophy of Mathematics: Selected Readings.Paul Benacerraf (ed.) - 1964 - Englewood Cliffs, NJ, USA: Englewood Cliffs, N.J., Prentice-Hall.
Aristotle's Criticism of Plato and the Academy.Harold F. Cherniss - 1944 - Baltimore: Johns Hopkins University Press.