Tag: Mathematics

Does the Third Man Argument Refute Mathematical Platonism?

Probably not.

Mathematical Platonism asserts that numbers are Forms, “the non-physical, timeless, absolute, and unchangeable essences of all things, of which objects and matter in the physical world are merely imitations”. Numbers fit seamlessly into the theory of Forms, as any given collection of objects imitates the number of objects in that collection; a number never can be actualized physically.
The most commonly accepted mathematical conception of numbers, Zermelo-Fraenkel set theory (ZFC), employs the language of sets and set theory to describe numbers as sets containing nothing but other sets.
The Third Man argument refutes Plato’s theory of the Forms. I won’t explicate the argument here, but suffice to say the argument results in a contraction where an arbitrary Form, F, is both “one” and “many”, which are contrary properties. By “purity”, a Form cannot have contrary properties, hence a contradiction. Furthermore, the theory of Forms results in an infinite regress where a form, F1, must partake in a form greater than itself, F2, which must again partake in a greater form, F3, and so on.
Numbers are unique from the broader, more general theory of Forms, in that the greater form, F2, is no more abstract than the given form, F1. In fact, F2 is already defined by ZFC as the number n+1. As an illustrative example, consider a common ZFC definition for zero. Let zero be the null set. Then, one is the set containing the null set, i.e zero. So, when the Form of zero must partake in a greater Form, it simply partakes in the Form of one.

Future Questions
The question remains whether a number (n) partaking in its successor (n+1) contradicts the principle of Purity, but I think not. I am tempted then by the thought of how, if correct, Mathematical Platonism coexists and otherwise interacts with Aristotle’s Four Causes.