Q&A: Mathematical Platonism
Mathematical Platonism
Question
I listened to the series on Platonism and enjoyed it מאוד. I want to ask: what is the nature of the mathematical Idea?
In mathematics we have several ways to construct a mathematical object. Take the real numbers, for example. I can define them axiomatically, I can use Dedekind cuts, and I can use sequences of decimal digits or the completion of a collection of Cauchy sequences.
After careful treatment I can show that all these definitions are isomorphic, even though each of them describes an object that at first glance seems fundamentally different.
Does the world of Ideas contain “real numbers,” and all these methods are just different realizations of the same Idea? Or is each one an Idea in its own right, but mathematics has shown me that they are isomorphic? In other words, what is the ideal mathematical entity: the mathematical structure, or the concept that I define?
Does proving an isomorphism show me that this is the same Idea, or are these still different Ideas?
Thank you
Answer
Definitions do not necessarily capture the thing-in-itself. It is worth seeing my article on Zeno’s arrow, where I showed that defining velocity as the derivative of the position function with respect to time is an operational definition, not an essential one. The same is true in mathematics. Different definitions can hit upon the same Platonic object. But there can also be a situation in which these are different objects that stand in an isomorphic relation to one another. Think about the Cartesian and polar representations of points in the two-dimensional plane. The term “angle” appears only in the polar representation, yet there is an isomorphism between these representations. That does not mean there is no Platonic object of angle.