Platonism

Platonism is a philosophy of mathematics, which holds that mathematical concepts are objects ("forms"), existing independently of man's mind, that cannot be perceived through the senses.

Platonism dates back to Plato and Socrates. The term "Platonism" is also sometimes used to refer to the philosophy of Plato generally, which applies to much more than just mathematics.

Platonism is one of the three major philosophies which dominate the minds of modern mathematics, the others being Intuitionism and Formalism/Logicism. Very few working mathematicians adhere consistently to any of those three philosophies. Almost all of them tacitly accept the (mutually inconsistent) premises of all three.

Examples

In this section, I will show some examples of implicitly Platonic things written by working mathematicians.

The purpose of this section is to justify my claim that the premises of Platonism are widespread.

It is easy to find examples, because almost every single sentence of modern math is Platonic.

On the Intuitionism and Formalism/Logicism pages, I will quote from the same papers written by the same mathematicians.

At first glance, these quotes will appear to be innocuous. They have the form of statements about real things. They are Platonic because they are not about real things.

In the quotes, I have bolded some of the words which I think reflect Platonism

From Jacob Lurie, Higher Topos Theory:

For larger values of $n$, even the language of stacks is not sufficient to describe the nature of the sheaf ${\mathcal {F}}$ associated to the fibration ${\tilde {X}}\rightarrow X$. To address the situation, Grothendieck proposed [...] that there should be a theory of $n$-stacks on $X$ for every integer $n\geq 0$.

From [TODO]

Refutations

Aristotle's refutation

[TODO I heard this argument once and it was great, but I forget it. Something about how the idea of objects "participating in" a form leads to an infinite regress.