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.

Platonism is one of the ~three philosophies which dominate modern mathematics, the others being Intuitionism and Formalism/Logicism. However, few if any mathematicians adhere consistently to any of those three philosophies.

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.

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 $n\geq 0$.

-- Jacob Lurie, Higher Topos Theory