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 Platonic things written by leading 20th or 21st-century mathematicians.

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

From one perspective, it is easy to find examples, because almost every single sentence of a modern math paper sounds Platonic. From another perspective, it is difficult to find examples, because it's hard to know if any given mathematician literally believes what he's saying, or if he just sees it as more of a convention to which he is adhering.

In some cases, for the benefit of the reader, I have bolded words which I think reflect Platonism.

From a book by 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 a memoir by Alexandre Grothendieck, Récoltes et Semailles:

À vrai dire, c’est la “tour de Teichmüller” dans laquelle la famille de toutes ces multiplicités s’insère, et le paradigme discret ou profini de cette tour en termes de groupoïdes fondamentaux, qui constitue l’objet unique le plus riche, le plus fascinant que j’aie rencontré en mathématique. Le groupe Sl(2), avec la structure “arithmétique” du compactifié profini de Sl(2,Z), (consistant en l’opération du groupe de Galois ${\textstyle {\text{Gal}}({\overline {\mathbb {Q} }}/\mathbb {Q} )}$ sur celui-ci), peut être considéré comme la principale pierre de construction pour la “version profinie” de cette tour.

From an interview with Peter Scholze:

So actually, I'm not a creative person at all. I'm a hardcore mathematical Platonist, so I believe that there is a mathematical world for us to discover. And in this way, I'm not creative. It's like, when I am trying to learn math, I am trying to learn what's there. It's that I'm trying to give it names, but that's all.

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. Something like: if an object participates in its form, what sort of thing is the participation? There needs to be another form of that type of object participating in that type of form. And so on.