Intuitionism

Intuitionism is a philosophy of mathematics, which holds that mathematical objects exist in intuition.

Intuitionism is based, fairly directly, on the philosophy of Immanuel Kant. Kant thought that all the ideas of mathematics (and indeed, all ideas generally) are not descriptions of things in themselves (noumena), but rather are structures imposed by our mind on reality (phenomena).

Examples

Bold mine.

From Jacob Lurie, Higher Topos Theory:

Unfortunately, not every $\infty$-topos ${\mathcal {X}}$ can be obtained as topological localization of an $\infty$-category of presheaves. Nevertheless, in §6.2.4 we will construct $\infty$-categories of sheaves which closely approximate ${\mathcal {X}}$ using the formalism of canonical topologies.