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 (noumenal), but rather are structures imposed by our mind on sensible inputs (phenomenal). [TODO elaborate on Kant's theory]

Here's a quote from Popper talking about Brouwer in Epistemology Without a Knowing Subject

In his Inaugural Lecture (1912) Brouwer starts from Kant. He says that Kant's intuitionist philosophy of geometry—his doctrine of the pure intuition of space—has to be abandoned in the light of non-Euclidean geometry. But, Brouwer says, we do not need it, since we can arithmetize geometry: we can take our stand squarely on Kant's theory of arithmetic, and on his doctrine that arithmetic is based upon the pure intuition of time.

Intuitionism is one of the three major philosophies which dominate the minds of modern mathematics, the others being Platonism 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.

It would

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.

From a talk by David Nadler, Dequantizing Symplectic Geometry:

Main question: Can we construct Lagrangian submanifolds from branes?

Much of math consists of procedures for starting with some mathematical objects, and using them to "construct" other mathematical objects.

A consistent Platonist would interpret these procedures differently. He would say that technically, rather than using some mathematical objects to construct others, we are using some mathematical objects to guide our intuition to a place where we can perceive the existence of others, and thereby discover them. However, as indicated by their choice of wording ("construct" rather than "discover"), working mathematicians are usually not consistent Platonists in this respect. I think the reason why is that often, working mathematicians sense (correctly) that some of the things with which they are dealing feel very man-made.

Most of the objects that working mathematicians consider cannot be constructed in perceptible reality (for example, one can't produce a Lagrangian submanifold by using a computer program or a block of wood). And one can't construct anything in the Platonic realm, because the objects there are supposed to be non-interactive and acausal. Therefore, the only place where one of these constructed objects could actually exist is: in man's intuition. Working mathematicians are thus forced to accept something along the lines of Intuitionsm, albeit unconsciously and inconsistently.

Working mathematicians have commonly-used words that differentiate constructed objects from discovered objects. An object which is discovered is labeled as "natural," or "canonical," or (less frequently) "perfect." An object which is constructed is labeled as "unnatural," "non-canonical," "messy," etc.

The idea of construction doesn't have to be intuitionist. For example, it is valid to talk about constructing an algorithm for---say---solving a linear equation. In this situation, "constructing" is used synonymously with creating or coming up with.