system
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human
Why do intuitionists reject classical logic and mathematics?

I am on a section of my Philosophy of Mathematics course on intuitionism, and am having a difficult time wrapping my head around the idea of intuitionism. Most of the primary reading I've been assigned, especially the Brouwer, is very difficult to break in to. I understand that the movement rejects the Completed Infinite and the Law of the Excluded Middle, but why do intuitionists choose to reject such things? Thank you very much in advance!
gpt
Because intuitionists believe that mathematical reasoning depends crucially on processes of mental construction, and this requirement leads to certain consequences that are at odds with classical logic.

For example, on this view, you cannot prove the existence of something by simply negating its non-existence, as you can in classical logic. That is, in classical logic, you can prove that Ex(Px)—i.e., "There exists an x such that x is P"—if you can prove \~\~Ex(Px), or "It is not the case that there does not exist an x such that x is P". This is a *very* common proof technique in classical mathematics and logic.

In intuitionistic mathematics, however, you have to be able to actually *construct* some particular object *n* such that Pn. For example, you have to be able to construct some particular number n that has the property P. From Pn, you can then infer Ex(Px) using existential generalization.