Asklemmy

I would say mathematics is a consequence of, or branch of philosophy in its own right. The name intuitionism derives from the source of this branch of mathematics - "2 primal intuitions".

1. Twoity - we are able to perceived time, and are thus able to split the universe into two, three, four etc parts. Counting is not something we just learn, it is something built into us as humans.

2. Repetition - we can repeat operations and not stop, just as we can never stop counting.

From these two (heavily paraphrased) ideas we can derive all of mathematics.

The first is actually enough to give us everything up to the rationals, the second grants us the reals and beyond.

While we lose excluded middle, we gain things such as "all total real functions are uniformly continuous on the unit interval" (Brouwer), the removal of the information paradox in physics (someone used Posey's take on intuitionism to rewrite all physics to see where it led), and the wonder of lawless sequences (objects we cannot predict entirely, but still work with).

The intuitionist is very very formal "you are either alive or not alive" is a very nice statement to make, but entirely worthless if one cannot tell which you are! Excluded middle is not universally false in intuitionism; it is true for decidable statements, of which having an apple or not does seem to fall within (though here we can question how "apple-like" must something be to be considered an apple if we wish to be peverse). However, to argue it is true for any statement means your disjunct (or) must be very weak indeed - the classical mathematician is happy with this, the intuitionist demands that a disjunct not only present two options, but provide a way of determining which if the two applies on a case to case basis (hence excluded middle applying for decidable things).

Simplifying your example of an apple, you can think of it as a Platonist just having the statement that everything is either and apple or not. Meanwhile, the Intuitionist also demands there be a guide on how to sort everything into "apple" or "not apple" before they make that statement.

Classical mathematics does also have a huge unintuitive step - mathematics must exist independently of humanity. Every theorem ever proved, and ever to be proved, exists somewhere. Where you ask? The platonic plane of ideal forms beckons, with all the madness it entails!