Intuitionism in "Philosophy of Math"

[Kratzer]

I am taking a philosophy of math class currently, and the past few days of lecture we have delved into intuitionism. We've been discussing Kronecker and Brauer(sp?), and my professor said intuitionism is mainly based on the concept of "intuitively founded". I feel i don't understand it very well, the way he explained it was: it is when one experiences something, before they register it in their mind...or something like that. Can someone help me out with this?

[bunny]

When I studied the philosophy of maths we referred to Brouwer as a constructivist rather than an intuitionist. I dont know if that represents a real division or just a naming convention.

My rather limited understanding of the constructivist position is that it proceeds from the belief that only the natural numbers (ie the "objects" of counting) are given to us by a fundamental intuition. He required (philosophically, not in his mathematical work) that all maths be built up constructively and that mathematical objects should not be considered meaningful unless they can, in principle, be given by a construction from the natural numbers.

The constructivists didnt accept many proofs that formalist or platonist mathematicians would. In particular they didnt accept the validity of proof by contradiction - it is not sufficient to establish an object's existence based on an assumption of non-existence leading to a contradiction.

Rather a waffle and we focussed more on Brouwer than Kronecker (who I didnt think many people took seriously at all.) Nonetheless, it might mean something. /images/graemlins/tongue.gif

[Sharkey]

It is an irony of the attempt at objectivity that, in the final analysis, the known properties of any “object” exist only insofar as they can be mentally constructed.

[guesswest]

I believe intuitionism is a specific (and controversial, so I'd guess distinct) branch of constructivism, but I can't remember what differentiates it from garden variety constructivism. My poor memory coupled with a general lack of talent at philosophy of math.......I do remember Brouwer wrote 'the' book on intuitionism though. Perhaps you could pick that up.