One of the methods of proof that Intuisionists/Constructivists reject is that of the excluded middle. This law implies that everything is either true or false. Typical examples of this can be found in propositional logic. However this excludes the possibility of "maybe" or "I don't know" or "uncertain" or "undetermined".
Many of Brouwer's work on constructivism come across as fairly mystical writings. One of the few "assumptions" that he allows is the existence of the natural numbers, which he believes we all have some underlying conciousness of.
An interesting result of having to be constructive in proofs is that all functions end up being continuous. (Something many students I know would be very happy about;) Other important consequences include not being able to use the Axiom of Choice and hence Zorn's Lemma, which is generally used to give the natural topology on the reals. This also disallows proofs of the existence of non-principal ultrafilters.