Far from what the piddling "internet" may lead you to believe, Intuitionism is not a reaction to or even in opposition towards Neoplatonist tendencies in mathematics; rather, it opposes Formalism (and to some extent Logicism) on the syntax-semantics divide. Where formalist mathematicians would have you believe that mathematics can be reduced to a formal system, where truth, pattern-recognition, and symbol manipulation are all coeval, the intuitionist argues that meaning is inextricable from mathematics.

The formalists were birthed by those who in the early 20th century shrank with horror from set-theoretic antimonies: "give me consistency, or give me death!" they cried, and they offered up synthetic a priori mathematics in exchange for ignorant bliss. The intuitionists are the children of Brouwer, who abandoned the Law of Excluded Middle (only in theory and not in practice, it should be said) and championed the a priori intuition of time, its repetition and difference, to uphold meaning in mathematics. Away with Cantor and realized infinite progressions, away with Hilbert's insipid program!