Mathematics as Mental Acts

Intuitionism accuses formalism of confusing mathematics with formal systems. Mathematics is a metal act, and not a string of symbols. According to intuitionism, many of the proofs used in mathematics are not valid. The paradoxes that brought mathematics into a foundational crisis would all evaporate if the process of proof had been “constructive”. This brings intuitionism to articulate a logic that is different from classical logic; for instance, the principle of the excluded middle is not valid in intuitionistic logic. In classical logic the meaning of the logical connectives can be clarified in terms of truth tables, but this is not so in intuitionistic logic. Through a discussion of language, mathematics, and meaning, Brouwer inspired Wittgenstein to return to philosophy, and he inspired Freudenthal to see mathematics as a human activity.