Kant’s Mathematical Realism

Though my title speaks of Kant’s mathematical realism, I want in this essay to explore Kant’s relation to a famous mathematical anti-realist. Specifically, I want to discuss Kant’s influence on L. E. J. Brouwer, the 20th-century Dutch mathematician who built a contemporary philosophy of mathematics on constructivist themes which were quite explicitly Kantian.1 Brouwer’s theory (called intuitionism) is perhaps most notable for its belief that constructivism (whatever that means) requires us to abandon the traditional (classical) logic of mathematical reasoning in favor of a different canon of reasoning, called intuitionistic logic. Brouwer thought that classical logic is intrinsically bound up with a nonconstructive (or “realistic”) view of mathematics. This means that, according to Brouwer, when we do mathematics we must give up bivalence (the principle that a given sentence either is true or is determinately false), we must no longer use such familiar logical laws as excluded middle, and we must sometimes forebear from the classic method of reductio ad absurdum. All of these are intuitionistically invalid classical principles.