In the modern discussions about possibility of synthetic a priori propositions, the theory of definition has a fundamental importance, because the most definition’s theories hold that analytic judgments are involved by explicit definition . However, for Kant –first author who pointed out the distinction between analytic and synthetic propositions–many analytic judgments are made by analysis of concepts which need not first be established by definition. Moreover, for him not all a priori knowledge is analytic. The statement that not all analytic (...) judgment is derived from definition and possibility of synthetic a priori knowledge, indicates Kant didn’t believe, contrary to modern theories about analytic judgment, the definition is an essential ground of knowledge. (shrink)

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. Brouwer’s theory is perhaps most notable for its belief that constructivism requires us to abandon the traditional logic of mathematical reasoning in favor of different canon of reasoning, (...) called intuitionistic logic. Brouwer thought that classical logic is intrinsically bound up with a nonconstructive view of mathematics. This means that, according to Brouwer, when we do mathematics we must give up bivalence, 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. (shrink)