Develops a structuralist understanding of mathematics, as an alternative to set- or type-theoretic foundations, that respects classical mathematical truth while ...

Anti-realism is a doctrine about logic, language, and meaning that is based on the work of Wittgenstein and Frege. In this book, Professor Tennant clarifies and develops Dummett's arguments for anti-realism and ultimately advocates a radical reform of our logical practices.

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)