Mathematical Knowledge and Mathematical Objects

Abstract

This chapter is about the ontology and epistemology of mathematical objects. The core problem for an empiricist is that conceiving mathematical objects as existing independently of human thinking makes it impossible to understand how we can have mathematical knowledge, while the alternative, a constructivist conception, resolves the epistemological problem, but entails the identification of truth with provability. That entails that the law of excluded middle must be dismissed as a generally valid logical principle. The identificaiton of truth with provability is furthermore problematic when taking into account Gödel’s first incompleteness theorem. The chapter ends with suggesting a modified constructivism, which keeps the distinction between truth and provability, thus avoiding counter arguments based on Gödel’s theorem.