Epistemology, Ontology and the Continuum

Abstract

Standard wisdom has it that mathematical progress has eclipsed Kant’s view of mathematics on three fronts: intuition, infinity and the continuum. Not surprisingly, these very areas define Brouwer’s own relation to Kant, for Brouwer attempted to recreate the Kantian picture of the continuum by updating Kant’s notions of infinity and intuition in a set theoretic context. I will show that when we look carefully at how Brouwer does this, we will find a certain internal tension (a “disequilibrium”) between his epistemology of intuition and the ontology of infinite objects that he must adopt. However, I will also show that when we search the corresponding Kantian notions for that same disequilibrium, then — despite first impressions — we will find equilibrium instead. Because of this, I will suggest at the end that the basic components of Kant’s eighteenth century view provide a foundation for important parts of classical rather than intuitionistic mathematics. This, in turn, will lead to a reassessment of Brouwer’s Kantianism, and the way that mathematics has progressed from Kant’s time.