The Phenomenological Foundations of Mathematics: Introducing a Phenomenological Intuitionism

Abstract

AbstractThe aim of this chapter is to establish a phenomenological mathematical intuitionism that is based on fundamental phenomenological-epistemological principles. According to this intuitionism, mathematical intuitions are sui generis mental states, namely experiences that exhibit a distinctive phenomenal character. Here three main questions arise: What are mathematical intuitions? What does it mean to undergo a mathematical intuition? What role do mathematical intuitions play in mathematical reasoning? The first two questions have already been addressed below, particularly in Chap. 3. The focus of this chapter is on the third question. My understanding of mathematical intuitions emerges naturally from my phenomenological epistemology. In Sect. 14.1, I will briefly recapitulate my conception of mathematical intuition. The focus, then, is on the role mathematical intuitions play in mathematical practice and reasoning. Section 14.2 addresses the role mathematical intuitions play in so-called picture proofs. In Sect. 14.3, I elaborate on the role mathematical intuitions play in symbolic proofs. Section 14.4 states the building blocks of my phenomenological mathematical intuitionism and in Sect. 14.5 I will address the relationship between axioms and intuitions, making a smooth transition to phenomenological reflections on physics. While I crucially draw on Husserlian principles and adopt ideas we find in phenomenologically minded mathematicians such as Hermann Weyl and Kurt Gödel, the overall objective is systematic in nature: to offer a plausible approach towards mathematics.