The Intentionality of Mathematical Abstraction
Abstract
Although Edmund Husserl was trained as a mathematicians and wrote about mathematical thinking in most of his published works, little attention has been paid to his late phenomenological philosophy of mathematics. Among those who have studied Husserl's philosophy of mathematics, there is a widespread impression that his account is either intractably intuitionistic or else proven unworkable by later developments in mathematical logic. In this dissertation, I develop and defend a phenomenological description of the intentionality of mathematical abstraction aimed at knowing based on Husserl's account. A phenomenological approach to the philosophy of logic and mathematics asks us to describe what must feature in our orientation to the world and implicit norms of recognition for us to have an experience of coming to see mathematical truths for themselves. I describe the core of mathematical thinking as eidetic intuition—a disciplined imaginative achievement—founded on systematic disregard for our ordinary concern with truth, yielding insight into the essential features belonging to categoriality as such, the same categoriality that structures everyday perception. I defend an interpretation of Husserl's key concept of definiteness that shows how it can bridge inquiry into pure, algebraic, possibly intuitionistic mathematics with inquiry into realistic, classical mathematics. Then, I develop an intentional analysis of diagonalizing, an achievement that can be performed both in and outside the algebraic attitude proper to pure mathematics. This leads to the conclusion that Husserl's reliance on definiteness does not put a phenomenological account of mathematics at odds with Gödel's incompleteness theorems. Instead, by paying attention to the intentional structures involved, we can understand both the purely mathematical content of the theorems and their material application found in the logician's study of formal theories and their objects. Along the way, I illustrate many insights a phenomenological approach to mathematics can provide, helping us understand some features of mathematical theorizing that make it such a distinctive discipline. In particular, we we can understand reasons for peculiar qualities of mathematical innovation, foundations, interderivability, and proof, without needing to resolve debates that have dominated traditional philosophical approaches to mathematics.
Subject Area
Philosophy|Mathematics|Philosophy of Science
Recommended Citation
Reppert, Justin, "The Intentionality of Mathematical Abstraction" (2021). ETD Collection for Fordham University. AAI28496512.
https://research.library.fordham.edu/dissertations/AAI28496512
