Browsing by Subject "Foundations of mathematics"

This thesis addresses the concept of mathematical foundations from a mathematics-first perspective. The primary goal of this thesis is to understand the mathematics-first reasoning for the search of foundations of mathematics and to understand how the concept of a foundation should be interpreted in a mathematics-first approach to the philosophy of mathematics. The central finding of my thesis is that the value of foundations of mathematics for a mathematics-first philosopher is found in the ability of the foundation to serve the mathematical community and push mathematics forward. The first chapter provides historical context on the question of foundations, covering the primary schools of thought, formalism, intuitionism, and logicism, as well as discussing the arguments for and against set theory as a foundation in mathematics. The second chapter presents taxonomies by Shapiro, Marquis, Maddy, and Ladyman & Presnell that classify the senses in which a foundation is used or should be used. The chapter also discusses differing attitudes toward the philosophy of mathematics, including philosophy-first, philosophy-last-if-at-all, and mathematics-first approaches. The third chapter explores the value of a foundation for a mathematics-first philosopher, with Maddy's taxonomy serving as the basis for understanding how foundations can benefit the mathematical community. The value of having foundations is found in the security, maximality, and unity they provide to the discipline. The thesis concludes by reflecting on the metaphors used in the foundational debate and considering the notion of mathematical progress as it relates to the value of a foundation from a mathematics-first perspective.