Saunders Mac Lane: From Principia Mathematica through Göttingen to the Working Theory of Structures

Saunders Mac Lane heard David Hilbert’s weekly lectures on philosophy and utterly believed Hilbert’s declaration that mathematics will know no limits. He studied algebra with Emmy Noether, and both mathematics and philosophy with Hermann Weyl. As a young algebraist he created today’s standard working method for mathematical structure: category theory, with topologist Samuel Eilenberg. As one step, they created the now standard definition of “isomorphism.” They originally saw categories as just a working tool. But in the 1950s, Mac Lane saw Alexander Grothendieck and others radically extend the range of the theory, and in the 1960s, he took up William Lawvere’s idea of categorical foundations. The essay relates all of this to current philosophical structuralism, especially concerning isomorphisms and automorphisms of structures. It concludes by comparing Mac Lane’s motives for structuralist working mathematics with current philosophical motives for structuralist ontology.

Intuitionism in the Philosophy of Mathematics: Introducing a Phenomenological Account

ABSTRACT The aim of this paper 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. The focus is on two questions: what does it mean to undergo a mathematical intuition and what role do mathematical intuitions play in mathematical reasoning? 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.

WEYL REEXAMINED: “DAS KONTINUUM” 100 YEARS LATER

Hermann Weyl was one of the greatest mathematicians of the 20th century, with contributions to many branches of mathematics and physics. In 1918, he wrote a famous book, “Das Kontinuum”, on the foundations of mathematics. In that book, he described mathematical analysis as a ‘house built on sand’, and tried to ‘replace this shifting foundation with pillars of enduring strength’. In this paper, we reexamine and explain the philosophical and mathematical ideas that underly Weyl’s system in “Das Kontinuum”, and show that they are still useful and relevant. We propose a precise formalization of that system, which is the first to be completely faithful to what is written in the book. Finally, we suggest that a certain set-theoretical modern system reflects better Weyl’s ideas than previous attempts (most notably by Feferman) of achieving this goal.