“Algebraic” Approaches to Mathematics

Abstract

At the turn of the twentieth century, philosophers of mathematics were predominantly concerned with the foundations of mathematics. This followed the so-called “crisis of foundations” that resulted from the apparent need for infinitary sets in order to provide a proper foundation for mathematical analysis, and was exacerbated by the discovery of both apparent and actual paradoxes in naïve infinitary set theory (most famously, Russell’s paradox). Philosophers and mathematicians at this time saw their job as to place mathematics on firm, and indeed certain, axiomatic foundations, so as to provide confidence in the new mathematics being developed. Thus, the “big three” foundational programmes of logicism, formalism, and intuitionism were established, each providing a different answer to the question of the proper interpretation of axiomatic mathematical theories. Notes

I am extremely grateful to participants in the New Waves in Philosophy of Mathematics workshop in Miami for their helpful discussion of this paper, and particularly to Roy Cook, Mark Colyvan, and Øystein Linnebo for thoughtful written comments.