Modern mathematics and its philosophy

Philosophy of mathematics and set theory can be approached from several perspectives. In this course, we will use my current grant  "The role of set theory in modern mathematics" and use it as a guideline to identify specific topics of interest.

Organization of classes.

We will proceed by reading relevant articles and discussing them from various angles. The class will be organized as follows: I will introduce a certain philosophical topic and possibly add some necessary mathematical content. Then there will be a 60 minute part for reading papers (you can bring your own notebooks or tablets or use the computers in the adjacent rooms). In the final part we will be discussing the results.

Additionally, each student will pick an article and present it to the other student during the semester. The presentation should be at least 45 mins long.

How do we identify papers we will read?

As an exercise in actual research, we will start with three texts which will guide us to further papers:

1] Grant proposal "The role of set theory in modern mathematics". Basic context which suggests how to connect philosophy and mathematics today.

2] Paper "Set theory and structures" by Barton and S-D Friedman, from the book "Reflections on the foundations of mathematics" (Eds. Centrone, Kant, Sarikaya), Synthese Library 407, Springer, 2019. Well-written exposition written jointy by a mathematician and a philosopher. It deals with foundational issues, with the focus on the difference between set theory and category theory.

3] Paper "Does mathematics need new axioms?" by Feferman, H. Friedman, Maddy, Steel", The Bulletin of Symbolic Logic, vol 6, no 4, 2000. Written by several mathematicians and one philosopher (Maddy), it looks at the problem of new axioms from the viewpoint of actual  applications and mathematics.