Philosophy of set theory and mathematics
Topic outline


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 SD Friedman, from the book "Reflections on the foundations of mathematics" (Eds. Centrone, Kant, Sarikaya), Synthese Library 407, Springer, 2019. Wellwritten 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.

In this section, there will be articles and books which we will read (and possibly many others which you can read out interest).
The first three files are the basic text which are mentioned in the previous paragraph.
The collection "Philosophy of mathematics" (Eds. Benacerraf, Putnam) contains "old" influential papers, some of them are certainly worth reading. We will read some of them.
The collection "The Oxford Handbook of Philosophy of Mathematics and Logic" (Ed. Shapiro) contains more recent papers, some of them again worth reading. We will read some of them.

Please read the following two papers and think about summarizing them both. Indentify key ideas (according to you), in both of them.
1] Benacerraf, "What number could not be".
2] Hilbert, "On the infinite".
Both to be found in Philosophy of Mathematics (Benacerraf, Putnam, Eds).

Please read Sections 10.1 and 10.2 from Barton and Friedman, "Chapter 10: Set theory and structures", page 223 in "Reflections on the foundations of mathematics".
Note: If you want to read more, you can read the whole paper. We will see next class how it goes.

We will discuss some of the differences between category theory and set theory as they are viewed by Mac Lane (category theory) and Mathias (set theory). You can read Mac Lane's paper "Categorical algebra and settheoretic foundations" as a concise statement of Mac Lane's position. But more interesting is the exchange between Mac Lane and Mathias given it the collection Set theory of the continuum (Judah, Just, Woodin Eds) in the sections "What is Mac Lane missing?" on page 113 and "Is Mathias an Ontologist?" on page 119. All materials can be downloaded in the section "Recommended reading" above.

Read the paper by Macintyre "Model theory: Geometrical and settheoretic aspects and prospects". Think how it connects with our discussions of the role of set theory vs. category theory. For understanding Macintyre's position, read first abou this life and mathematical interests, for instance here:
https://en.wikipedia.org/wiki/Angus_Macintyre
Compare with the paper of Baldwin "The Dividing Line Methodology: Model Theory Motivating Set Theory".
Focus on the fact how both see Shelah's contributions as essential, but interpret them differently. 
Based on the previous discussions you will probably anticipate that the answer to this question will depend on the interests of the given mathematician, but it is still good to see this in more detail. Read "Does mathematics need new axioms?" above.
For extra reading, you may read Maddy's "What Do We Want a Foundation do?" in the book "Reflection on the foundations of mathematics".