Some thoughts about teaching introductory courses in set theory

Dianna Crown, the physics woman on YouTube, has posted a video where she is interviewed by her editor about why and how she found herself majoring in physics in MIT.

Here is the video:

One of the question is how she found herself studying physics. The answer was that she had very enthusiastic teachers when it came to physics, and that caught on to her.

For the past few years, I’ve been telling people that undergraduate students are very impressionable. They follow the teachers that seem to be most excited about the material they teach. You need a bit of charisma, of course. But if you have that, and you teach with excitement, and with awe at all the awesome things that you teach, students are going to follow you. Sure, only about 10 people finished the axiomatic set theory course I gave last year. But the first lecture had about 30-35. And almost every year I’ve had someone student (or more) ask me about axiom of choice related problem they can write a paper for some seminar.

Because I enjoy teaching set theory. I think it’s awesome. I love it. And I want to pass this enthusiasm forward. And for this, I am willing to match my students in effort. If someone needs extra help, I am always willing to give them the extra help. Crash course in proof writing before an exam? Sure. Meeting outside of office hours to talk about the material in class? My pleasure. As long as I feel that the student is making an effort, I will put the same effort.

Now. I am not writing this to glorify myself as a teacher. I am writing this to make a point. Some years ago, if my memory serves me right, there was some panel about what to do regarding set theoretic education. One of the thing is to revive some of that energy. In a university where nobody is teaching set theory properly, no student is going to want to study set theory (well, there might be some exceptions, but these are rare). In a university where the set theory teachers are terrible teachers, there won’t be many people who would want to pick up the mantle and do research.

Dianna, just as well, mentioned that one of the things she realized is that learning physics is very different from research in physics. The same can be said about math (and probably about anything, because you usually learn something as a student after the research is done. Research takes tenacity, patience, more tenacity. Research is draining the belief out of you, and reaching into the bottom of your emotions for failure and resignation. And if you’re doing it right, research is about bouncing back from these feelings, and finding the right path to actually get things done.

And set theory is no different. Only that compared to other mathematical branches, set theory is the “odd kid out” when it comes to undergraduate material. Even as students. Basic calculus has examples, basic analysis has examples, basic linear algebra has examples, basic combinatorics has many examples. And all of these examples come from real life. They are easy to explain. Set theory, however, doesn’t have examples. It has motivations, it has “prettifications of concepts”; but set theory does not have concrete examples you can explain. We cannot comprehend infinitude, so we cannot understand intuitively why there are more real than rational numbers. This requires relinquishing intuition in favor of definitions, so new intuition can be built.

So getting students interested in basic set theory is even more complicated than it is to get them to stick with math. But it is not impossible. We’re all here, and there are grad students in set theory.

What is missing, I guess, is better, more enthusiastic, and more giving, and more willing to walk the extra mile for their students. If we cannot make that happen, then set theory students are going to dwindle down, and then the demand for set theorists in departments will dwindle down, which will cause set theory education as a whole to dwindle even more… and eventually, we will go extinct.

Let’s not go extinct. Let’s find our inner motivation, and let’s get some students interested in set theory!

4 thoughts on “Some thoughts about teaching introductory courses in set theory”

  1. Set theory is not going extinct (except at Yiannopoulos University). The best way to motivate people to work on set theory is to make it applicable to the real world (people are more likely to believe in set theory and large cardinals if they see practical applications of them). Does set theory research give people skills which are directly applicable in the real world? Do set theorists program computers while researching set theory? Do cardinals above $2^{\mathbf{c}}$ have any direct practical applications? Large cardinals improve the consistency strength far beyond ZFC, so there is no reason why this extra theorem proving power cannot have practical applications.

    Does anyone even feel remotely embarrassed saying that their set theory research has no foreseeable practical applications? Does anyone feel uncomfortable with graduate students studying set theory while many of these students will eventually need to work in the private sector?

    1. @Joseph
      I agree that finding applications would make set theory more popular and as a by-product increase the number of pure set theorists, purely by making a larger number of people realise that it’s a thing that exists. But I doubt that the applications themselves are what will motivate students to become interested in set theory and do “pure” set theory though, but instead see it as a necessary tool to do what they really want to do. I would guess an analogue is algebraic geometry and the abstract notion of schemes – they’re necessary to analyse what (some) algebraic geometers want to work with (generalised surfaces). But in any case, applications would still be great for set theory of course, but I think purely for the increased exposure.

      To your last question on the private sector: what I’ve been told is that mathematicians aren’t hired in the private sector because of their choice of courses, but because learning these courses has ‘sculpted’ their thinking into becoming highly efficient problem-solving machines. Set theory still accomplishes that goal, as any other mathematical discipline.

    2. I’m puzzled by this post. You seem to hint that there are clear answers to your questions – answers that we should provide our students with to make set theory more appealing to them. But I, for one, not only don’t know reasonable answers to many of those questions, I even fail to see how possible answers might help our students.

  2. In my opinion you are correct almost in every point. You need passionate teachers to make the topic more interesting and to make students stay with the subject. Me myself got really interested in the foundations of mathematics exactly because my teacher was really friendly and passionate about the subject.

    This however wasn’t eventually enough to make stick and the reason was I wanted to go more applied and maybe one day work on the private sector (the set theory I was into was maybe more philosophical than mathematical as my teacher worked at the department of philosophy). Also the funding is not that easy to get for the foundations as it is for many other subjects. But I admit that it was a close call and would not have given it a try without a teacher that was so excited.

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