In praise of some history

Teaching pure mathematics is not a trivial thing. You have to overcome the several barriers that were constructed by the K12 education that mathematics is a bunch of “fit this problem into that mold”.

I recently had a chat with James Cummings about teaching. He said something that I knew long before, that being a good teacher requires a bit of theatricality. My best teacher from undergrad, Uri Onn, had told me when I started teaching, that being a good teacher is the same as being a good storyteller: you need to be able and mesmerize your audience and keep them on the edge of their seats, wanting more.

My answer to James was something that I had in mind for a while, but never put into words until then. You should know a bit of history of the topic you’re teaching.

If you look at [pure] mathematical education—at least undergrad—it is quite flat. You just have a list of theorems, each extending the previous, building this wonderful structure. But it’s a flat building, it’s a drawing. The theorems come one after the other, after the other… Historically, however, there were many decades (if not centuries) between one theorem to the next. Rolle’s theorem came about the late 17th century, but Lagrange’s theorem came only in the mid-19th century. So in one lecture, we covered some 150 years of mathematical progress. That is amazing, if you can point this out properly to your students. Not to mention the oppositions that people had to infinitesimal calculus in Rolle’s days, which makes it interesting, and contributes to the definitions given by Cauchy as solid foundations for analysis.

Similarly, the history of the Cantor-Bernstein theorem is incredible. As in the history behind’s König’s theorem (about cardinal arithmetic). Those things are amazing, what sort of motivations and mistakes people had made back then, when these fields were fresh.

The more I thought about it, the more I realized that there are two important reasons that one should always spice up their teaching with some historical facts.

  1. The first is that mathematical education is flat, as I remarked above. We learn the theorem, one after another. In one lecture, you can cover decades or even centuries of mathematical progress. And it feels dry, it feels like “why are you teaching me all this?” sort of thing. I still remember that as a student, I do.

    But with a bit of history, suddenly everything becomes three-dimensional, it becomes something that had actual progress. It shines a light on “there is a notion of mathematical progress”. Something that engineering students, for example, often baffled by.

  2. The second reason is that often theorems and motivations were coming from attempts to disprove something. König, if we mentioned him already, proved his lemma as an attempt to prove that the real numbers cannot be well-ordered. Baire, Borel and Lebesgue rejected the axiom of choice because they felt it is preposterous that there are non-measurable sets.

    When you explain this to students, you show them that their natural confusion about a topic, especially abstract and confusing topic, is natural. You show them that a lot of smart people made the same mistake before. And while today we know better, their instinctive recoil makes sense. This reinforces the idea that they didn’t misunderstand something, that they are not stupid, and that mathematics is often surprising (at least when dealing with the infinite).

So we have these two reasons. And I think these are excellent reasons for adding some historical references when talking about mathematics. Of course, you shouldn’t put more than a pinch of cumin in your stew, because cumin is not the main part of your dish, it’s just what makes a good meal into a great meal (well, at least good cumin). You shouldn’t talk only about history in a mathematical course. This should be the slight addition that gives taste, flavor and volume to your material.

Historical anecdotes are what turns a flat material into a fleshed form of progress, from one theorem to the other. Use them sparsely, use them well. But use them.

Constructive proof that large cardinals are consistent

I am not a Platonist, as I keep pointing out. Existence, even not in mathematics, is relative and confusing to begin with, so I don’t pretend to try and understand it in a meaningful way.

However, we have a proof, a constructive proof that large cardinals are consistent. And they exist in an inner model of our universe.

Recall that $0^\#$ exists if and only there exists a non-trivial mouse. Now recall that such mice exist. Vacanati mice.

I’m sorry to all those who claim that inaccessible cardinals are inconsistent. Your claim is that reality is inconsistent. Which might just be the case…

Now you can ask whether or not large cardinals are a human construct. Here we run into a problem, as these non-trivial mice are a human construct themselves…