Motivated History of Mathematics

Originally Posted 09-07-2012

The UNI Math Dept has been doing a curriculum discussion cycle lately, so I have lots of “broad stroke” ideas floating around in my head. Today, I had a thought about our History of Mathematics course. As a disclaimer: I have never taught this course, and I am not sure I will ever get the chance.

I had always thought of the history of mathematics courses as something squishy. What is the purpose of such a course, really? On one hand it can be used as a multicultural filler. “See, many different civilizations contributed to our understanding of mathematics. Appreciate these innovative ancient non-westerners!” On the other hand, it could be some sort of “greatest hits of the last two millenia.” At UNI it is part of what our pre-service teachers take to help them get certification. It is not at all clear to me what the content of such a course should be.

Another disclaimer: of course people from all over the world are clever enough to get involved and do modern mathematics, even those raised in a different cultural setting from me. I’ll even go so far as to say that I am not all that great at math, so most countries of the world have lots of people who are better at it than I am. Let’s stipulate that I don’t have an axe to grind, and though I am a white, educated, western civilization male, I am really not trying to put anyone down. I am thinking about how to do a better job as a teacher. And we move on…

So, what is to become of “math history” as a course of study? I find the history of mathematics interesting from two perspectives: First, I think history is interesting, and the history of mathematics gives me an entry point to learning a new discipline. How is it that a historian goes about his/her business? What counts as a question worth asking? How do you go about finding answers? What counts as an acceptable argument in this context? This is really the Liberal Arts way of thinking of an education. Professors love this, but students don’t always agree. Math undergrads are not usually all that jazzed about what counts as “how to do history.” I absolutely love the work of Jeremy Gray, for instance. But I don’t think my students are ready to appreciate it.

The second way I find the history of mathematics interesting is as a practitioner. I have learned more mathematics than the average person, and some of it is not particularly well-motivated. Where did the cross-product come from anyway? Who thought this crazy thing up, and why? [By the way, I was looking for the answer to that question earlier this week. It seems like the answers should be something like “Lagrange to measure volumes of tetrahedra” or “Hamilton and other physicists to handle vector equations, like Maxwell’s equations of electrodynamics”.] Now, I am not sure that this motivates most of the undergraduates that I have met. They really don’t care for the subject quite so deeply as I do.

Today, I saw a paper discussing “who was the first ‘Modern Mathematician’?” This got me thinking. The paper was about the change in attitude and technique in mathematics found in the work of Viete and Descartes. It is neat, but I don’t know enough to judge right now. But it reminded me of a talk I saw at MathFest in 2011 by Phil Kutzko (University of Iowa). Kutzko’s main point was that students raised in cultural settings which are not mine (basically) may not have been exposed to the raw, stripped-down, rationality-is-the-be-all-end-all way of thinking characteristic of Descartes’ work. This world view is sort of a prerequisite for how modern mathematics is done, so if you haven’t experienced it you will have an extra hurdle in your study. At least, that I what I took away from his talk.

So, is it possible to make a manageable list of major cultural shifts in the practice of mathematics over recorded human history? If so, maybe a useful “math history” course could be structured to show students the before-and-after comparisons for these shifts. This might be useful for all of the pre-service teachers at UNI. They might learn to recognize when a student has missed a shift or two, and then have an idea of what needs to be addressed to help the student along.

Anyway, I was thinking. Here are some candidates for major cultural shifts. At least, these are my favorites in the undergraduate curriculum.

  • Euclid, as representative of the shift from practical to abstract mathematical work. No measuring fields of grain anymore, instead, thinking about the platonic ideal of a rectangle.

  • Descartes, as representative of the meaningful use of formal symbols and algebraic reasoning to solve lots of problems.

  • Newton, introduction of “motion” as a way to understand lots of geometric properties

  • Leibniz very formal algebraic “calculus of logic”

  • the introduction of hyperbolic geometry, and the destruction of a single ideal universe

  • Cantor’s investigations of the infinite.

  • Cauchy’s work on the foundations of calculus

  • Fourier and the idea of what constitues a function

  • Abel, Klein, etc, on the introduction of algebraic structures as organizing principles for mathematical thought and work.

As you can see, this ad hoc list is very much tailored to the kinds of things I know about. I am sure that it is insufficient or unworkable in any number of ways. I am happy to take comments on the extent to which I am an idiot.

Still, the point remains: is it feasible to organize a useful and interesting course introducing undergraduates to the major milestones in mathematical history by picking out exactly the “paradigm shifts” and cultural changes that lead to the modern conception of what it is to do mathematics? How would that really work?