Some stories about becoming a mathematician

I have a peculiar little day with neither an A-to-Z, a recap, or a Reading the Comics post to publish. It feels illicit somehow. Well, let me share something which I ran across recently. It’s an e-book published by the American Mathematical Society, Living Proof: Stories of Resilience Along the Mathematical Journey. Editors Allison K Henrich, Emille D Lawrence, Matthew A Pons, and David G Taylor.

It’s a collection of short autobiographical essays from mathematicians. The focus is on the hard part. Getting to an advanced degree implies some a lot of work, much of it intellectually challenging. I felt a great relief every essay I found where someone else struggled with real analysis, or the great obstacle of qualifying exams. And I can take some comfort, thinking back at all the ways I was a bad student, to know how many other people were also bad students in the same ways. (Is there anyone who goes on to a doctoral program in mathematics who learns how to study before about two years into the program?)

There are other challenges. It’s not a surprise that American life, even academic life, is harder if you can’t pass as a heterosexual white male. A number of the writers are women, or black people, or non-heterosexual people. And they’re generous enough to share infuriating experiences. One can admire, for example, Robin Blankenship’s social engineering that convinced her algebraic topology instructor that she did, indeed, understand the material as well as her male peers did. One can feel the horror of the professor saying he had been giving her lousy scores because his impression was that she didn’t know what she was doing even as she gave correct answers. And she wasn’t the only person who struggled against a professor grading from the impression of his students rather than their actual work. There is always subjectivity and judgement in grading, even in mathematics, even in something as logically “pure” as a proof. But that …

I do not remember having a professor grading me in a way that seemed out of line with my work. The grade to expect if the grading were done single-blind, with no information about the identity of the exam-taker. But, then, I’m white and male and anyone looking at me would see someone who looks like he should be a mathematician. I think my presentation is, to be precise, “high school physics teacher who thinks this class just might be ready to see an amazing demonstration about something we call the Conservation of Angular Momentum”. That’s near enough to “mathematician” for most needs. This makes many of these essays, to me, embarrassing eye-openings.

So I think the book’s worth reading. And it is a great number of essays, most of them two or three pages. So it’s one you can pick up and read when you just have a few free minutes.

Looking to Euler

I haven’t forgotten about writing original material here — actually I’ve been trying to think of why something I’ve not thought about a long while is true, which is embarrassing and hard to do — but in the meanwhile I’d like to remember Leonhard Euler’s 306th birthday and point to Richard Elwes’s essay here about Euler’s totient function. “Totient” is, as best I can determine, a word that exists only for this mathematical concept — it’s the count of how many numbers are relatively prime to a given number — but even if the word comes only from the mildly esoteric world of prime number studies, it’s still one of my favorite mathematical terms. It feels like a word that ought to be more successful. Someday I’ll probably get in a nasty argument with other people playing Boggle about it.

Apparently, though, Euler didn’t dub this quantity the “totient”, and the word is a neologism coined by James Joseph Sylvester (1814 – 1897). That’s pretty respectable company, though: Sylvester — whose name you probably brush up against if you study mathematical matrices — is widely praised for his skill in naming things, although the only terms I know offhand that he gave us were “totient” and “discriminant”. That b^2 - 4ac term in the quadratic formula which tells you whether a quadratic equation has two real, one real, or two imaginary solutions, was a name (not a concept) given by him, and he named (and extended) the similar concept for cubic equations. I do believe there are more such Sylvester-dubbed terms, just, that we need a Wikipedia category to gather them together.

I’m amused to be reminded that, according to the St Andrews biographies of mathematicians, Sylvester at least one tossed off this version of the Chicken McNuggets problem, possibly after he’d worked out the general solution:

I have a large number of stamps to the value of 5d and 17d only. What is the largest denomination which I cannot make up with a combination of these two different values.