Add It Up

In lauding the discipline of mathematics, two avenues of approach are well established and perhaps exhaustive; if a third exists, I have not come across it. Both appear throughout the pages of Anthony Bonato’s Limitless Minds: Interviews With Mathematicians (American Mathematical Society). The editor and interviewer is a professor at Ryerson University, in the department you can probably guess.

The first way of celebrating mathematics describes its concepts and proofs using words like “beauty” and “elegance,” with “rigor” pretty much a given. Here perfection is very much in the mind of the beholder. We, the mathematically ungifted, must take the adepts’ word for it that what looks like a page of hieroglyphics is not just meaningful but, in its own ineffable way, satisfying and perfect -- the score for a music very few can hear.

As for the other way of talking about math’s dominion, the most forceful expression of it appears as the title of a paper from 1960 by the theoretical physicist Eugene Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” The phrase has taken on a life of its own. One of Bonato’s interview subjects refers to it in passing as an almost proverbial tribute to the power of applied mathematics. The beauties of pure theory elude the lay public, but the effectiveness of mathematical tools is tangible and inescapable -- baked into every piece of technology, for one.

Now, equating the value of mathematics with its practical usefulness misses Wigner’s point entirely. We’ll come back to that in a minute. But the two perspectives on math just sketched -- as an expression of creativity at its most abstract, on the one hand, and as an extremely effective tool kit, on the other -- seem to be antithetical. They imply different motivations and point to different career paths.

And, as noted, they come up repeatedly in Limitless Minds, which collects a dozen interviews with colleagues from Canada and the United States that originally appeared on Bonato’s blog, The Intrepid Mathematician, in 2016 and ’17. The line of questioning is broadly similar from one interview to the next. They cover when and how the subjects became interested in mathematics, the course of their education and career, and what problems or areas of specialization they have pursued. (They are invited to explain their research in terms accessible to the nonmathematician, which seldom goes very far.) The interviews supply an overview of the profession that will inspire anyone on the fence about pursuing a math major to go for it, as is undoubtedly the point of compiling them into a book.

The pure/applied math distinction is a thread running throughout the text. Asked what inspires her mathematical ideas, the graph theorist Maria Chudnovsky frames the contrast as real and significant:

It could be a problem that seems beautiful, or a concept that seems beautiful. Or it could be someone else’s proof that seems beautiful and I want to see what else I can do with it … There is math that is motivated by physics, chemistry or engineering. That is somehow separate. In much of math, you are just looking for the most beautiful thing you can think of. And only that determines if something is interesting or not. Beauty is also subjective. What I think of as beautiful someone else might think is ugly.

Decidedly less of an aesthete is Izabella Laba, who works on geometric measure theory, additive combinatorics and mathematical physics. “I don’t set out to produce beautiful mathematics,” she says. “I want it to be correct and to answer questions that people might be interested in. Making it appealing and beautiful can be directly at odds with the requirement to keep it correct and complete. I try to reconcile these competing demands where possible, and that often improves my work. But where they can’t be reconciled, I have to stay with the correct and ugly.”

But in many of the interviews, the dichotomy seemed almost irrelevant. Emphasis falls repeatedly on the need to remain open, flexible and ready to make far-fetched connections. “One thing that I think is important,” says Federico Ardila, known for his work in combinatorics, “is disrespecting every border that people have tried to draw in mathematics. An ambitious student who is just starting out might try to find two fields that people think are unrelated and discover the relationship between them. Mathematics is interconnected in unexpected ways. I think the most interesting work comes from taking two islands in mathematics, and showing how they’re connected … I think that always leads to very interesting mathematics.”

The algebraic topologist Alejandro Adem makes a similar point about the benefit of being able to change hats in the course of one’s work life. “Young people should also understand that what you are doing now is not what you will do in 10 years,” he says. “There will be times when you will teach more, do more research, or do more administrative work, or even outreach or industrial connections. If you are open-minded, then it can be enriching and gives you more mathematical ideas and keep you intellectually awake.”

The potential to collaborate across differences in cognitive styles comes through in an anecdote by Richard J. Nowakowski, who works on graph theory and combinatorial game theory. He recalls discussing a problem with a colleague who “thought in terms of formulas” while his own approach was visual:

I said, “Do you mean the following?” drawing something rough on the board. He looked at it for 30 seconds and said, “Do you mean the following?” I asked the same again, and we went back and forth like this for four hours. At the end, we had a paper! It wasn’t a major result, but it was interesting to see how two people could do this back-and-forth. It was like playing tennis.

It would be possible to go back through and tag each mathematician as either pure or applied, but I finished the book thinking that the distinction was not always all that salient. And to follow up on an earlier point, Eugene Wigner’s oft-repeated comment on “the unreasonable effectiveness of mathematics in the natural sciences” does not celebrate application over theory, however entrenched that dichotomy remains in our thinking. What impressed him (and what can inspires a kind of awe to think about for very long) is that an enormous body of knowledge had been generated by mathematicians concerned only with developing concepts that emerged from within their own field, with no regard for application -- but which then proved to be exactly what physicists needed to make models and solve problems. Within its beauty, mathematics generated a kind of power.

“You spend ages trying to figure things out,” Ingrid Daubechies, a mathematical physicist, tells Bonato. “When you figure it out, you are overjoyed, but then you realize you want to understand it better. As you understand it better, you feel like a moron for not having seen it sooner. Research has fantastic highs that are not very frequent, and they don’t last a long time. If you love mathematics, then that is good enough for you.”