Noah Giansiracusa explains how the exploration of interdisciplinary mathematics appears particularly relevant in moments where beauty must shine through in the darkest of times
I was recently asked to speak in a Zoom-based online conference (https://unwin.people.uic.edu/SIDEMath2020) with an unusual title: Selected Interdisciplinary Directions of Entertaining Mathematics. To those outside the mathematics community, the phrase “Entertaining Mathematics” might seem an oxymoron, while to those of us in the community it might seem a tautology! But what does this really mean and what is this conference about? Well, I must admit I was invited because the organizers are close friends—a dynamic wife and husband mathematician-physicist team, Laura Schaposnik and James Unwin—rather than because my interdisciplinary mathematical work has somehow attracted such a large following that speaking invitations flood my email inbox (ha! I think my papers in these topics generally have more authors than readers).
The first aspect of this conference title that I’d like to clarify is subtle but, in my opinion, very important: what’s the difference between interdisciplinary mathematics and applied mathematics? Both of these deal with mathematics in the “real world” rather than the pure mathematics of theorems and proofs populating a vast and intricate landscape that resides primarily in our minds. Certainly, applied mathematics is the more ubiquitous of the two and, admittedly, in some settings the two terms might be used interchangeably. But to me there is a tremendous difference between them—or at least, there is a distinction that I hope to draw attention to and encourage cognizance of. Applied mathematics uses mathematics as a powerful set of tools to influence and/or understand the world around us. For instance, applied mathematics is used to model physical and sociological entities, and to predict the behavior of complex phenomena; it provides a powerful backbone to almost all corners of science and technology. The emphasis here is that in applied maths we use mathematics to study something in the real-world, and often to help us make real-world decisions about it. The elegance of the maths involved is irrelevant, the measure of success is instead almost entirely outcome-based. This does not mean applied math is devoid of beauty—quite the opposite, admittedly!—just that beauty is not the driving factor. Perhaps a collection of differential equations allows us to predict economic crashes; there may well be a certain beauty to these equations, but that is not the point in them, the point is that they do in fact make reasonably accurate predictions, and the beauty is almost a by-product of this.
What then of interdisciplinary mathematics? At least the way I’m interpreting it in the context of this conference on entertaining mathematics—though I encourage this use and distinction more broadly—is that interdisciplinary mathematics is less about using maths to influence the world and more about using the world to influence maths itself. What I mean is that interdisciplinary mathematics often happens when we find curious phenomena occurring in other, ostensibly non-mathematical, disciplines that we recognize as being capable of interpretation and expression in mathematical language, especially when this latter manifestation has a certain mathematical depth and intrigue to it. Recognizing that marginalization in economics is essentially differentiation in calculus is perhaps an amusing (or bemusing) observation for a mathematician, but most professional mathematicians do not spend their time reading research papers about elementary calculus. On the other hand, discovering that symmetries in certain ancient cultural art reflect the representation theory of infinite groups, say, is potentially very exciting to a mathematician. It is extremely unlikely that one will discover or prove a new theorem by finding maths in art, but being able to understand certain aspects of certain art by using sophisticated mathematical language and ideas can feel extremely rewarding after years of studying this mathematics in the often monastically-isolating setting of academic pure mathematics.
So, to me, interdisciplinary mathematics is really about uncovering sophisticated mathematics in the real world, especially in places where it is not expected (so political science is a more plausible place for it than physics). Finding math like this can feel like an archeological adventure, and it can be quite exhilarating if one keeps an open mind. The main thing to remember is that one isn’t aiming to resolve deep controversies in other fields or even to advance to the state of knowledge in these fields significantly by using mathematical tools (that would be more on the applied math side of things); the audience is more often the mathematician than the scholar in the field in question, though admittedly it is difficult to suppress the quixotic fantasy that the mathematician will swoop in and solve problems in other fields using the most wondrous high level pure mathematics. (It is ok to harbor this fantasy privately, just don’t let it leak into your paper’s abstract or you’ll likely paint yourself as a charlatan.)
Now that this distinction between applied mathematics and interdisciplinary mathematics has been drawn, a natural question is why is this conference on entertaining interdisciplinary mathematics taking place during our current time of a pandemic? And why are some mathematicians drawn to this kind of work, even though the papers it produces often fall into the unfortunate interstices of academic publishing (not deep/novel enough maths for a maths journal, not useful/understandable enough for a journal in the relevant discipline)? I cannot speak for others, but I can tell you what has grabbed me about this kind of work and why I’m so excited to be part of this conference, and also why I think it resonates particularly strongly right now.
As a pure mathematician I certainly love the theoretical work I engage in professionally most of the time, but I also find it extremely removed from the world around me. With the coronavirus pandemic and all the suffering it has unleashed (both medically and socially) I often find myself wishing that my work was more relevant, that I could use my technical expertise to help guide society through this dark chapter of world history—but, realistically, I know that my mathematical knowledge leaves me ill-equipped to the tasks that need to be done now, such as epidemiological modeling. There are plenty of people doing this already, people who are far more qualified than I am, and this is absolutely an area where applied mathematics has truly been shining. It would be foolish to think I, a pure mathematician, could quickly leap into doing such work, and likely pernicious for me to try. On the other hand, during this time of despair and isolation we have seen in many ways how important art and culture is to so many people—virtual tours of museums, impromptu neighborhood musical performances, live-streamed theatrical and musical events of all kinds. We need beauty in our life right now, even more than we normally do. And while the vast majority of the population understandably does not find beauty in mathematics, it is absolutely true that mathematicians, especially pure mathematicians, feel a deep, powerful, aesthetic beauty when studying high level mathematics. To me, what makes the talks in this conference “entertaining” is the very fact that they are interdisciplinary in the sense I’ve described above, namely, they introduce a mathematically-literate audience to vignettes of mathematical beauty residing in art, in society, in the “real world”. It is hard for me to focus right now on my usual pure mathematics research; somehow for me personally it feels a bit strange to be continuing it while so much of the world has ground to a halt. On the other hand, just as seeing videos of Italians singing opera together from their balconies during the long weeks of lockdown has brought a much needed spark of joy and optimism to the world, being able to share my work on using geometry to conceptualize the voting patterns of judges in the U.S. Supreme Court with a small sliver of the mathematics community, and being able to learn from others in this community what exciting mathematics people have found in ostensibly non-mathematical realms, provides a social connection that is much needed at this moment—and it encourages a feeling that intellectual beauty will pull us through these difficult times, as it has so many times throughout the history of humanity.