Finding, and sharing, mathematical beauty in the world

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.)