Defining Mathematics
What is mathematics? If you’re like me, you may be used to expanding others’ understanding of what is included under the umbrella of mathematics. “It’s not just numbers!” we assure our children, “it’s not just Calculus!” we assure our students, “yes, there are still areas of mathematics left to research!” we assure our friends. We feel like we understand the expansiveness of the term “mathematics.” And yet, if you’re like me, you may yourself need to be reminded that mathematics reaches areas beyond those that you work with on a regular basis.
Take gerrymandering, for example. Drawing districting maps to disenfranchise a political party or racial group is clearly in the realm of political science and the law. But mathematicians, statisticians, and computer scientists have shown that we can also use our skills to address the problem of gerrymandering. We have studied metrics to detect gerrymandering, used Markov Chain processes in an effort to study the space of potential redistricting maps, and created new algorithms to assess the limits of extreme gerrymandering. And many of these studies are not just academic; mathematicians and computer scientists have written amica briefs, given expert witness testimony, and had their scholarly work cited in court case after court case (see the MGGG, ALARM, and Quantifying Gerrymandering sites for some samples). My own work (with collaborators Campisi, Ratliff, and Somersille) on the GEO metric is being used by the National Democratic Redistricting Committee in their map assessments. When considering what “applied mathematics” encompasses, frequently mathematicians first think of the sciences such as physics and biology. We should expand this understanding to include applications within the social sciences, such as the detection of gerrymandering.
But we don’t even have to go to the extreme of looking at the social sciences to see how mathematicians (such as myself) may consider some areas of study as “other.” As I found myself expanding my understanding of applied mathematics to encompass the study of gerrymandering, I also found myself using more computational techniques in that study. In fact, I became so interested in bolstering my computational foundation that I decided to get a Master’s degree in Computer Science . . . a field which I believed was clearly distinct from Mathematics.
And yet, I saw mathematics everywhere in each course that I took for my degree. Who knew that Principal Component Analysis, Convex Geometry, and Measure Concentration would pop up all over my classes? Not only was this a surprise to me, but it was thrilling and empowering to realize that I already had a significant foundation for and understanding of the key tools used in my courses such as Applied Machine Learning, Data Mining, and Deep Learning for Healthcare. Given how many of my mathematician friends have moved into these and related fields, it should not have been a surprise. But my previous understanding of these fields was that they were adjacent to mathematics, not that they were a part of mathematics . . . which is how I have increasingly come to view them.
And this is how I encourage you, dear reader, to view them too . . . although this may be a selfish request. Indeed, my interest in the computational realm of mathematics has expanded far enough that I recently decided to join a Computer Science department. For many of my colleagues, this move has seemed natural. After all, my study of combinatorics, graph theory, discrete geometry, algorithms, and computational complexity easily fits in the realm of Computer Science. But some of my colleagues (myself included!) have been surprised to see a mathematician moving into what we had previously thought of as a completely different field. Like that XKCD comic about fields arranged by purity, it can be hard for some of us mathematicians to see mathematics as being submerged within other fields, as opposed to mathematics being a separate, more pure entity. And while I’m more than happy to gain the title of “computer scientist,” I want to (and I’d like others to!) continue to acknowledge that I’m still a “mathematician,” and that mathematicians can work and teach in departments outside of math departments.
In going back to graduate school, I became a student again in more ways than one. Yes, I completed assignments that I didn’t write, I went to office hours that were held by someone else, and I took exams that were graded. But I also needed to be schooled in the expansiveness of mathematics. Like some of my own students, I needed to (re)-learn the fact that the study of mathematics includes much more than I had previously understood.
Ellen Veomett earned her PhD in Mathematics from the University of Michigan, and her background is in Combinatorics and Discrete Geometry. Her recent research focuses on mathematical and computational techniques to detect gerrymandering. She is currently an Associate Professor in the Computer Science department at the University of San Francisco.
Credit: Randall Monroe at https://xkcd.com/435/