What is mathematical research?

by Willie Wong

Recently I sat on a panel of “research scientists” for Career Day at the New Jersey Governor’s School in the Sciences. During the discussion, I was pointedly asked, by a young woman who aspires to be a research mathematician, what “mathematicians do for research?” Much to my own dismay, I fumbled around for a short answer, but ended up giving a shpiel that is much too long for the occassion and which ended up mostly sound and fury, and probably confused everyone in the room except for myself.

Contrary to the impression I have just given, I have thought quite a bit on how to answer the question “what is it that you do?” I am even quite good at tailoring the answer to the audience. If that were the question asked, I would have told the audience of high school students that I study the long term behaviour of the universe using general relativity, that this falls in the realm of mathematics and not physics because it is patently impossible to recreate large scale structures of the universe, such as nebulae or black holes, in a laboratory on Earth, and so I study the mathematical consequences of the assumptions underlying general relativity. If time allowed or if I were further prompted, I would have told them about the propagation of waves and other fancy things from partial differential equations.

But that was not the question I was asked.

I was asked to describe, in generality, what mathematicians, a diverse group of people of various interests and goals, do. And I still don’t have a good answer after thinking about it for several days. The problem is one of scope. For example, suppose I ask the question, “what do doctors do?” The simple answer would be, “Doctors cure diseases.” I can analogously answer the question posed by saying that “mathematicians look for universal truths that can be deduced from a set of axioms,” or that “mathematicians examine relationships between figures, forms, and quantities expressed symbolically.” But these are just dictionary definitions. A better analogy for the question posed to me is, “what do doctors do to cure diseases?” The question now admits a lot more answers! A surgeon treats diseases by mechanically repairing or excising the damaged tissue; an internist treats diseases by exploiting biochemical pathways that either kill a foreign irritant or stimulate a biological response within the patient; an endocrinologist regulates abnormal biological responses using hormones; an ophthalmologist may prescribe corrective lenses to enhance vision; a prosthetist replaces lost body parts with biomechanical tools. And the list goes on and on: there is no one simple description of what doctors precisely do. The same can be said of any profession with sufficiently diverse specialties–engineers, politicians, and artists are just a few more examples of these kinds of broad labels.

The label “mathematician” is yet another. I happen to be an analyst (one who studies the properties of functions using tools which find their roots in calculus) specializing in partial differential equations, a mathematical physicist (one who studies mathematical relationships between physical phenomena), and a geometer (one who studies the shape of figures–how they are curved and how far apart are the constituent points). Other people may be interested in number theory, topology, abstract algebra, or the myriad subdivisions and combinations thereof, to say nothing of the various specialties in applied mathematics. To ascribe one fixed research interest to mathematicians is folly.

Yet perhaps I was looking at the question wrong, or maybe searching for an answer too rigidly. It is certainly not the case that universal or nearly universal attributes cannot be given to mathematicians. When confronted with the question, “what mathematicians do for research?” I looked for factual answers about the fruits of their labour. The fruits are of course, as diverse as the seeds that were sowed. Instead, what if I searched for procedural answers about how mathematicians work? Rather than drawing my analogies with doctors, engineers, and artists, I can also draw the analogy to farmers. Some farmers plant corn, some cabbages, some raspberries and blueberries, some apples and peaches. Each different produce is planted on a different schedule, in a different soil, watered a different amount, fertilized with different mixes. But even in this statement we glimpse the universality of a procedure to farming.

To put it another way, what if instead of focusing on “what mathematicians study”, I focused on “how mathematicians study” when trying to expand on the dictionary definition of a mathematician? (Arguably, this was not the intent of the question posed to me by the high school student; to answer it like this is akin to how politicians smoothly deflect the question into a favourable segue.)

It has been almost a month since I started composing this essay, and ten days since I stopped the thought up there. I have a few ideas how I can continue this essay, but all of them divagates from what I originally intend to write about. The end of this string of thoughts seems already frayed. In the meantime, however, I’ve started reading a book. And in the book, is a description of mathematicians more eloquent then I can conceive myself

Translating a thing into the analytical language is akin to what the alchemist does when he extracts, from some crude ore, a pure spirit, or virtue, or pneuma. The feces–the gross external forms of things–which only mislead and confuse us–are cast off to reveal the underlying spirit. And when this is done we may learn that some things that are superficially different are, in their real nature, the same.

Those were words of Isaac Newton imagined by Neal Stephenson in Quicksilver. In the book Newton was speaking about how to apply mathematics, especially calculus and analytical geometry, to the study of nature, especially physics. Yet this process of discarding the “gross external forms” to reveal the “underlying spirit” is also very much what modern mathematics is like. Much of modern mathematics involve finding the minimal requisite set of predicates from which a certain logical conclusion can be drawn. The goal is to peal away the inessentials so we can understand why the essential are what they are.

I think I am just talking around in circles, so I better just stop here.