Popular writings about mathematics, and autobiographies by eminent mathematicians, provide some sense of what doing research-level mathematics is like. In this article I seek to convey the underlying implicit “flavor” of the profession, not the day-to-day explicit activity. Let’s work by analogy. That is, I will make 4 “doing research mathematics is like X” assertions, intended as a starting point for a “compare and contrast” discussion. These analogies come from my personal experience doing mathematical theory [probability, specifically] a few steps away from real world questions. No doubt someone doing highly abstract pure mathematics or domain-specific data analysis would offer different views.
Doing research mathematics is like chess
That is, as an intellectual game. Seeking criteria for which human activities should be called “games” presents an interesting challenge, but for the first three criteria one might formulate — “voluntary; agreed rules; no immediate consequences outside the activity” — theoretical mathematical research fits just as well as does chess. The fact of all theorem-proof mathematics being consequences of agreed axioms resembles the given rules of chess. Of course a key difference is that mathematics is open-ended — one can create definitions, that is axioms, for a new particular type of structure — so maybe better to view mathematics as an entire genre of games. Some of these structures clearly relate to the physical world — the axioms of finite group theory derive from symmetries of physical objects — but others are “deep” in that they lie many layers — structures defined in terms of previous structures — deep beneath the “surface” structures with tangible real world interpretations. Do these structures reflect some Platonic reality, or are they just a human choice of what seems interesting? This has led to IMHO endless vague philosophy.
My point here is to compare with chess: the rules of chess are completely arbitrary from a real-world-model viewpoint, but also are fine-tuned to make an interesting game.
A less-discussed connection concerns “white to play and mate in 2” chess problems. These can be educational for a novice, and of course require great ingenuity to devise, but they involve very unnatural configurations which would not arise in an actual game. By analogy, in my field of Probability, one sees puzzles like the Boy or Girl paradox (A two-child family has at least one boy. What is the probability that it has a girl?). There is no natural context in which one would have precisely this information available; you can make up various unnatural stories, and then the answer depends on how (in the story) you obtained the information. Here is a serious analysis of a similar problem.
Doing research mathematics is like mountain climbing
One obvious analogy is strenuous intellectual activity versus strenuous physical activity. But what I actually have in mind is the famous “because it’s there” justification for mountain climbing. I guess that most researchers in theorem-proof mathematics would also answer the “why?” question with some version of “because it’s there” and adding “contributing to the body of mathematical knowledge”. That is, like mountain climbing, new mathematics is judged worthwhile in itself. Note that the “it’s there” description is a little unconvincing because young researchers often seek advice on how to find problems to study. Mathematical problems are less visibly “there” than are mountains.
As another analogy, being the first to climb a notable mountain gains you a modicum of fame — see this Wikipedia list of around 300 first ascents.
And theorems that become especially useful or noteworthy to the community are often named after the first prover (Do mountain climbers regret that mountains are not named after the first climber?). I have no idea how many mathematicians have had their names thus attached, but perhaps a larger number have retired disappointed about not having a theorem named after them. Recall G.H. Hardy’s comment: “Immortality” may be a silly word, but probably a mathematician has the best chance of whatever it may mean.
Doing research mathematics is like composing music
There are many connections between music and mathematics and here are just two.
Composing new music is clearly a creative act, and research in theorem-proof mathematics is (most mathematicians feel) a similarly creative activity.
The proof of a theorem of the format “properties A and B together imply property C” is presented in a textbook as a linear argument from assumptions to conclusion. But a theorem that was initially readily proved in this way
— that is with some clear “direction” to follow — is an “easy” theorem. A “hard” theorem is one where you don’t initially see a direction. Here you can try a proof by contradiction — suppose A and B were true while C was false —
then argue in some direction until you reach a contradiction. And one often first comes up with some “stepping stone” scheme: maybe I can prove this and then maybe I can prove that …… In occasional conversions with composers they tell a similar story: start by imagining a few snatches of tune and a general theme, and then figure out how to put things together.
Another analogy is that music has genres — here is a list of several hundred genres. The different fields of mathematics — here is a widely-used list of 64 fields — are very much like genres. In both music and mathematics, most people enter and stay in one or several related genres, though some become famous by virtue of crossing boundaries. To what extent the choice of genre is influenced by exposure to that genre, versus some psychological predisposition, is likely an unanswerable question. In both contexts a minority of people are chauvinistic about their own genre and dismissive of the value of other contemporary genres as being merely fashionable.
Doing research mathematics is like a half-time job for people predisposed to focus
Some of the stereotypes about mathematicians — unfashionably dressed and socially awkward — are alas rather accurate. More significant is a “focus” aspect. In his famous essay “The mathematician’s art of work”, Littlewood wrote
Either work all out or rest completely. I recommend four hours a day or at most five, with breaks about every hour (for walks perhaps).
In our own era of constant distraction, the past era in which one could just choose when to work undisturbed sounds delightful! Anyway, my point concerns a little known side effect of “desire to focus”. Many mathematicians, by also applying focus to some other activity, become very good amateurs in that other activity. Amongst people I know well, such activities include
(i) bridge, chess, go, crosswords(not so surprising, being other intellectual games);
(ii) music (performing);
(iii) sports such as cricket, field hockey, marathon running, volleyball, fencing, tennis;
(iv) magic, juggling.
Not the absent-minded stereotype, in this regard.