What a Mathematican Does 101

My girlfriend is a biologist. I've noticed that introducing oneself as a biologist is a fairly good conversation starter. You don't have to be an expert in genetics or cellular biology to have an opinion about evolution or cloning. Even talking about your pet's strange behavior is fair game.

I am a mathematician. When I introduce myself as a mathematician, the response usually falls into one of three categories:

1. The "Oh, you must be pretty smart" response. I know people mean well, but I did not go into mathematics to impress people. Indicative of a larger problem, specifically the impression that you have to be a genius (usually some kind of detached savant) to work or take an interest in the subject. Also, hard to respond to. (I lean towards "Smart enough to work at Starbucks.")

2. The "I was bad at math in school" response. One of my professors told us once of a time when someone (I think a delivery man) nearly broke into tears when the professor introduced himself as a mathematician. The delivery man had such a traumatic and humiliating algebra class that just the mention of the subject sent him into shivers. It's true that most people are not as emotional about it. They mention their time in school because that is their only exposure to math, and the only way they can relate to me.

I can only sympathize and explain that math is a lot more stimulating when it's not a set of tools to be applied to boring problems. Though I did not struggle at math in high school, I didn't really enjoy it either, and I have a lot of sympathy with this perspective. I was a political science major until I realized how different math is in college (at least at Reed, where I was not forced to take two years of calculus before I could study something more fun).

3. Quietly and quickly moving to a different subject. This is by far the most common response.
Sad, but preferable to the first two.

A response I have never gotten, though I would love to get it, is "What does a mathematician actually do?" People do not usually ask this question, either because they are not interested or too embarrassed, although there is nothing to be embarrassed about. I didn't know the answer to this question until after I took two years of college math. Math professors don't usually tell you, because they're more concerned with helping you pass whatever class you're taking (and I'd never been brave enough to ask). However, it is a really important question, partly because there are so many misconceptions, many of which I fell victim to.

How would I answer this question? The short answer that I lean towards lately is "I study worlds that aren't real."

Most colleges group math with other sciences, and I understand the logic behind this. Besides the fact that science uses tools from math, both mathematicians and scientists observe things and make and test hypotheses. There are two differences.

Firstly, the things I observe are not facts about the world in which we live, but about worlds which are purely hypothetical. I study worlds like the world of natural numbers, the world of geometry, or the world of set theory. It is okay to call them models, but they don't necessarily "model" anything about the real world, or at least that is not the context in which I, as a mathematician, study them. Certainly, physicists or computer scientists might be interested in the conclusions I draw, but those are just applications and not the subject itself.

Secondly, unlike a scientist's conclusions, my conclusions can be completely verified, such that they can never be refuted. This is because certain axioms are true in these worlds, as well as certain rules of logic which let me derive things from them. Most people would say I assume the axioms, but I have a slight problem with the word "assume". I prefer to say that I am studying precisely that world in which these axioms are true. How do I know that there is such a world? Because I know that there are statements that can be derived from these axioms; in a certain sense, I am only dealing with language. (I choose to sidestep the Platonist-Antiplatonist debate about mathematical reality that pops up in math philosophy.)

Of course, what I have just stated sounds tremendously boring. It needs to be noted that a mathematician is not interested in all statements that can be derived from these axioms; one could tell a computer to derive statements for me, but the vast majority of these would be meaningless gibberish that no one could care less about. The statements that I am interested in are ones that are either useful for their applications, or just aesthetically pleasing (and the two go together far more often than not).

What does it mean for a fact about a hypothetical world to be aesthetically pleasing? Often, the worlds we're interested in can be visualized in interesting ways. Not just in terms of a picture, but in terms of examples from real life. But visualizations alone don't quite do it justice. Some facts we just want to be true; something we expect should be true by thinking of examples, but can't quite figure out why, until we do, which is gratifying. Some facts are very surprising, and they might help us conceptualize something we thought we had no hope of grasping. Some facts make connections between two worlds that seemed separate.

Part of being a mathematician is deciding what deserves to be studied; and like any scientist, we should be able to explain why the topic we've chosen is interesting, or just beautiful. There is an artistic aspect to math, just as there is a philosophical aspect.

Anyways, I hope to say more about certain aspects of the mathematician later. Comment if you have any questions.