Over on Google+, a computer scientist at McGill named Artem Kaznatcheev passed on this great description of what it’s like to learn math, written by someone who calls himself ‘man after midnight’:

The way it was described to me when I was in high school was in terms of ‘levels’.

Sometimes, in your mathematics career, you find that your slow progress, and careful accumulation of tools and ideas, has suddenly allowed you to do a bunch of new things that you couldn’t possibly do before. Even though you were learning things that were useless by themselves, when they’ve all become second nature, a whole new world of possibility appears. You have “leveled up”, if you will. Something clicks, but now there are new challenges, and now, things you were barely able to think about before suddenly become critically important.

It’s usually obvious when you’re talking to somebody a level above you, because they see lots of things instantly when those things take considerable work for you to figure out. These are good people to learn from, because they remember what it’s like to struggle in the place where you’re struggling, but the things they do still make sense from your perspective (you just couldn’t do them yourself).

Talking to somebody two or levels above you is a different story. They’re barely speaking the same language, and it’s almost impossible to imagine that you could ever know what they know. You can still learn from them, if you don’t get discouraged, but the things they want to teach you seem really philosophical, and you don’t think they’ll help you—but for some reason, they do.

Somebody three levels above is actually speaking a different language. They probably seem less impressive to you than the person two levels above, because most of what they’re thinking about is completely invisible to you. From where you are, it is not possible to imagine what they think about, or why. You might think you can, but this is only because they know how to tell entertaining stories. Any one of these stories probably contains enough wisdom to get you halfway to your next level if you put in enough time thinking about it.

What follows is my rough opinion on how this looks in a typical path towards a Ph.D. in math. Obviously this is rather subjective, and makes math look too linear, but I think it’s a useful thought experiment.

Consider the change that a person undergoes in first mastering elementary algebra. Let’s say that that’s one level. This student is now comfortable with algebraic manipulation and the idea of variables.

The next level may come somewhere during a first calculus course. The student now understands the concept of the infinitely small, of slope at a point, and can reason about areas, physical motion, and optimization.

Many stop here, believing that they have finally learned math. Those who do not stop, might proceed through multivariable calculus and perhaps a basic linear algebra course with the tools they currently possess. Their next level comes when they find themselves suffering through an abstract algebra course, and have to once again reshape their whole thought process just to squeak by with a C.

Once this student masters all of that, the rest of the undergraduate curriculum at their university might be a breeze. But not so with graduate school. They gain a level their first year. They gain another their third year. And they are horrified to discover that they are expected to gain a third level before they graduate. This level is the hardest of them all, because it is the first one that consists in mastering material that has been created largely by the student.

I don’t know how many levels there are after that. At least three.

So, the bad news is, you never do see the whole picture (though you see the old picture shrink down to a tiny point), and you can’t really explain what you do see. But the good news is that the world of mathematics is so rich and exciting and wonderful that even your wildest dreams about it cannot possibly compare. It is not like seeing the Matrix—it is like seeing the Matrix within the Matrix within the Matrix within the Matrix within the Matrix.

As he points out, this talk of ‘levels’ is too linear. You can be much better at algebraic geometry than your friend, but way behind them in probability theory. Or even within a field like algebraic geometry, you might be able to understand sheaf cohomology better than your friend, yet still way behind in some classical topic like elliptic curves.

To have worthwhile conversations with someone who is not evenly matched with you in some subject, it’s often good for one of you to play ‘student’ while the other plays ‘teacher’. Playing teacher is an ego boost, and it helps organize your thoughts – but playing student is a great way to amass knowledge and practice humility… and a good student can help the teacher think about things in new ways.

Taking turns between who is teacher and who is student helps keep things from becoming unbalanced. And it’s especially fun when some subject can only be understood with the combined knowledge of both players.

I have a feeling good mathematicians spend a lot of time playing these games—we often hear of famous teams like Atiyah, Bott and Singer, or even bigger ones like the French collective called ‘Bourbaki’. For about a decade, I played teacher/student games with James Dolan, and it was really productive. I should probably find a new partner to learn the new kinds of math I’m working on now. Trying to learn things by yourself is a huge disadvantage if you want to quickly rise to higher ‘levels’.

If we took things a bit more seriously and talked about them more, maybe a lot of us could get better at things faster.

Indeed, after I passed on these remarks, T.A. Abinandanan, a professor of materials science in Bangalore, pointed out this study on excellence in swimming:

• Daniel Chambliss, The mundanity of excellence.

Chambliss emphasizes that in swimming there really are discrete levels of excellence, because there are different kinds of swimming competitions, each with their own different ethos. Here are some of his other main points:

1) Excellence comes from qualitative changes in behavior, not just quantitative ones. More time practicing is not good enough. Nor is simply moving your arms faster! A low-level breaststroke swimmer does very different things than a top-ranked one. The low-level swimmer tends to pull her arms far back beneath her, kick the legs out very wide without bringing them together at the finish, lift herself high out of the water on the turn, and fail to go underwater for a long ways after the turn. The top-ranked one sculls her arms out to the side and sweeps back in, kicks narrowly with the feet finishing together, stays low on the turns, and goes underwater for a long distance after the turn. They’re completely different!

2) The different levels of excellence in swimming are like different worlds, with different rules. People can move up or down within a level by putting in more or less effort, but going up a level requires something very different—see point 1).

3) Excellence is not the product of socially deviant personalities. The best swimmers aren’t “oddballs,” nor are they loners—kids who have given up “the normal teenage life”.

4) Excellence does not come from some mystical inner quality of the athlete. Rather, it comes from learning how to do lots of things right.

5) The best swimmers are more disciplined. They’re more likely to be strict with their training, come to workouts on time, watch what they eat, sleep regular hours, do proper warmups before a meet, and the like.

6) Features of the sport that low-level swimmers find unpleasant, excellent swimmers enjoy. What others see as boring – swimming back and forth over a black line for two hours, say – the best swimmers find peaceful, even meditative, or challenging, or therapeutic. They enjoy hard practices, look forward to difficult competitions, and try to set difficult goals.

7) The best swimmers don’t spend a lot of time dreaming about big goals like winning the Olympics. They concentrate on “small wins”: clearly defined minor achievements that can be rather easily done, but produce real effects.

8) The best swimmers don’t “choke”. Faced with what seems to be a tremendous challenge or a strikingly unusual event such as the Olympic Games, they take it as a normal, manageable situation. One way they do this is by sticking to the same routines. Chambliss calls this the “mundanity of excellence”.

I’ve just paraphrased chunks of the paper. The whole thing is worth reading! I can’t help wondering how much these lessons apply to other areas. He gives an example that could easily apply to mathematics—a

more personal example of failing to maintain a sense of mundanity, from the world of academia: the inability to finish the doctoral thesis, the hopeless struggle for the magnum opus. Upon my arrival to graduate school some 12 years ago, I was introduced to an advanced student we will call Michael. Michael was very bright, very well thought of by his professors, and very hard working, claiming (apparently truthfully) to log a minimum of twelve hours a day at his studies. Senior scholars sought out his comments on their manuscripts, and their acknowledgements always mentioned him by name. All the signs pointed to a successful career. Yet seven years later, when I left the university, Michael was still there-still working 12 hours a day, only a bit less well thought of. At last report, there he remains, toiling away: “finishing up,” in the common expression.

In our terms, Michael could not maintain his sense of mundanity. He never accepted that a dissertation is a mundane piece of work, nothing more than some words which one person writes and a few other people read. He hasn’t learned that the real exams, the true tests (such as the dissertation requirement) in graduate school are really designed to discover whether at some point one is willing just to turn the damn thing in.