How I realised I can't do math

Pictured: one of many unsuccessful attempts at learning number theory.

For most of my life, I thought I was could do mathematics with enough effort.

This was not an unreasonable conclusion. I liked science. I got a PhD in molecular biology. I spent years working in quantitative biology, where mathematics was not exactly optional. I have happily wandered into statistics, statistical mechanics, linear algebra, probability, information theory, and various other mathematical forests without immediately turning around and running home.

More importantly, I liked what I thought mathematics was.

I liked symmetry. I liked abstraction. I liked the idea that a handful of axioms could generate an enormous intellectual structure. I liked mathematical physics in particular, where apparently unrelated things could turn out to be different faces of the same object. Disclaimer here: I liked what I thought these things were, which is not necessarily the thing itself.

So when I decided, relatively late in life, that I wanted to actually learn mathematics, I assumed I knew roughly what would happen.

I would work hard. I would read the books. I would do the problems. Eventually, I would become good at it.

This turned out to be slightly optimistic.

If you are conscientious and reasonably intelligent, you can become good at a lot of things. You learn the material. You recognize the kinds of questions that are likely to be asked. You learn the techniques. You get better at exams.

And mathematics, particularly the mathematics most of us encounter in school, can be approached this way.

There is an enormous amount of mathematics that can be learned procedurally.

Here is a derivative. Here is a matrix. Here is a probability distribution. Here is a differential equation. Here is the technique you use to solve it.

You learn the machinery.

And if you are good at learning machinery, you can become very good at mathematics without ever asking yourself the slightly more uncomfortable question:

Could I have invented any of this?

I didn't ask that question for a long time.

Then I started learning mathematics properly.

The first thing I noticed was that there was a difference between understanding a proof and being able to find one. It is not obvious when you're experiencing it. I could read a proof and think, Yes, of course. Every step followed. The definitions made sense. The argument was logically sound. Sometimes the proof was even beautiful.

Then I would close the book and look at the problem. Nothing. Not “I need another twenty minutes.” Not “I vaguely know which theorem to use.” Just a rather alarming expanse of intellectual fog.

I began to notice this happening repeatedly.

A mathematician would produce an argument that seemed almost inevitable once I saw it. I would understand why it worked. I could explain it afterward. I could sometimes reconstruct parts of it.

But I could not have found it.

This distinction became increasingly difficult to ignore.

There is a particular humiliation in seeing a solution to a problem and realizing that the solution is not merely something you hadn't thought of yet.

It is something you would never have thought of.
Surely I just need more background

Naturally, I responded to this in the most sensible way possible.

I assumed I needed to learn more mathematics.

This is an extremely convenient hypothesis because it can keep you occupied indefinitely.

Maybe I needed more linear algebra.

So I studied linear algebra.

Maybe my problem was probability.

So I started working through probability theory.

Maybe I needed real analysis.

So I went to real analysis.

Then measure theory.

Then more probability.

Then number theory.

At each stage there was a brief period of optimism.

Perhaps this was the missing piece.

Perhaps mathematical ability was just a sufficiently large pile of prerequisites, and I had simply not yet accumulated the pile.

This is a particularly seductive idea for someone coming from science. There came a point when I had to consider the possibility that I might simply not be very good at mathematics.

I resisted this for quite a while.

It felt absurd.

I have spent most of my life being the person who could learn difficult things.

I had survived a PhD. I had changed fields. I had learned enough quantitative biology to work with physicists and mathematicians. I could read papers containing mathematics that would have been completely opaque to me ten years earlier.

Surely I could learn mathematics.

But I was beginning to realize that this was not quite the same question.

The question was not:

Can I understand mathematics?

The answer to that is, within limits, yes.

The question was:

Can I do mathematics?

And that answer appeared to be considerably less flattering.
Understanding is not doing.

It's time for me to honestly accept my shortcomings without having it be a judgement on my character.