In school, I often wasn’t sure when I was allowed to think.
In many subjects, we learned simplified versions of things. This was necessary, I’m sure, but it made it hard to meaningfully think about those things.
Despite being fascinated by animals and plants (fish, in particular), biology at school never resonated with me. I put this down to the fact that we weren’t really allowed to think in biology. Or rather, we couldn’t really think in biology.
We were learning about a bunch of things — how photosynthesis works, what the parts of a cell are, how antibodies have particular shapes to “fit” together with antigens. But these all felt like very high level explanations, so much so that we couldn’t really apply any thinking. There was never a scenario in biology where we learned a set of facts from which we could correctly deduce a new fact. Or I couldn’t, at least.
We were working at too high a level of abstraction, so I never felt like I developed any understanding.
Maths was one of the few subjects that felt at the right level of abstraction — close enough to the metal to be able to exercise some thinking — and I think that’s why I chose to study it at university.
A year into my degree, however, I learned that I’d been thinking at too low a level of abstraction all along.
For a long time, I thought maths was about manipulating symbols in the right way to get to a result.
This got me through school, but at university I found it really hard. The turning point was a few weeks before my first-year exams, when I was studying with a friend, Adrien. He seemed to really get the subject. I asked him how he managed to remember so many proofs so effortlessly.
“Well… you just have this picture in your head, right? Then you construct the proof from that.”
Uhhh, what? There were no maths pictures in my head, that was for sure.
I asked him to explain one particular theorem I was struggling with — the colourfully named Scenic Viewpoint Theorem: “Every sequence of real numbers has a monotonic subsequence.”
This says that if I come up with a sequence of numbers, then you’ll always be able to pick, just from that sequence, a new sequence of numbers that only ever increases, or only ever decreases.
Here’s what the formal proof looks like:
This was one of the most basic proofs in our first-year Analysis course, basic enough for any lay person to understand conceptually. But I struggled with it because, on paper, it had a lot of symbols. Look at the the akr bit — a subscript within a subscript! So many complicated-looking symbols.
When Adrien explained it to me, however, he didn’t use any symbols at all. He drew a picture on the whiteboard, and spoke in simple English (albeit with an endearing French accent) about what was going on. If you heard his explanation, you might not even have guessed we were talking about maths.
(My iPad is out of battery right now — I’ll draw the picture and explain the proof non-mathematically later this week!)
This blew my mind. But it wasn’t because I finally understood the Scenic Viewpoint Theorem.
It was because I realised I’d been thinking about maths completely wrong my whole life.
I’d been thinking in terms of markings on a page, when I should have been thinking in terms of pictures in my head — my level of abstraction was too low. I was getting bogged down in the symbols and formalities, while Adrien was thinking at a higher level — one in which he could reason fluently about concepts. I was looking at the trees, while Adrien was seeing the forest.
Part of what it means to really understand something, I think, is to be able to move freely between levels of abstraction as and when you need to.
In writing the proof for the Scenic Viewpoint Theorem in an exam, I was working at a very low level of abstraction. In explaining the intuition to me with a picture, Adrien was working at a higher level. And if you wanted to draw inspiration from the theorem and apply it to a completely different domain (as you often hear “geniuses” do), you’d be working at a higher level still.
When learning something new, I’ve found it’s really useful to hear experts engaging with one-another about the topic. Not to hear what they think — as a novice, a lot of it will go over your head — but to hear how they think, and in particular, the level of abstraction in which they’re working.
Thinking is easy. What’s hard is figuring out the right level of abstraction in which to do it.
Well, okay — thinking is hard, too.