It’s so hard to explain what math research is like. For some reason, people seem to assume it involves doing long, complicated calculations; perhaps because that was what they found difficult about math in school. In reality, long and complicated calculations are performed by computers. We’ve gotten much better at automating those processes – it’s really stunning what computers can do these days. Every once in a while I stop taking it for granted, just for a moment, and remember how powerful these ubiquitous tools really are. Anyway, this post is going to be a bit of a philosophical ramble about the nature of math reasoning followed by a (charming) anecdote about how I proved my first little theorem of my graduate career.
In fact, the difficulty of math is much more about figuring out how to prove something, and less about the actual argument. Consider any of the theorems you learned in school – the Pythagorean Theorem for triangles, for example. This proof is taught to, and understood by, children under the age of eighteen! The argument itself is not so difficult – you follow some logical steps and end up at the result. There are literally hundreds of proofs. But now imagine that you are discovering this theorem for the first time. You have noticed a pattern in the relationships between the lengths of sides of a triangle. You can’t find any counterexamples, so you think that it might be universally true. But you don’t know, so you set out to prove the general concept. How do you do it? If someone from the future were to show up and explain why it’s true, you would understand, just like the kids in school do. But there isn’t a time-travelling teacher – and you have this niggling worry in the back of your mind that it might not be true at all. The difficult part of the proving something is not following the logic; it’s figuring out which logic to follow.
I spent the past few days working feverishly to try to complete a proof. As usual in math research, this was an emotional rollercoaster. Every supposed breakthrough is followed by the discovery of an error. But one nice thing about struggling with how to prove something is that as you play with a mathematical object, you start to understand more and more about it. Eventually you can usually convince yourself that something is true, even if you have no formal proof for it. You get excited, and then discouraged, but even when you realize your argument is flawed, you can see that a similar argument ought to work.
This kept happening to me while I was solving this problem. I had one insight – that graphs can be either fat and round, or long and skinny. As obvious as that sounds, after that realization I knew that I would be able to solve my problem. I’m trying to generalize an argument made by two other mathematicians, and I knew that this was the way to do it – it was too similar, too elegant not to be. I did pages and pages of calculations and estimates, drew diagram after diagram. But I couldn’t quite get what I was looking for.
On Tuesday, I thought I had it. I tested all the cases, did all the calculations, checked for obvious errors – it looked good, so I emailed my advisor that I had something to show him. On Tuesday night, I found the first error. Talk about emotional turmoil! So I spent another few hours brainstorming ways to fix it. None of them seemed to work. They were all reasonable, but just didn’t work out in the end, or relied on a claim that seemed like it should be true but that I couldn’t prove. I went to bed disappointed (and couldn’t sleep, so I got up and did another hour of work before going back to bed). The next morning, I went back to some of those ideas. “Maybe one of these can be massaged until it works,” I thought. They’re all reasonable; surely something can be fixed. But to my chagrin, none of them worked. When I went to bed on Wednesday, I thought about my problem and discovered another error. (Luckily, when I got up to write it down, I realized that it didn’t matter, and actually made my life easier.) On Thursday, I decided to brute-force one method; to apply as much heavy machinery to it as it took to make the argument work. Up to a single claim, I had the argument. But I couldn’t prove this claim. It seemed reasonable, but that wasn’t enough. I took it to a professor, who agreed that it seemed reasonable but didn’t have an argument for it. I took it to a fellow graduate student, who said that he might have a sketch of an argument, but wasn’t sure. I started to feel more confident. I took it to two post-docs, who agreed that it seemed reasonable, and then stood at a chalkboard with me for 40 minutes trying to prove it. We couldn’t do it. I asked my advisor, who told me that it was indeed an interesting question (which was a relief, to know that it wasn’t completely trivial!), but that he didn’t know the answer. I started to get hopeful. But then one of the post-docs came and showed me a counter-example. My claim was not true, my argument failed. Ten minutes later, I had the actual solution. It was easy, straightforward, elegant. I wrote it up, checked it, and sent it to my advisor.
My understanding is that this is normal – that math research is full of ups and downs, false starts and blind alleys. I think it also highlights two things in particular: the first is the importance of collaboration. Throughout the process, I talked about my problem with friends. Explaining my ideas to someone else helped crystallize them, and they could help point out potential flaws or confusions. And for this final claim that my friend disproved, I wouldn’t have found that counterexample without her. The second thing is the importance of perseverance. There were a lot of setbacks in this process – and I’m not done with my proof yet. I’ve done half of the first step of (at least) three. It’s great, and I’m incredibly pleased, but it’s just a battle, not the whole war. Luckily, I enjoy this process and I think my problem is interesting, so perseverance isn’t much of a challenge. It’s easy to get discouraged of course, but as soon as one method fails another idea pops into my head, and I have to start exploring it right away. And it helps that I can feel it as I get closer to a solution, so that even if I hit a setback I know that the right answer is just around the corner.