I think many people have been put off mathematics as young people. But actually what you find with children is they really enjoy it before they've had some adverse experience. A bad experience [is] probably because you were taught or you were in an environment where people were afraid of it. But the natural state I found in most children [is that] they find it very exciting. Children are born curious, exploring the outside world. I'm trying to explain to them [that] for people who carry on, [doing maths is] really an enjoyable experience – it's very exciting.
Now what you have to handle when you start doing mathematics as an older child or as an adult is accepting this state of being stuck. People don't get used to that. Some people find this very stressful. Even people who are very good at mathematics sometimes find this hard to get used to and they feel that's where they're failing. But it isn't: it's part of the process and you have to accept [and] learn to enjoy that process. Yes, you don't understand [something at the moment] but you have faith that over time you will understand — you have to go through this.
It's like training in sport. If you want to run fast, you have to train. Anything where you're trying to do something new, you have to go through this difficult period. It's not something to be frightened of. Everybody goes through it.
What I fight against most in some sense, [when talking to the public,] is the kind of message, for example as put out by the film Good Will Hunting, that there is something you're born with and either you have it or you don't. That's really not the experience of mathematicians. We all find it difficult, it's not that we're any different from someone who struggles with maths problems in third grade. It's really the same process. We're just prepared to handle that struggle on a much larger scale and we've built up resistance to those setbacks.
Yes, some people are brighter than others but I really believe that most people can really get to quite a good level in mathematics if they're prepared to deal with these more psychological issues of how to handle the situation of being stuck.
What do you do when you get stuck?
The process of research mathematics seems to me [to be] that you absorb everything about the problem, you think about it a great deal, all the techniques that you use for these things. Usually [the problem still] needs something else – so yes, you get stuck.
Then you have to stop, let your mind relax a bit and then come back to it. Somehow your subconscious is making connections and you start again, maybe the next afternoon, the next day, the next week even and sometimes it just comes back. Sometimes I put something down for a few months, I come back and it's obvious. I can't explain why. But you have to have the faith that that will come back.
The way some people handle this is they work on several things at once and then they switch from one to another as they get stuck. I can't do that. I get manic about it. Once I'm stuck on a problem I just can't think about anything else. It's more difficult. So I just take a little time off and then come back to it.
I really think it's bad to have too good a memory if you want to be a mathematician. You need a slightly bad memory because you need to forget the way you approached [a problem] the previous time because it's a bit like evolution, DNA. You need to make a little mistake in the way you did it before so that you do something slightly different and then that's what actually enables you to get round [the problem].
So if you remembered all the failed attempts before, you wouldn't try them again. But because I have a slightly bad memory I'll probably try essentially the same thing again and then I realise I was just missing this one little thing I needed to do.
How important is creativity in mathematics?
Well, creativity is what it's all about. I think outside there are different reactions to mathematics, one is [that the] general public think "Isn't it all known already?", or that it's somehow machine-like.
But no, it's extremely creative. We're coming up with some completely unexpected patterns, either in our reasoning or in the results. Yes, to communicate it to others we have to make it very formal and very logical. But we don't create it that way, we don't think that way. We're not automatons. We have developed a kind of feel for how it should fit together and we're trying to feel, "Well, this is important, I haven't used this, I want to try and think of some new way of interpreting this so that I can put it into the equation," and so on.
We think of ourselves as very creative. I think that's sometimes a little frustrating for mathematicians because we're thinking in terms of beauty and creativity and so on, and of course the outside world thinks of us as much more like a computer. It's not how we think of ourselves at all.
It could be a little like music. In some sense, music, you can just write it out in terms of numbers. I mean, they're just notes. It's up, down, up, down, put a rhythm in. It could be written out completely digitally, and it is. But you listen to Bach or Beethoven, that's not a series of numbers, there's something else there. It's the same with us. There's something very, very creative that we get very passionate about.
- Andrew Wiles: What Does It Feel Like To Do Maths?