- Solve a known open problem that people care about.
- Solve a problem you made up and convince others that it’s interesting / important / impactful.

So research requires some confidence, if not arrogance, to attempt. Successful researchers clearly need to find this confidence, and it’s not something that comes immediately. So I thought it might be helpful, especially to beginning graduate students (assuming any are reading), for me to spell out why I think succeeding in research is not as hopeless as it might first seem.

Lots of people think in the following chain, especially regarding solving known open problems: 1) Famous researcher X attempted this problem and didn’t get anywhere. 2) They’re clearly better / more experienced than I am. 3) Hence, I won’t be able to solve it.

The worst part of thinking this way is that it’s self-fulfilling – if you never make a serious attempt at a problem believing you can solve it, you probably never will. The second worst part is that this reasoning is flawed.

Lets assume that experienced researcher X attempted (but failed to solve) the problem you’re working on; it doesn’t mean that you can’t. There are a couple reasons for this.

- Famous researcher X is probably working on lots of problems and doesn’t have nearly as much time to devote to your problem as you do. By seriously applying yourself, you might succeed where X hasn’t.
- The process of doing research is partly random. You might just have an idea that X hadn’t.
- Some advances may have come out since X seriously attempted your problem. The average graduate student can solve lots of problems Gauss couldn’t solve, and it’s because science has made progress since then. This type of reasoning holds true on much smaller time scales.

- Work on a problem that you’re actually excited about. This will make it much easier to put in the work needed to be successful. If research feels more like work than fun, then you might be doing it wrong.
- Find a problem that you think you have a chance in tackling. You should be able to feel in your gut if the problem “feels right.” Many problems are indeed too hard to attempt for beginning researchers.
- Don’t forget you have access to experienced person Y (your advisor) who might give you some insight that experienced person X didn’t have.

So what's an example of a problem that's easy today but Gauss couldn't solve?

ReplyDeleteFrom the top of my head, there are lots. We can solve many questions in probability by applying Bernstein inequalities or Chernoff bounds. We don't even think of their proofs as difficult, but they were unknown to Gauss. We know a lot about undecidability, but Gauss didn't even know if uncomputable functions existed (or how to properly formulate these questions). We can compute the value of lots of games using minimax theorems or prove the existence of an equilibrium for many games. We can prove the existence of lots of objects using the probabilistic method. None of these things are at all difficult for us (or would be for Gauss), but this "technology" was unavailable to him.

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