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Principles of Effective Research: Part XI

by Michael Nielsen on July 27, 2004

This is the final installment in my series on the principles of effective research.

This installment was, in some ways, the most fun to write: it’s some thoughts on the difficulties involved in solving the most important research problems, and how to address those difficulties.

Now, I’ve had less success than I would like in this regard (I doubt I’m alone), so this should all be taken with a big grain of salt. Nonetheless, like so many other research scientists, I’d like to solve some big problems, and this post is an attempt to describe a bit of a model for how to go about doing so.


Working on important problems

It’s important that you work towards being able to solve important problems. This sounds silly, but people don’t do this for any number of reasons. I want to talk a little about those reasons, and how to avoid them.

Reason 1: Lack of self-development. Many people don’t spend enough time on self-development. If you stop your development at the level which resulted in your first paper, it’s unlikely you’ll solve any major problems. More realistically, for many people self-development is an incidental thing, something that happens while they’re on the treadmill of trying to solve problems, generate papers, and so on, or while teaching. While such people will develop, it’s unlikely that doing so in such an ad hoc way will let them address the most important problems.

Reason 2: The treadmill of small problems. Social factors such as the need to publish, get grants, and so on, encourage people to work only on unimportant problems, without addressing the important problems. This can be a difficult treadmill to get off.

My belief is that the way to start out in a research career is by working primarily on small and relatively tractable problems, where you have a good chance of success. You then continue the process of self-development, gradually working up to more important problems (which also tend to be more difficult, although, as noted above, difficulty is most emphatically not the same as importance). The rare exception is important problems that are also likely to be technically easy; if you’re lucky you may find such a problem early in your career, or be handed one. If so, solve it quickly!

Even later on, when you’ve developed to the point that you can realistically expect to be able to attack important problems, it’s still useful to tackle a mixture of more and less important problems. The reason is that tackling smaller problems ensures that you make a reasonable contribution to science, and that you continue to take an active part in the research community. Even Andrew Wiles continued to publish papers and work on other problems during his work on Fermat’s Last Theorem, albeit at a rather low rate. If he had not, he would have lost contact with an entire research community, and losing such contact would likely have made a significant negative difference to his work on Fermat’s Last Theorem.

Reason 3: The intimidation factor. Even if people have spent enough time on self-development that they have a realistic chance of attacking big problems, they still may not. The reason is that they have a fear of working on something unsuccessfully. Imagine Andrew Wiles feeling if he had worked on Fermat’s Last Theorem for several decades, and completely failed. For most people, the fear of ending up in such a situation is enough to discourage them from doing this.

The great mathematician Andrei Kolmogorov described an interesting trick that he used to get around this problem. Rather than investing all his time and effort on attacking the problem, he’d put the problem into a larger context. He’d announce a seminar series in which he’d lecture on material that he thought would be related to the problem. He’d write a set of lecture notes (often turning into a book) on material related to the problem. That way, he lowered the psychological pressure on himself. Rather than investing all his effort in an attack on the problem – which might ultimately be a complete waste of time – he knew that he’d produce something of value. By making the research process part of a larger endeavour, he ensured that the process was a success no matter how it came out, even if he failed to solve the problem, or was scooped by someone else. It’s a case of not putting all of one’s psychological eggs in one basket.

Richard Feynman described a related trick. He said that when he started working on a problem he would try to convince himself that he had some kooky insight into the problem that it was very unlikely anybody else had. He admitted that very often this belief was erroneous, or that, even if original, his initial insight often wasn’t very good. But he claimed that he found that he could fool himself into thinking that he had the “inside track” on the problem as a result, and this was essential to getting up the forward momentum necessary to really make a big dint in a difficult problem.

Committing to work on an important problem: For the difficult problems, I think commitment is really a process rather than a moment. You may decide to prepare a lecture to talk about a problem. If that is interesting, you enjoy it, and you feel like you have some insight, you might decide to prepare a few lectures. If that goes well, perhaps you’ll start to prepare more lectures, write a review, and maybe make a really big contribution. It’s all about building up more and more insight. Ideally, you’ll do this as part of some larger process, with social support around you.

People who only attack difficult problems: There is a converse to the problem I’ve been talking about, which is people who are only interested in attacking problems that are both difficult and important. This affliction can affect people at any stage of their career, but it manifests itself in somewhat different ways at different stages.

In the case of the beginner, this is like a beginning pole vaulter insisting on putting the bar at 5 meters from the time they begin, rather than starting at some more reasonable height. Unless exceptionally pigheaded, such a person will never learn to vault 5 meters successfully, simply because they will never learn anything from failure at a more realistic starting height. This sounds
prima facie ridiculous, but I have seen people burn out by following exactly this strategy.

The case of the more experienced researcher is more difficult. As I’ve emphasized, once you’ve reached an appropriate level of development I think it’s important to spend some time working on the most important problems. But if that’s all you do, there are some very significant drawbacks. In particular, by attacking only the most important and most difficult problems an experienced researcher (a) takes themselves out of circulation, (b) stops making ongoing contributions, (c) loses the habit of success, and (d) risks losing morale, which is so important to research success. I think the solution is to balance one’s work on the more and less important problems: you need to schedule time to do the more important stuff, but should also make sure that you spend some time on less high-risk activities.

In both cases, the explanation is often, at least in part, intellectual macho. Theorists can be a pretty judgmental lot about the value of their own work, and the work of others. This helps lead some into the error of only working on big problems, and not sometimes working on little problems, for the fun of it, for the contact it brings with colleagues, and for the rewarding and enjoyable sense of making a real contribution of some significance.

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