Learn the limitations of your tools

An education isn’t how much you have committed to memory, or even how much you know. It’s being able to differentiate between what you do know and what you don’t.

Anatole France

Mathematical education (and research papers) tends to focus, naturally enough, on techniques that work. But it is equally important to know when the tools you have don’t work, so that you don’t waste time on a strategy which is doomed from the start, and instead go hunting for new tools to solve the problem (or hunt for a new problem).

Thus, knowing a library of counterexamples, or easily analysed model situations, is very important, as well as knowing the type of obstructions that your tool can deal with, and which ones it has no hope of resolving. Also it is worth knowing under what circumstances your tool of choice can be substituted by other methods, and what the comparative advantages and disadvantages of each approach is.

If you view one of your favorite tools as some sort of “magic wand” which mysteriously solves problems for you, with no other way for you to obtain or comprehend the solution, this is a sign that you need to understand your tool (and its limitations) much better.

This point is particularly worth keeping in mind if you think that you have just used one of your favorite tools to prove some impressive result (such as the proof of a major unsolved problem). When this happens, you should try to see if there is a way to rewrite your argument in such a way that the tool is not used. If you understand your tool well, then you should be able to do this, though probably at the cost of making the argument significantly longer and messier. But if you do not see any way to proceed without the tool at all, then you should take this as a warning sign that you might not be using the tool properly.

While unintuitive, one should also discuss the known limitations of a tool when writing a paper that uses that tool. At first glance, this may seem to undercut the value of the paper; but it helps identify precisely what classes of related problems might also be suitable for trying the tool on, and also by demonstrating awareness of such limitations, one gains more credibility and appearance of objectivity when promoting the strengths of the tool, or comparing it to competing methods. It also helps identify the direction in which new breakthroughs will need to come from in order to go beyond what can be done with that tool alone. Finally, if one conceals the limitations of a tool from readers, and the work is interesting, eventually some followup work will run into the same limitations, and discuss them in their work; and it will appear that such issues were totally overlooked in your previous paper.

See also “[learn the power of other mathematician’s tools](learn the power of other mathematician’s tools)” and “[learn and relearn your field](learn and relearn your field)”.