Let me tell you about a mistake I made recently. Some months ago, I made a few calculations which seemed promising, and I thought they could lead to some new results. When I showed the calculations to my supervisor, he was skeptical, and rightly so. Our project involved some fairly abstract concepts in the theory of operator semigroups, so his immediate advice was to check the calculations against some simple, concrete examples . So of course I did that, and quickly realised that the calculations were totally wrong! Moreover, they were wrong for a rather silly, elementary reason — a pitfall that I thought I should be capable of detecting with my current mathematical experience.
To clarify what I mean by “elementary”: no, it wasn’t a silly error in the sense of a missing minus sign or multiplying two numbers incorrectly. Mathematicians say that something in their field of study is “elementary” if it is a basic fact in that particular field. However, to paraphrase my supervisor, “basic does not mean easy”, so if I were to describe the mistake to a non-mathematical audience, it would still be extremely difficult. Another interesting point is the following. When you do research in pure mathematics, quite often you deal with abstractions upon abstractions, and it is easy to forget the humble origins of these abstract concepts. For example, the concept of a linear operator is fundamental to many areas of maths, and as cool as it is to be working with linear operators on infinite dimensional Banach spaces or whatever, it is also worth checking a statement about abstract linear operators on, let’s say, a 2-by-2 matrix as encountered in 1st year university maths (or even high school, depending on where you were educated). This comic sums up the sentiment quite well (excuse the pun).
Now this brings us to the main point of this blogpost. The wrong calculation I described above is not the mistake I want to write about; rather, I want to focus on what happened next. I assume that most people do not enjoy looking at their failures. I am not sure what I did with those few pages of calculations: either I threw them out, or I shelved them away somewhere I had forgotten. In any case, recently I wanted to return to the same problem with a different perspective, but now I have lost my point of reference. Because I had misplaced or thrown out the incorrect calculations, I cannot really remember what I had got wrong the first time! To be fair to myself, I have a vague idea of the kind of error it was, and it is very unlikely I will make exactly the same error again, but nevertheless it would have been more efficient to keep the original calculations, wrong as they are, so that I can remember exactly what the lapse of judgment was. As I have learned by now, mathematical work is not only about putting one true statement after another. It is equally important and even illuminating to know what kind of result cannot be true. In hindsight, throwing away those calculations was a bigger mistake than the calculations themselves.
There are certain similarities with the performing arts, although I have to admit that the comparison gets fuzzy on closer inspection. For a start, mistakes in mathematics are objective errors: you are objectively wrong if you claim that 1 + 1 = 3, for instance. On the other hand, speaking from a background in Western classical music, even though there are certain objective errors (e.g. playing a wrong note), usually we are unhappy with a performance because we feel that we could have done certain things better. One could say this is a “subjective” error. Despite this fundamental difference though, both mathematicians and performing artists undeniably improve by learning from their mistakes, objective or subjective. Indeed, I would say that learning from mistakes is an essential part of development in the two disciplines.
As a violin student, I used to dislike making recordings — what a way to highlight one’s faults and weaknesses! However, in hindsight I should have taken more opportunities to record my playing, and to embrace whatever mistakes (objective or subjective) occurred during a performance. One should not take this viewpoint too pessimistically. The idea is not to focus only on failures; rather it is about using those failures strategically to improve one’s understanding or problem solving technique in the case of mathematics, or one’s expression capabilities in the case of the performing arts.
I end this post with a nice little comic, which comes with some wise words from the great mathematician Paul Halmos: