I finally got a copy of the Princeton Companion to Mathematics in my hands today, and within a depressingly (but not unexpectedly) short time found my first mistake, in an article written by me. I’ve established that small corrections can almost certainly be made in time for the next printing, so this post is to invite anybody who happens to spot an error to let me know in a comment on this post. (That way, all the corrections that are needed will be in one convenient place.) I’m particularly interested in mathematical mistakes, though typos are also good to know about. Just to get the ball rolling, here’s the one I spotted, together with a reflection on how it arose, since it’s relevant to mathematical writing in general.
Incidentally, before I say any more, I want to say that a huge amount of proofreading has gone into the book, so I expect the density of mistakes to be pretty small. But because the book is a big one, I also expect the number of mistakes to be not all that small.
The mistake occurs on page 24 of the book, where I give a formula for the product of two elements of the dihedral group of order 8. I am discussing semidirect products of groups, so I use the notation to stand for the element that would normally be written where is a quarter turn and is a reflection. I point out that and then claim that it follows from that that . Of course, the power of on the right-hand side should in fact be .
How could I write something that jumped out at me as obviously false when I came back to it just now? Probably because I was focusing on the interesting case when . But it was still pretty careless. And the general point I want to make here is that such carelessness is incredibly tempting. Several years ago, David Preiss and I were colleagues at University College London, and he said something that I’ve never forgotten, which was that if you are suspicious of a mathematical argument and want to find the mistake, look for anywhere where it says something like, “An easy argument shows that …” or “It is not hard to prove that …” I dare say he was not the first person to make this observation, but it doesn’t seem to be quite as standard as it might be.
For the purposes of an expository book like the Princeton Companion, it was quite common, and the right thing to do, to miss out details of arguments, and to say things like, “If you do this calculation, you will find that …” I was inclined to trust authors (including myself) when they wrote things like this, but at a late stage I found one or two that were embarrassingly false, and began to think that it would be a good idea to check them systematically. Vicky Neale, a graduate student at Cambridge, very kindly agreed to do a lot of this task (though unfortunately she came in at such a late stage that this final check did not cover the whole book), and she discovered that an extraordinarily high percentage of these not fully justified assertions were wrong. In most cases, most readers wouldn’t notice, because all that really mattered was that the calculation had an answer: it wasn’t so important what the answer was. Unfortunately, the one I’ve just discovered will be genuinely confusing to the reader for whom that particular article was intended. And I’d very much like to eliminate as many as possible of the ones that remain.
The general moral I would draw from this experience is that in expository writing one should be especially scrupulous about checking calculations, because there tends to be a higher density of assertions that are not fully justified (because they are plausible and their justifications would get in the way of the exposition). To put it in a punchier way: if you haven’t checked it, then it is wrong. That’s particularly true if you worked it out in your head, as I think I did in this instance.
As for research articles, I think my advice here is pretty hard line: proofs should be written out in full, or, failing that, precise algorithms should be specified for generating such proofs. To explain what I mean with an example: instead of saying, “An argument similar to the proof of Lemma 2.3 shows that …” one should say, “The proof of this fact is similar to the proof of Lemma 2.3; the differences are that in this case we replace by , and …” etc. Of course, another important algorithm for generating part of a proof is something like “Insert here the main result of .” I just mean that “It is not hard to show that” should appear only if giving the proof would be an insult to the intelligence not just of experts but even of mathematicians who are not familiar with your area. Doing this not only forces you to deal with the weak parts of your arguments, but it also makes what you write easier to read. After all, if it’s true that “An easy argument shows that …” then you should be able to give the argument reasonably elegantly, and the reader is free to ignore it. If it’s not true, then the reader may well waste time trying to see the obviousness of what you claim.