“An Editor Recalls Some Hopeless Papers”, Wilfrid Hodges1998-03 (math, epistemology, logic)⁠:

I dedicate this essay to the two-dozen-odd people whose refutations of Cantor’s diagonal argument (I mean the one proving that the set of real numbers and the set of natural numbers have different cardinalities) have come to me either as referee or as editor in the last 20 years or so. Sadly these submissions were all quite unpublishable; I sent them back with what I hope were helpful comments. A few years ago it occurred to me to wonder why so many people devote so much energy to refuting this harmless little argument—what had it done to make them angry with it? So I started to keep notes of these papers, in the hope that some pattern would emerge.

These pages report the results. They might be useful for editors faced with similar problem papers, or even for the authors of the papers themselves. But the main message to reach me is that there are several points of basic elementary logic that we usually teach and explain very badly, or not at all.

…§3. Why this target? Cantor’s argument is short and lucid. It has been around now for over a hundred years. Probably every professional mathematician alive today has studied it and found no fallacy in it. So there is every temptation to imagine that anybody who writes a paper attacking it must be of dangerously unsound mind. One should resist this temptation; the facts don’t support it. On a few occasions I was able to speak to the authors of these papers; one or two were clearly at sea, but others were as sane as you or me. In the course of researching this paper I came across statements by two of the leading logicians of this century, which-read literally-were just as crazy as anything in these attacks on Cantor’s argument. Read on and judge.

There is a point of culture here. Several of the authors said that they had trained as philosophers, and I suspect that in fact most of them had. In English-speaking philosophy (and much European philosophy too) you are taught not to take anything on trust, particularly if it seems obvious and undeniable. You are also taught to criticize anything said by earlier philosophers. Mathematics is not like that; one has to accept some facts as given and not up for argument. Nobody should be surprised when philosophers who move into another area take their habits with them. (In the days when I taught philosophy, I remember one student who was told he had failed his course badly. He duly produced a reasoned argument to prove that he hadn’t.)

…It’s nothing more than a guess, but I do guess that the problem with Cantor’s argument is as follows. This argument is often the first mathematical argument that people meet in which the conclusion bears no relation to anything in their practical experience or their visual imagination…nothing like its conclusion was in anybody’s mind’s eye. And even now we accept it because it is proved, not for any other reason.

…§8. Conclusion. First, contrary to what several critics of Cantor’s argument suggested in their papers, at least one mathematician was prepared to look at their refutations with some care and sympathy.

Second, a small number of the criticisms are fair comment on misleading expositions. A much larger number of the criticisms are fair comment on some serious and fundamental gaps in the logic that we teach. Even at a very elementary level—I’m tempted to say especially at a very elementary level—there are still many points of controversy and many things that we regularly get wrong.

Third, there is nothing wrong with Cantor’s argument.