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In a nice and witty lecture titled "how to write mathematics badly" (available on YouTube at https://www.youtube.com/watch?v=ECQyFzzBHlo&t=23s), Jean-Pierre Serre describes various ways in which a paper can be poorly/confusingly/inaccurately written.

Around min 34:00 in the previous link, he criticizes the use of the word "constant", in particular in inequalities. The example he provides is of the type:

$$\|Af\|\le C\|f\|$$ for some constant $C$

where $A$ is a complicated operator depending on many parameters. In this case, he says, usually the only thing that the writer means is that $C$ does not depend on "some of the data" of the problem. He adds that this attitude "caused lots of mistakes".

What are examples of these mistakes? Has any significant piece of mathematics been rewritten or erased altogether because of some problem with proofs invoking "constants" too nonchalantly?

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    $\begingroup$ I can't cite any mistakes caused by this practice, but I have no doubt there have been many. When I first ran into this when studying PDEs, it made me very uncomfortable, so I always wrote explicitly what parameters $C$ did depend on. So my $C$ always looked something like $C(n,a,b,P, Q, \beta)$. If I could write an explicit formula for $C$, I often did, since the dependence of the parameter sometimes mattered, too. $\endgroup$ – Deane Yang Dec 21 '20 at 17:37
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    $\begingroup$ It is the kind of mistake students make every day, and I have certainly seen it in submitted manuscripts, which were insufficiently memorable to recall specifics. I do not see the point of creating a list here. $\endgroup$ – Michael Renardy Dec 21 '20 at 18:18
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    $\begingroup$ I, a nephew of Jean-Pierre Serre, am certainly guilty of invoquing such constants in my research activity. But the inequality I am proud of (Compensated Integrability) has an explicit and accurate constant. $\endgroup$ – Denis Serre Dec 21 '20 at 19:26
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    $\begingroup$ In the rest of mathematics, this problem is solved by introducing quantifiers at the right place in the statement (usually written out in words). I've never fully understood why this is not [always] used when doing estimates... $\endgroup$ – R. van Dobben de Bruyn Dec 21 '20 at 19:28
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    $\begingroup$ @R.vanDobbendeBruyn In any field of math, if quantifiers were introduced separately for each statement, the exposition would become very cluttered (esp. if there are a lot of nonce variables that occur in just one line). Outside of "hard" analysis, though, there are often convenient definitions that hide a lot of the quantifiers (e.g., continuity hides epsilon and delta quantifiers, a universal property in a category hides universal and existential quantifiers, etc.). But in the "fuzzy" world of estimates we often don't have this luxury (unless one uses something like nonstandard analysis). $\endgroup$ – Terry Tao Dec 22 '20 at 1:31
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Edit: The original answer below refers to Nelson's attempt from 2011. Upon a cursory look at the afterword by Sam Buss and Terence Tao to Nelson's paper placed in arxiv in 2015 (after his death), it seems he later attempted to address the error referred to in the original answer below; it would be interesting to know what the experts think on how successful his efforts were or potentially can be.

Original Answer: Edward Nelson's recent project on finding inconsistency of arithmetic (which was the subject of a MathOverflow Question) might be pertinent. The error, discovered by Terence Tao, seems to be the dependence of a constant on the underlying theory that Nelson did not account for.

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    $\begingroup$ This is fascinating. The afterword by Buss and Tao indicates that the final paper (while not establishing what it set out to do) contains many interesting ideas worthy of further exploration. It was also incomplete at the time of Nelson's death, containing only 6 of a planned 10 parts. Does anyone know if there has been any work done since 2015 in separating the wheat from the chaff in Nelson's paper? $\endgroup$ – John Coleman Dec 24 '20 at 14:15
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It came to my mind what's perhaps the oldest example of this kind of mistake, so I add an answer to my own question: in 1821 Cauchy 'proved' that convergent sums of continuous functions are continuous, and later on Abel found counterexamples (see [1] for historical details). Of course Cauchy implicitly assumed uniform convergence, which means that he treated his $\delta$ as "more constant" than it was...

[1]: Sørensen, H. K. (2005). Exceptions and counterexamples: Understanding Abel's comment on Cauchy's Theorem. Historia Mathematica, 32(4), 453-480.

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    $\begingroup$ The nLab article on the Cauchy sum theorem gives a thorough discussion of this theorem, including various suggestions for how to argue that Cauchy's argument was correct after all. But someone is confused here— if not Cauchy, then his critics. $\endgroup$ – Timothy Chow Dec 22 '20 at 17:24
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A good example (of a somewhat different kind though) was given by Adian in the introduction to his book "The Burnside problem and identities in groups" (1975, English edition 1979), where he refutes the proof of the main result from his rival's book by stating that

``the conditions $$ \begin{aligned} &u_4 = u_1 +r_{25} \; \text{(p.145, line 10 from below)} \\ &r_{25} \ge u_{37} +54/e \; \text{(p.283, line 4 from below)} \\ &u_{37} >14\alpha +214/e, \; \text{where}\; \alpha=\varepsilon_{30}+u_{13} +6u_4 \; \text{(p.221, lines 11 and 12 from below)} \end{aligned} $$ give an obvious contradiction $u_4>r_{25}>u_{37}>u_4$.''

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  • $\begingroup$ Well if we want to be pedantic, $u_1<0$ would solve everything here, and so far I don't see that excluded. $\endgroup$ – Torsten Schoeneberg Dec 22 '20 at 18:37
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    $\begingroup$ Isn't the implied inequality $u_4 > r_{25} > u_{37} > 84\cdot u_4$? Am I being dense? (Assuming all constants are positive) $\endgroup$ – o r Dec 22 '20 at 19:35
  • $\begingroup$ @or, re, of course $84u_4 > u_4$ if everything is positive. $\endgroup$ – LSpice Dec 22 '20 at 19:48
  • $\begingroup$ This is an exact quote $\endgroup$ – R W Dec 22 '20 at 21:54
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    $\begingroup$ @LSpice ok thanks I was being dense, I don't even know what I mixed up here ... $\endgroup$ – o r Dec 22 '20 at 23:21
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There are periodically false proofs of the rapid decay (RD) Property (see Chatterji and Saloff-Coste - Introduction to the property of rapid decay) for cocompact lattices in semisimple Lie groups (Valette's conjecture). This is a functional-analytic property of these groups. As far as I understand, wrong proofs typically make use of some quantitative decreasing of coefficients, and this involves a "constant". The issue (sorry if I'm a bit imprecise) being that this "constant" actually depends on the dimension of something related to the induced representation restricted to a maximal compact subgroup.

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