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Examples of common false beliefs in mathematics.

Hopefully this is not overly controversial, but I thought it would be instructive to compile a list of errors which are commonly (or at least not too uncommonly) made in higher level mathematical research and published mathematical works (i.e. research papers, books, etc). If nothing else this could serve as a warning to the rest of us.

Here I would like to keep the focus on mathematical errors, oversights, or misunderstandings, and not those which are related to typographical, grammatical, or purely historical issues. To start the list and to give some idea what types of errors I am thinking of, here are a few examples:

    1. Confusing the finite field $\mathbb{F}_{p^k}$ for $k>1$ with the ring $\mathbb{Z}/p^k\mathbb{Z}$.
    2. Assuming that every open set in $\mathbb{Q}_p$ is also closed (true for balls but not in general).
    3. Assuming that the hypothesis in a conditional statement is a necessary condition, just because a weaker hypothesis does not necessarily imply the conclusion.

One remark, since this is intended solely for our own edification, it is probably better to avoid mentioning specific references, for the obvious reason.

Also, if anyone can think of better tags for this question please go ahead.

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marked as duplicate by Martin Brandenburg, Gerald Edgar, Dan Petersen, Terry Tao, Suvrit Jan 15 '13 at 20:55

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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I think this question overlaps a lot with mathoverflow.net/questions/23478 –  Dan Petersen Jan 15 '13 at 15:48
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I agree with Dan. Especially if one is being asked not to name published references of such mistakes, the list will look very similar. Except maybe for one thing: the other question seemed to encourage answers that had an element of humility: here is something I once believed, but now recognize as being a dumb mistake. Whereas here it could become a list of other people's dumb mistakes. :-) –  Todd Trimble Jan 15 '13 at 16:32
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This also overlaps with mathoverflow.net/questions/35468 "Widely accepted mathematical results that were later shown wrong". –  Robert Bryant Jan 15 '13 at 19:19
    
Robert: I think this question is distinct from the one that you link to, but I fear that there will be a latge overlap/duplication of answers –  Yemon Choi Jan 15 '13 at 19:28
    
Sorry about that, I wasn't aware of the other question. –  Alan Haynes Feb 6 '13 at 9:32

4 Answers 4

Perhaps the canonical example is when Suslin, as a student of Lusin, was asked to read a paper of Lebesgue where the latter asserted that the image of a Borel set under a continuous mapping is again Borel. This marked the beginning of analytic sets (and eventually the projective hierarchy).

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Lamé's wrong proof of Fermat's Last Theorem, which led to the fact that cyclotomic fields do not have unique factorization.

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One very famous error was made by H. Dulac in his 1923 paper which was supposed to prove that a polynomial system of differential equations in the plane has finitely many isolated limit cycles. (This was one of the Hilbert problems). It was believed until 1980-s that he proved the result.

The error was the following. He proved that certain function, say analytic on $(0,1]$ has an asymptotic expansion of the form $$f(x)\sim \sum_{n=0}^\infty a_n x^n,$$ in the usual sense that asymptotic expansions are understood.

He concluded that $f$ has finitely many zeros on $(0,1]$.

The mistake was found in the early 80-s and corrected. The new proofs of the finiteness theorem for limit cycles, which are accepted as correct are very long and difficult. But there are at least two different published proofs, one that I know is by Ecalle, and another by Ilyashenko.

A related problem is to estimate the number of isolated limit cycles in terms of the degree of the equation. This is still unsolved. And there were published wrong solutions.

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There have been many false proofs of the Jacobian Conjecture, and it might be interesting to have a survey of the attempts and why they ultimately failed (i.e., what holes there were in the attempted proof or what counterexamples to steps in the attempted proof were found).

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I think Arno van den Essen has a book about the Jacobian conjecture which does this in part; I heard him give a talk some years ago about this. –  Mariano Suárez-Alvarez Jan 15 '13 at 19:37

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