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We've all done it: we have a "proof" that is so pretty... but we know it's wrong. Either it proves something we know is false, or the proof doesn't use one of the hypotheses you know it needs to, or we assume something we can't assume... and the whole pretty argument goes away. I had a professor who said that he sometimes convinces himself that some proof works right before he goes to bed, only to wake up and find out that it's total nonsense. He says it's probably some coping mechanism his body has so he can get some sleep!

Though the proofs often end up unsalvageable (is that a word?), I think they are an important part of learning mathematics and being creative. I think it was Sir Ken Robinson (not a mathematician, but a good thinker) who said something like "Being wrong isn't always good, but if we don't have the capacity to be wrong we can never be creative." (If anyone knows the precise quote, do fill it in!)

So let's hear them- your pretty proofs that turn out to be nonsense. Like the well-written "How not to prove the Poincare Conjecture," these can be well thought out proofs that have a small hidden assumption that makes everything blow-up, or they can be totally silly like the time I thought I proved Brouwer's fixed point theorem using the Baire Category Theorem during my freshman year...

Bonus points if anyone's "false" methods (or a close adaptation) ended up working for a different problem later on!

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closed as not a real question by Mariano Suárez-Alvarez, Andrea Ferretti, Pete L. Clark, José Figueroa-O'Farrill, Harry Gindi Jul 17 '10 at 13:25

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

-1 because I'm ornery and think that MO has more than enough of this type of question. –  Theo Johnson-Freyd Jul 17 '10 at 2:23
Agreed. It's not clear to me what this question is covered which isn't covered by either "interesting mathematical mistakes" or "common false beliefs," except insofar that the answers are going to be less universal. –  Qiaochu Yuan Jul 17 '10 at 3:43
+1 because, this question is about something that we thought to be pretty, but it turned out to be wrong. This is an interesting phenomena, that (I guess) every mathematician experiences in his life, (I guess) not only when he is a student. Maybe this question is quite personal, so it is hard to give really genuine answers, but the question is not empty. It is different from the question of Gowers (common faults believes) -- Gowers's question is about common things, and this question is about individual. "most interesting math mistakes" is also different. –  Dmitri Jul 17 '10 at 9:22
As someone who isn't keen on this question but has not yet voted to close, I've started a thread on meta tea.mathoverflow.net/discussion/515/… inviting those who see merit in the question to come and leave their reasons. So if you're registered on meta, Dmitri, your comments would be welcome there. –  Yemon Choi Jul 17 '10 at 9:26

4 Answers 4

Here are four tempting fallacies that I have seen, that are in my opinion are all teachable and interesting:

  1. A finite covering space of a disk with finitely many holes, is again a disk with finitely many holes; in particular it is still planar.

  2. If you lengthen all three edges of a triangle, its area increases.

  3. If $F$ is a field with two finite-index subfields $K$ and $L$, then $K \cap L$ also has finite index in $F$.

  4. There are exactly two Lie groups up to isomorphism that are diffeomorphic to a pair of circles.

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What would be a counterexamples for 3? In 4, what is the second Lie group that is diffeomorphic to pair of circles? (the first one is the direct sum of S^1 and Z/2, I assume.) –  Anweshi Jul 17 '10 at 13:17
3. $F = \mathbb{C}(x)$, $K = \mathbb{C}(x^2)$, $L = \mathbb{C}((x+1)^2)$. Actually there are tons of examples in function field theory. 4. $S^1 \times C_2$, $O(2)$, and (the one that is the easiest to miss) $\text{Pin}^-(2)$, the normalizer of a circle in $\text{SU}(2)$. –  Greg Kuperberg Jul 17 '10 at 13:28
But where are the pretty proofs that go away? –  Victor Protsak Jul 17 '10 at 19:25
@Victor Yes, I suppose I did not answer that part of the question. 2 is from a failed proof of the Kepler conjecture, not exactly a pretty proof but at least a proof. 3 shows up from time to time in function field problems. –  Greg Kuperberg Oct 20 '10 at 7:44

For some problem in Algebraic Topology (presumably related to homotopy groups or similar, with free groups): I thought that two groups $G$, $H$ were isomorphic, because $G \approx A \subseteq H \approx B \subseteq G$, where $\approx$ means "is isomorphic to" and $\subseteq$ means "is a subgroup of".

However, I was very shocked when informed that this does NOT imply that $G \approx H$ in general! (I thought this was true, and spent a long time trying to prove it; but I knew that I hadn't succeeded. So I suppose this doesn't qualify, but anyway).

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This precisely repeats an answer in the "common false beliefs" thread: mathoverflow.net/questions/23478/… . Can someone explain what about this question is not covered in either that thread or the "mathematical mistakes" thread? –  Qiaochu Yuan Jul 17 '10 at 3:42
To be fair, in the other thread, it was another user making this point. Furthermore, Zen wasn't even registered on MO when that other thread started! While I understand the desire not to repeat ourselves, there has to be, I think, some flexibility to allow people to join the community here... –  Matthew Daws Jul 17 '10 at 8:39
Sorry if I wasn't clear. My point isn't that Zen should've been aware that he was repeating an answer. My point is that the fact that a repeated answer was possible at all casts doubts on the question. –  Qiaochu Yuan Jul 17 '10 at 18:30

I did think that Burnside's pq theorem was very simple to prove. You just use that if a finite group has a Hall-subgroup of each possible order then it is solvable. When I finally bothered to look at the actual proof of this fact it of course turned out to use Burnside as basis for induction.

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I was once given a question (in a take home exam) along the following lines: A sequence of complex-valued bounded measurable functions $f_n (x)$ on a probability space $X$ was given, with the functions $f_n$ satisfying some conditions which I won't specify here. The problem was to show that the $L^2 (X)$ limit of the sequence is a certain given function $f(x)$. After some thought, I came up with the following "solution": I defined a sequence of functions $g_n (x,y) \in L^2 (X \times X)$ and a function $g(x,y) \in L^2 (X \times X)$ such that $g_n (x,x) = f_n (x)$ and $g(x,x)=f(x)$ for all $x \in X$, and I proved that $g_n (x,y) \to g(x,y) \;$ in $L^2 (X \times X)$. I was sure that I had, in fact, proved a generalization of the proposition I was given.

Embarrassingly, it took me some time to realize my mistake. I tried to salvage my proof using continuity arguments and what not, but in the end I gave up and managed to concoct a different approach, which unfortunately was a lot more complicated and messy.

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