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Any pure mathematician will from time to time discuss, or think about, the question of why we care about proofs, or to put the question in a more precise form, why we seem to be so much happier with statements that have proofs than we are with statements that lack proofs but for which the evidence is so overwhelming that it is not reasonable to doubt them.

That is not the question I am asking here, though it is definitely relevant. What I am looking for is good examples where the difference between being pretty well certain that a result is true and actually having a proof turned out to be very important, and why. I am looking for reasons that go beyond replacing 99% certainty with 100% certainty. The reason I'm asking the question is that it occurred to me that I don't have a good stock of examples myself.

The best outcome I can think of for this question, though whether it will actually happen is another matter, is that in a few months' time if somebody suggests that proofs aren't all that important one can refer them to this page for lots of convincing examples that show that they are.

**Added after 13 answers:** Interestingly, the focus so far has been almost entirely on the "You can't be sure if you don't have a proof" justification of proofs. But what if a physicist were to say, "OK I can't be 100% sure, and, yes, we sometimes get it wrong. But by and large our arguments get the right answer and that's good enough for me." To counter that, we would want to use one of the other reasons, such as the "Having a proof gives more insight into the problem" justification. It would be great to see some good examples of that. (There are one or two below, but it would be good to see more.)

**Further addition:** It occurs to me that my question as phrased is open to misinterpretation, so I would like to have another go at asking it. I think almost all people here would agree that proofs are important: they provide a level of certainty that we value, they often (but not always) tell us not just *that* a theorem is true but *why* it is true, they often lead us towards generalizations and related results that we would not have otherwise discovered, and so on and so forth. Now imagine a situation in which somebody says, "I can't understand why you pure mathematicians are so hung up on rigour. Surely if a statement is obviously true, that's good enough." One way of countering such an argument would be to give justifications such as the ones that I've just briefly sketched. But those are a bit abstract and will not be convincing if you can't back them up with some examples. So I'm looking for some good examples.

What I hadn't spotted was that an example of a statement that was widely believed to be true but turned out to be false is, indirectly, an example of the importance of proof, and so a legitimate answer to the question as I phrased it. But I was, and am, more interested in good examples of cases where a proof of a statement that was widely believed to be true *and was true* gave us much more than just a certificate of truth. There are a few below. The more the merrier.

"Sufficient unto the day is the rigor of the arguement."-old Historian of Mathematics saying,authorship unknown – Andrew L 0 secs ago – The Mathemagician – 2010-12-06T20:47:36.210

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I hope I'm not flogging a dead horse, but there is a great discussion on cstheory that was spawned by this thread and fits with @gowers 'further addition' section. In particular, it is a list of cstheory results where the rigorous demonstration of an 'obviously true' statement resulted in interesting insights.

– Artem Kaznatcheev – 2014-01-27T06:54:46.877@KConrad: I have been told the story of a mathematician who, while making a talk and being asked about a proof of some results he had claimed, exclaimed: “Why do you need a proof? It's a theorem!” – ACL – 2015-10-24T11:21:25.270

+1 I'm very glad someone asked this. As someone who works closely with physicists and engineers, I struggle to justify my desire for precision and proof sometimes. – icurays1 – 2017-12-13T19:02:03.877

The tag "gn.general" is a mistake, coming from the addition of a space in "general topology." Right now, yours is the only post using it. Perhaps retag? – JBL – 2010-09-03T13:18:36.670

12There's a clear advantage to knowing a 'good' proof of a statement (or even better, several good proofs), as it is an intuitively comprehensible explanation of why the statement is true, and the resulting insight probably improves our hunches about related problems (or even about which problems are closely related, even if they appear superficially unrelated). But if we are handed an 'ugly' proof whose validity we can verify (with the aid of a computer, say), but where we can't discern any overall strategy, what do we gain? – Colin Reid – 2010-09-03T13:53:39.983

I couldn't find an appropriate tag but will definitely retag if anyone has a good suggestion. (Or am happy to see it retagged if anyone with power to do so wants to do so.) – gowers – 2010-09-03T14:43:29.653

<< statements that lack proofs but for which the evidence is so overwhelming that it is not reasonable to doubt them.>> Examples? – Sergei Tropanets – 2010-09-03T15:26:20.913

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How about Merten's Conjecture? It was verified in a very large number of cases before it was disproved. http://mathworld.wolfram.com/MertensConjecture.html

– Skip – 2010-09-03T15:27:10.6074I think a fairly good example is that the end-to-end distance of a typical self-avoiding walk of length n is n^{3/4}. Another is Goldbach's conjecture. (In both cases, there are powerful heuristic justifications that are backed up by computational evidence.) – gowers – 2010-09-03T15:30:12.853

Sorry -- I should make clear that that n^{3/4} was an approximation. – gowers – 2010-09-03T15:31:00.137

11What kind of person do you have in mind who would suggest proofs are not important? I can't imagine it would be a mathematician, so exactly what kind of mathematical background do you want these replies to assume? – KConrad – 2010-09-03T15:33:20.647

2Proofs (hopefully) provide understanding. Nobody doubts RH is true but we don't know why. – Felipe Voloch – 2010-09-03T16:07:30.923

9Colin Reid- I think one can differentiate between a person understanding and a technique understanding. The latter applies even if we cannot understand the proof. We know that the tools themselves "see enough" and "understand enough", and that in itself is a significant advance in our understanding. But we still want a "better proof", because a hard proof makes us feel that our techniques aren't really getting to the heart of the problem- we want techniques which understand the problem more clearly. – Daniel Moskovich – 2010-09-03T16:26:59.557

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Here is some interesting and provocative reading with some opposition to rigor, by a mathematician: http://www.math.rutgers.edu/~zeilberg/Opinion111.html

– Jonas Meyer – 2010-09-03T20:54:12.060@Voloch, there are still people who doubt the proof of the Poincare conjecture. So it seems doubtful that it could be true that "nobody doubts RH". Without even a proposed proof, why shouldn't people doubt it? – Ryan Budney – 2010-09-03T22:47:43.493

2This is a bit off topic, but related to gowers last sentence. I don't really like the idea of having to justify that proof is important. I think asking the question

`Why?' is natural and important in its own right. Curiosity appears to be built in to us. 'Why?' drives a significant portion of the sciences and humanities and everyday life. It would be hard to deny that our active desire to satisfy this inbuilt curiosity is at least partially responsible for human advance (whatever that is). For me, trying to find specific examples where having a proof`

saves lives' cheapens the whole process. – Robby McKilliam – 2010-09-04T00:26:19.217Still, I did like Daniel's plane story :) – Robby McKilliam – 2010-09-04T00:29:55.077

6I just want to make it clear that I do NOT dispute the importance of proofs, or intend this question to be a discussion of that. Rather, I want to take it for granted that proofs are important, but have a good supply of examples that ILLUSTRATE their importance. Also, I'm much more interested in examples that show that finding proofs yielded far greater insights into the theorems than in the question of whether we can ever be fully certain of a statement that is not yet proved (not that I deny the importance of the latter). – gowers – 2010-09-04T14:43:26.423

1@KConrad -- Zeilberger was one example, mentioned by Jonas Meyer above. I can imagine that there might be physicists who would not understand why we are so interested in making their work rigorous, but I can't give actual examples there. – gowers – 2010-09-04T14:45:21.363

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Gyorgy Sereny http://mathoverflow.net/questions/37610/demonstrating-that-rigour-is-important/37747#37747 raised the point that for mathematics (or at least large parts of mathematics) we cannot even separate the "theorems" and the "proofs" and, in fact, it is the proofs that gives the theorems their importance.

– Gil Kalai – 2010-09-04T21:40:44.7533In any case, while the role of rigor is an interesting question and also what substitute do we have to rigorous proofs, i think that it is usually much easier to explain to an outsider the importance of rigorous proofs than to explain to an outsider the importance of the statements we are proving. – Gil Kalai – 2010-09-04T21:48:18.827

15Concerning the Zeilberger link that Jonas posted, sorry but I think that essay is absurd. If Z. thinks that the fact that only a small number of mathematicians can understand something makes it uninteresting then he should reflect on the fact that most of the planet won't understand a lot of Z's own work since most people don't remember any math beyond high school. Therefore is Z's work dull and pointless? He has written other essays that take extreme viewpoints (like R should be replaced with Z/p for some unknown large prime p). – KConrad – 2010-09-05T01:39:40.283

By the way, here's a related post http://mathoverflow.net/questions/29104/why-are-proofs-so-valuable-although-we-do-not-know-that-our-axiom-system-is-cons

– Kevin H. Lin – 2010-09-06T09:05:05.52021Every proof has it's own "believability index". A number of years ago I was giving a lecture about a certain algorithm related to Galois Theory. I mentioned that there were two proofs that the algorithm was polynomial time. The first depended on the classification of finite simple groups, and the second on the Riemann Hypothesis for a certain class of L-functions. Peter Sarnak remarked that he'd rather believe the second. – Victor Miller – 2010-09-06T15:56:15.913

5This question is an excellent example how MO can be effective and useful for a question which, while having strong academic merit, also has strong discussion-flavor, and lead to a discussion that can be rather subjective and even argumentative – Gil Kalai – 2010-09-07T08:01:27.203

Tim: I changed essentially just one letter of the tag to be "gm.general-mathematics", to bring it in line with the arXiv classification scheme. – Willie Wong – 2010-09-08T15:18:31.657

Willie: thanks for that -- it looks like a much more appropriate tag. – gowers – 2010-09-08T16:16:19.060