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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.

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    $\begingroup$ There'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? $\endgroup$
    – Colin Reid
    Sep 3, 2010 at 13:53
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    $\begingroup$ What 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? $\endgroup$
    – KConrad
    Sep 3, 2010 at 15:33
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    $\begingroup$ Colin 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. $\endgroup$ Sep 3, 2010 at 16:26
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    $\begingroup$ Concerning 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). $\endgroup$
    – KConrad
    Sep 5, 2010 at 1:39
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    $\begingroup$ Every 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. $\endgroup$ Sep 6, 2010 at 15:56

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In response to the request for an example of a statement that was widely but erroneously believed to be true: does Gauss's conjecture that $\pi(n) < \operatorname{li}(n)$ for every integer $n \geq 2$, disproved by Littlewood in 1914, qualify?

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    $\begingroup$ It's a good example of the need for rigor. But I've always been skeptical of the story that it was widely believed to be true, since any competent mathematician familiar with Riemann's 1857 explicit formula for π(n) would have realized that there are almost certainly going to be occasional exceptions. (Littlewood removed the word "almost" by a more careful analysis.) $\endgroup$ Sep 8, 2010 at 19:58
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Richard Lipton recently blogged about this question in the context of why a potential proof of $P \neq NP$ would be important. I am probably bastardizing his words, but one of the reasons he gives is that a proof may give new insight and methods of attack to other problems. He cites Wiles' proof of Fermat's Last Theorem as an example of this phenomenom.

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In my experience, the two greatest difficulties in mathematics are:

  1. The obvious is not always true.

  2. The truth is not always obvious.

Rigour is the essence of mathematics. A rigorous proof provides an explanation of why a particular mathematical statement is true, and, at the same time, takes care of all the "Yes, but what if"s.

Rigour and proof provide the guarantee of correctness and reliability.

Rigour and proof refine our mathematical insights and instincts so that the superficially "obvious" misleads us less frequently.

When I pose the problem "1, 2, 3, x Find x." the initial response is usually a derisory laugh, of disbelief that I am serious, because "the answer is obviously 4". It is easy to demonstrate using practical examples that this statement is, as it stands, nonsense. A rigorous analysis is required.

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Michael Atiyah's discussion of the "proof" and it role seems to relevant to be posted here.

Taken from: "Advice to a Young Mathematician in the Princeton Companion to Mathematics." http://press.princeton.edu/chapters/gowers/gowers_VIII_6.pdf This link was provided by "mathphysicist" in answer on another MO question: A single paper everyone should read?

Quotation from M. Atiyah:

"In fact, I believe the search for an explanation, for understanding, is what we should really be aiming for. Proof is simply part of that process, and sometimes its consequence."

"... it is a mistake to identify research in mathematics with the process of producing proofs. In fact, one could say that all the really creative aspects of mathematical research precede the proof stage. To take the metaphor of the “stage” further, you have to start with the idea, develop the plot, write the dialogue, and provide the theatrical instructions. The actual production can be viewed as the “proof”: the implementation of an idea.

In mathematics, ideas and concepts come first, then come questions and problems. At this stage the search for solutions begins, one looks for a method or strategy. Once you have convinced yourself that the problem has been well-posed, and that you have the right tools for the job, you then begin to think hard about the technicalities of the proof."

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    $\begingroup$ I have described my search in 1965-74 for higher order Seifert-van Kampen theorems as: "An idea for a proof in search of a theorem." It took 9 years to find the gadgets to get a theorem. I think it is quite good to look at the work we do as a search for theorems. There is an apochryphal dedication of a PhD Thesis: "I am deeply grateful to Professor X, whose wrong conjectures and fallacious proofs, led me to the theorems he had overlooked." Sounds like excellent supervision!! It also leads to the question: What is a theorem? $\endgroup$ May 3, 2012 at 17:35
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I'm not sure it answers the question, but for me, providing (and understanding) a rigorous proof is a way to be sure you understood the deep thing behind what may seems "obvious", for good or wrong reasons: An example I like to give to non-mathematicians, is the Cantor's proof that the cardinality of $\mathbb [0,1]$ is strictly bigger than the one of $\mathbb N$.

The next metaphor is, I can imagine, well known, but that serves the purpose of the question: You start by asking what "two sets have the same cardinality" means. For that, I like to tell the story about the shepherd which can count until, say, 100, but has 1000 sheeps. Every year they go away to pasturelands, and come back. The clever shepherd uses small rocks and associate in his head "one rock = one sheep" to see if he lost, or won, sheeps the year after. Once this notion of bijection is somehow acquired, you ask about the bijection between $\mathbb N$ and $2\mathbb N$, and uses a piece of paper to illustrate. Finally, you go absurd and prove you can't have "one rock = one sheep" for $\mathbb [0,1]$ and $\mathbb N$. Surprisingly, even the more mathematophobics always feel their intuition is wrong and they seem to enjoy this feeling of admitting to themselves it was something deep; infinity is a complicated notion, and they were able to touch that. Maybe a mathematical proof is something which helps to transform a philosophical discussion into a statement nobody accepting elementary axioms can't argue against.

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Gowers is particularly interested in cases in which 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." Reading through the answers, I am struck by a recurring theme. There are many cases where

  • something seemed impossible;
  • it was impossible;
  • a lot of people didn't see the point of proving impossibility;
  • the search for an impossibility proof led to the discovery of structures that arguably would not have been discovered otherwise.

Just to list a few examples explicitly:

  • It seemed impossible to prove the parallel postulate from the other axioms. The proof of impossibility led directly to non-Euclidean geometries.
  • It seemed impossible to "square the circle." The rigorous proof (Lindemann–Weierstrass) forms the foundation of modern transcendental number theory.
  • It seemed impossible to solve all polynomial equations using radicals. Rigorous investigation of this impossibility led to what we now call Galois theory.
  • (Daniel Moscovich's answer) It seemed impossible to unknot a trefoil. The search for a rigorous proof led to the discovery of all kinds of knot invariants. More generally, topology and geometry are rife with examples of "obviously inequivalent" structures, and the search for rigorous proofs has uncovered all kinds of important invariants.
  • It seemed impossible to write down a procedure that would mechanically determine the truth or falsity of an arbitrary mathematical question. The rigorous proofs (incompleteness/undecidability) revealed fundamental limits to human knowledge.

In some cases, you could perhaps quibble with my claim that most if not all people thought these things were impossible, or that people didn't see the point of trying to prove impossibility. In this regard, let me mention the article Why was Wantzel overlooked for a century? The changing importance of an impossibility result by Jesper Lützen. Lützen persuasively argues that Wantzel's proofs of the impossibility of trisecting the angle and duplicating the cube were all but ignored because people just weren't that interested in an impossibility proof of something that everybody already believed, or suspected, was impossible. So even mathematicians are not immune to the tendency to undervalue negative results.

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    $\begingroup$ I do have some quibbles with the last one. First, didn't Hilbert rather famously claim that it was possible to do this? Second that "revealed fundamental limits to human knowledge" is sort of underselling what Gödel did, since if you already believed those limits exist, it's just a version of confirming what people already believed. So an answer along the line of the others is it led to an understanding of hierarchies of stronger and weaker theories, tools for constructing new models, understanding of the relationship of computation and logic, etc. $\endgroup$
    – Will Sawin
    Dec 3, 2021 at 4:24
  • $\begingroup$ @WillSawin If you're referring to "Wir müssen wissen; wir werden wissen" then note that that was in 1930, after he had formally posed the Entscheidungsproblem. So I regard that famous statement as aspirational (and phrased colorfully in order to contrast with Ignoramus et ignorabimus), not a claim that the Entscheidungsproblem was definitely solvable. In any case, I think that Hilbert was in the minority. Countering the majority attitude, he was trying to convince people that if you can't find a solution then it is equally important to try to prove that no solution exists. $\endgroup$ Dec 3, 2021 at 4:52
  • $\begingroup$ The prevailing view was certainly not that mathematics could be reduced to a mechanical procedure. I expect that if any mathematicians even paused to ask the question, they mostly would have dismissed it quickly; the Entscheidungsproblem was a bold concept at the time. As for how to describe the achievements of Gödel, Church, Turing, etc., I have no argument with what you said. $\endgroup$ Dec 3, 2021 at 4:57
  • $\begingroup$ @TimothyChow Do you have a reference for the statement about the prevailing view? My impression was that it went well beyond Hilbert specifically and that there was quite a bit of belief in the notion that Science (and Mathematics) Could Solve Everything. $\endgroup$ Dec 3, 2021 at 5:08
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    $\begingroup$ @StevenStadnicki There is some information at the end of Lützen's article. Most of the evidence is indirect: (1) we can infer from the way Hilbert spoke about the Entscheidungsproblem and related topics that he was rowing against the prevailing current, and (2) we don't have records of other people (other than maybe Ackermann) stating a firm belief in a positive resolution. Maybe there's some confusion here: there may have been a fairly widespread belief in scientism (an excessive belief in the power of science). That's different from a belief in a decision procedure for mathematics. $\endgroup$ Dec 3, 2021 at 13:52
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Allow me to quote part of the introduction of chapter 9 of Lovász: Combinatorial Problems and Exercises.

The chromatic number is the most famous graphical invariant; its fame being mainly due to the Four Color Conjecture, which asserts that all planar graphs are 4-colorable. This has been the most challenging problem of combinatorics for over a century and has contributed more to the development of the field than any other single problem. A computer-assisted proof of this conjecture was finally found by Appel and Haken in 1977. Although today chromatic number attracts attention for several other reasons too, many of which arise from applied mathematical fields such as operations research, attempts to find a simpler proof of the Four Color Theorem is still an important motivation of its investigation.

So here it's not so much the proof but the search for a proof that has given something extra over just believing the theorem. Does that still count as an answer to this question?

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The fundamental lemma is an example that most believed and on whose truth several results depend. According to Wikipedia, Professor Langlands has said

... it is not the fundamental lemma as such that is critical for the analytic theory of automorphic forms and for the arithmetic of Shimura varieties; it is the stabilized (or stable) trace formula, the reduction of the trace formula itself to the stable trace formula for a group and its endoscopic groups, and the stabilization of the Grothendieck–Lefschetz formula. None of these are possible without the fundamental lemma and its absence rendered progress almost impossible for more than twenty years.

and Michael Harris has also commented that it was a "bottleneck limiting progress on a host of arithmetic questions."

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    $\begingroup$ Fundamental lemma is not even 50% obvious by any stretch of imagination. Like most of the Langlands program, it is not a specific result that admits a precise, uniform statement; rather, it is a guiding principle that needs to be fine-tuned in order to be compatible with other things that we, following Langlands, would like to believe in $-$ then, and only then, does it becomes meaningful to talk about proving it. $\endgroup$ Sep 6, 2010 at 21:35
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    $\begingroup$ Thank you, Victor. I'm not proposing that the Fundamental Lemma is obvious, but it seems that is was accepted as likely to be true, because others based new work on it. PC below gives the example of Skinner and Urban, and Peter Sarnak says here time.com/time/specials/packages/article/…, that "It's as if people were working on the far side of the river waiting for someone to throw this bridge across," ... "And now all of sudden everyone's work on the other side of the river has been proven." $\endgroup$ Sep 6, 2010 at 22:45
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    $\begingroup$ I was under the impression that the fundamental lemma, at least, is a set of statements clear enough to be amenable to proof attempts. I think we have on page 3 here arxiv.org/abs/math/0404454 and Theorems 1 and 2 here arxiv.org/abs/0801.0446 precise statements for the various fundamental lemmas... It's possible I'm misunderstanding you. $\endgroup$ Sep 6, 2010 at 22:55
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    $\begingroup$ The fundamental lemma had a precise formulation, due to Langlands and Shelstad, in the 1980s (following earlier special cases). It is a collection of infinitely many equations, each side involving an arbitrarily large number of terms (i.e. by choosing an appropriate case of the FL, you can make the number of terms as large as you like). It was universally believed to be true because otherwise the theory of the trace formula (some of which was proved, but some of which remained conjectural), as developed by Langlands and others, would be internally inconsistent, something which no-one ... $\endgroup$
    – Emerton
    Sep 7, 2010 at 4:05
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    $\begingroup$ ... believed could be true. This a typical phenomenon in the Langlands program, I would say: certain very general principles, which one cannot really doubt (at least at this point, when the evidence for Langlands functoriality and reciprocity seems overwhelming), upon further examination, lead to exceedingly precise technical statements which in isolation can seem very hard to understand, and for which there is no obvious underlying mechanism explaining why they are true. But one could note that class field theory (which one knows to be true!) already has this quality. $\endgroup$
    – Emerton
    Sep 7, 2010 at 4:13
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Mathematics wasn't that rigorous before N. Bourbaki: in the Italian school of Algebraic Geometry of the beginning of the XXth century the standard procedure was Theorem, Proof, Counterexample. Also at the time of Cauchy some theorems in analysis began like "If the reader doesn't choose a specially bad function we have..."

The use of rigour in analysis, which Cauchy began, avoided that by being able to explain what was a "good function" in each case: analytic, $C^{\infty}$, being able to do term-by-term derivation in its expansions series...

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  • $\begingroup$ -1: Bourbaki seems rather late for an estimated transition to modern standards of rigor. Would you say that Hilbert's work was lacking in rigor? $\endgroup$
    – S. Carnahan
    Oct 27, 2010 at 7:02
  • $\begingroup$ You're completely right in that Hilbert's work wasn't lacking rigour, but it wasn't the standard, think of H. Poincaré $\endgroup$ Oct 27, 2010 at 18:59
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    $\begingroup$ @Gabriel : This seems like a delicate historical claim. I would actually be interested to see some sources on the evolution of standards of rigor within the mathematical community. $\endgroup$ Dec 6, 2010 at 23:11
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    $\begingroup$ @Andres www-history.mcs.st-and.ac.uk/Biographies/Weil.html "Nicolas Bourbaki, a project they began in the 1930s, in which they attempted to give a unified description of mathematics. The purpose was to reverse a trend which they disliked, namely that of a lack of rigour in mathematics. The influence of Bourbaki has been great over many years but it is now less important since it has basically succeeded in its aim of promoting rigour and abstraction." $\endgroup$ Dec 7, 2010 at 8:26
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    $\begingroup$ The unified description of mathematics was not the initial intent of the Bourbaki project. By all accounts, it was to write an up to date analysis textbook (losing a whole generation to World War 1 had left a gap). Of course, the whole thing got out of hand pretty quickly... $\endgroup$ Mar 19, 2011 at 5:15
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This question already has an amazing amount of great answers. Being a physicist with very limited knowledge of mathematics, I certainly cannot expect to contribute something of equal value, however after reading through the answers I'm missing a certain aspect. The missing issue is: "What is a proof"? The "rigorization" process of Calculus with $\epsilon$/$\delta$ proofs was mentioned as a major progress. However I saw nothing doubting that the current foundations of mathematics might still be "improved", where it is of course an interesting question what that would mean. A while ago, when I was trying to find the answer to above question, I came across the following story:

Fields medalist Vladimir Voevodsky, when working on rather scary stuff that I do not begin to understand (motivic cohomology...), came across many cases where ground-breaking published papers with proofs contained errors, which would only be noticed years after. Sometimes this would render a lot of later work worthless. Errors were not noticed, although people were studying the papers in seminars. When it happened to him, it genuinely scared him and got him to start working on computer-assisted proof, as well as an axiomatization of mathematics that goes beyond ZFC, called "Univalent Foundations". It has several conceptual advantages but apparantly is very unknown (and not complete yet!). The aim is to produce a framework where computer verified proofs are a practical option (contraty to now, where such proofs are extremely cumbersome and impossible to use on a regular basis in publications).

Returning closer to the question at hand: A quote from his motivation (a lengthy article containing many references and statements relevant to this question) for the foundational work of his is:

A technical argument by a trusted author, which is hard to check and looks similar to arguments known to be correct, is hardly ever checked in detail.

Granted, the results I'm describing here are far removed from practical applications, such as would interest an engineer. However it isn't entirelly unreasonable that fields like theoretical physics would eventually come into contact with parts of mathematics which seem similar in terms of obscurity. The verdict is: Even in the 21st century, having a published result with a proof is not enough to be sure it's true, not even in the most renowned journals and by the most trusted authors. To achieve that, one would need to push the limits of rigour even further, thus acknowledging its importance (where of course it is up to debate whether and how we want to do this).

By the way, Voevodsky has a recorded talk of his where he considers the question "what if the current foundations of mathematics are inconsistent?", where he tries to imagine how one would work in a framework known to be inconsistent, rather than one where one hopes but can never prove that verything is fine

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    $\begingroup$ This doesn't seem to me to be an answer to the question. The points made here might be worth raising in a new question. $\endgroup$ Apr 24, 2017 at 5:58
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    $\begingroup$ I thought this was answering the question because it "Demonstrat[es] that rigour is important". It also points out cases where having a (correct!) proof for something turned out to be very important for salvaging some results, which lost credibility and also the interest of the mathematical community, when their fundamentals turned out to be wrong. If more people think my impression was wrong, I strongly encourage someone to delete the answer. $\endgroup$ Apr 24, 2017 at 6:10
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