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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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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? – Colin Reid Sep 3 '10 at 13:53
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? – KConrad Sep 3 '10 at 15:33
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. – Daniel Moskovich Sep 3 '10 at 16:26
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). – KConrad Sep 5 '10 at 1:39
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. – Victor Miller Sep 6 '10 at 15:56

38 Answers 38

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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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.) – Richard Borcherds Sep 8 '10 at 19:58

This is not strictly an answer to the question since it is a hypothetical rather than an example, but perhaps relevant nonetheless: Suppose that you have some computer program-say for keeping an airliner stable under gusts of wind-and it relies on a numerical algorithm, proved to converge under reasonable assumptions about air pressure and wind velocity, so on. A faster and more stable algorithm is developed for which no proof of convergence is known, though all the researchers in the field assure you that it always converges and that they are certain that it will always converge given the plausible scenarios your code is likely to be used for. I think it is clear that you should not trust their judgment, but rather retain the old code, despite the clear desirability of having a faster numerical algorithm.

So from the perspective of the researchers, a proof might not be all that important; it may have only told them what they already knew and generated no new insights in the process. But from the perspective of the consumers of mathematics, knowledge that a proof exists may lead to incremental improvements in technology that would otherwise not happen. Of course, at this point we've come full circle and it becomes important to the researchers to supply a proof.

My second point again distinguishes between consumers of mathematics and researchers: It is sometimes much easier to become 100% certain of something than 99% certain. 99% certainty that a given statement is true requires thinking about many concrete examples and developing intuition, whereas 100% certainty requires logically assenting to the statements contained in a proof. By this standard, I would say that I am 100% certain about the bulk of my mathematical knowledge, and not 99% certain. Perhaps this is a lamentable state of affairs, but time is finite. We cannot hope to develop intuition about all the statements we wish to use while working on problems we do wish to develop intuition about. In that sense, proofs encode in a few kb's the vast amount of information stored as the intuitions of all the researchers working on a particular problem. Again under this model, the purpose of proofs is for the convenience of non-researchers.

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I doubt that an airline would not choose greater efficiency (which usually comes with lower costs) even if there is a small percentage of risk left. – Johannes Hahn May 4 '11 at 14:25

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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-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? – S. Carnahan Oct 27 '10 at 7:02
You're completely right in that Hilbert's work wasn't lacking rigour, but it wasn't the standard, think of H. Poincaré – Gabriel Furstenheim Oct 27 '10 at 18:59
@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. – Andrés E. Caicedo Dec 6 '10 at 23:11
@Andres "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." – Gabriel Furstenheim Dec 7 '10 at 8:26
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... – Thierry Zell Mar 19 '11 at 5:15

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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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. – Victor Protsak Sep 6 '10 at 21:35
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…, 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." – Anthony Pulido Sep 6 '10 at 22:45
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 and Theorems 1 and 2 here precise statements for the various fundamental lemmas... It's possible I'm misunderstanding you. – Anthony Pulido Sep 6 '10 at 22:55
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 ... – Emerton Sep 7 '10 at 4:05
... 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. – Emerton Sep 7 '10 at 4:13

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." 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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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? – Ronnie Brown May 3 '12 at 17:35

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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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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