"No". That was my answer till this afternoon! "Mathematics without proofs isn't really mathematics at all" probably was my longer answer. Yet, I am a mathematics educator who was one of the panelists of a discussion on "proof" this afternoon, alongside two of my mathematician colleagues, and in front of about 100 people, mostly mathematicians, or students of mathematics. What I was hearing was "death to Euclid", "mathematics is on the edge of a philosophical breakdown since there are different ways of convincing and journals only accept one way, that is, proof", "what about insight", and so on. I was in a funny and difficult situation. To my great surprise and shock, I should convince my mathematician colleagues that proof is indeed important, that it is not just one ritual, and so on. Do mathematicians not preach what they practice (or ought to practice)? I am indeed puzzled!

Reaction: Here I try to explain the circumstances leading me to ask such "odd" question. I don't know it is MO or not, but I try. That afternoon, I came back late and I couldn't go to sleep for the things that I had heard. I was aware of the "strange" ideas of one of the panelist. So, I could say to myself, no worry. But, the greatest attack came from one of the audience, graduated from Princeton and a well-established mathematician around. "Philosophical breakdown" (see above) was the exact term he used, "quoting" a very well-known mathematician. I knew there were (are) people who put their lives on the line to gain rigor. It was four in the morning that I came to MO, hoping to find something to relax myself, finding the truth perhaps. Have I found it? Not sure. However, I learned what kind of question I cannot ask!

Update: The very well-known mathematician who I mentioned above is John Milnor. I have checked the "quote" referred to him with him and he wrote

"it seems very unlikely that I said that...".

Here is his "impromptu answer to the question" (this is his exact words with his permission):

Mathematical thought often proceeds from a confused search for what is true to a valid insight into the correct answer. The next step is a careful attempt to organise the ideas in order to convince others.BOTH STEPS ARE ESSENTIAL. Some mathematicians are great at insight but bad at organization, while some have no original ideas, but can play a valuable role by carefully organizing convincing proofs. There is a problem in deciding what level of detail is necessary for a convincing proof---but that is very much a matter of taste.

The final test is certainly to have a solid proof. All the insight in the world can't replace it. One cautionary tale is Dehn's Lemma. This is a true statement, with a false proof that was accepted for many years. When the error was pointed out, there was again a gap of many years before a correct proof was constructed, using methods that Dehn never considered.

It would be more interesting to have an example of a false statement which was accepted for many years; but I can't provide an example.

(emphasis added by YC to the earlier post)

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    $\begingroup$ On my opinion, this is a legitimate and important question. These discussions are common, and sometimes even happen on the pages of BAMS. I propose to reopen. $\endgroup$ – Alexandre Eremenko Apr 18 '13 at 3:59
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    $\begingroup$ I find it useful to distinguish between pre-rigorous thinking, rigorous thinking, and post-rigorous thinking (see my essay on this at terrytao.wordpress.com/career-advice/… ). It is desirable to transition from a rigorous mindset to a post-rigorous one, but it is not desirable to transition from a rigorous mindset to a pre-rigorous one. $\endgroup$ – Terry Tao Apr 18 '13 at 4:31
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    $\begingroup$ I agree that this is a legitimate and important question, but not for MO. I've started a meta thread: tea.mathoverflow.net/discussion/1579/… $\endgroup$ – Qiaochu Yuan Apr 18 '13 at 7:56
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    $\begingroup$ The answer to this question is indeed "no". The correct analogy with physics is not whether physicists wish to get rid of relativity. It is whether physicists wish to get rid of physical reality. If they did rid of it, then physics would die (or morph into something not-physics), because physical reality is the final test of good physics. If mathematicians got rid of rigor, then mathematics would die (or morph into something not-mathematics), because rigorous proof is the final test of good mathematics. That will be my (community wiki) answer if this gets re-opened. $\endgroup$ – Lee Mosher Apr 18 '13 at 14:05
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    $\begingroup$ There has been a closing war on this question. The war is now over. tea.mathoverflow.net/discussion/1579/… $\endgroup$ – François G. Dorais May 1 '13 at 0:39

19 Answers 19


I was not going to write anything, as I am a latecomer to this masterful troll question and not many are likely going to scroll all the way down, but Paul Taylor's call for Proof mining and Realizability (or Realisability as the Queen would write it) was irresistible.

Nobody asks whether numbers are just a ritual, or at least not very many mathematicians do. Even the most anti-scientific philosopher can be silenced with ease by a suitable application of rituals and theories of social truth to the number that is written on his paycheck. At that point the hard reality of numbers kicks in with all its might, may it be Platonic, Realistic, or just Mathematical.

So what makes numbers so different from proofs that mathematicians will fight a meta-war just for the right to attack the heretical idea that mathematics could exist without rigor, but they would have long abandoned this question as irrelevant if it asked instead "are numbers just a ritual that most mathematicians wish to get rid of"? We may search for an answer in the fields of sociology and philosophy, and by doing so we shall learn important and sad facts about the way mathematical community operates in a world driven by profit, but as mathematicians we shall never find a truly satisfactory answer there. Isn't philosophy the art of never finding the answers?

Instead, as mathematicians we can and should turn inwards. How are numbers different from proofs? The answer is this: proofs are irrelevant but numbers are not. This is at the same time a joke and a very serious observation about mathematics. I tell my students that proofs serve two purposes:

  1. They convince people (including ourselves) that statements are true.
  2. They convey intuitions, ideas and techniques.

Both are important, and we have had some very nice quotes about this fact in other answers. Now ask the same question about numbers. What role do numbers play in mathematics? You might hear something like "they are what mathematics is (also) about" or "That's what mathematicians study", etc. Notice the difference? Proofs are for people but numbers are for mathematics. We admit numbers into mathematical universe as first-class citizen but we do not take seriously the idea that proofs themselves are also mathematical objects. We ignore proofs as mathematical objects. Proofs are irrelevant.

Of course you will say that logic takes proofs very seriously indeed. Yes, it does, but in a very limited way:

  • It mostly ignores the fact that we use proofs to convey ideas and focuses just on how proofs convey truth. Such practice not only hinders progress in logic, but is also actively harmful because it discourages mathematization of about 50% of mathematical activity. If you do not believe me try getting funding on research in "mathematical beauty".
  • It considers proofs as syntactic objects. This puts logic where analysis used to be when mathematicians thought of functions as symbolic expressions, probably sometime before the 19th century.
  • It is largely practiced in isolation from "normal" mathematics, by which it is doubly handicapped, once for passing over the rest of mathematics and once for passing over the rest of mathematicians.
  • Consequently even very basic questions, such as "when are two proofs equal" puzzle many logicians. This is a ridiculous state of affairs.

But these are rather minor technical deficiencies. The real problem is that mainstream mathematicians are mostly unaware of the fact that proofs can and should be first-class mathematical objects. I can anticipate the response: proofs are in the domain of logic, they should be studied by logicians, but normal mathematicians cannot gain much by doing proof theory. I agree, normal mathematicians cannot gain much by doing traditional proof theory. But did you know that proofs and computation are intimately connected, and that every time you prove something you have also written a program, and vice versa? That proofs have a homotopy-theoretic interpretation that has been discovered only recently? That proofs can be "mined" for additional, hidden mathematical gems? This is the stuff of new proof theory, which also goes under names such as Realizability, Type theory, and Proof mining.

Imagine what will happen with mathematics if logic gets boosted by the machinery of algebra and homotopy theory, if the full potential of "proofs as computations" is used in practice on modern computers, if completely new and fresh ways of looking at the nature of proof are explored by the brightest mathematicians who have vast experience outside the field of logic? This will necessarily represent a major shift in how mathematics is done and what it can accomplish.

Because mathematicians have not reached the level of reflection which would allow them to accept proof relevant mathematics they seek security in the mathematically and socially inadequate dogma that a proof can only be a finite syntactic entity. This makes us feeble and weak and unable to argue intelligently with a well-versed sociologist who can wield the weapons of social theories, anthropology and experimental psychology. So the best answer to the question "is rigor just a ritual" is to study rigor as a mathematical concept, to quantify it, to abstract it, and to turn it into something new, flexible and beautiful. Then we will laugh at our old fears, wonder how we ever could have thought that rigor is absolute, and we will become the teachers of our critics.

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    $\begingroup$ Thank you for this tremendous answer, Andrej! It practically made my day. $\endgroup$ – Todd Trimble May 8 '13 at 23:07
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    $\begingroup$ The real problem is that mainstream mathematicians are mostly unaware of the fact that proofs can and should be first-class mathematical objects. This is why I fought the meta-war. Unfortunately, the people who need to hear this message have probably moved on now. Thanks, Andrej. $\endgroup$ – Paul Taylor May 9 '13 at 7:48
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    $\begingroup$ @Paul: I am not sure what your sentence in bold means in practical terms. Take my proof with Higgins in 1978 of a 2-d van Kampen theorem. How far is it from a first class mathematical object, and how would one go about turning it into one? I see a proof as a path through a mathematical landscape; and a landscape of new concepts is one which has to be mapped out and charted. In other words, I am talking about mathematical structures as the landscape in which a proof resides. I would like an analysis which tells me if proofs are "equivalent"! $\endgroup$ – Ronnie Brown Jun 30 '13 at 21:38
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    $\begingroup$ @AndrejBauer Dear Andrej. Now after more than a year I'm coming back to this question and your wonderful answer. In particular, this time what has attracted my attention is the possibility of an educational use of this claim that "proofs and computation are intimately connected, and that every time you prove something you have also written a program, and vice versa". Is there any good expository read that you might suggest? I want to explore the potential of using that claim when we teach proof. $\endgroup$ – Amir Asghari Jan 16 '15 at 17:51
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    $\begingroup$ @AmirAsghari: This is late, but that proof-program correspondence that Andrej was referring to is the Brouwer-Heyting-Kolmogorov interpretation, which is actually easy to understand once you read about it. It fits nicely with intuitionistic logic, but the underlying idea has a similar application to classical logic, such as in the link between the arithmetical hierarchy and polynomial hierarchy where the quantifiers corresponds to oracles). $\endgroup$ – user21820 Dec 29 '15 at 8:05

While I would be one of the last people to agree that proof is not important in mathematics, I will say that when I hear a one hour colloquium type lecture about a difficult new theorem, I do prefer to be told the insights and intuitions that go into the proof rather than a board full of gory details. Then I can go home better prepared to try to slowly read and understand those gory details.

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    $\begingroup$ Of course. But I think the issue here rather is about published mathematical papers telling insights and intuitions yet without the gory details. $\endgroup$ – Abdelmalek Abdesselam Apr 17 '13 at 23:29
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    $\begingroup$ This will become more relevant when (if) machine-verified proofs become more common. If I present you with a piece of arithmetic, I do not need to justify my working because you can equally well feed it into your calculator. Similarly, it would be nice to be able to omit gory details from proofs where a machine could fill them in. Coq sort of does this, except that the insight lies in the human choice of the statements of the lemmas: the details of the proofs are terse and opaque. $\endgroup$ – Paul Taylor Apr 21 '13 at 14:53
  • $\begingroup$ This answer hit the nail on its head. We are machines, we trust other people, and prefer to hear what ideas they have than why they think those ideas might be true. $\endgroup$ – Andrej Bauer May 9 '13 at 6:16
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    $\begingroup$ We are not machines. Not. Damn you Freud! $\endgroup$ – Andrej Bauer May 9 '13 at 6:16

I guess the question "Is the rigor just a ritual" has got enough answers, so I'll address another one:

Has something happened in the world of mathematics that I am not aware of?

My answer is: yes, if you replace "aware of" by "consciously aware of". Of course, what I'll say will be "subjective and argumentative".

1) There are far too many people that call themselves "mathematicians" or "mathematical education specialists". Many of them are just street peddlers who make their living by selling their "results" and "theories" and whose mentality is that of an egg seller on the flea market. The goal is to get as good price as possible keeping the production costs as low as possible. One also has to maintain good relationships with nearby sellers and with market authorities and to keep an eye on the latest consumer trends. It would be nice to get a better place for the stand, etc. The question of the quality of eggs has to be addressed only if an angry mob of people is approaching. Otherwise, everything that is oval-shaped and white or brown in color will do.

2) The professor-student relationship is no longer that of a master and an apprentice but that of a service person and a client. The result is the most abominable. I'll abstain from discussing what it means for professors but for the students it ultimately means that they are treated as subhuman beings, i.e., they are considered as having almost no intelligence whatsoever, so instead of lifting the students to the level of the craft, the craft is lowered to their level. This happened in the arts when primitive ancient drawings were declared masterpieces alike to the paintings of Renaissance masters. Like the primitivization of arts led to all monstrosities that fill the "modern art" museum halls, which make me doubt that most modern artists can draw or sculpt at all, this primitivization of mathematics (whose main expression is presenting the mathematics as a mere taxonomy, a bunch of simple algorithms, and the art of pushing calculator buttons) will inevitably lead to reverting the craft to its pre-Greek level. Moreover, I have read a couple of math. education papers that, after you remove all fancy buzzwords from them, advocate exactly this transition.

3) Many mathematicians lost all pride and turned into mere beggars for money (grants, salary increases) and recognition (competition for prizes, publications in top journals, etc). I've recently heard some amazing new terminology like "the submission-rejection cycle" (you submit to a journal, get rejected, submit to another one, get rejected, etc.).

4) There is no hope for fundamentally new weapons that can be developed soon using further advances in pure math. This removed the need for rigorous mathematical education for military purposes and made the math. education a purely political issue. Despite all my disgust towards the wars, I have to grant the military the basic common sense: they have a clear goal to beat the enemy and whatever can serve this goal will be promoted and maintained at the operational level. The politicians need only to please the electorate for whom they coined the wonderful name "taxpayers". It doesn't matter how much a "taxpayer" knows about the science. As long as it is done on his money, he is the boss and he is the one to tell the right from the wrong. Moreover, even when the taxpayers do have common sense, their representatives in the legislature usually don't.

5) The Platonic idea of mathematics as an objective (super)reality was replaced by the idea of mathematics as sociological and cultural phenomenon. Note the words "mathematics is on the edge of a philosophical breakdown since there are different ways of convincing and journals only accept one way, that is, proof". They show clearly that the person saying them has lost all sense of an explorer of an unknown land whose task is to find out what is there and to make sure that what he sees is not a fata morgana. His goal now is merely to "convince other people of something".

I can continue, but I guess you got the idea by now. We are no longer viewed as high priests, or explorers, or technical experts, but rather as street sellers of strange and hardly digestible goods by the general public (which would be still tolerable) and by ourselves (which is suicidal, IMHO).

There is still a simple remedy: behave with pride and teach the craft properly whenever you can do it without losing your means of living immediately. I have little hope that this remedy will be applied widely, but you can always do it locally. And the last piece of advice: do not lose your sleep over the opinions of other people and do not argue with them. Look at what real results they achieved with their approach instead. If they have nothing to put on the table, just consider them a bunch of flies. The fly buzz can be quite irritating and some flies can deliver a venomous bite, but still a fly is a fly and a human is a human (not because a human has two eyes and a nose and the fly has a pair of wings as the modern humanists try to convince us, but because a human can absorb the whole Universe and to transcend his temporal and spatial limits and his self-centeredness, while the fly will always see only the piece of honey or dung it can feed on at the next moment).

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    $\begingroup$ fedja, there are courses en.wikipedia.org/wiki/Assertiveness#Training where you can learn to deal with your crippling inhibitions, then speak as you truly feel. $\endgroup$ – Will Jagy May 6 '13 at 5:28
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    $\begingroup$ Fedja, too venomously in the last paragraph. In my opinion. $\endgroup$ – Sergei Akbarov May 6 '13 at 6:06
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    $\begingroup$ As for 4), I’m pretty sure the military (or other government agencies) are involved in research in cryptography. $\endgroup$ – Emil Jeřábek May 6 '13 at 10:12
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    $\begingroup$ Entartete Mathematik? $\endgroup$ – Rbega May 6 '13 at 11:11
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    $\begingroup$ I too come late to this thread and for me fedja's essay is fully comprehendable and I agree even with that topics which vote for critical reflection of what and how mathematicians are doing. I think the tone of "sour-ness" does not revert the meaningfulness of the said, but might be a result of being eremite with that thoughts. $\endgroup$ – Gottfried Helms Nov 15 '14 at 7:00

I disagree a little with Greg Martin's answer. In my mind, the correct analogy with physics is the question: is physical reality just a ritual that most physicists wish to get rid of if they could? Physics is at its root a laboratory science where good results must accord with physical reality. Of course there is a lot of creative activity in physics which does not take place in the laboratory, during which physical theories are developed outside the setting of the laboratory; sometimes people who practice this are called "theoretical physicists". But in the end, physical theories that contradict physical reality either die or undergo changes that put them back in accord with physical reality. Physical theories that survive are ones which are actually verified to be in accord with physical reality; the people who do this part of physics are sometimes called "laboratory physicists". Even a theoretical physicist worth his or her salt needs to have a good "physical intuition" in order not to spin out physical nonsense.

I like to think that mathematics at its root is a laboratory science where good results must accord with logic. Of course there is a lot of creative activity in mathematics where logic is set aside, where one instead uses intuition or analogy or common sense or beauty or naturality or one of many other "illogical" activities in order to discover a solution to a mathematical problem. But in the end, once a potential solution has been discovered, it must be tested by logic, that is to say, it must be proved correct.

Extending what Milnor is quoted by the OP as saying, regarding "Some mathematicians... while some...", I can imagine a world where mathematical activity is divided into "laboratory mathematics" and "theoretical mathematics": the theoretical mathematicians just do the creative part, developing solutions of problems; the laboratory mathematicians do the grunt work to provide the actual proofs. My wording is chosen as a way to play devil's advocate, I'm not sure I see any actual value in such a division. At the very least, a mathematician worth his or her salt needs to have a good "logical intuition" in order not to spin out mathematical nonsense.


Highly recommended is this article by the late Vladimir Arnold, in which he talks of a "strong mafia of left-brained mathematicians" who "succeeded in eliminating all geometry from the mathematical education [...] replacing the study of all content in mathematics by the training in formal proofs and the manipulation of abstract notions." (Page 3 and the first half of page 4 are relevant to the question.)

Update (following Misha's comment):

Rigor is often mistaken with excessive formalism and voiding of arguments from intuitions to the extent that the proofs are more suitable for computers than humans. Arnold's article (and indeed most attacks on "rigor" such as the one the OP is referring to) criticize excessive formalism. The real rigor, on the contrary, has no conflict with intuition. Far from it, rigor (which is the basis of mathematics) is the refinement of intuition to the point that it is free from logical sloppiness. Rigor therefore should enhance intuition rather than abolishing it. Where to set the threshold of sloppiness? Arnold would probably set it on the basis of practicality and in connection with the originating real-world problems.

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    $\begingroup$ Dear Algernon: How do you think Arnold's (subjective and argumentative) article answers OP's question? Do you mean that Arnold's viewpoint is that "rigor is unimportant" and "Death to Euclid!"? Or do you mean that Arnold's article takes "rigor is unimportant" to its logical absurd conclusion and thereby proves the opposite? Or, maybe, "let a hundred flowers bloom" and, thus, rigor is both important and unimportant? $\endgroup$ – Misha Apr 25 '13 at 20:20
  • $\begingroup$ @Misha: Thanks for the comment. I updated my answer to elaborate the connection to OP's question. I also added reference to the relevant part in Arnold's article. $\endgroup$ – Algernon Apr 25 '13 at 23:24
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    $\begingroup$ Practicality can vary wildly. Who would care about continuous functions that are nowhere differentiable, let's just do real analysis with piecewise C^1 functions. Oops, here's a sample path of Brownian motion $\endgroup$ – Yemon Choi Apr 28 '13 at 19:14
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    $\begingroup$ The update is the real answer to the question. I do not really care about a proof if it does not enhance my understanding. $\endgroup$ – Steve May 3 '13 at 13:47

Here is an "uncommon" answer in favor of rigour from an old paper of Atiyah (Bull. Inst. Math. App. 10 (1974), 232-234): "How research is carried out?" In particular, I like the first part of the quote that is less heard (uncommon).

Now you may well ask what is the point of rigour? Some of you may define rigour as "rigor mortis" and believe that pure mathematics comes along to stifle the activities of people who really know how to get the answers. Again, I think, we ought to bear in mind that mathematics is a human activity and our aim is not only to discover things but to pass this information on. Now somebody like Euler, who knows how to write down divergent series and get correct answers, must have a good feeling of what ought to be done and what ought not to be done. Euler had an intuition built up out of a great variety of experience, and this kind of intuition is very hard to convey. So the next generation will come along and will not know how it is done, and the point of having a rigorous mathematical statement is so that something which in the first place is subjective and depends very much on personal intuition, becomes objective and capable of transmission. I have no wish at all to deny the advantages of having this kind of intuition, but only to emphasis that in order for this to be conveyed to other people it must eventually be presented in such a way that it is unambiguous and capable of being understood by someone who does not necessarily have the same kind of insight as the originator. Beyond this, of course, as long as you deal with a certain range of problems then your intuition is quite capable of leading to the right answer although you may not be sure how to justify it. But when you go to the next stage of development and start to build a more elaborate problem on the structure you already have, it becomes more and more important that the initial groundwork should be fairly firmly understood. So the necessity for having rigorous arguments is again because you are going to be building, and if you do not build on solid foundations the whole structure will be in danger.

  • $\begingroup$ Please let me know if I wasn't allowed to answer my own question, in particular, since the answer is essencially somebody else's idea. I thought the body of the post is too long at the moment, and it won't be a good idea to make it longer. Moreover, I am about to report to our next meeting what I have learnt so far. Again, I thought it is also a good idea to let the viewers see what I saw so far in a neutral way. I have some non-neutral comments that I will write them in meta as soon as they find a reportable form. $\endgroup$ – Amir Asghari May 3 '13 at 8:55
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    $\begingroup$ It's completely fine to answer your own question. I set off the quotation of Atiyah to make it clear that it's his (and gave the source). I think also you should put your non-neutral comments here where they belong, not meta, where the discussion was about whether this question was appropriate for MO. $\endgroup$ – Todd Trimble May 3 '13 at 11:33

An analogous question in physics might be: Is relativity just a ritual that most physicists wish to get rid of if they could? When we're going about our daily lives, most of the time people don't care about relativity: Newtonian physics explains everything we're going to see, it's simpler, and it's intuitive. We wouldn't bother to set up a relativistic calculation to decide when the bus is going to arrive. But in situations where our intuition is lacking, and/or it's really important to us that our answer is correct, then we need to incorporate relativity (and sometimes we learn that our intuition isn't always dependable!).

In math, when we're going about our daily lives, most of the time people don't care about rigor: intuitive arguments, exhibiting a few terms in a pattern, and arguing from experience and approximation work pretty well. We wouldn't bother setting up an integral to calculate how far our car gets on a tank of gas. But in situations where our intuition is lacking, and/or it's really important to us that our answer is correct, then we need to incorporate rigor (and sometimes we learn that our intuition isn't always dependable!).

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    $\begingroup$ To that last remark, it is often the case that our intuition is not dependable; at least in the parts of abstract mathematics which deal with intangible objects (read: infinite objects). $\endgroup$ – Asaf Karagila Apr 17 '13 at 23:08
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    $\begingroup$ Proof is not just there in order to be sure that our answer is correct. Proof is about convincing others that our answer is correct. $\endgroup$ – André Henriques Apr 21 '13 at 20:01

A pillar of modern science is the notion of falsifiability.
A theory/result (theorem) that is not in principle falsifiable is not part of science.

In experimental sciences, a researcher will describe their experimental setup in as much detail as possible, in order to allow others to reproduce them. New results are only accepted once they are reproduced by independent research groups.

In mathematics, things are slightly different.
The process of giving a "seal of approval" comes through people reading a proof. A mathematician who comes up with a new result presents his/her proof to the community. Then, a (possibly small) number of mathematicians will read though the details of the proof, and will let others know whether they believe in the proof or not. Once enough mathematicians declare that they believe in the proof of a new result, that result is accepted by the community.

As you can see, the notion of proof is essential for the good functioning of the above process.


There is the case of Hilbert's 16th problem in which errors were found in some proofs. See "Centennial History of Hilbert's 16th Problem", Yu. Ilyashenko, Bull. Amer. Math. Soc. 39 (2002), 301-354. One result which was published in 1923 was found to be faulty in 1981 over 50 years later.

There is another thread about rigor in mathematics which has some examples: Demonstrating that rigour is important


Eugenia Cheng has written an interesting essay that is relevant to this discussion in which she examines the notion of "moral" as used by mathematicians: http://cheng.staff.shef.ac.uk/morality/morality.pdf


You might be interested in

"THEOREMS FOR A PRICE: Tomorrow’s Semi-Rigorous Mathematical Culture" by Doron Zeilberger http://arxiv.org/abs/math/9301202


"The Proof is in the Pudding: A Look at the Changing Nature of Mathematical Proof" by Steven G. Krantz http://users.cs.dal.ca/~jborwein/Preprints/Books/MbyE/Second-Ed/Material/krantz-proof.pdf

  • $\begingroup$ Here are two quotes from "Theorems for a price". I have chosen the first quote for the language used, that is highly related to the form of the question posed. The second quote comes to give a glimpse of Zeilberger's "answer" to the question (i.e, assuming he was answering the question). Thanks to "unknown" and John. Both quotes have been just chosen to give a quick view of the paper to the viewers. $\endgroup$ – Amir Asghari Apr 27 '13 at 20:24
  • $\begingroup$ First quote: There are writings on the wall that, now that the silicon savior has arrived, a new testament is going to be written. Although there will always be a small group of “rigorous” old-style mathe- maticians(e.g. [JQ]) who will insist that the true religion is theirs, and that the computer is a false Messiah, they may be viewed by future mainstream mathematicians as a fringe sect of harmless eccentrics, like mathematical physicists are viewed by regular physicists today. [JQ: Jaffe and Quinn] $\endgroup$ – Amir Asghari Apr 27 '13 at 20:26
  • $\begingroup$ Second quote:I can envision an abstract of a paper, c. 2100, that reads : “We show, in a certain precise sense, that the Goldbach conjecture is true with probability larger than 0.99999, and that its complete truth could be determined with a budget of $10B.”...As absolute truth becomes more and more expensive, we would sooner or later come to grips with the fact that few non-trivial results could be known with old-fashioned certainty. Most likely we will wind up abandoning the task of keeping track of price altogether, and complete the metamorphosis to non-rigorous mathematics. $\endgroup$ – Amir Asghari Apr 27 '13 at 20:30
  • $\begingroup$ Who knows, perhaps some of the younger viewers find a chance to check Zeilberger's claim in 2100 :) $\endgroup$ – Amir Asghari Apr 27 '13 at 20:54
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    $\begingroup$ By 2100, 10 billion dollars will be the price of a loaf of bread. $\endgroup$ – Gerry Myerson May 6 '13 at 5:25

There is a very nice large audience essay by Caroline Chen on this topic, entitled The Paradox of the Proof. Centered around Mochizuki's claim of a proof of the abc conjecture, I think that it gives a great insight on how we do mathematics (or, at least, try to).

Here are a few quotes.

What we do:

For centuries, mathematicians have strived towards a single goal: to understand how the universe works, and describe it. To this objective, math itself is only a tool — it is the language that mathematicians have invented to help them describe the known and query the unknown.

But sometimes we can't:

“I decided, I can’t possibly work on this. It would drive me nuts,” he [De Jong] said.


Kim sympathizes with his frustrated colleagues, but suggests a different reason for the rancor. “It really is painful to read other people’s work,” he says. “That’s all it is… All of us are just too lazy to read them.”

Anyway we should keep tryin':

“You don’t get to say you’ve proved something if you haven’t explained it,” she [Cathy O'Neill] says. “A proof is a social construct. If the community doesn’t understand it, you haven’t done your job.”


Having digested previous answers, I think this point is mostly new.

Imagine a paper with a large computer-found-and-verified proof that is mostly incomprehensible to humans. The body of the paper, freed from the obligation to provide full rigor and proofs, consists of intuition and informal arguments, perhaps graphs and so on, explaining "why" the theorem is true.

Sound nice? But now another paper is published on the same theorem. It argues that most of the intuition advanced in the previous paper is irrelevant to this theorem. It suggests another completely different line of informal argument that also seems to completely explain why the theorem must be true. Etc.

This is like the world before the scientific method (and sadly often after as well). We have some phenomenon that is observed to be true. Humans are notoriously good at making up convincing stories for why it is true. And we are notoriously prone to accepting flawed explanations. The point of the rigorous scientific method is to cure us of this.

Similarly in mathematics, rigor and proofs are required to connect "understanding" or "convincing" to actual ground truth. Otherwise (as in Milnor's cautionary example!), someone can tell you a fact is true and give seemingly very convincing reasons why, yet it turns out to be true for totally unrelated reasons, or even false.


So while mathematics may be considered a social process of gradually convincing ourselves and each other what is true, rigor is still a crucial part of that process in connecting what we are saying to what is actually true. Otherwise it is too easy to convince humans of false or irrelevant explanations.


Please would somebody better qualified than me to do so write accounts here of Reallisability and Proof Mining.

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    $\begingroup$ Is proof mining something like coal mining in a Dickens novel? Starved kids working themselves to near death, just so they could feed their sickly mothers... getting into the dark mines by dawn and leaving them after the sun has set, and spending their days in the darkness of the mines, looking for proofs... letter after letter, word after word, string after string... $\endgroup$ – Asaf Karagila Apr 26 '13 at 9:57
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    $\begingroup$ Come on Todd, this site is getting far too serious. I was the first to up-vote Asaf's comment because it was amusing and I suspect that the other three votes were for the same reason. As for writing about Realisability myself, I spelt out the basics of the Curry-Howard "isomorphism" in mathoverflow.net/questions/128478/intuitionistic-logic-as-quantization-of-classical-logic/128489 but I don't know anything at all about Proof MIning. $\endgroup$ – Paul Taylor Apr 26 '13 at 21:06
  • $\begingroup$ Alright then, so I misread Asaf's comment. Stuff happens. I'm happy to remove my own comment, but I still encourage you (Paul) to improve your answer by explaining how it relates to the question. $\endgroup$ – Todd Trimble Apr 26 '13 at 23:15
  • $\begingroup$ Todd, no harm done! All in good spirit... :-) $\endgroup$ – Asaf Karagila Apr 26 '13 at 23:23
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    $\begingroup$ The answers that have been given to the questions are "soft" but professionally valid ones about the importance of proof and rigour because intuition is not reliable, etc. The value of MathOverflow in my opinion is that it is interdisciplinary. I wanted to point out that there are disciplines which can give a "hard" answer to the question, although I am not qualified to report on them. However, this is unlikely to happen given the amount of ill feeling that has been generated. $\endgroup$ – Paul Taylor Apr 28 '13 at 21:04

Intuition is important. It is how we "do" mathematics. It is how we "feel" it and "see" it. It is our eyes, ears, and hands. What are proofs then? Proofs are how we build mathematics.

Intuition will basically always give a reasonable mathematical result. However, it does not accumulate. If you build everything with intuition, you quickly wind up in contradictory circles. You can fix these contradictions, but you'll end up spending most of your time doing so, since every time someone has a new intuition, you have to check it with all the other results.

For example, take the Banach-Tarski paradox. Intuition would have quickly eliminated it as a possibility. Then when Banach and Tarski came along, any chunk of mathematics based on the intuition that it was impossible would need to be rewritten. It could be done, but I hope its clear that this would be infeasible to do regularly.

Of course, this would probably also happen with Fractals, probability (many times), set theory (more than necessary), topology, computer science (many times), mathematical logic, etc...

How do proofs solve this? Because we know if a proof is wrong very quickly (usually). We do not have to wait to see if someone has a different intuition/proof, we can just check the proof. Sometimes proofs can go a while without being corrected, but they are usually repairable. Therefore, when using proofs, we can fairly safely build up mathematics without worrying too much about lasting errors. In fact, if we agree on some formal system (like ZFC), it becomes even easier. Of course, we are not writing completely formal proofs (yet), but proofs are less likely to conflict if everyone has the same theory in the back of their head. Then the only results that are only decidable with intuition are the axioms, which is usually a small list of statements.

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    $\begingroup$ "For example, take the Banach-Tarski paradox. Intuition would have quickly eliminated it as a possibility." That would depend on how well-developed your intuition was. From a sufficiently advanced viewpoint, the "paradox" is quite intuitive. $\endgroup$ – Gerry Myerson Feb 28 at 11:31
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    $\begingroup$ @GerryMyerson That is true, but we reached that level of advancement using proofs. In particular, it took us quite a while to get to it, I imagine that it would have been eliminated quite early on, before we had a theory of cardinals. $\endgroup$ – PyRulez Feb 28 at 12:03

Intuition and insight are complementary to rigorous calculations and proofs. Mathematics needs both of them. But they are not equal...

Mathematicians need insight and intuition, because otherwise automatic generation of proofs would be enough, journals would be written by computers.

Rigorous proofs are also needed, because otherwise we would just emit conjectures and it would be enough. (Conjectures are important, and sometimes a paper can be based solely on a conjecture, without proof. But even such a paper contains arguments supporting the conjecture, proofs of particular cases, etc.)

Rigorous calculations, even the symbolic ones, (a particular kind of proof) are more and more delegated to computer software like Maple and Mathematica. And, for who can afford, to PhD students and other juniors.

In time, the proof will be more and more the job of computer software. The mathematician will cook-up conjectures, and simply ask the computer to prove them, or to find couterexamples. Probably the computer will be able to estimate a potential impact of the result, and inform the mathematician of the profitability of the article.

Probably it will become a norm that the proof will be formalized and separated in a special format, so that it can be checked easily by computer programs, so this part of the peer review will be automatic. The other part of the peer review will be to evaluate the importance of the result, for example by the impact in solving or simplifying other problems. Much of this can also be done automatically, maybe by some algorithm similar to Google's page rank algorithm or the citation factors, but before being actually cited, just because it provides key elements in other proofs which are not yet finished.

But even if the scientific software will become that smart, a tiny amount of rigor will still be necessary, to be able to tell it the tasks.

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    $\begingroup$ And probably there will be speculative predictions. $\endgroup$ – Martin Brandenburg Apr 26 '13 at 13:55

The most important, I think, is to realize that rigorous proof not is a question about truth, but about a struggle to convince. The element of struggle will remain as well as the probability $\epsilon > 0$ of resulting errors.

I think there should be a mathematical theory of mathematical proofs, with elements of "dependency" and "nearness". And in the future maybe there will be both regular and irregular proofs depending on their formulation with respect to the current supervising theory.


At the risk of stirring the pot further:

"All physicists and a good many quite respectable mathematicians are contemptuous about proof."
-G.H. Hardy (Wikiquote)

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    $\begingroup$ But Hardy goes on to say "I have heard Professor Eddington, for example, maintain that proof, as pure mathematicians understand it, is really quite uninteresting and unimportant, and that no one who is really certain that he has found something good should waste his time looking for proof.... [This opinion], with which I am sure that almost all physicists agree at the bottom of their hearts, is one to which a mathematician ought to have some reply." $\endgroup$ – John Stillwell May 3 '13 at 3:21
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    $\begingroup$ I don't agree with Hardy, but just wanted to provide some talking point. $\endgroup$ – David Roberts May 3 '13 at 3:40
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    $\begingroup$ Thanks to John Stillwell for providing a fuller quotation -- the final words are something that most mathematicians would agree with. And while he exaggerates when he says "all physicists", he's right that many (most?) physicists have such attitudes, perhaps even moreso in his day. $\endgroup$ – Todd Trimble May 3 '13 at 13:47
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    $\begingroup$ When talking informally, Woodin sometimes tells you that such and such is true. If this is the first time he does that with you, he may stop at some point and clarify: "Note that I said that it is true, not that we have a proof." $\endgroup$ – Andrés E. Caicedo May 3 '13 at 18:33
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    $\begingroup$ Does he say things like "the negation of CH is true"? :-P $\endgroup$ – David Roberts May 3 '13 at 21:35

I apologize, this should be a comment to unknown's posting of "THEOREMS FOR A PRICE", but I lack sufficient reputation.

Doesn't a proof that says "Goldbach is over 99.999% likely to be correct" have to be 100% correct? In other words, doesn't even a post-rigorous proof have to be rigorous somewhere?

I think this is precisely what Terry Tao is getting at with his comment to the original question. In fact, I'll go further: it feels like some people who are questioning the need for rigor want to jump immediately to Tao's "post-rigorous" thinking without having gone through the "pre-rigorous" and "rigorous" phases. This may work for some prodigies, but I doubt it's a good method in general. But then, I believe in rigor!

  • $\begingroup$ Yes, a proof that certifies 99.999% likelihood of validity should itself be impeccable. I think Terry Tao was driving at something slightly different, about arguments being both correct but perhaps more crucially for the advancement of understanding, letting the high-level ideas shine through without being cluttered with details at a routine and professionally trivial level. But I agree that everyone, and I would include prodigies, has to go through these stages. That said, I feel as though some of this discussion is academic: which mathematicians nowadays publicly wish to get rid of rigor? $\endgroup$ – Todd Trimble May 3 '13 at 13:21
  • $\begingroup$ @Todd Trimble, I agree about mathematicians, but the problem seems to be non-mathematicians who nonetheless think they need a say in this. And while I don't have much sympathy, I suppose we ought to ask working client scientists about how academic is the response from climate change deniers. If the methods of mathematics become completely incomprehensible, it arms the critics, unfortunately. $\endgroup$ – user5794 May 3 '13 at 21:10
  • $\begingroup$ Thanks for clarifying your comment, trb456. I largely agree, but I thought the general discussion here was to be about mathematicians (e.g., the author of Theorems For a Price). I didn't mean to sound insulting by using the word 'academic', and I'm sorry if I did. $\endgroup$ – Todd Trimble May 4 '13 at 11:12
  • $\begingroup$ @Todd Trimble: I don't think you sounded insulting. If you go all the way back to the top of the OP's original question, you will see discussion of some panel where these sympathies were being expressed. So I guess someone feels this way, though perhaps not so many to be a problem, yet! $\endgroup$ – user5794 May 4 '13 at 14:49
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    $\begingroup$ @Amir: I am utterly confused about what actually happened at that discussion. I don't think anyone who wasn't there can really comment on that discussion, and certainly not on the basis of such a skeletal report. It might help if the discussion were videotaped and made public, so that you could point to specific moments of the discussion that trouble you, although it's my opinion that MathOverflow (and its particular functionalities) are in fact poorly adapted for such a discussion about a discussion. I have just entered a vote to close. $\endgroup$ – Todd Trimble May 8 '13 at 0:05

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