Is rigour just a ritual that most mathematicians wish to get rid of if they could?

Question

"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)

Answer 1

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.

Answer 2

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.

Answer 3

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).

Answer 4

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.

Answer 5

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.

Answer 6

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.

Answer 7

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.

Answer 8

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!).

Answer 9

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

Answer 10

Another MO question about rigor got me thinking about this old question again. One valuable feature of rigor, which I don't think has been said explicitly in the other answers, is that rigor allows me to be confident, in the privacy of my own study, that my argument is correct.

Much has been said by philosophers of mathematics about how a person working in isolation can easily make mistakes without realizing it, and how a proof is of little value unless it is absorbed by the mathematical community at large. All that is true, but should not be allowed to obscure the fact, which is nearly unique to mathematics, that because there is such a thing as mathematical rigor, I have the ability to verify on my own that an argument that I've come up with on my own is objectively correct. In the natural sciences, for example, a hypothesis that I come up with has to be checked against the empirical facts, and that might cost millions of dollars.

Euler was mentioned in another answer. Someone of Euler's caliber can, for example, manipulate divergent series without rigorously defined rules and not get into trouble, because he knows of many ways to sanity-check his calculations. However, what is a mere mortal to do? If rigorous definitions and proofs are not clearly laid out, then the average mathematician or student has no reliable way of telling whether their calculation with (say) infinite series yields a correct conclusion or is nonsense. They have to ask an expert and accept the verdict of the expert.

Let me emphasize that I am focusing on the value of rigor in the setting of a single individual working alone. The Atiyah quote in the other answer I mentioned above emphasizes the importance of rigor in allowing knowledge to be objectively codified and transferred to other people. I fully agree that rigor plays a vital role here, but I am saying something more. Rigor also plays a vital role when I am sitting quietly at my desk trying to come up with new mathematics. It lets me tell when I have a complete solution and when I have an incomplete solution. It gives me confidence that my rigorously proven lemma can be used as a solid foundation for further investigation.

Rigor therefore contributes to making mathematics more "democratic." I don't deny that the mathematical community has hierarchies and non-democratic features. Nevertheless, it is rigor that makes it possible for an independent researcher to build something of permanent value with limited contact with the larger mathematical community. Even for those who are "plugged in" to the main community, a large proportion of creative mathematics is initially generated by individuals coming up with their own ideas privately, and testing and validating them before socializing them. Rigor plays an absolutely fundamental role in guiding and shaping this private thinking process. It lets me have a very good sense ahead of time, before I say anything to my colleagues, whether my argument is going to be accepted. If I'm on reasonably good terms with my colleagues then I know that either they will be convinced, or they will point out my mistake and I will agree that I erred. Without rigor, there is no way I can enjoy this kind of confidence.

I sometimes wonder what it would have been like to try to do research in analysis in the days before calculus was put on a rigorous footing. It's hard for me to imagine. I think I would always be unsure whether my arguments were really correct. People who pooh-pooh rigor have, I think, been "spoiled" by the fact that nowadays everything can be made as rigorous as anyone cares to make it. Next time someone tries to downplay the importance of rigor, ask them what they think it would be like to do research in mathematics in the absence of rigor. I'm reminded of the famous quote by Kant, "The light dove, cleaving the air in her free flight, and feeling its resistance, might imagine that its flight would be still easier in empty space."

Answer 11

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: https://eugeniacheng.com/wp-content/uploads/2017/02/cheng-morality.pdf (Eugenia Cheng: Mathematics, morally)

Answer 12

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.

Answer 13

You might be interested in

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

and

"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 - DOI: 10.1007/978-0-387-48744-1

Answer 14

Here's one thing that can happen when you view proof as a mere ritual that is stealing the time you could be using to experience beautiful ideas. You get so wrapped up in your self-confidence to immediately perceive mathematical truth that you end up espousing something false and harming younger mathematicians by misleading them.

This happened to me. A well-respected mathematician with a prestigious position published a very nice paper, full of lovely arguments, but containing one astounding unproved claim. I saw an application for this claim and, being unable to prove it myself, eventually asked him for the argument. He waived his hands vaguely, and did so again with slightly decreasing levels of vagueness as I pressed for more details and raised objections. Because of the pandemic and the fact that we were on different continents, this ended up going on for about a year. Finally I was able to come up with a definitive counterexample to his claim. He was so sure that he could just see something and that a proof just required a dash of cleverness and a little bit of hard work and technical prowess to put together. Well no, he was deluded, and he wasted my time.

That is why we actually need proof.

Answer 15

There is a very nice large audience essay by Caroline Chen (Wayback Machine) 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.”

Answer 16

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.

Answer 17

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

Answer 18

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!

Answer 19

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.

Answer 20

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.

Answer 21

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)