This question strikes me as too subjective and argumentative for MO. I think a more acceptable version could be written (e.g. a reference request asking for essays that mathematicians have written about rigor), but that it is not the current version.

I'm not seeing how the question could be re-written to be anything more than a wide-ranging discussion. I suspect Terry Tao's comment is the closest to a good response to that (not yet written) discussion topic.

Could someone propose a concrete rewriting of the question?

Have you read the FAQ and the "How to ask" documents, Ricardo? It might be helpful to put your discussion in the context of those guiding documents, rather than via anecdotal examples. There is a certain aspect of randomness to community moderation, so we should try to avoid the relativist drift of anecdotes.

@Ricardo Andrade: some general and some specific remarks.

You asked how much lattitude the moderators give. I am not sure if this is just a slightly imprecise wording or if there is some misunderstanding: the moderators (in a strict sense; that is the six mentioned in the FAQs) are not really involved in most closing decisions, or at least they are not involved as moderators, but only as 'normal users'. The closing (and reopening) is handled on a day to day basis by at the moment about 300 users that have 3000+ points, so you are to be one of them soon.

Completely practically speaking a question is closed if 5 out of these 300 decide it should be (and reopened if 5 decide this way). This accounts already for a lot of fluctuation. This is not quite as strong as I put it as many out of the 300 hardly ever vote and some are even already inactive, but still there is a considerable pool of frequent voters that do vote.

Regading 'soft' questions in general: this is particullarly difficult to say what is the criterion here. I think I can claim that there are likely only very few people that followed this more closely than me over the last two years or so. I think I have meanwhile some general feeling what will work and what will not work, but still there are many surprises. I do not consider this as a big problem, since as you remarked, what is written in the official description concerns only mathematical question. I thus take the opinion everything else is a priori off-topic and sometimes an expception is made. And essentially by definition there cannot be preceises rules governing exceptions.

Regarding 'subjective and argumentative': what I consider as typically problematic is when answers (necessarily due to the question) will (essentially) purely be based on personal opinion as opposed to at least also containing a factual component. This is always problematic, but even more if the subject is potentially controversial.

Regarding 'not a real question': this can be used if the question is not really a question, as the reason says. But rather some attempt to start a discussion couched into the form of a question.

For me this is often a reason for closure: it seems OP does not really want to ask a question but rather means to start a discussion via some almost or sometimes truly rhetorical question. Some questions to me really read like: This and that. Surely, you agree?! That and this. Clearly this is scandalous. Isn't it?!

And there are still many more things. If you really want to get a feel, you could read old meta-discussion in the category 'Is this question acceptable?' You will find pages over pages of discussion. There is no rule not even a clear concensus, but mainly just a group of users each having their opinion on the matter. And then "we fight it out" (mainly in a friendly way though :-) ) on a case by case bases.

For the present question: let us just look at what is actually asked. Some sometimes find this a stupid way to proceed as anyway it is clear what is meant, but I do not follow this, since if it were so clear why not say it clearly.

There is the title question:

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

As OP said this seems like a non-question, but now he had some experience that makes them doubt that (but nobody shared this specific experience, this is a problem! see below). So, then:

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

Hard to tell. For one thing, how should I know what OP is aware of. And also the preceeding description of the event is quite unclear to me. There are many many details missing to allow me to form a mental image of the situation to say anything in a meaningful way.

Do mathematicians not preach what they practice (or ought to practice)?

I do not even quite understand what this means. At least it seems based on OP's (personal and not communicated) idea what mathematicians ought to practise.

So, to me clearly there is no specific question to be answered here. The best one could do is start some general discussion about the importance or not of prrofs in mathematics. And, discussion is something that in general is something that is tried to be avoided on MO. Sometimes it still happens for one reason or another but there are various users that agree that this should be minimized and vote like so.

Final note: my paraphrase of the current question could be read as quite dismissive. This is not the intent. But I considered it as useful to exagerate a bit to better get across what are problems one can see in this question.

This question reminds me of an MO-inappropriate question that's been bubbling around in my brain for a long time. It is, roughly:

"What does 'rigor' actually mean to adult mathematicians?"

[I don't mean "adult mathematician" to mean anything profound here and I certainly don't mean use it in an elitist way. I am just describing something that, sociologically speaking, seems to happen to students of mathematics and seems to stop happening at a certain point in someone's mathematical career.]

The idea behind the question is this: we throw the word "rigor" around a lot, but mostly in conversations with people who are not full-fledged mathematicians. Especially, the last time someone told me that my argument was "not rigorous", I was an undergraduate and the person was grading my homework. And I think that he really meant, "Sorry, but your argument is incomplete in a significant way. To complete it would require not just more lines of text but actually more work / an additional idea beyond what you have done."

I don't think that the reason that no one has said "not rigorous" to me in my adult mathematical life is that I've become so damned rigorous. I think it's because if I have a gap in my reasoning they so more directly and specifically. (On the other hand it is true that I still get referee reports calling passages in my writing "unclear", despite the fact that, you know, I'm such a clear writer. Often I want to reply in frustration "Just saying 'this is unclear' is not very clear! Please tell me more specifically what the problem is.") Nor do I use the word "rigorous" myself, either with adult mathematicians or with students or other outsiders. As above, it feels a little lazy to me: surely I can explain better what the problem is.

Nevertheless sometimes I hear people, including adult mathematicians, using "rigor" when they talk about mathematics. Often one hears rigor described as a continuum, alongside similarly ephemeral but generally believed to exist concepts like depth, elegance, power and so forth. Increasingly I suspect that I don't believe in rigor on a sliding scale. If you show me a proof in an area in which I am sufficiently qualified to follow it, I will either see that it is correct, see that you've made a mistake, see that you have a gap in your reasoning that is not routine to fill, or find your writing so unclear that I just can't follow your argument. Which of these means you're not being rigorous?

I did, by the way, read Jaffe-Quinn and the responses to it. I think they're about something different (which is much more relevant to the question being discussed in the thread than the present rant), namely they want to introduce a new kind of proof, so they rename standard mathematical proof "rigorous proof". This seems clearly in line with what I said above: conventional mathematics -- what do they call it in their article? Charles Mathematics?? -- is rigorous by its nature.

Nor do I use the word "rigorous" myself, either with adult mathematicians or with students or other outsiders. As above, it feels a little lazy to me: surely I can explain better what the problem is.

Yes, it's certainly important to explain the problem in more detail than just saying an argument isn't rigorous. However, I think rigor is an important explanatory concept for distinguishing between arguments that could be refined to a formal proof and arguments that could not without a lot of extra work. For example, heuristic or simplifying assumptions, models that are literally incorrect yet have explanatory power (e.g., random models for the distribution of primes), uncontrolled approximations, techniques that work generically but fail in critical cases, failure to consider non-obvious cases as possibilities in the first place, etc.

There are students, amateurs, and people in other fields who genuinely do not understand this distinction, and it's valuable to be able to explain that there are all sorts of arguments, which may be illuminating or convincing without being fully rigorous, while only rigorous arguments count as a mathematical proof.

Hmm, that's a good question, and I was overlooking this ambiguity. I meant the latter (i.e., unrigorous = not formalizable), but I agree that in undergraduate education calling an argument unrigorous often means "I know perfectly well how to fill in the details, but part of the exercise here was for you to demonstrate that you could fill them in, and you haven't done that convincingly enough."

"Death to Euclid!" is a very funny thing to say.

A web search reveals that Basil of Caesarea, in his Address to Young Men on the Right Use of Greek Literature, refers to someone vowing death to Euclid, but it's actually Euclid of Megara rather than Euclid of Alexandria.

I think perhaps the most important aspect of rigor, beyond its precise meaning is it's an act where we demand that we translate our perhaps rather nebulous and personal ideas into something that is for all purposes a completely artificial and foreign language. Its the act of translation into this neutral language that allows us to expose flaws in the core idea. It's similar to the act of converting pseudo-code into code and then hitting the compile button.

Pete @ Henry: Interestingly, "Death to Euclid!" was also one of the Bourbaki's battle cries; obviously, for a different reason than the one I have mentioned!

Many mathematicians will be slightly irritated with these kinds of points. But not only are they valid, they are more valid than often acknowledged. It is easy to start to think "well yes, technically we don't formalise our meta descriptions, but that is because everyone understands what is going on and we do, in fact, get repeatability and we know what we are talking about in an ideal sense outside of physics or physical processes." But this is because most mathematicians don't study the foundations of social semantics and the philosophy of language. Not everyone knows what everyone means by the application of axioms. Students get this wrong all the time. Training never gets to the point where there is absolute certainty or consistency, and this training is entirely focused on repetition and understanding how to reach more consistent repetition, just as one does in rituals (so it's not out of the blue where such a characterisation might come from). Computers also don't reach consistency. And underlying this all is a folk philosophy conception of idealism that is not explored or confronted. There is this belief that even without all of these "technicalities", underlying it all is at least some meaning that is eternal, unchanging, and outside of the physical world. That belief on the semantics of formal mathematics is of course completely unproven, likely unprovable, often wrong in the details when made explicit, and in the end not in any way "formal".

Okay, so lets say that someone is willing to concede this. It's still true that you get better engineering tolerances within the framework of formalisation, right? Well, this is an empirical proposition so it's something that requires more than just an a priori waving of the hands and some formal argument. It certainly isn't something that you can just say is obvious! That is not how science is done (in fact, it is quite the opposite), and at a minimum that makes it something worthy of discussion. And discussion might point to there being a number of cases where most early development of mathematics occurred in informal frameworks, and in many cases informal arguments advanced us much further and in ways that formal arguments have yet to reach. Wiles argument on Fermat is not formal (an indeed originally contained errors) - but it has been drilled down enough that those familiar in the fields of galois representation, modular forms, and the other various fields needed for the counting work feel it is solid. This touches on some of Professor Tao's blog comments here, but he lives in a whole different sphere of understanding than I, so I wouldn't want to co-opt his points here.

And I don't want to beat this point too much, but when you focus on formality as the primary requirement, it tends to breed overconfident nonformal feelings towards the results. For instance, the success in limit-based analysis bred a contempt for infinitessimal calculations that can be seen in a number of late 19th century and early 20th century texts towards such arguments. There was a belief that because they were not yet formalised, they were in fact wrong. The later formalisation by Robinson actually showed that much of that was thought informal did indeed have rigorous foundation. In the other direction, one purely mathematical result in physics by Von Neumann purported to show that it was impossible to have a mathematical foundations of quantum mechanics using "hidden variables" (his famous No-Go theorem). So when later deBroglie and Bohm showed that there were indeed such models of quantum mechanics with completely bisimulating ("isomorphic" with scare quotes) observable predictions, they were actually openly derided in conferences.

So there might be a good argument that the focus of mathematics should be more than formality, and formality should maybe seen as supporting role for making the semantics of the arguments more explicit and shedding light on the underlying form of the argument. In many cases today it is the case that those who formalise an argument are not the same as those who make it originally. And there needs to be the understanding that formalisation does not help ensure arguments are consistent, either - Godel bites any hope of that. It's just - "more explicit" that helps make things "more consistent" by weeding the obvious inconsistencies. Unhealthy focus on formality, though, can make it easy to be abusive and harder to make progress.

(continued)

[Retracted.]

quid - fair point. That'll teach me not to read the whole thread! I retract my previous objection.

(Added: on reflection, I do think there are some differences. Pete's comment is phrased as a question, whereas ex0du5's are polemical. But I agree that there are several comments here that may be 'off topic', and I don't particularly want to get into an argument about it.)

I want to add that, as quid suggested, my posts weren't intended to be polemical. I would not have posted that to MathOverflow, either, as it needs scholarly attribution, quotes, and more context before it would be appropriate there. I only intended to show on Meta why I thought the question made sense and to give some frameworks that shows consonance with the ideas in the question. I did not intend to try to convince of any of the positions stated, only to show they exist and have validity within their respective systems.

I also know that the word "ritual" can be taken as an insult, particularly among the strongly rational mathematical community. It is also clear that sometimes it's use is _intended_ as an insult. But there is a sense that is common in certain disciplines (like anthropology and psychology) where ritual is often only meant to describe the process of making actions repeatable through a practiced process - e.g. are meant to be free from any judgment-laden evaluation of the ceremony's cultural or religious meaning. I only meant to show that such descriptions (which also are given for the scientific process and other repeatable process of knowledge acquisition) have some interpretation with evidence backing them. I apologise if anything I wrote implies the insulting characterization of rigor; I have a deep respect for rigor, and it is one of the main reasons I have such a fondness for foundational issues.

Scott, about mathematician X, your scenario illustrates why many responsible mathematicians don't value complete rigor above all else. It's the kind of example that agrees with the idea that there is value outside complete rigor. Do you think what I have written disagrees with that? Or possibly do you think what I have written was attempting to disparage the reasonableness of many mathematicians' pragmatic views towards rigor? My points were not to make caricatures of mathematician's attitudes here. The original statement (not posted by me but which I was responding) suggested many mathematicians don't hold rigor in as high esteem as one might expect, and I was giving reasons why some foundational reasoning might support that. You seem to be suggesting I might not agree with that, and I'm not sure how to respond.

Concerning how requirements of rigor might hold back progress, your example is as good as any I might provide. Obviously, if mathematicians did have unreasonable expectations of rigor and nonrigorous reasoning was overly disparaged and had no place in math, then such bold steps would take years to finalize, possibly lifetimes. It is fortunate that rigor is not a requirement for participating in the mathematical process, as such advances do take us far.

On harms, I am not sure what you don't agree with about my two examples. Maybe if you describe why the examples of Robinson and Von Neumann don't provide what you are looking for in harms, I might be able to respond better. The examples were meant to show that overzealous belief in the advantages of rigor can sometimes lead to various "blindnesses" that can work against progress. There are many such examples throughout the history of mathematics. It's a very human argument about our psychologies. I've tried to state in multiple places that none of that should be construed as being against rigor having an important role in math. It is only an argument that reasonable mathematicians perhaps shouldn't value rigor higher than other important values. I can explain more why I think incidences surrounding those examples were due to such overzealousness appreciation of rigor, or I can give other examples if it helps in understanding. I am not sure if this would help the meta discussion on the merits of the OP, but if some believe it can clarify, I can definitely respond with whatever details are asked.

Showing that a conjecture would imply X, and later showing that X is inconsistent (with mathematics as we know it) is a huge thing.

Not often, I think, these two steps were combined together. It would usually go that someone note the implication, and later someone would show the inconsistency.

@avocat I am really confused. As I mentioned in the post, I came to MO to find the truth. I always consider myself a true student of knowledge, so to accept an answer for the sake of avoiding the deletion of the post is not something I would do. I respect MO and I would not certainly consider it as a game. And I expect the moderators also do not think so. As a newcomer to MO, upon the first closure of the post, I was nearly convinced that It is not a MO question. But then, we have those interesting comments... and you know the rest of story. That means, there are something in the question that makes it a "real question" for many MO users (as it was and is for me). Unfortunately, I start agreeing with you saying "I consider deleting a question with this much activity ...". That's a pity!

Do you suggest that I write something in the body of the post? You know, as far as my interest concerns, I understood that there is a real issue in the rigor, otherwise the question just remained unnoticed. Thus I may seek for an answer somewhere else. However, I still believe that keeping the question open would be beneficial to me and all those MO viewers interested to the issue (and even those who voted to close the post), since at the end, there would be a possibility to see many different ideas in one place. Now, what shall I do?

There has only been one delete vote; they don't get reset when a question gets reopened. This question will not be deleted anytime soon.

Open again. But it seems to me that the raison d'etre of the question has been undermined, being to a large degree based on a misunderstanding of Milnor's point of view (as indicated by his quoted correction). There seems to be essentially zero support in the answers, and in Milnor's response, for the notion that mathematicians would gladly dispense with rigor if they could, the importance of rigor being acknowledged all around. Thus it feels to me that the answers, as elaborated so far, are mostly "preaching to the choir" (Zeilberger's speculations notwithstanding, which were offered merely as one data point, without any particular endorsement in the answer).

Interesting topic, perhaps, but IMO not a good fit for MathOverflow.

@ashgari.amir: on a practical note, could you please try to use less metaphoric and more direct formulations (or at least provide the latter in addition). I find it somewhat non-trivial to figure out what precisely you mean with your last comment, for example.

Let's call this one a draw.

I cleared all the remaining votes. The question should remain open until the time where it starts attracting junk answers, where it will be closed for that reason.

I am in full agreement with Henry, and my hand had in fact wavered over the close button before François called it a draw. I wished I had had greater courage in my convictions.

@Henry Cohn: Let me try an explanation of what I think is present in favor of this question (while I am certainly not in favor of it; the only reason I did not vote to close it is that I do not vote to close anything at all at the moment).

The closest to an explicit argument in favor of the question is I think this relatively early comment (with 15 upvotes):

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. – Alexandre Eremenko

The first question to me is a bit what "this" [question] is but I would assume in the end the general subject of the importance and/or status of rigor and proof in mathematics is meant. Since this is actually a relevant and interesting topic one can be of the opinion that some answering/discussion/talk on it should happen on MO. What precisely is asked for or even meant by rigor is not so important; everybody could interpret it a bit as they see it personally and say something (interesting); the range of the topics in the answers given also supports this.

To me this is vaguely reminiscent of the situation in some earlier questions like http://mathoverflow.net/questions/101420/music-mathematical-point-of-view-revised-closed where in particular early versions of the question did not make much sense as asked and/or the given answers had little to do with the question as asked but some found it a good occassion to talk a bit about math and music in general.

In my observation/opinion this is not rarely a key-point in disagreement: the ones say/think the question is not good at least not as asked, the others say/think even if the question is not so good as asked one can/could interpret it in an interesting way so that the answers will/could be interesting.

Or even more roughly, the divide between: "let us answer questions" and "let us talk about subjects".

That there was so much voting on this one has I think also practical reasons not only intrinsic to the question; it was visible a long time (on main and on meta), and reappeared repeatedly. Yet it is also a general subject that many/most mathematicians care about in one form or another and so it is not surprising there was a big pool of people contributing their opinion (whereas perhaps the relation of math and music is of less universal interest in the community).

@bsteinberg: I didn't enter a vote to close lightly. But looking back at François's comment from seven days ago here in meta, it is my opinion that the question has begun to attract answers which, though I wouldn't term them "junk" exactly, reflect poorly on MathOverflow. I really strongly feel it is time that this question be closed. (Edit: I hadn't seen his stronger declaration "the war is now over" in the comments under the OP until just this moment.)

@quid: yes, I mainly meant fedja's answer. While I can sympathize with some of his sentiments, the discussion does not seem to be constructive (to put it very mildly). And some of the other recent answers seem either not very well thought through, or not all that relevant. (Oh, and I agree that "Entartete Mathematik" isn't helping matters at all.)

I am sorry to have to rehash this, but I think the problem begins with the question. Expanding on what Ryan Budney said way up top: the description of the panel discussion which forms the basis of the question is vague and anecdotal (and IMO, slightly alarmist), and it's basically impossible to know what happened at that event. As a result, the question became a Rorschach which invited an overly wide and meandering discussion.

Also: sorry to the moderators if I'm causing a pain in the neck by casting another vote to close after they wanted to stop the "war". But I think this merits reconsideration.