TITLE: Is rigour just a ritual that most mathematicians wish to get rid of if they could? 
QUESTION [73 upvotes]: "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)

REPLY [10 votes]: 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."