"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)