Story where professor claims a step in a proof "is obvious" when it is far from obvious

Question

I picked up a funny story about a professor who stated that something was "obvious" after taking a long time to think about it: the story on the Wiktionary user page of 'DCDuring':

A mathematics professor is giving a lecture and has made an assertion as part of his presentation. A student, not understanding the basis for the assertion, asks why it is true. The professor responds that "It is obvious." Then the professor steps back, stares at the board and ponders for several minutes. Then he turns and walks out of the lecture hall. He is absent for a fairly long time and finally one of the students goes to look for him. He sees the professor in his office working on the blackboard which he has covered with mathematics. The student returns and reports to the class. Finally, just before the class is scheduled to end the professor reappears, and announces "Yes, it is obvious."

I asked 'DCDuring' where the story came from, and he said (not very convinced): "I've read that the mathematician was Norbert Wiener" (see here). What is the actual origin of this tale?

Answer 1

This is a well-known piece of mathematical folklore that has been “told about most teachers of any great reputation” (Norman Beers, quoted below) and so we cannot take seriously any particular assignment, whether to Wiener as in the question, or to Hardy as suggested by Lazerowitz (quoted below).

The earliest printed versions of the tale that I was able to find are from 1950, but both Norman Beers and R. B. Kershner make it clear that by this date the story was already a popular piece of oral folklore. Possibly earlier versions had different wording, making them hard for me to trace.

The most obvious of the many reasons why we cannot agree that secrecy is obviously necessary is illuminated by a story told about most teachers of any great reputation, especially in mathematics and in mathematical physics. For, like the student who asked Professor Jones, “Is the answer to that (partial differential equation) really obvious?,” we sincerely do not understand the reasons for the assertion. And we, like the student, are not helped if the Professor requires to be excused from the classroom and given the balance of the lecture period in his private office before returning to the class with a flat, “Yes. It is obvious.”

_{Norman R. Beers (1950). ‘The Atomic Industry and Human Ecology’. Nucleonics 6:5 (May 1950), p. 22–3.}

There is a story that G. H. Hardy passed over a step in the proof of a mathematical theorem he was lecturing on to his class with the comment, “This is obvious”. His students objected that it was not obvious to them, and asked for the explanation. Hardy, who apparently was taken aback by the objection, left the lecture room to look over the omitted step and after a few minutes’ deliberation returned and announced to the class, “Yes, it is obvious. We shall go on.”

_{Morris Lazerowitz (1950). ‘Strong and Weak Verification II’. Mind 59:235 (July 1950), p. 345.}

A professor lecturing to a graduate class said, “This is obvious—or is it?” then went into his office and returned after 30 minutes and continued, “Yes, it is obvious.”

_{Charles K. Robbins (1950). ‘Mathematical Miscellany’. In Mathematics Magazine 24:2 (November–December 1950), p. 115.}

Now the word “obvious” is a rather dangerous one. There is an incident, which has become something of a legend in mathematical circles, that illustrates this danger. A certain famous mathematician was lecturing to a group of students and had occasion to use a formula which he wrote down the remark, “This statement is obvious.” Then he paused and looked rather hesitantly at the formula. “Wait a moment,” he said. “Is it obvious? I think it’s obvious.” More hesitation, and then, “Pardon me, gentlemen, I shall return.” Then he left the room. Thirty-five minutes later he returned; in his hand was was a sheaf of papers covered with calculations, on his face a look of quiet satisfaction. “I was right, gentlemen. It is obvious,” he said, and proceeded with his lecture.

_{R. B. Kershner (1950). The Anatomy of Mathematics, p. 77. New York: Ronald Press.}

Answer 2

Richard P. Feynman tells a similar story (from personal experience at Princeton graduate school) in his book Surely You're Joking, Mr Feynman (in the section headed ‘A Different Box Of Tools’):

I still remember a guy sitting on the couch, thinking very hard, and another guy standing in front of him, saying, “And therefore such-and-such is true.”

“Why is that?” the guy on the couch asks.

“It’s trivial! It’s trivial!” the standing guy says, and he rapidly reels off a series of logical steps: “First you assume thus-and-so, then we have Kerchoff’s this-and-that; then there’s Waffenstoffer’s Theorem, and we substitute this and construct that. Now you put the vector which goes around here and then thus-and-so…” The guy on the couch is struggling to understand all this stuff, which goes on at high speed for about fifteen minutes!

Finally the standing guy comes out the other end, and the guy on the couch says, “Yeah, yeah. It’s trivial.”

We physicists were laughing, trying to figure them out. We decided that ‘trivial’ means ‘proved’. So we joked with the mathematicians: “We have a new theorem — that mathematicians can prove only trivial theorems, because every theorem that’s proved is trivial.”

Answer 3

Gareth Rees has done a excellent job of finding similar jokes, as well as the first time they were written down. However, he doesn't really tell you the reason the joke is funny.

In mathematical papers, you often find sentences like "it is obvious that ...". And relatively often, the thing that they assert is obvious is not obvious at all unless you are an expert in the subfield of mathematics that the paper deals with (and occasionally, not even then). For example (from The Canadian Journal of Mathematics, 1959):

With F(Y) and (Y,T) as described, it is obvious that (Y,T) is a zero-dimensional Hausdorff space and thus completely regular.

It seems improbable that an actual mathematician would have done something like this, so I suspect that this started as a joke, with the subject being "a mathematician" or something similar, and the names of actual famous mathematicians were apocryphally added later. What an actual mathematician would most likely have done¹ is explain why the fact was true, since it probably would have been clear to them but not at all clear to their students.

As an aside, recent sources of advice on mathematical writing tend to warn against doing this. And my impression is that "it is obvious that ..." is becoming less frequent in mathematical papers, although it's not clear to me whether this is a consequence of the joke or of the advice.

¹ There are a few mathematicians whose lecturing style is confrontational enough that they might have done something like this, but I think it's much more likely that the story started as a joke.

Answer 4

I remember that during a lecture by Prof. Cassels at Cambridge (famous for lecturing without notes) he said "Well, from here on it's obvious [short pause] but I can't quite see why at the moment."

Not really an answer to the question, but a good (and true) story.