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.