The obvious and the obviously obvious

This quote (cited by Andrew Gelman via a badly chosen URL), "The necessary conceit of the essayist must be that in writing down what is obvious to him he is not wasting his reader’s time." got me to thinking about what we mean when we say "obvious" or, more precisely, the different things that different people mean when they use the word in different contexts.

The obvi-... er, first example that comes to mind is this anecdote I first encountered in The World of Mathematics."

A famous math professor was giving a lecture during which he said "it is obvious that..." and then he paused at length in thought, and then excused himself from the lecture temporarily. Upon his return some fifteen minutes later he said "Yes, it is obvious that...." and continued the lecture.

A slightly different form of this anecdote was cited by Paul Renteln and Alan Dundes in their essay on mathematical folklore, unfortunately, it seems fairly obv-... make that, fairly clear to me that they missed the point of the joke:

This metajoke says a lot about mathematicians. First, they are often very quick thinkers, able to reach conclusions far faster than others. Second, they can see the humor in some jokes but are easily bored by the routine or familiar. Third, they often dismiss results that are obvious to themselves as “trivial”, even though the results may not be trivial to others. The following joke vividly illustrates this penchant.

A mathematics professor was lecturing to a class of students. As he wrote something on the board, he said to the class “Of course, this is immediately obvious.” Upon seeing the blank stares of the students, he turned back to contemplate what he had just written. He began to pace back and forth, deep in thought. After about 10 minutes, just as the silence was beginning to become uncomfortable, he brightened, turned to the class and said, “Yes, it IS obvious.”

The problem with this interpretation is that the students' confusion is not only not a central feature of the joke; it's not even a standard element (note that it doesn't appear at all in the previous version).

The joke here isn't that what's obvious to a mathematician may not be obvious to mere mortals. Instead, it's the far more interesting point that mathematicians and their ilk (and if you're reading this...) often use the words like "obvious" in a way that, though relatively precise, is very different from the way normal people use them. In common usage, being obvious is itself obvious. Normal people sometimes wonder if something that seems obvious is really true but they never spend time wondering if something that is true is really obvious.

At the risk of speculating on the motives of the apocryphal (and keeping in mind that I haven't taken a pure math class in more than a decade), I'd say that 'obvious' in this context means 'does not require a lemma.' You will hear mathematicians use the word in this sense, even though the question of whether or not a proof is complete is often far from obvious inn the traditional sense.

You could make a similar point about the way economists use 'rational' but that's a topic for another post.