When is something "obvious"?
I try to be a good student but I often find it hard to know when something is "obvious" and when it isn't. Obviously (excuse the pun) I understand that it is specific to the level at which the writer is pitching the statement. My teacher is fond of telling a story that goes along the lines of
A famous maths 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.
My teacher's point is that this only comes with a certain mathematical maturity and even eludes the best mathematicians at times.
I would like to know :
Are there any ways to develop a better sense of this, or does it just come with time and practice ?
Is this quote a true quote ? If so, who is it attributable to and if not is it a mathematical urban legend or just something that my teacher likely made up ?
Like Florian, I really like Gowers' definition of obvious. Of course this is a very personal definition. A proof that instantly springs to mind for one person may not spring to mind for another. I am not really sure what there is to say at this level of generality beyond that.
Really phrases like "it is obvious that..." and "clearly..." are bad habits. In a mathematical argument they are the places you should look at first for possible errors.
Perhaps another story will be illuminating: a professor of mine once made an assertion in lecture that I didn't quite see instantly. I asked him "is that obvious?" and he replied "yes." I asked him "is it obvious that that's obvious?" and, after a short pause, he replied "no."
I really like the following definition (here given by fields medalist Timothy Gowers, and he credits his former colleague):
A statement is obvious if a proof instantly springs to mind.
However, for many mathematicians and teachers the meaning of "obvious" unfortunately is much broader.