When is something "obvious"?

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.


I believe the famous mathematician was G. H. Hardy, and I am sure he had a different view as to what is "obvious" compared to lesser mortals (like myself).

I would say that "obvious" should only ever be applied to things that the speaker believes that his listener should find "simple" ..

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