This should be required reading for anyone concerned with... well, just about anything. It is about as thought-provoking as anything I have ever read, its imagery applicable to all kinds of disciplines, not just mathematics.
Take music theory, for example. According to music theory, there is such a thing as a pitch class, by virtue of which an A is in the same pitch class as all other As. But what if it isn't?
How could it not be? If some composer uses A in a way that robs it of all its ordinary characteristics it is, strictly speaking, no longer a member of a class but an event unto itself.
- Recalling some specific instances of this: The G minor triad on the piano with which Debussy opens the Violin/Piano Sonata
- The opening unison G of the Mozart G minor Piano Quartet
- The famous B# on the cello C string in Mendelssohn's D minor Piano Trio - don't let anyone ever tell you that B# and C are the same.