The purpose of abstraction is not to be vague, but to create a new semantic level in which one can be absolutely precise

(written by lawrence krubner, however indented passages are often quotes). You can contact lawrence at:


What is the black king in chess? This is a strange question, and the most satisfactory way to deal with it seems to be to sidestep it slightly. What more can one do than point to a chessboard and explain the rules of the game, perhaps paying particular attention to the black king as one does so? What matters about the black king is not its existence, or its intrinsic nature, but the role that it plays in the game.

The abstract method in mathematics, as it is sometimes called, is what results when one takes a similar attitude to mathematical objects. This attitude can be encapsulated in the following slogan: a mathematical object is what it does. …
—Tim Gowers, A Very Short Introduction to Mathematics