Friday, January 2, 2009

Silly details about numbers

The ancient Greeks found numbers fascinating in an unprecedented way (or so Morris Kline's Mathematics for the Nonmathematician informs me). Rather than adopting the more practical attitude of the Egyptians and Mesopotamians toward numbers, the Greeks recognized a conceptual beauty in the ability use the self-same methods on any possible collection of objects. That is to say, abstraction, and the universality thereof.

To assign a single, unique label to every quantity, and then to perform mental operations upon them which, amazingly, correctly modeled comparable situations in reality--astonishing. And I do mean that without irony.

Take a pie or any appropriately divisible object. Cut it into halves, then cut each half into thirds. We take it for granted today that the answer may be computed easily through a mechanical procedure: (1 * 1/2) * 1/3 = 1/6. This is to say that, without having made any cuts or further measurements, we already know with certainty what size the smaller pieces will be! (Or the size that they will approximate, because it is not an ideal world.)

But this is a fantastic discovery, for by beginning from only a few known facts, we discover what must be the case for physical objects after manipulating them--all without having left our armchairs.

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