Sunday, July 10, 2005

Numbers and Experience

It is often argued that while geometry is unable to adequately describe the world around us, numbers are more reliable, more certain. 2 cows plus 2 cows, always equal 4 cows. Unlike non euclidean geometries, there are no non standard arithmetics. Gauss, for a time at least, believed that ‘truth resides in number.’ In a similar vein Jacobi said “God ever arithmetizes” (as opposed to eternally geometrizing).
However, as Kline observes in Mathematics, The Loss of Certainty, the sharpest attack on the truth of arithmetic came from Hermann von Helmoholtz, a superb physicist and mathematician. In his Counting and Measuring he observed that the problem in arithmetic lay in the automatic application of arithmetic to physical phenomena. Some kinds of experiences suggest whole numbers and fractions, while others don’t: one raindrop added to another does not make two raindrops. Two pools of water, one at 40◦ another pool of water at 50◦ when mixed together do not make a pool of water at 90◦. Lebesgue facetiously pointed out that if one puts a lion and a rabbit in a cage, one will not find two animals an hour later! Helmoholtz gives many (more serious) examples but his overarching point is that only experience can tell us where to apply, and not apply, standard arithmetic.

Like Euclidean Geometry, arithmetic is not absolutely applicable to the physical world.