A Paradoxical Subject

Daniel N. Rockmore reviews Mathematics and Common Sense: A Case of Creative Tension by Philip J. Davis, in American Scientist:

These essays offer, among other things, a bird’s-eye view of the world of professional mathematics. The more interesting material derives from the fertile, if somewhat uneasy or even confusing, relationship mathematics has with the real world and real people. In essence, Davis makes the point Screenhunter_03_nov_14_1017that mathematics and common sense spring from the same source—a human, if not primal, inclination to organize and communicate experience—but that mystery, confusion and even magic can arise out of these humble human origins.

For example, the property of “fiveness,” which could be common to a small flock of sheep, the members of the shepherd’s family and the fingers on the shepherd’s hand, is more generally a concept of number that comes out of the penchant for and necessity of identifying one-to-one correspondences. Geometry can be seen as the natural outcome of the search for a means of communicating size and shape. The irony is that from such “common sense” and concrete inclinations, mysteries are born. Considerations of number lead naturally to the primes, still a source of simple-to-state but difficult-to-solve problems. Contemplating distance, we come quickly to irrational numbers (note the name!) and, over time, to the mind-bending pursuits and puzzles of modern geometry and topology.

More here.