The Essence of Mathematics : Through Elementary Problems

Interested students of mathematics, who seek insight into the \essence of the discipline", and who read more widely with a view to discovering what the subject is really about, may emerge with the justifiable impression of serious mathematics as an austere, but distant mountain range { accessible only to those who devote their lives to its exploration. And they may conclude that the beginner can only appreciate its rough outline through a haze of unbridgeable distance. The best popularisers sometimes manage to convey more than this { including hints of the human story behind recent developments, and the way different branches and results interact in unexpected ways; but the essence of mathematics still tends to remain elusive, and the picture they paint is inevitably a broad brush substitute for the detail of living mathematics. This collection takes a different approach. We start out by observing that mathematics is not a fixed entity { as one might unconsciously infer from the metaphor of an \austere mountain range". Mathematics is a mental universe, a work-in-progress in our collective imagination, which grows dramatically over time, and whose eventual extent would seem to be unconstrained { without any obvious limits. This boundlessness also works in reverse, when applied to small details: features which we thought we had understood are repeatedly _controlled in, or reinterpreted, in new ways to reveal _finer and _finer micro-structures.