Mathematics

Whither mathematics? . . . or wither mathematics? These are the questions that preoccupy mathematician Morris Kline (New York University) in this gloomy contemplation of mathematics' fate. From first to last he focuses on truth as it is - or is not - established in mathematics. As a lucid commentator Kline breezes along, discoursing on the Babylonians and Pythagoreans, the worthies of the Middle Ages, the Renaissance, and the modern era. He is quick to point out how the "queen of the sciences" served the religious and cultural ideals of men in a God-centered society. Mathematics could demonstrate the divine and absolute harmony of nature, reflecting the glory of God. Of course a Galileo or a Descartes, while still devout, offered-slightly different views, seeing in mathematics a method to be exploited and explored without reference to the deity. In time, other mathematicians and philosophers would question the very notion of truth and logical proof. With the discovery that there could be such entities as non-Euclidean geometries or whole classes of "transfinite" numbers, notions of reality - and of mathematics as mirroring that reality - began to fade. Later, the works of Kurt Godel and others shook the very foundations of mathematics, establishing that no mathematical system could be consistent or complete, no proof certain; there would always be undecideable propositions and the like. What is so surprising is Kline's acute emotional reaction to these developments: mathematics for him appears to be suffering a fate worse than death. His only solution - clearly reflecting his strong applied math bias - is a pragmatic one. If mathematics works as an aid in explaining the natural world, fine. Since it so often has worked, we can assume that it will continue to work, go on to refine the methods, and ignore questions of truth and axiomatics. Many mathematicians may find Kline's concern bewildering and beside the point; they will continue to "do" mathematics - inspired, if not by truth, then by beauty or by the very pleasure of the doing. The bright young buff, however, might enjoy the exposition and shrug off the dire conclusions. (Kirkus Reviews)