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)
This work stresses the illogical manner in which mathematics has developed, the question of applied mathematics as against 'pure' mathematics, and the challenges to the consistency of mathematics' logical structure that have occurred in the twentieth century.
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