A snag in the fabric of our most rigorous science: Mathematics: The loss of Certainty, by Morris Kline. New York: Oxford University Press. $19.95.
If you are like me, the above-mentioned loss of certainty befell you five minutes into the first morning of high school calculus, and to this day your eyes slide over clots of mathematical equations, when they rudely interrupt good friendly English prose text, like a cheap ball point skidding across oilpaper. Yet despair not of Morris Kline's book. Think of it as a rich adn endlessly reticulating survey through the history of idea -- mathematical ideas mainly, but with continual interpenetration into natural science and logic -- marred only occasionally by skid-patches glistening with algebraic Xs and logs and those Greek letters of whic you never learned the names, and you may find large parts of interesting, it not easy.
Overlook also the bad writing, the dizzying mixture of metaphors, the annoying repetitions of incidental information, the unpersuasive attempt to claim grand intellectual breadth through trite quotes from Shakespeare, Hobbs, Goethe, and what you will be left with is really a very impressive history of the philosophy of mathematics.
It's an area of endeavor, as it turns out, in which some fascinating questions have been up for debate: You and I, in our childlike credulity, may still believe that 2 plus 2 definitely equals 4, that the sum of the angles in any triangle definitely equals 180 degrees, and that there can definitely be such a thing in our universe a two parallel lines. But word from the experts declares that the truth isn't so simple. Yes, mathematics still aspires to be the most "rigorous" of sciences, as it always has, but the problem, Kline observes, is that "there is no rigorous definition of rigor."
It was during the early 19th century, in the new philosphical climate brought to fullness by Hume and Kant, that the fabric of mathematical certainty began to unravel. In 1999 Karl Friedrich Gauss voiced his doubt that Euclid's so-called "paralled axiom" could be proved deductively from the other Euclidean axioms - which had been a crucial supposition throughout classical mathematics -- and from 1813 gauss began inventing the first non-Euclidean geometry.
Before the Georg Bernhard Riemann, a student of Gauss, was also creating a non-Euclidean geometry, one that was even perhaps more applicable to actual physical space than was Euclid's. About the same time augustin-Louis Cauchy was visited with the disconcerting conviction that all of classical mathematics, passed off for 2,000 years of mankind's supreme rational achievement, was actually, in Kline's words, "a wilderness of intuitions, plausible arguments, inductive reasoning and formal manipulation of symbolic expressions."
So Cauchy set out to rigorize mathematics, calculus first and then the other branches, and thereby launched a movement -- the axiomatization movement -- seeking to prove that all mathematical theory and practice follows indisputably from indisputable first axioms. This effort occupied many of the best mathematical minds of the Victorian years, and by 1900 Henri Poincare could announce, with ripe hubris: "One may say today that absolute rigor has been attained." He was, Kline tells us, more than a little premature.
Far fro being settled, the struggle to make mathematics sound and respectable went deeper still during the first decade of this century, with not just the axiomatic structure but the basic intellectual foundation of mathematics at issue, and two diametrically opposed schools of thought offering systems of justification. The logicists, most prominently and ably represented by Bertrand Russell, claimed all mathematics to be derivable from logic, and therefore, in turn, from a body of manifest truths.
Against that view intuitionist school argued, sometimes perversely, its Kantian position that derivation of mathematics from logical principles was less trustworthy than fabricating it from firect mental intuitions. Mathematics, it said, was utterly independent of the real world, an activity of and solely within the human mind. Or as Leopold Kronecker quipped one evening after dinne: "God made the integers; all the rest is the work of man.c
Into the fray against both these factions eventually came two further schools of thought: the formalists, led by David Hilbert, who personally viewed the intuitions much the way Glenn Gould must view punk rock; and those in the set-theoretic school, who for some unexplained reason write under the collective pseudonym "Nicholas Bourbaki," and look disdainfully upon all the others. And so the controversy continues today, among four irreconcilable viewpoints, over the very significant questions, "Can the entire edifice of mathematics be proved in any objective sense to be true? And if so, how?" -- with no sus in failure.
You may not be susceptible to the odd, quiet charms of mathematical ratiocination. Neither am I. But what we have here, what animates Morris Kline's book, is a first-class and impassioned war of ideas.