Mathematical proof is foolproof, it seems, only in the absence of fools
These incidents and others like them have led to talk of a crisis in mathematics, and to fears that proof cannot be trusted to lead us to eternal and indubitable truth. Already in 1972 Philip J. Davis of Brown University was writing:
The authenticity of a mathematical proof is not absolute but only probabilistic.... Proofs cannot be too long, else their probabilities go down, and they baffle the checking process. To put it another way: all really deep theorems are false (or at best unproved or unprovable). All true theorems are trivial.
A few years later, in Mathematics: The Loss of Certainty, Morris Kline portrayed mathematics as a teetering superstructure with flimsy timbers and a crumbling foundation; continuing with this architectural conceit, he argued that proofs are "a façade rather than the supporting columns of the mathematical structure."
Davis and Kline both wrote as mathematical insiders—as members of the club, albeit iconoclastic ones. In contrast, John Horgan positioned himself as a defiant outsider when he wrote a Scientific American essay titled "The Death of Proof" in 1993. "The doubts riddling modern human thought have finally infected mathematics," he said. "Mathematicians may at last be forced to accept what many scientists and philosophers already have admitted: their assertions are, at best, only provisionally true, true until proved false."
My own position as an observer of these events is somewhere in the awkward middle ground, neither inside nor outside. I am certainly not a mathematician, and yet I have been an embedded journalist in the math corps for so long that I cannot claim detachment or impartiality. I report from the no man's land between the fences.
I do believe there is a kind of crisis going on—but only because the entire history of mathematics is just one crisis after another. The foundations are always crumbling, and the barbarians are always at the gate. When Haken and Appel published their computer-aided proof, it was hardly the first time that a technical innovation had stirred up controversy. In the 17th century, when algebraic methods began intruding into geometry, the heirs of the Euclidean tradition cried foul. (Hobbes was one of them.) At the end of the 19th century, when David Hilbert introduced nonconstructive proofs—saying, in effect, "I know x exists, but I can't tell you where to look for it"—there was another rebellion. Said one critic: "This is not mathematics. This is theology."
All in all, the crisis of the present moment seems mild compared with that of a century ago, when paradoxes in set theory led to Gottlob Frege's lament, "Alas, arithmetic totters." In response to that crisis, a rescue party of ambitious mathematicians, led by Hilbert, set out to rebuild the edifice of mathematics on a new foundation. Hilbert's plan was to apply the process of proof to proof itself, showing that the axioms and theorems of mathematics can never lead to a contradiction—that you can never prove both "x" and "not x." The outcome is well known: Kurt Gödel proved instead that if you insist on consistency, there are true statements you can't prove at all. You might think that such a Tower of Babel catastrophe would scatter the tribes of mathematics for generations, but mathematicians have carried on.
That some of the latest proofs from the frontiers of mathematical research are difficult and rely on novel tools seems to me utterly unexceptional. Of course the proofs are hard to digest; they were hard to create. These are solutions to problems that have stumped strong minds for decades or centuries. When Perelman's proof of the Poincaré conjecture defeats my attempts at understanding, this is a disappointment but not a surprise. (I can't keep pace with Olympic marathoners either.) If I have a worry about the state of mathematics, it's not the forbidding inaccessibility of the deepest thinkers; rather it's my own clumsiness when I tackle perfectly humdrum problems, far from the frontiers of knowledge.
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