American Scientist

To the Editors:

I think the pessimism of Brian Hayes's column on mathematical proof ("Foolproof," Computing Science, January-February) is unwarranted. Morris Kline, mentioned in the article, was known for his extreme pessimism, and yet he wrote a magnificent history of the subject without worries about its certainty. There is no corpus of knowledge more certain than mathematical knowledge. If there are difficulties therein, they are understandable, since they exist in two regions: foundations and the cutting edges of the field.

Gödel's undecidability theorem isn't a bunker-buster as so much popular literature claims. The nonscientist should see it as eradicating forever all mathematical hubris—something no other endeavor can claim, unfortunately. The problem now moves on to propositions that may indeed be provable, but do not seem so because it would take more than the lifetimes of the best mathematicians to prepare themselves for the task (and they don't know that, of course).

There is no reason for Hayes to call an axiomatic structure brittle. It is an un-breachable fortress against falsehood precisely because no false theorem can occur within its walls. A false theorem is as absurd in such a system as is moving a rook diagonally. In fact, mathematicians assume absurdities in order to prove that the negation of the absurdity is correct. G. H. Hardy puts it so well when considering proof by contradiction: "It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game." On what other stage is such a delicate ballet between truth and falsehood ever played?

And why split hairs about the reason for proof? To differ with David Bressoud's point, establishing the truth of a proposition is exactly the same as explaining why the statement is true (except for pedagogical issues). Thus, the two functions are of identical importance. A proposition's proof is so much a part of the proposition itself that we understand them to be an ordered pair. As such, the proof need be displayed just once, but it always lies behind the assertion of the proposition, as if it were a hyperlink.

I do agree with Hayes's summary of mathematical history as a series of catastrophes, but the resolution of such disasters always adds more truth to mathematics, as time has witnessed. Please return the pessimism to Samuel Beckett!

Luis F. Moreno
Broome Community College
Binghamton, New York

Mr. Hayes responds:

"Pessimism" is not at all how I would describe my attitude toward mathematical proof. I think it is a difficult art—but that's part of what makes it worth practicing. As for Samuel Beckett, I'll let him speak for himself. "Fail again," he said. "Fail better."

To the Editors:

Brian Hayes's interesting essay suggests that the polemically correct haggling going on in the mathematics community about "proof" may be only the latest manifestation of a general philosophical paradigm shift that was also manifest in late 19th-century physics and psychology. Kurt Gödel's thinking about consistency mirrors Heisenberg's uncertainty principle, Pauli's exclusion principle and the probabilistic nature of quantum theory. And the multiple realities of Freudian psychology echo in the relativism of deconstruction and postmodernism in the "lit crit" courses of university English departments. Mathematicians' near-fetishization of chaos theory and fractals says more about "what's hot and what's not" (i.e. fractionalization, uncertainty and relativism) than it does about the pursuit of ultimate knowledge. Of course, mathematicians might not deign to any such "lumping together" of their quest for Truth with that of the experimentalists of science and the egalitarians in the humanities. All of these epistemological struggles suggest that there's as much to learn about the search for "truth" in mathematics from the study of the anthropology and sociology of academia as there is from the navel-gazing subtleties of what constitutes mathematical "proof" and what doesn't.

Alan Hull
Conyers, GA