LETTERS TO THE EDITORS
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