March
2, 2005 Edition > Section: Arts and
Letters
Kurt Godel's Astounding
Achievement
Books
BY JOHN DERBYSHIRE
March 2, 2005
Out in the remotest regions of mathematics, far from the
bustling and long-populated center, out where this great
thriving empire adjoins the windswept badlands of philosophy,
is the topic called Foundations. Here mathematicians use the
techniques of their discipline to inquire into the nature of
that discipline itself, into the very fundamentals of math:
number, set, proof, and contradiction. The ideas that underlie
Foundations can be traced back to Leibniz and Descartes, but
as a coherent subject of study it is quite new, having
properly begun only with the work of Giuseppe Peano in the
1890s.
To this day the most sensational achievement in Foundations
remains the 1931 paper "On Formally Undecidable Propositions
of 'Principia Mathematica' and Related Systems" by the
Austrian Kurt Godel. In that paper Godel (1906-78) brought to
an abrupt end the program of strict formalism in Foundations -
the program, that is, to reduce mathematics to a content-free
game in which arbitrary symbols are combined according to
arbitrary rules. "Principia Mathematica" was one of the great
early texts of Foundations. Bertrand Russell, one of its
authors, had declared that: "Mathematics is that subject in
which we do not know what we are talking about, nor whether
what we say is true." That was the keynote of formalism; that
was the conception of mathematics shattered by Godel's great
paper. Thanks to Godel we now know - with mathematical
certainty! - that any attempt to reduce math to pure formalism
must fail, just as the attempt to comb down a sphere covered
with hair must always leave a "whorl point" where the combing
collapses into uncertainty.
In fact, as Rebecca Goldstein says in "Incompleteness"
(Atlas Books, 304 pages, $22.95), Godel's paper " is the third
leg, together with Heisenberg's uncertainty principle and
Einstein's relativity, of that tripod of theoretical
cataclysms that have been felt to force disturbances deep down
in the foundations of the 'exact sciences'" during the 20th
century. The results in that paper, she goes on to say, "range
far beyond their narrow formal domain, addressing such vast
and messy issues as the nature of truth and knowledge and
certainty."
That last assertion is open to question. There are
philosophers who will deny that Godel's results have any
consequences at all for epistemology (that is, the
philosophical theory of knowledge). If the point is arguable,
though, Rebecca Goldstein is just the person to argue it.
Professor of philosophy at Trinity College, Hartford, and the
author of several novels and a short-story collection, Ms.
Goldstein has had a long-standing interest, apparent in her
fiction, in the nature of truth and in the connections between
thought, language, and the external world. In this new book
she offers a discursive account of Godel, his influences, his
life, and the contents of that stunning 1931 paper.
"Incompleteness" is a difficult book to categorize. It is
not really a biography, though it includes a sufficient
account of Godel's rather eventless life and highly peculiar
personality. (The standard biography, published in 1997, is by
John W. Dawson.)
Nor is it merely a popularized account of Godel's 1931
result, though a sketch of that result and the method Godel
used to prove it is included. Ms. Goldstein, rather, uses
Godel as a frame on which to hang some commentaries on
epistemology and related matters.
Her central concern is the nature of mathematical truth.
What does a mathematician actually mean by saying that such
and such a proposition is true? The strict formalists of the
early 20th century would have replied: He means only that the
proposition can be derived from agreed axioms by agreed rules
of deduction. This was, to borrow a phrase from logician
Martin Davis, the applecart that Godel overturned.
Godel was a firm Platonist, who believed that mathematical
objects, while of course they do not belong to the physical
world, none the less have a reality that we can "trust," in
the same sense that we "trust" what our senses tell us about
physical objects. He wrote: "I don't see any reason why we
should have any less confidence in this kind of perception,
i.e. in mathematical intuition, than in sense perception,
which induces us to build up physical theories and to expect
that future sense perceptions will agree with them and,
moreover, to believe that a question not decidable now has
meaning and may be decided in the future."
This clear Platonism put Godel at odds with the foremost
mathematician of his time, David Hilbert, who had written
that: "Mathematics is a game played according to certain
simple rules with meaningless marks on paper." In a landmark
address in 1900, Hilbert had proposed 23 problems for
mathematicians to concentrate on in the new century. Problem
no. 2 was to prove that the rules of arithmetic, properly
stated, would suffice to prove every conceivable arithmetic
statement either true or false. Hilbert had then gone on to
launch a formalist program, devising a means to investigate
the nature of mathematical proof by dint of techniques he
named "metamathematics." Godel turned these same techniques
against the formalist program and exposed its inherent
limitations.
Godel's Platonism in fact went against all the main
currents of thought in his time, not only in mathematics but
also in philosophy and in physics. From 1926 to 1928 Godel was
a regular at the weekly meetings of the Vienna Circle, a
discussion group led by the philosopher Moritz Schlick.
Already a Platonist, Godel must have found the radical
empiricism of the Circle uncongenial. As Ms. Goldstein says:
"He could not have been more at odds with their
metaconvictions." It is something of a mystery why he stuck
with them for so long. Ludwig Wittgenstein was a hero to the
Circle (though he declined an invitation to join), and Ms.
Goldstein has some very interesting things to say about the
parallels between Wittgenstein's "Unsayable" and Godel's
"Undecidable."
The connection with physics is even more intriguing. It
would have been striking enough at any time to say, as Godel
did, that mathematical objects possess real existence
independent of human minds. It was doubly so to say this when
the physical world of solid objects was itself evaporating
away into a cloud of quantumtheoretical abstractions.
Prominent physicists of Godel's time - notably Werner
Heisenberg - denied the existence of any real-world
independent of observers. Godel's first intention, on entering
the University of Vienna in 1924, had been to study physics,
and he maintained a keen interest in the subject all his
life.
Albert Einstein was Godel's dearest friend, and the two of
them spent many hours together in the 1940s and early 1950s,
strolling the lanes and pathways around Princeton's Institute
for Advanced Study, where they were both employed. Ms.
Goldstein's book is particularly illuminating on the
friendship between these two men of genius, and on their
common aversion to the denial of objectivity, both physical
and mathematical, so characteristic of their time - that cast
of mind that later seeped into the humanities and fine arts,
poisoning them with the absurdities of "postmodernism."
"Incompleteness" is a difficult book, but not unnecessarily
so. This is difficult material, at the borders of what we
understand about human knowledge. The author has skillfully
humanized it by showing us Godel, Wittgenstein, and Einstein
in their work, their friendships, and their disagreements.
Perhaps only a novelist could have done this. Rebecca
Goldstein has, in any case, done it superbly well.
Mr. Derbyshire last wrote in these pages on Pol
Pot.
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