BOOKS
509
Or, restating the Last Theorem in even fewer words than its Latin original:
"There are no positive, whole numbers
a,
hi
and
c
for which
an
+
h
n
=
en
when
11
>
2." Fermat's annotation had an addendum: "I have a truly mar–
velous demonstration of this proposition which this margin is too narrow
to contain."
For more than three centuries, mathematicians tried in vain to redis–
cover Fermat's lost proof of his Last Theorem (or to disprove the theorem
by finding a counterexample, i.e., turning up a Pythagorean triple
a,
hi
and
c
for which it is the case that
a
11
+
hit
=
C
It
when
n
>
2.) Eventually, in
1908, a rich German amateur mathematician, Paul Wolfskehl, endowed a
prize of 100,000 marks to be awarded by the Gottingen Academy of
Science for the first presentation of a valid proof-but apparently not for a
valid disproof or counterexample. (Based on a comparison of the price of
a double room, breakfast included, in a five-star Berlin hotel in 1908 [five
marks per night] and in 1997 [400 marks per night], I reckon that the
Wolskehl Prize was worth about five million dollars at the current rate of
exchange of 1.7 marks to the dollar.) Countless proofs were submitted to
the Academy over the years (621 alone during the first year after the
announcement of the Prize), most of them sent in by cranks and crackpots,
and everyone of them invalid. In view of these perennial failures, doubts
arose whether Fermat had, in fact, found a "marvelous demonstration" for
his theorem, despi te the later validation by prominent mathematicians of
many of Fermat's other brilliant theorems whose proofs he had kept to
himself.
At about the time that Wolfskehl offered his prize, the very resistance
of Fermat's Last Theorem to proof or disproof began to take on a philo–
sophical significance on its own that transcended the parochial problem of
Pythagorean triples. Was it possible that the theorem was an instance of a
mathematical proposition that can be stated in clear and unambiguous
terms and whose truth or falsity is, nevertheless, undecidable? This was a
deeply disturbing question because mathematics provides us with the most
reliable rational descri ption of the mental patterns (a.k.a. "reali ty") that we
impose on the world by our brain's selective culling and processing of the
welter of sensory data we constantly take in. Thus any inadequacy of the
descriptive power of mathematics would limit our rational understanding
of the outer world.
To eliminate that ominous possibility, David Hilbert set out in the
early years of this century to demonstrate the descriptive adequacy of
mathematics, by reducing it to an axiomatic system that it is both consis–
tent (in the sense that it is free of contradictions) and complete (in the
sense that the truth or falsity of any meaningful proposition could be
derived logically from the axioms.)
