Full Text of Savant FLT Parade Article
William G. Dubuque
Fermat's Last Theorem, as scanned by a HPIIcx scanner and OCRed with
Xerox's Textbridge. Hopefully this should make the article available
to the international audience.
[Parade Magazine, November 21, 1993, page 10]
Ask Marilyn
by Marilyn vos Savant
When a British professor solved a problem that
had been puzzling mathematicians for centuries,
newspapers reported the news on their front
pages. But for some of us, the mystery had just
begun. Who was this fellow Fermat, anyway,
and why the headlines over some notes scribbled
in a margin? Marilyn vos Savant sorts it all out
in this response to a reader's question and in a
book now out, titled "The World's Most Famous
Math Problem: The Proof of Fermat's Last
Theorem and Other Mathematical Mysteries"
(St. Martin's Press).
----
Andrew Wiles, a Princeton mathematics pro-
fessor, claims to have solved the most famous
problem in mathematics. The theorem--actu-
ally a conjecture until proved--was stated by
the French mathematician Pierre de Fermat
in the 17th century. It went this way: Can it
be proved that in the equation x^n + y^n = z^n
there is no solution if "n", represents any
whole number larger than 2? There are plen-
ty of solutions if "n" represents 2. For ex-
ample, 3^2 + 4^2 = 5^2. But if "n" represents 3
or more, according to Fermat, there are no
solutions. Have you ever tried to prove it, and
if so, did you succeed?.
-Joseph McGriff, Odessa, Tex.
No, I've never tried, and I don't think I would have
succeeded even if I had. Moreover, I don't think the
current work succeeds in proving "Fermat's last
theorem" either--even if no mathematical errors
are discovered in it. Here's why:
More than 350 years ago, Pierre de Fermat wrote
down his apparently simple little "theorem" in the
margins of a mathematical book he was reading,
adding that he had discovered a remarkable proof for
it but that there was no room to include the proof in
the margins too. He died without ever presenting
the step-by-step logic to substantiate this tantalizing
claim, confounding the best of mathematicians ever
since in their efforts to do so.
Since the arrival of computers, it has been shown
that the theorem clearly holds true, even for ex-
tremely high numbers. That might seem proof
enough for the general public, but for mathemati-
cians it's no proof at all. Finally, many of them came
to the reluctant conclusion that Fermat had made a
mistake and didn't have a proof after all.
But on June 23, 1993--at the end of a three-day
lecture series at the Isaac Newton Institute for Math-
ematical Sciences at Cambridge University in Eng-
land--Dr. Andrew Wiles, a British mathematician
who teaches at Princeton University, made a sur-
prise announcement that he had proved Fermat's
last theorem (also known as F.L.T.). Almost at once,
telephones began to ring, faxes churned out copy,
electronic mail zapped into computers all over the
world, and the communications satellites went into
overdrive.
Very few people knew what Wiles had been do-
ing for seven years in his little third-floor attic of-
fice at home, where he worked away in secret at the
problem--and he wanted it that way, for good rea-
son. After all, what would people think? Worse,
what would they think if he had worked on it for a
lifetime and failed? The assessments would proba-
bly not be charitable, especially for a man with a
wife and children and a house with an average as-
sortment of squeaky screen doors, leaf-filled gut-
ters and dandelions in the backyard. It would surprise
no one if, after seven years of this, every window in
the house eventually had become stuck shut.
But there's more to the story than what appeared
in the news. This is where I think it all goes astray.
The mathematics of today is a far cry from the
mathematics of Fermat's time. In brief, here's what
has happened in the field and how this relates to the
current work on Fermat's last theorem. The Euclid-
ean geometry of Fermat's day is a set of principles
derived by rigorous logical steps from the axioms
detailed by Euclid, the Greek mathematician of the
third century B.C. The fifth axiom is known as "Eu-
clid's parallel postulate," and it can be rephrased
this way: If a point lies outside a straight line, one
(and only one) straight line can be drawn through that
point that will be parallel to the first line.
Some mathematicians in the 19th century began
to disagree with the "parallel postulate," and a few
of them even invented their own geometries, called
non-Euclidean geometries, of which there are two
important forms--both of which replace the fifth
postulate with alternatives. One of the two main
alternatives allows an infinite number of parallels
through any outside point, from which was devel-
oped "hyperbolic" geometry. The other main alter-
native allows no parallels through any outside point,
from which was developed "elliptic" geometry.
Superficially, this seems ridiculous to the non-
mathematician, but the new systems of geometry
have their own definitions and systems of logic.
The best-known example of a non-Euclidean idea
is Einstein's general theory of relativity, which has
little valldity outside elliptic geometry. Most people
are unaware of this. If elliptic geometry is in error,
so is Einstein's world.
Wiles' proof is also non-Euclidean. The chain
of proof is based in hyperbolic geometry, which one
of its founders himself named "imaginary geome-
try." Here's the crux of the matter.
Three of the oldest problems in mathematics-
all more than 2000 years old-are known as "Dou-
bling the Cube," "Trisecting the Angle" and "Squar-
ing the Circle." (All constructions were to be
accomplished using only a ruler--as a straight edge,
not as a measuring device--and a compass.) The
problem of doubling the cube is to construct a cube
with twice the volume of a given cube; the prob-
lem of trisecting an arbitrary angle is to construct a
method to divide any given angle into three equal
parts (it must work for every angle); and the prob-
lem of squaring the circle is to construct a square
with an area equal to that of a given circle.
It wasn't until the 19th century that the problems
were all proved impossible to solve, and they are
now considered "famous impossibilities." Scientif-
ic American noted that "Fermat's last theorem dif-
fers from circle-squaring and angle-trisecting in that
those tasks are known to be impossible, and so any
purported solutions can be rejected out of hand."
Bearing all this in mind, what would we think if
it were discovered that Janos Bolyai, one of the
three founders of hyperbolic geometry, managed to
"square the circle"--but only by using his own hy-
perbolic geometry? Well, that's exactly what hap-
pened. And Bolyai himself said that his hyperbolic
proof would not work in Euclidean geometry.
So one of the founders of hyperbolic geometry
(the geometry used in the current proof of Fermat's
last theorem) managed to square the circle?! Then
why is it known as such a famous impossibility?
Has the circle been squared, or has it not?
Has Fermat's last theorem been proved, or has it
not? I would say it has not; if we reject a hyperbolic
method of squaring the circle, we should also reject
a hyperbolic proof of Fermat's last theorem. This is
not a matter of merely changing the rules (for ex-
ample, using a ruler as a measuring device instead
of a straight edge). It's much more significant than
that. Instead, it's a matter of changing whole defi-
nitions. And, regardless, it is logically inconsistent
to reject a hyperbolic method of squaring the circle
and accept a hyperbolic method of proving F.L.T.!
----
If you have a question for Marilyn vos Savant, who is
listed in 'The Guinness Book of World Records" Hall
of Fame for "Highest IQ," send it to: Ask Marilyn, PA-
RADE, 750 Third Ave., New York, N.Y. 10017. Because
of volume of mail, personal replies are not possible.