Ask Marilyn ® by Marilyn vos Savant is a column in Parade Magazine, published by PARADE, 711 Third Avenue, New York, NY 10017, USA. According to Parade, Marilyn vos Savant is listed in the "Guinness Book of World Records Hall of Fame" for "Highest IQ."
In her Parade Magazine column of November 21, 1993, Marilyn reports that she does not believe that Andrew Wiles has succeeded in proving Fermat's Last Theorem, because the proof relies on non-Euclidean (hyperbolic) geometry.
Note that in the discussion that follows, the notation x**n is used to represent x to the nth power.
Just because a tool is inappropriate for one task does not mean that tool is inappropriate for all other tasks. If we reject Wiles' proof of Fermat's Last Theorem, must we also reject Einstein's General Theory of Relativity?
When we are asked to solve a problem using "only a ruler and a straightedge," I think it's inappropriate to rely on non-Euclidean geometry. While Wiles' proof is clearly not the same as Fermat's "remarkable" proof, I don't understand why it's invalid simply because it relies on non-Euclidean geometry.
As a non-mathemetician, this is the strongest argument I can make. I'll surely be hearing from my creative readers.
As an aside, I seem to remember reading that the audience realized where Wiles' lecture was headed long before Wiles made any announcement.
Thanks to Charlie Kluepfel <ChasKlu@aol.com>, who wrote to clarify my explanation:
Doing constructions in Euclidean geometry with only a straightedge and compass is a construction problem. Fermat's Last Theorem is an existence problem. Anything goes in proving existence, and one is not limited in the tools that are used so long as they are correct. Just because one cannot construct a square with the same area as a given circle, or a cube with the twice the volume of another given cube, does not mean that these abstract spaces (the cubes or square and circle in question) do not exist as regions of space.
The key to understanding her Fermat's Last Theorem gaffe, which in my opinion is by far the worst of her errors, is to realize that geometry used to be the foundation of mathematics, in the following sense.
The foundation of mathematics consists of basic axioms and logical rules. All mathematics is built on this foundation in the sense that to demonstrate the validity of a mathematical result, one must show that it follows from the axioms via the logical rules.
Clearly, the axioms play an absolutely critical role in this setup. In the early days, the foundational axioms were geometric axioms. Nowadays, however, the most widely accepted foundational axioms are set-theoretic axioms. In particular, "ZFC" (standing for Zermelo-Fraenkel-Choice) is the "standard" foundation for mathematics today.
The underlying kernel of truth in vos Savant's argument is that if I make a claim and ask for a proof of it, I implicitly am asking for a proof from the standard axioms. If you then give a proof using nonstandard axioms, then you're not really answering my question.
Vos Savant made several errors. First, she assumed that geometry still forms the foundation of mathematics today, which it does not. Second, she assumed that the axioms of Euclidean geometry are the "standard" axioms of mathematics and that non-Euclidean geometries are "nonstandard" foundations for mathematics. Third, she assumed that since Wiles used "hyperbolic geometry" in his proof, he must have been giving a proof using a set of non-Euclidean, and therefore nonstandard, axioms.
If we grant these three errors, then her argument is a reasonable one. (Indeed, if Wiles had used a nonstandard axiom of set theory in an essential way his proof, most mathematicians would not have regarded Fermat's Last Theorem as having been resolved satisfactorily.) But since mathematics is based on set theory today, and since standard set theory is powerful enough to incorporate both Euclidean and non-Euclidean geometries, her argument is a non-starter.
The main mistake in Marilyn's argument was the implication that the Wiles' proof is wrong because he used some math outside the Arithmetics.
Marilyn's argument is wrong because it goes against a basic tenet of modern mathematics:
It is useful to remember a basic law of logical thinking:
So the question is, was the meaning of Fermat's theorem changed when Wiles proved it by using Hyperbolic geometry?
I don't think so!
P.S. None of the previous writers listed the important restriction that x, y, and z must be positive whole numbers! Without this restriction, there would be an infinite number of solutions, all of the form
z = (x**n+y**n)**(1/n)
The point is well taken that restrictions need to be placed on the variables, but the restrictions listed are incomplete. Adding Alexandre's restriction to the one in the question (that n must be a whole number larger than 2), we arrive with what Fermat undoubtedly considered to be the restrictions on the values in his theorem. It turns out, though, that these restrictions can be relaxed quite a deal, and FLT still holds.
Working under the assumption that FLT is true with the above restrictions (and from what I know, those who understand Dr. Wiles' proof are convinced it is), we can extend them to the following:
Part I: x, y, and z are rational numbers; x*y*z != 0. It should be fairly obvious that FLT will not hold if x, y, or z are allowed to be zero, so I'll skip ahead to the other half of the problem.
There are actually two pieces to this part: show that FLT holds when x, y, and z are something other than integers, and show that FLT holds if one or more of them are negative.
Rational numbers are defined to be numbers that can be expressed as a fraction of integers. So we can let x, y, and z be rational numbers, say a/b, c/d, and e/f, where all six variables are integers, and get: (a/b)**n + (c/d)**n = (e/f)**n
Now we simply multiply through by (b*d*f)**n and get: (a*d*f)**n + (b*c*f)**n = (b*d*e)**n
Since the product of integers is an integer, we have transformed the problem into one that has already been taken as true.
Next, we deal with the sign of x, y, and z. If n is even, then x**n, y**n, and z**n will always be positive, so any combination of signs are permitted. If n is odd, then the terms retain their sign, and we get one of the following:
1. x**n + y**n = z**n 2. x**n + y**n = -z**n 3. x**n - y**n = z**n 4. x**n - y**n = -z**n 5. -x**n + y**n = z**n 6. -x**n + y**n = -z**n 7. -x**n - y**n = z**n 8. -x**n - y**n = -z**n
(Note that I've made x, y, and z positive and inserted the negative sign where needed. This may appear wrong, but it can be shown that it results in an equivalent set of equations.)
It's now just a simple matter of rearranging the equations until all terms are positive. Doing this reduces the eight above equation forms to two: either there are two terms added and set equal to the third term (1, 3, 4, 5, 6, & 8), or all three terms are added to give us zero (2 & 7). The first is equivalent to FLT, and the second supports Fermat's conjecture since 3 positive numbers cannot add up to zero.
Part II: n is an integer; |n| > 2. Since we already assume that FLT holds for n > 2, we need only show that it holds for n < -2. This can be done when we realize that negative exponents merely create reciprocals of their opposite powers. Thus, x**(-n) + y**(-n) = z**(-n) becomes (1/x)**n + (1/y)**n = (1/z)**n. We need only multiply through by (x*y*z)**n. This gives us (y*z)**n + (x*z)**n = (x*y)**n. Since the product of any two rational numbers is another rational number, we have transformed the problem into a case that is already held to be true. Thus, FLT holds for n < -2.
The last thing to show is that FLT does not hold if n equals -1 or -2. In the interest of brevity, I will merely give examples showing that this is so: 6**(-1) + 3**(-1) = 2**(-1), and 20**(-2) + 15**(-2) = 12**(-2).
I suppose that technically speaking, FLT holds if n=0, since that will always reduce the equation to 1 + 1 = 1. Most mathematicians, however, add the restriction that n cannot equal zero, but I have yet to receive an explanation as to why, except for one that reduces to "because I said so."
Von Savant's confusion is truly fundamental. She does not appear to understand the difference between a postulate (ala Euclid) and an axiom (in modern mathematics). A postulate is something that one must assume is true, in order to use the theorems that follow from those assumptions. If we were trying to prove something about the physical world, we would first have to commit to a specific set of postulates about the real world.
An axiom (or set of axioms), on the other hand, is something which once proved to hold, allows all theorems proved from those axioms to be applied. Wiles showed that the axioms of hyperbolic geometry holds in a specific scenario related to Fermat's last theorem, and that is why he was then free to use the theorems of hyperbolic geometry which followed from those axioms in that specific scenario.
Wiles has no need to depend upon unsubstantiated postulates such as Euclid's, since he is not proving a fact about the physical world, but rather a fact about numbers.
In her book, Von Savant is confused about one other issue which is not described in this article. In particular, she doesn't understand the distinction between that mathematical principle of induction and the less formal use of the term induction used in other realms. She argues that induction is an invalid technique for proving a fact, since in induction one argues about all cases from just a few. In other words, she is confusing the following two definitions from Webster:
1: inference of a generalized conclusion from particular instances -- compare deduction
2b: mathematical demonstration of the validity of a law concerning all the positive integers by proving that it holds for the integer 1 and that if it holds for all the integers preceding a given integer it must hold for the next following integer
Definition (2b) is used by mathematicians to write proper and conclusive proofs, while definition (1) is used by expositors and writers in the hopes of constructing reasonably convincing arguments.
Jud McCranie <firstname.lastname@example.org> points out that axioms and postulates are synonyms. Axioms aren't proven -- they are assumed to be true.
David Wolfe <email@example.com> responds:
I propose the following simple textual changes to the first two paragraphs which attempt to circumvent the issue of distinguishing the terms "axiom" and "postulate" per se. Instead, I try to highlight how mathematics is interpreted and applied in modern vs. Ancient times.
Von Savant's confusion is truly fundamental. She does not appear to understand the difference between Euclid's postulates and a theory's axioms. Euclid's postulates were statements that one must assume is true, in order to use the theorems that follow from those assumptions. Since he was trying to prove facts about the physical world, he first committed to a specific set of postulates about the physical world.
An axiomatic theory, on the other hand, begins with a set of axioms which once known to hold, allows all theory proved from those axioms to be applied. Wiles showed that the axioms of hyperbolic geometry hold in a specific scenario related to Fermat's last theorem, and that is why he was then free to use the theorems of hyperbolic geometry which followed from those axioms in that specific scenario.
I can add a more simple explanation of why Marilyn is wrong. As is often the case with these kinds of things, she has not clearly stated (and clearly does not understand) exactly what has been proved impossible. What has been proved, using the theory of field extensions in abstract algebra, is that in Euclidean Geometry, one cannot construct a square whose area is the same as a given circle. The proof depends on features of Euclidean geometry that are simply false for other geometries, like hyperbolic geometry. For example, in hyperbolic geometry, the sum of the angles of a quadrilateral is always less than 360 degrees, and varies with the size of the quadrilateral. Theorems about similar triangles fail spectacularly in hyperbolic geometry.
Is hyperbolic geometry "real?" One can ask the same question about Euclidean geometry; after all, there is nothing so flat and perfect as the Euclidean plane in the "real world." The existence of Euclidean geometry follows from the usual axioms for the real numbers that are the basis for most of mathematics, including elementary calculus. The existence of hyperbolic geometry follows from those same axioms. One is as "real" as the other. If one believes in the integers (which one must if one is to consider even the statement of FLT) then one must believe in the real numbers, which can be constructed from the integers, and so one must believe in hyperbolic geometry, which is constructed using the real numbers.
When I was in 7th grade, I worked very hard trying to find a way to construct the trisection of an angle, despite having been told that the problem had been studied for centuries and finally proved impossible in the last century. Apparently Marilyn vos Savant, despite her high IQ, has never quite gotten past that adolescent state of mind that places one at the center of the universe, superior and invincible.