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From the issue dated August 3, 2007

Ambiguity and Paradox in Mathematics

By WILLIAM BYERS

Many people believe that mathematics provides a model of what thinking is, or should be. They imagine that mathematical thinking always proceeds in a logically rigorous, step-by-step fashion from one truth to another, like a formal proof or a computer program. In fact, insights in mathematics — whether they are the scholar's breakthroughs or the student's leap to a new level of understanding — involve a different mode of thinking that is essentially nonlogical.

Ambiguity and its cousins, contradiction and paradox, are everywhere in mathematics, both in content and thinking. Strangely, the subject that appears to be the very paradigm of reason, and that is therefore the model for many other disciplines, contains as an irreducible element exactly what reason ostensibly does away with. Mathematical thought has nonrational, though not irrational, components.

Let me be more precise about what I mean by ambiguity: It is the existence of multiple, possibly inconsistent, points of view. In other words, an ambiguous situation is a single situation or idea that can be seen from two or more conflicting viewpoints.

A good metaphor for ambiguity is binocular vision. When you look at things out of one eye, the world seems flat and two-dimensional. However, when you use both eyes, the inconsistent viewpoints registered by each eye combine in the brain to produce a unified view that includes something entirely new: depth perception. In the same way, the conflicting points of view in an ambiguous situation may give birth to a new, higher-order understanding.

The metaphor reveals that ambiguity refers not only to an objective description of a situation, but also to the manner in which situations of inconsistency and even conflict can be resolved. Thus, ambiguous situations contain the potential for change; they are dynamic and can be creative. Ambiguity points to a valid way of thinking involved in mathematics, one that needs to find its place alongside logic if we are to account for the power and effectiveness of the discipline.

Ambiguity, as I use the term, is no great mystery; it is present in every joke. A joke typically has two conflicting points of view, one of which is not explicit. The conflict is resolved by "getting the joke." A joke is analogous to a moment of creativity precisely because you have to "get it"; an explanation will not do. When you do get the joke, the tension that arose because of the initial conflict dissolves in laughter.

Not only is ambiguity essential to humor, it is common in poetry and the fine arts. Leonard Bernstein said that it "is one of art's most potent and aesthetic functions. The more ambiguous, the more expressive." To go even further afield, ambiguity is the heart of Zen Buddhism. Koans, those verbal puzzles that are the basis of Zen training, always contain an ambiguous core. For example: "What is the sound of one hand clapping?" In contrast with our normal tendency to avoid the paradoxical, the point of a koan is to focus on its ambiguities.

In many mathematical situations, ambiguity is an essential ingredient. A list of such situations would include: addition, decimal numbers, variables in algebra, the square root of two, equations and functions, and even the ideas behind one of the recent triumphs of theoretical mathematics, the proof of Fermat's Last Theorem [see below].

Take, for example, the equation 1 + 1 = 2. The statement seems to be clear and precise. We feel that we understand it completely, and that nothing further needs to be said about it. Yet Bertrand Russell and Alfred North Whitehead spent an entire volume of their monumental Principia Mathematica just getting to that point. The numbers 1 and 2 are in fact extremely deep and important ideas. They are basic to science and religion, to perception and cognition. While 1 represents unity, 2 represents duality. What could be more fundamental?

The equation also contains an equal sign. The notion of equality is another very basic idea, whose meaning grows the more we think about it. Then we have the equation itself, which states that the concepts of unity and duality have a relationship with one another that we represent by equality — in other words, there is unity in duality, and duality in unity. That deeper structure in the equation is typical of an ambiguous situation.

Ambiguity is also present in physics. Take Einstein's famous equation E = mc[squared]. We usually think of that equation as saying that there are two things, matter and energy, which are related. But I prefer to think that it means there is fundamentally one ambiguous essence, which we call matter in certain situations and energy in others. The idea of ambiguity captures what is really going on in many other situations in physics, such as complementarity in quantum mechanics — which holds that an electron, for example, is paradoxically both a localized particle and a probability wave, even though those two descriptions are normally mutually exclusive.

Bringing the ambiguous nature of mathematics to the fore should not be construed as an attack on its logical structure. Logic is indispensable to mathematics in many ways. But logic is only one element, and a rather superficial one at that.

The true essence of mathematics is found in its ideas. Ideas are a way of categorizing facts or data by revealing relationships that may have gone unacknowledged. In short, an idea is an organizing principle.

Are ideas logical? Logic is involved in verifying an idea's validity or in relating one idea to others, but forcing an idea into a logical mold may have the unwanted effect of killing the idea. Think of a metaphor like Shakespeare's "All the world's a stage ... ." Of course the world is not actually a stage, so logically the statement is false. Yet on another level, the metaphor reveals a deeper meaning. Even a simple equation like 1 + 1 = 2 can be thought of as a metaphor in that way, with a deeper meaning waiting to be revealed.

Mathematical research often starts with an idea and attempts to prove its validity by putting the idea into a logically precise form. That often leads to a problem, which is resolved by modifying the original idea or coming up with a new one. The process continues until the mathematician reaches a satisfactory result.

Many great mathematical ideas involve paradoxes. Zero, or the nothing that is, is a classic example. Even some valid proofs, like the famous incompleteness theorem of Kurt Gödel, have a paradox as their central idea. Ideas can come from anywhere, even from mistakes. Goro Shimura said of his colleague Yutaka Taniyama — with whom he developed a mathematical conjecture that had profound implications for reformulating certain problems — that "he was gifted with the special capability of making many mistakes, mostly in the right direction."

Why is it important to think about the nature of mathematics? What follows from basing our view of the discipline on what mathematicians actually do and how they actually think, instead of mistaken preconceptions about them?

Mathematics integrates the strictest logical criteria with the deepest creativity. When we teach and apply the discipline, when we use it as a model for other fields or for the way human beings think, we tend to focus on its logical or algorithmic dimension. But that approach, like using the computer as a model of the human mind, ignores the ambiguity that is the foundation of creative thinking; it focuses on the superficial. The creative uses of the mind — even in mathematics — cannot be reduced to the algorithmic.

The tension between logical precision and creative openness plays out every day in mathematics classrooms all over the world. Success in math requires the integration of logic and ambiguity, yet ambiguous aspects of the discipline are often ignored because many mathematicians are uncomfortable with ambiguity.

In addition, many students who are highly skilled in mathematics are reluctant to give up a way of thinking that has brought them success in the past. To use an example from elementary school, a child can be introduced to multiplication as repeated addition, but if she holds on to that way of thinking, she will never truly master multiplication. Poor performance in mathematics courses may not be a function of raw intelligence as much as an inability to be flexible in thinking — an inability to grasp ambiguity.

Flexible thinking is discontinuous. It cannot be mastered by breaking a situation down into a series of small steps; it requires a breakthrough. The teacher must not impose some supposedly correct point of view, but must guide students as they confront and see past obstacles.

Our whole society is captivated by a mythology of reason, which has come down to us from the ancient Greeks. Reason is one of the most influential ideas in history, but it has unfortunate side effects when it is considered to be the unique, objective truth. In the teaching of mathematics, it gives the subject a rigidity that has traumatized countless individuals who are otherwise highly intelligent and competent. It has also contributed to the division between the arts and the sciences. Overemphasizing reason has led to the establishment of a series of false dichotomies: the content versus the practice of mathematics, the human brain versus the human mind, and even the human being versus nature. Ambiguity is a vehicle for putting those pieces back together.

Mathematics, the sciences, and the arts are all ways in which people try to understand themselves and the world around them. Like the other disciplines, mathematics is fundamentally an exercise in human creativity and needs to be evaluated as such, not merely on how precise or consistent it is. Studying the nature of mathematics from that point of view would give us important insights into the nature of the human mind.

William Byers is a professor of mathematics at Concordia University, in Montreal, and author of How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics, published by Princeton University Press in June.

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Fermat's last theorem is a theorem first proposed by Fermat (in 1637) in the form of a note scribbled in the margin of his copy of the ancient Greek text Arithmetica  by Diophantus. The scribbled note was discovered posthumously, and the original is now lost. However, a copy was preserved in a book published by Fermat's son. In the note, Fermat claimed to have discovered a proof that the Diophantine equation x^n+y^n==z^n has no integer solutions for n>2 and x,y,z!=0.

The full text of Fermat's statement, written in Latin, reads "Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet" (Nagell 1951, p. 252). In translation, "It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain."

As a result of Fermat's marginal note, the proposition that the Diophantine equation
x^n+y^n==z^n,
(1)

where x, y, z, and n are integers, has no nonzero solutions for n>2 has come to be known as Fermat's Last Theorem. It was called a "theorem" on the strength of Fermat's statement, despite the fact that no other mathematician was able to prove it for hundreds of years.

Note that the restriction n>2 is obviously necessary since there are a number of elementary formulas for generating an infinite number of Pythagorean triples (x,y,z) satisfying the equation for n==2,
x^2+y^2==z^2.
(2)

A first attempt to solve the equation can be made by attempting to factor the equation, giving
(z^(n/2)+y^(n/2))(z^(n/2)-y^(n/2))==x^n.
(3)

Since the product is an exact power,
{z^(n/2)+y^(n/2)==2^(n-1)p^n; z^(n/2)-y^(n/2)==2q^nor{z^(n/2)+y^(n/2)==2p^n; z^(n/2)-y^(n/2)==2^(n-1)q^n.
(4)

Solving for y and z gives
{z^(n/2)==2^(n-2)p^n+q^n; y^(n/2)==2^(n-2)p^n-q^nor{z^(n/2)==p^n+2^(n-2)q^n; y^(n/2)==p^n-2^(n-2)q^n,
(5)

which give
{z==(2^(n-2)p^n+q^n)^(2/n); y==(2^(n-2)p^n-q^n)^(2/n)or{z==(p^n+2^(n-2)q^n)^(2/n); y==(p^n-2^(n-2)q^n)^(2/n).
(6)

However, since solutions to these equations in rational numbers are no easier to find than solutions to the original equation, this approach unfortunately does not provide any additional insight.

If an odd prime p divides n, then the reduction
(x^m)^p+(y^m)^p==(z^m)^p
(7)

can be made, so redefining the arguments gives
x^p+y^p==z^p.
(8)

If no odd prime divides n, then n is a power of 2, so 4|n and, in this case, equations (7) and (8) work with 4 in place of p. Since the case n==4 was proved by Fermat to have no solutions, it is sufficient to prove Fermat's last theorem by considering odd prime powers only.

Similarly, is sufficient to prove Fermat's last theorem by considering only relatively prime x, y, and z, since each term in equation (1) can then be divided by GCD(x,y,z)^n, where GCD(x,y,z) is the greatest common divisor.

The so-called "first case" of the theorem is for exponents which are relatively prime to x, y, and z (p∤x,y,z) and was considered by Wieferich. Sophie Germain proved the first case of Fermat's Last Theorem for any odd prime p when 2p+1 is also a prime. Legendre subsequently proved that if p is a prime such that 4p+1, 8p+1, 10p+1, 14p+1, or 16p+1 is also a prime, then the first case of Fermat's Last Theorem holds for p. This established Fermat's Last Theorem for p<100. In 1849, Kummer proved it for all regular primes and composite numbers of which they are factors (Vandiver 1929, Ball and Coxeter 1987).

The "second case" of Fermat's last theorem is "p divides exactly one of x, y, z. Note that p|x,y,z is ruled out by x, y, z being relatively prime, and that if p divides two of x, y, z, then it also divides the third, by equation (8).

Kummer's attack led to the theory of ideals, and Vandiver developed Vandiver's criteria for deciding if a given irregular prime satisfies the theorem. Genocchi (1852) proved that the first case is true for p if (p,p-3) is not an irregular pair. In 1858, Kummer showed that the first case is true if either (p,p-3) or (p,p-5) is an irregular pair, which was subsequently extended to include (p,p-7) and (p,p-9) by Mirimanoff (1905). Vandiver (1920ab) pointed out gaps and errors in Kummer's memoir which, in his view, invalidate Kummer's proof of Fermat's Last Theorem for the irregular primes 37, 59, and 67, although he claims Mirimanoff's proof of FLT for exponent 37 is still valid.

Wieferich (1909) proved that if the equation is solved in integers relatively prime to an odd prime p, then
2^(p-1)=1 (mod p^2).
(9)

(Ball and Coxeter 1987). Such numbers are called Wieferich primes. Mirimanoff (1909) subsequently showed that
3^(p-1)=1 (mod p^2)
(10)

must also hold for solutions relatively prime to an odd prime p, which excludes the first two Wieferich primes 1093 and 3511. Vandiver (1914) showed
5^(p-1)=1 (mod p^2),
(11)

and Frobenius extended this to
11^(p-1),17^(p-1)=1 (mod p^2).
(12)

It has also been shown that if p were a prime of the form 6x-1, then
7^(p-1),13^(p-1),19^(p-1)=1 (mod p^2),
(13)

which raised the smallest possible p in the "first case" to 253747889 by 1941 (Rosser 1941). Granville and Monagan (1988) showed if there exists a prime p satisfying Fermat's Last Theorem, then
q^(p-1)=1 (mod p^2)
(14)

for q==5, 7, 11, ..., 71. This establishes that the first case is true for all prime exponents up to 714591416091398 (Vardi 1991).

The "second case" of Fermat's Last Theorem (for p|x,y,z) proved harder than the first case.

Euler proved the general case of the theorem for n==3, Fermat n==4, Dirichlet and Lagrange n==5. In 1832, Dirichlet established the case n==14. The n==7 case was proved by Lamé (1839; Wells 1986, p. 70), using the identity
(X+Y+Z)^7-(X^7+Y^7+Z^7)==7(X+Y)(X+Z)(Y+Z)x[(X^2+Y^2+Z^2+XY+XZ+YZ)^2+XYZ(X+Y+Z)].
(15)

Although some errors were present in this proof, these were subsequently fixed by Lebesgue (1840). Much additional progress was made over the next 150 years, but no completely general result had been obtained. Buoyed by false confidence after his proof that pi is transcendental, the mathematician Lindemann proceeded to publish several proofs of Fermat's Last Theorem, all of them invalid (Bell 1937, pp. 464-465). A prize of 100000 German marks, known as the Wolfskehl Prize, was also offered for the first valid proof (Ball and Coxeter 1987, p. 72; Barner 1997; Hoffman 1998, pp. 193-194 and 199).

A recent false alarm for a general proof was raised by Y. Miyaoka (Cipra 1988) whose proof, however, turned out to be flawed. Other attempted proofs among both professional and amateur mathematicians are discussed by vos Savant (1993), although vos Savant erroneously claims that work on the problem by Wiles (discussed below) is invalid. By the time 1993 rolled around, the general case of Fermat's Last Theorem had been shown to be true for all exponents up to 4x10^6 (Cipra 1993). However, given that a proof of Fermat's Last Theorem requires truth for all exponents, proof for any finite number of exponents does not constitute any significant progress towards a proof of the general theorem (although the fact that no counterexamples were found for this many cases is highly suggestive).

In 1993, a bombshell was dropped. In that year, the general theorem was partially proven by Andrew Wiles (Cipra 1993, Stewart 1993) by proving the semistable case of the Taniyama-Shimura conjecture. Unfortunately, several holes were discovered in the proof shortly thereafter when Wiles' approach via the Taniyama-Shimura conjecture became hung up on properties of the Selmer group using a tool called an Euler system. However, the difficulty was circumvented by Wiles and R. Taylor in late 1994 (Cipra 1994, 1995ab) and published in Taylor and Wiles (1995) and Wiles (1995). Wiles' proof succeeds by (1) replacing elliptic curves with Galois representations, (2) reducing the problem to a class number formula, (3) proving that formula, and (4) tying up loose ends that arise because the formalisms fail in the simplest degenerate cases (Cipra 1995a).

The proof of Fermat's Last Theorem marks the end of a mathematical era. Since virtually all of the tools which were eventually brought to bear on the problem had yet to be invented in the time of Fermat, it is interesting to speculate about whether he actually was in possession of an elementary proof of the theorem. Judging by the tenacity with which the problem resisted attack for so long, Fermat's alleged proof seems likely to have been illusionary. This conclusion is further supported by the fact that Fermat searched for proofs for the cases n==4 and n==5, which would have been superfluous had he actually been in possession of a general proof.

In the Homer^3 episode of the television program The Simpsons, the equation 1782^(12)+1841^(12)==1922^(12) appeared at one point in the background. Expansion reveals that only the first 9 decimal digits match (Rogers 2005). The episode The Wizard of Evergreen Terrace mentions 3987^(12)+4365^(12)==4472^(12), which matches not only in the first 10 decimal places but also the easy-to-check last place (Greenwald).

SEE ALSO: abc Conjecture, Beal's Conjecture, Bogomolov-Miyaoka-Yau Inequality, Euler System, Fermat-Catalan Conjecture, Fermat's Theorem, Generalized Fermat Equation, Mordell Conjecture, Pythagorean Triple, Ribet's Theorem, Selmer Group, Sophie Germain Prime, Szpiro's Conjecture, Taniyama-Shimura Conjecture, Vojta's Conjecture, Waring Formula. [Pages Linking Here]

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