From the issue dated August 3, 2007

Ambiguity and Paradox in Mathematics


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  

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  

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  

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.

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

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,

A first attempt to solve the equation can be made by attempting to  
factor the equation, giving

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.	

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,	

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).	

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

can be made, so redefining the arguments gives

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  

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  

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).	

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

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),	

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

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),	

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)	

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

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  

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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LAST MODIFIED: February 15, 2006


Weisstein, Eric W. "Fermat's Last Theorem." From MathWorld--A Wolfram  
Web Resource.

s. e. anderson (author of "The Black Holocaust for Beginners" -  
Writers + Readers) +