# How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics

# How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics

## Synopsis

To many outsiders, mathematicians appear to think like computers, grimly grinding away with a strict formal logic and moving methodically--even algorithmically--from one black-and-white deduction to another. Yet mathematicians often describe their most important breakthroughs as creative, intuitive responses to ambiguity, contradiction, and paradox. A unique examination of this less-familiar aspect of mathematics, *How Mathematicians Think* reveals that mathematics is a profoundly creative activity and not just a body of formalized rules and results.

Nonlogical qualities, William Byers shows, play an essential role in mathematics. Ambiguities, contradictions, and paradoxes can arise when ideas developed in different contexts come into contact. Uncertainties and conflicts do not impede but rather spur the development of mathematics. Creativity often means bringing apparently incompatible perspectives together as complementary aspects of a new, more subtle theory. The secret of mathematics is not to be found only in its logical structure.

The creative dimensions of mathematical work have great implications for our notions of mathematical and scientific truth, and *How Mathematicians Think* provides a novel approach to many fundamental questions. Is mathematics objectively true? Is it discovered or invented? And is there such a thing as a "final" scientific theory?

Ultimately, *How Mathematicians Think* shows that the nature of mathematical thinking can teach us a great deal about the human condition itself.

## Excerpt

A FEW YEARS AGO the PBS program *Nova* featured an interview with Andrew Wiles. Wiles is the Princeton mathematician who gave the final resolution to what was perhaps the most famous mathematical problem of all time—the Fermat conjecture. The solution to Fermat was Wiles's life ambition. “When he revealed a proof in that summer of 1993, it came at the end of seven years of dedicated work on the problem, a degree of focus and determination that is hard to imagine.” He said of this period in his life, “I carried this thought in my head basically the whole time. I would wake up with it first thing in the morning, I would be thinking about it all day, and I would be thinking about it when I went to sleep. Without distraction I would have the same thing going round and round in my mind.” In the *Nova* interview, Wiles reflects on the process of doing mathematical research:

Perhaps I can best describe my experience of doing mathe

matics in terms of a journey through a dark unexplored

mansion. You enter the first room of the mansion and it's

completely dark. You stumble around bumping into the fur

niture, but gradually you learn where each piece of furni

ture is. Finally after six months or so, you find the light

switch, you turn it on, and suddenly it's all illuminated. You

can see exactly where you were. Then you move into the

next room and spend another six months in the dark. So

each of these breakthroughs, while sometimes they're mo

mentary, sometimes over a period of a day or two, they are

the culmination of—and couldn't exist without—the many

months of stumbling around in the dark that precede them.

This is the way it is! This is what it means to do mathematics at the highest level, yet when people talk about mathematics, the elements that make up Wiles's description are missing. What is missing is the creativity of mathematics—the essential dimen-