Embrace ambiguity in math and in life 

By Barbara Black

Bill Byers was taken aback by the critical reception to his first book: “It’s quite extraordinary!”

How Mathematicians Think: Using Ambiguity, Contradiction and Paradox to Create Mathematics was published by Princeton University Press in June. It was promptly taken up as the main selection of the Scientific American book club for June, and the Chronicle of Higher Education asked him to write a full-page article for their Aug. 3 review section. There was a glowing review in the British publication New Scientist.

In an article in the Notices of the American Mathematical Society, Reuben Hersh congratulates the author for stating an important truth: “You can’t quite say that nobody has said this before. But nobody has said it before in this all-encompassing, coherent way, and in this readable, crystal-clear style.”

What Byers says is that math needs a new image. Most of us, even mathematicians, think it provides the very model of linear, logical, rational thinking. Engineers, physicists, lawyers, psychologists all use numbers to make their arguments seem “more true.” Yet math is full of delicious ambiguity; it’s a field ripe for imaginative speculation.

Professor William Byers Magnifying glass

Professor William Byers

The equation 1 + 1 = 2 is a case in point. “The numbers 1 and 2 are in fact extremely deep and important ideas,” Byers said. “They are basic to science and religion, to perception and cognition. While 1 represents unity, 2 represents duality. What could be more fundamental?”

Doing mathematics often entails acknowledging ambiguity, and then trying to resolve it. Concordia’s own John McKay has become famous for identifying problems at the highest level, but even for a student struggling with an unfamiliar math concept, a little light bulb seems to go on when she or he “gets it.”

For Byers, “getting it” occurs in many ways, not only in mathematics. Once you start looking for ambiguity, it’s everywhere, from quantum physics to Shakespeare.

“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.’”

An interview with Byers about ambiguity in math soon turns into a lively discussion of politics, religion, marriage and poetry. For example, a metaphor like “All the world’s a stage” is ambiguous.

“Of course the world is not actually a stage, so logically the statement is false,” Byers explained. “Yet on another level, the metaphor reveals a deeper meaning. Even 1 + 1 = 2 can be thought of as a metaphor in that way, with a deeper meaning waiting to be revealed.”

That Byers should think this way is no surprise. Throughout his career he has demonstrated a healthy appetite for the humanities. “I have this strong romantic bent,” he admitted.

He was the principal for eight years of Lonergan College, an outgrowth of the former Loyola College. In the program, students studied one historical personality in depth throughout the year, such as Galileo or Dostoevsky. It was his long association with Lonergan that got him writing prose.

Byers intended this book for the intelligent layman, but also for his own colleagues, because he feels many of them need a wake-up call. Many of them are infecting their students with their own deadening rationality. “Mathematics is often badly taught,” he said.

When students can’t understand a concept, they and their teachers tend to think they’re just not very smart. They may have excelled in math early in their education, but faced with an unfamiliar idea, they freeze and retreat to the concepts they know. “But in order to learn something new, you have to give something up.”


Concordia University