Can maths be violent?

BOOK REVIEW: It may surprise most readers to learn that, just a century or so ago, some of the era’s greatest mathematical minds were enlisted in a debate about whether numbers exist.

- JORDAN ELLENBERG

You'd think, after millenniums, we'd have gotten that straight. But it’s not that simple, as Jason Socrates Bardi explains in his new book, The Great Math War.

In the late 19th century, Georg Cantor had proved, startlingly, that there wasn’t just one kind of infinity, but infinitely many kinds. And the infinity of the set of real numbers, while relatively modest, ‘was not the very smallest. Numbers are complicated!

A real number is traditionally described by an infinite sequence of decimal digits; that sequence might terminate, like 0.12, or it might obey some simple repetitive rule. It might be an irrational number, like pi, whose decimal expansion doesn’t repeat but which you can compute with a simple computer program.

Or it might be something worse. Cantor’s theorem shows that the infinity of describable numbers — such as 3/25, or — is much, much smaller than the infinity of all real numbers.

In other words? There are numbers we simply cannot describe. Can you give an example? By definition, no.

If that makes you uneasy, great. It made everybody uneasy. One of the generals in Bardi’s war, L E J Brouwer, advocated a radical solution: According to him, and to the “intuitionists” who shared his views, the only things that are real in mathematics are things human beings with a finite life span can describe, and the only things that are true are those that admit a proof of finite length.