The Magic of Numbers

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Summary: Excerpt from the Harvard Magazine
Review: On Mathematical Imagination
From whole numbers to infinity--and beyond

by Robert Kaplan and Ellen Kaplan

The lunatic, the lover, and the poet, said Shakespeare, are of imagination all compact. He forgot the mathematician, whose daily concerns are shapes in 27 dimensions, series that converge after more terms than there are particles in the universe, numbers larger than infinity, and others infinitesimally small as well as surreal and hyperreal and shapes in 27 dimensions, series that converge after more terms than there are particles in the universe, numbers larger than infinity, and others infinitesimally small as well as surreal and hyperreal and imaginary. The most monstrous thing about these fantasies, of course, is that they turn out to describe how our one and only universe works.
The people who intermediate between lunatics and the world used to be called alienists; the go-betweens for mathematicians are called teachers. Many a student may rightly have wondered if the terms shouldn't be reversed. Those wonderful outworks of the imagination that mathematicians produce rarely make it to the classroom, where mind-numbing drill in what seems a dead language tends to rule instead.

We are, however, at a fulcrum moment in the history of math teaching, with the balance tipping at last from oases of horror in a desert of boredom to epiphanies that make mathematics as appealing as is its sister-art, music. Outstanding examples of this revolution are The Magic of Numbers, by Benedict H. Gross and Joseph D. Harris (based on their Core course in quantitative reasoning) and Imagining Numbers (particularly the square root of minus fifteen), by Gade University Professor Barry Mazur-two books intended to draw lay readers into the secrets of what mathematics is really like and how our fellow humans invent it.

Gross, who is Leverett professor of mathematics and dean of Harvard College, and Harris, who is Higgins professor of mathematics, have managed a miracle: addressing an audience they rightly assume to be resistant, they lead it gently from utter scratch (how many whole numbers are there from 1 to 10?) to counting with Catalan numbers (in how many ways can n pairs of parentheses appear in a sentence?)-and they do this with bonhomie, grace, and humor. Can you imagine any textbook speaking to you like this?

Let's suppose you climb out of bed one morning, still somewhat groggy from the night before. You grope your way to your closet, where you discover that your cache of clean clothes has been reduced to four shirts and three pairs of pants. It's far too early to exercise any aesthetic judgment whatsoever: any shirt will go with any pants; you only need something that will get you as far as the dining hall and that blessed, life-giving cup of coffee. The question is,

How many different outfits can you make out of your four shirts and three pairs of pants?

Admittedly the narrative took a sharp turn toward the bizarre with that last sentence. Why on earth would you or anyone care how many outfits you can make? Well, bear with us while we try to answer it anyway.

This is a far cry from the standard word-problems that the Canadian humorist Stephen Leacock made such wonderful fun of: dashing old A rowing the best boat while B, as always, has all he can do to keep his leaky tub afloat-and you ask who won?

The Magic of Numbers takes us from counting to the heart of mathematics-number theory-where we feel the pulse and see the valves opening and closing. Like medical students, the lucky people in the Gross and Harris class, and now all readers, are shown the deepening layers of anatomy, the sense the systems make, and how they function; then, in carefully calibrated exercises, they have a chance to do probing of their own. What was impersonal becomes part of each reader's personality. The motto here could well be what medical students are told about operations: "Watch one, do one, teach one."

The book leads you so far toward the frontiers of mathematics that no one could be blamed for losing sight of everything else and waking up 20 years later in its mountains. But just when our heads begin to break through the clouds, Gross and Harris return us to a heightened reality by showing how relevant what we have learned is to a vital practical issue: public-key cryptography. Here the primes, whose tantalizing mysteries Gross and Harris have explored with their students, act as the covert keys to encrypted messages that are openly transmitted along with a public key: the product of two very large primes known only to the receiver. Why, then, can't a spy decipher the message? Because to work back from the product to its prime factors would take more time (even with the fastest computers in the world) than the enemy has to spare-and so a problem in the surreal world of espionage is solved by recourse to the more than real world of mathematics. We find ourselves initiates now in what, at the book's beginning, we knew about only by rumor.

Gross and Harris close their book by saying, "If you've stayed with us till now, you know a tremendous amount of real mathematics-more than all but a small fraction of people walking around.... [N]ow it's time for...you to close the book and get on with your life." What they don't say is that your life will be wholly different. Pindar praised the winners of Olympic events by telling them they had managed, once only and briefly, to touch the bronze sky. Those who win through to the end of The Magic of Numbers will be for the rest of their lives in touch with the accessible mystery of things.