Review of Prime Obsession

David Gelernter
(Professor of Computer Science at Yale University)

This is a striking and brilliant book, in many ways the most ambitious science-for-the-public attempt I have ever read.  John Derbyshire undertakes a task which is (we are more or less convinced by the end) impossible, and yet the book succeeds, and at its best it is beautiful.  It reads as if it were written not merely by a mathematics scholar but by a first-rate novelist — and that is what Derbyshire is.

An unmathematical reader might have difficulty in following it all but not in reading it.  If you can't stretch your concentration far and wide enough to cover the whole thing, there are nonetheless so many provoking and illuminating observations along the way that you will be scooped up and carried along whether or not you ever come to terms with the big picture.

Derbyshire sets out to explain "the greatest unsolved problem in mathematics," the Riemann hypothesis.  It is Derbyshire's bad luck that, over the last generation, two other problems that used to head up the Greatest Unsolved chart have been disposed of. Those two — the Four Color Theorem and the ever-popular Fermat's Last — each had the winsome appeal of a small cuddly animal.  Four colors are sufficient (says the Four Color Theorem) to let you tint any ordinary black-and-white map in such a way that no two adjacent countries or regions have the same color.  Sounds simple and is.  Anyone can heft it and (for that matter) cuddle it:  Nearly every math or science student figured he could discover a proof, and killed an enjoyable few days trying.  Such unsolved problems were no mere spectator sports.  Everyone could play.

But today, it is Riemann's turn atop the list.  Derbyshire shows us that the Riemann hypothesis is a broad, deep, and fascinating topic — but the hypothesis states (I regret to inform the reader) that  "All non-trivial zeros of the zeta function have real part one-half."  Which might possibly not strike you as the sexiest proposition you ever saw.

It takes a fair amount of mathematics even to say what the zeta function is, let alone why anyone should care.  But Derbyshire sets out to explain the hypothesis, how it relates to a deep and fascinating fact about prime numbers, and how it connects to many other far-flung parts of mathematics and physics.  Prime numbers are the heart of the story, and they (if not the hypothesis itself) are easily grasped.  As you count upwards from 1, prime numbers — which are divisible only by themselves and 1 (2, 3, 5, 7, 11, and so on) — keep showing up, but less and less often.  They grow ever scarcer, yet never peter out entirely.  Between 1 and 100 there are 25, between 901 and 1,000 only 14; in the last 100-count before one trillion, a mere 4.

In 1859 the great mathematician Bernhard Riemann published a paper about the Prime Number Theorem — which quantifies the Petering-Out of the Primes — and in the process invented "a mathematical object of great power and subtlety" . . . and then threw out a "casual, incidental guess" about this fine new object he had invented.  The new object was the zeta function, and his guess came to be known as the Riemann hypothesis — which has since become "an obsession" among mathematicians, says Derbyshire, "having resisted every attempt at proof or disproof.  Indeed, the obsession is now stronger than ever."

The hypothesis makes a rich and fascinating topic because it encapsulates so much and such varied mathematics, and has such wide-ranging implications.  It is a mathematical opal winking and shimmering with a million colors; contemplate this one gem long and carefully enough and you will see whole worlds without moving from the spot.  But (of course) this richness is exactly what makes the topic a potential killer to write about, especially (but not only) if your book is for laymen.

To explain the hypothesis, Derbyshire force-marches his readers through the fields of functions and limits and infinite series and natural logarithms, and across the complex plane; he allows us a refreshing quick dip into calculus and then (onward!) to "a little algebra" (but not too little), chaos theory, "the vis viva equation familiar to all students of elementary celestial mechanics," and beyond.

"A Little Algebra" is a chapter that shows the method's successes and failures.  Derbyshire's discussion of field theory (algebra over specially restricted sets of numbers or quasi-numbers) is quick but clear, bracing, and fairly easy to follow.  The author then spots himself eight pages for "operators" (which have to do with matrices, etc.) — and leaves a broken heap of dying half-explanations behind him as he marches inexorably forward.  Apologies to the "mathematically fastidious," instructions to go look it up "in any decent algebra textbook," "suffice it to say that they exist," "I'm afraid I can't explain just how you find . . ." — these statements all crop up within a single two-page stretch.  They are symptoms of an author's having tried manfully to run up a down escalator and failed.

But of course he faces the same difficulties all authors do when they try to bring science to the unmathed masses:  They leave the problem-solving, examples, and details out, so they can focus on the big ideas.  Fair enough.  But those edited-out details don't merely make textbooks boring and indigestible.  They slow them down too, to a human (or at least humanoid) pace.  One single ripe Derbyshire paragraph introduces the modulus and amplitude of complex numbers, the idea of measuring angles in radians, some handy facts about degrees in a radian, and the amplitudes of positive and negative real and imaginary numbers — together with, naturally, the standard notations for modulus and amplitude, and the formula for a complex number's amplitude as derived from Pythagoras.  Which should all be easy to follow, so long as you already know it.

And yet the book is compelling.  Compelling! — because Derbyshire writes with a novelist's eye and ear, and a novelist's feel for the concrete image, the telling detail, the come-hither sentence.  "Once again, log x looks as though it is trying to pass itself off as x⁰" — Derbyshire makes every part of this thought clear and fascinating; in fact it is a major sub-plot of his book.  Looking at a chart or table with powers of x displayed in some context or other, repeatedly you find the x-to-the-zero entry missing in action — and replaced by log x.  The implications are remarkable and profound.

Imaginary numbers "are no more imaginary than any other kind of number," the author briskly explains; "when was the last time you stubbed your toe on a seven?"  He seeks to recount "the great fusion between arithmetic and analysis — between counting and measuring, between numbers staccato and numbers legato."  He describes the graph of log x: At first "the ascent is very steep, and you need rock-climbing gear," but eventually "you can get upright and actually walk it."  Such crisp, clear, sparkling-as-a-mountain-brook sentences make plain why math books should be written only by distinguished novelists.  Perhaps we could enact a law to that effect.  (Or the Supreme Court could discover one.)

And so far I have described only half the book.  Prime Obsession amounts to a math book and a cultural history intertwined, math chapters alternating with history.  The history chapters deal with Bernhard Riemann (the hero), Leonhard Euler (the sub-hero), many other mathematicians, the monarchs of the states in which they lived, the appealing straightforwardness of Euler's Latin prose (brief sample and commentary included), the serious Protestant faith of Euler and Riemann, the fact that three of five Gφttingen mathematics professors were Jews at the time the Nazis took over and destroyed German intellectual life (evidently forever), but of those three only Edmund Landau belonged to the local synagogue; et cetera.

Prime Obsession is, in short, a learned man's labor of love.  And the learned man is a brilliant writer into the deal.  Such books don't crop up every day.  It must have been a daunting project to think about, and killingly hard to do. It is our good luck that he did it anyway.