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Music of the Primes

Another book on the Riemann Hypothesis! John Derbyshire's Prime Obsession (2004) does about as good a job on the theoretical side as Marcus du Sautoy's 2003 The Music of the Primes, even though Derbyshire is an amateur and du Satoy is an Oxford professor of mathematics. Derbyshire wins in his historical discussion of Bernhard Riemann. But du Satoy shines in his profiles of recent mathematicians. Clearly he has had personal contact with many of them, and it shows. In Chapter 9, for instance:

At the beginning of the 1970s, one mathematician stood at the head of this small band of sceptics. Don Zagier is one of the most energetic mathematicians on today's mathematical circuit, cutting a dashing figure as he sweeps through the corridors of the Max Planck Institute for Mathematics in Bonn, Germany's answer to the Institute for Advanced Study in Princeton. Like some mathematical musketeer, Zagier flourishes his razor-sharp intellect, ready to slay any passing problem. His enthusiasm and energy for the subject whisk you away in a whirlwind of ideas delivered in a rat-tat-tat voice and at a speed that leaves you breathless. He has a playful approach to his subject and is always ready with a mathematical puzzle to spice up lunch at the institute in Bonn.

Zagier had become exasperated by some people's desire to believe in the Riemann Hypothesis on purely aesthetic grounds, ignoring the dearth of real evidence to support it. Faith in the Hypothesis was probably based on little more than a reverence for simplicity and beauty in mathematics. ...

What's the Riemann Hypothesis? It's a statement about the distribution of prime numbers. A proof is worth a million dollars (see MillenniumMath), and worth a lot more in fame and glory. A huge amount of hard work has gone into studying the Riemann Hypothesis for more than 150 years. If it's true (or false) the implications for the rest of mathematics are immense.

Part of what du Satoy conveys, in elegant fashion, is the ultra-large-scale cooperative nature of the mathematical enterprise. In Chapter 8, discussing how work by Julia Robinson and Yuri Matijasevich solved another major challenge:

It is striking how mathematics has the ability to unite people across political and historical boundaries. Despite the difficulties posed by the Cold War, these American and Russian mathematicians would forge a strong friendship upon their obsession with Hilbert's inspirational problem. Robinson described this strange bond between mathematicians as being like 'a nation of our own without distinction of geographical origins, race, creed, sex, age or even time (the mathematicians of the past and of the future are our colleagues too) — all dedicated to the most beautiful of the arts and sciences.'

Matijasevich and Robinson would fight over credit for the proof, but not for self-aggrandisement — rather, each insisted that the other had done the hardest bit. It is true that, because Matijasevich ended up putting in the last piece of the jigsaw, the solution of Hilbert's tenth problem is often attributed to him. The reality of course is that many mathematicians contributed to the long journey from Hilbert's announcement in 1990 to the final solution seventy years later.

And as du Satory notes in an earlier paragraph:

... Robinson saw that the solution had been under her nose all the time, but it had taken Matijasevich to spot it. 'There are lots of things, just lying on the beach as it were, that we don't see until someone else picks one of them up. Then we all see that one,' she explained. She wrote to congratulate Matijasevich: 'I am especially pleased to think that when I first made the conjecture you were a baby and I just had to wait for you to grow up.'

Such a sweet image of collaboration! And just think: human society is a hugely larger example of working together, across space and time, as we figure things out and share know-how and make progress and help each other do better. Doesn't it make all the quibbles and selfishness over intellectual "property" seem infinitely petty?

^z - 2011-08-11