Prime Suspect: Number Theory And Politics
A few weeks ago I started reading a pretty terrific book on a very interesting subject: John Derbyshire's Prime Obsession: Bernhard Riemann And The Greatest Unsolved Problem In Mathematics. The book is a sort of combination of biography and technical writing, and takes as its subject something called the Riemann Hypothesis. The Riemann Hypothesis is one of the most important unsolved problems in number theory, and it's important not just as an abstraction – if proven, it would have possibly drastic repercussions in online security. The book is an enormous pleasure to read, but it turned out to be deeply disappointing as well, and in a very unexpected way.
There are an awful lot of number theory problems that can be easily understood by non-mathematicians, which is a big part of the appeal; if you're not a mathematician, but you have an interest in the subject, it's possible to understand the basics of many interesting problems without a lot of technical knowledge. One of the best examples is the so-called Collatz Conjecture, which was first formulated by Lothar Collatz in 1937. The Collatz Conjecture has to do with a seemingly banal sequence of operations you perform on any positive integer – 1,2,3, and so on. If the number is odd, you multiply it by 3 and add 1; if the number is even, divide by two. (The Conjecture is sometimes called the 3n+1 problem). The conjecture part of the Conjecture is that no matter what you start with, you'll always end up back at 1 and indeed, this seems to be true – it's proven to be true for any number anyone's ever tried – but so far there is no known proof for all cases. A proof for the Conjecture is so elusive that the great Hungarian mathematician Paul Erdõs said of it, "Mathematics may not be ready for such problems."
The Riemann Hypothesis is quite a lot more complicated, however. The Riemann Hypothesis has to do with the Riemann zeta function, which is based on a very simple function: f(n) = 1/1ⁿ + 1/2ⁿ + 1/3ⁿ ... and so on. In the zeta function, n is not just a simple integer – instead, it's a complex number.
Complex numbers have the form a + bi, in which a and b are real numbers, and i is the solution to x² = -1. There's no number that satisfies the condition for x, as there's no possible square root of negative numbers – hence the term "imaginary number" for any number of the form b x i (a real number multiplied by the imaginary unit i). It turns out that the name notwithstanding, you can do lots of very useful math with imaginary numbers, and you find them popping up in everything from quantum mechanics to electromagnetics, relativity, and fluid dynamics.
As you plug in different values for n into the zeta function, it spits numbers back. Some of those numbers are zeros. A bunch of these show up when a – the real part of a complex number; bi is the imaginary part – is an even negative number, like -1, -2, -4, and so on. These are called the trivial zeros of the zeta function. However, there are also non-trivial zeros, which appear when the complex number has a real component of 1/2.
The reason these are interesting to mathematicians is because the distribution of the non-trivial zeros seems to be connected to the distribution of prime numbers. A prime number is any number greater than 1 that you can't produce by multiplying two smaller numbers; 7 and 29 are examples. As primes get bigger, there start to be more and more numbers in between them, but you never quite run out; Euclid proved that there are an infinite number of primes more than 2000 years ago. The number of prime numbers less than a given number, plugged into a graph, gives you a curve that rises sharply at first, but then starts leveling off as the gaps between primes gets bigger and bigger.
Interestingly enough, you can approximate this curve with another function: π(N) ~ N/log(N), where π(N) is the so-called prime counting function. The funny thing about this function is that the match between the actual distribution of prime numbers, and the curve given by the prime counting function, isn't exact – but the gap closes as the number N (any positive integer) gets bigger and bigger.
The Riemann Hypothesis says that the mismatch between the two is related to the non-trivial zeros of the zeta function. This is a fairly complicated number theory problem – one thing's for sure, it's no Collatz Conjecture – and it seems like just the sort of abstruse and totally abstract problem that only a mathematician, and a mighty cold-blooded one at that, could love. But if the hypothesis is true and the relationship between the distribution of primes really is related to the zeta function's non-trivial zeros, it could have serious implications for something very real-world: cryptography. Prime numbers are essential to modern cryptography for the Internet, and if someone proves the Riemann Hypothesis it could have very problematic consequences. In an interview with The Guardian, Oxford University mathematician Marcus du Sautoy said, of one purported recent proof:
"The whole of e-commerce depends on prime numbers. I have described the primes as atoms: what mathematicians are missing is a kind of mathematical prime spectrometer. Chemists have a machine that, if you give it a molecule, will tell you the atoms that it is built from. Mathematicians haven't invented a mathematical version of this. That is what we are after. If the Riemann hypothesis is true, it won't produce a prime number spectrometer. But the proof should give us more understanding of how the primes work, and therefore the proof might be translated into something that might produce this prime spectrometer. If it does, it will bring the whole of e-commerce to its knees, overnight. So there are very big implications."
I've been trying to acquire a solid basic understanding of the Riemann Hypothesis for some time, so Derbyshire's Prime Obsession was exactly the sort of thing I'd been looking for. Reviews online praised it for its clarity, and Derbyshire for his ability to both convey complicated math clearly to non-mathematicians (anyone who got through high school algebra will be able to handle it with a little effort) and to put a human face on the enterprise of number theory. The book has wonderful mini-biographies of some of the greatest mathematicians in history, including Bernhard Riemann, whose rather sad life story stands in dramatic contrast to his achievements as a mathematician.
But I'm having a hard time enjoying the book. I found out, during the course of researching the author, that Derbyshire has made some rather unfortunate remarks about both women and people of color, and they're not just a few ill-considered late-night tweets – his articulated positions on both subjects led to his being dismissed from his position as a columnist for the National Review. I'm going to finish the book but I found the revelation about Derbyshire's views depressing. It's naive and rather romantic of me, I guess, but I have always liked to think that with intelligence, there comes a broader and more tolerant perspective as a necessary consequence, and seeing someone whose intelligence and ability as a writer I had come to respect so quickly, voicing such intolerant views, made me feel like he'd let down the side – let down the whole project of reason as a necessary facet of a life well lived.
He writes at one point in the book, about the hardscrabble environment in which Riemann grew up:
"Poverty aside, it needs an effort of imagination for us, living and working in a modern economy, to grasp the sheer difficulty of finding a job in those times and circumstances. Outside large cities the middle class barely existed. There was a scattering of merchants, parsons, schoolteachers, physicians, and government officials. Everyone else who did not own land was a craftsman, a domestic servant, or a peasant. The only respectable employment for women was as governesses; otherwise, they relied on their husbands or male family members for support."
That he can write with such natural sympathy about the oppressive conditions under which Riemann was born in the 19th century, and be so seemingly unconcerned with the oppressive conditions under which so many are born today, is puzzling. As I said, I'll finish the book – his ability to articulate complicated ideas clearly really is remarkable – but the joy's gone out of it, and I wish he had made the same "effort of imagination" to understand the living around him, as he has the dead.