Marcus du Sautoy is a professor at the Mathematical Institute, University of Oxford, and a Royal Society researcher. His work focuses on how prime numbers behave and the building blocks of symmetry. He also writes about maths for newspapers and academic journals, and has kept a journal of his work, recently published in Science, Not Art: Ten scientists’ diaries (Calouste Gulbenkian Foundation). His own book, The Music of the Primes (Fourth Estate), is a bestseller. It tells the story of mathematicians battling to understand how nature chose the primes.
What is a prime number? And a zeta function?
Primes are the atoms of arithmetic, numbers divisible only by themselves and one: 2, 3, 5, 7, 11, 13 and so on. The mystery lies in that tiny phrase “and so on” because mathematicians don’t know how to predict when the next prime will appear. It is one of the greatest teases of our subject. Mathematics is about the search for patterns, yet is built from a set of seemingly random numbers. In the middle of the 19th century, Bernhard Riemann came up with a new way to look at the primes. Using something he called a zeta function he found a mysterious pattern where others had found only disorder.
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Should we be teaching prime numbers to five-year-olds?
Why not? I went along to our local primary school and did a presentation about prime numbers. I bounced it off something they are interested in: football. When David Beckham went to Real Madrid he chose the shirt number 23, a prime number. So I said OK, why did he choose 23? And we went through some things about prime numbers: that they are building blocks, and Beckham is a building block, and the Chinese consider them macho numbers because you can’t break them down into factors. The children loved it. It can be done. I suppose that is what writing The Music of the Primes was about – trying to play people a bit of the music that is out there.
Of course, being an Arsenal supporter, I put Sol Campbell’s number 23 shirt rather than Beckham’s on the cover of the book!
You are a great populariser of maths. What exactly are you up against?
Too many people have the impression that what a mathematician does is long division to a lot of decimal places. What people don’t see is that there is really great stuff out there. I often compare it to the way you learn music. If you were just told to play scales, learn time signatures and arpeggios, and things like that, and no one played you any real music, then everyone would give up musical instruments immediately.
Why is it acceptable, trendy even, to say, “I’m useless at maths”?
We use reading so much that to say you can’t read would be an embarrassment. People feel they don’t really use maths very much, so it’s not a big deal, but mathematics is unique among school subjects in teaching you about analytic methods. People don’t see that to admit to being bad at mathematics is an admission that you are not very good at logical skills and putting things together. They think it is something that isn’t particularly useful, and that makes it is easy to dismiss.
In your diary you wrote about “the arrogant superior manner that mathematicians can have”. Does that make it harder to inspire people?
I regretted saying that, but it is quite hard to do popularisation when mathematicians look at it and say, “Well, you haven’t actually mentioned that there are some trivial zeros to the Riemann zeta function.” It is important to tell the truth to your audience; if they find out that you are lying to them, then you have undermined your credibility. But on the other hand you have to say, “let’s go for the strong thrust of this idea”, and not worry too much about all the details. Yet the details are what mathematicians spend their lives worrying about.
So how is your work going?
It’s going really well. I am very lucky to have this Royal Society research fellowship, which gives me 10 years of research time. It allows me the space to take risks, and that is quite a rare luxury these days.
I tend to work in phases – I will spend several years working intensively and then I might have several years that are fallow. It’s like sleep: the results might not be popping out, but when I come back to it, those fallow years will be incredibly valuable. One of the best results I came up with was after looking after my son when he was about one-and-a-half. My wife went back to work and I took over full-time childcare. I had a great time, going down to the Science Museum, going on the Thames – all the things I hadn’t done since I was a kid. That was quite a fallow period. Then when I went back to work it was like boof, boof, boof, all these things came out.
You work from home and you get to listen to music while your subconscious does the thinking. It sounds like the perfect job…
It is. There are a lot of similarities in the aesthetics of maths and music, and that’s why they work well together. When I am listening, it stimulates the right part of my brain that see sthings mutate and change.
I work from home partly because I find an office very oppressive – that environment of “Oh, I’ve got to be working.” When I am having an idea, that’s great. If I’m not, then I needn’t worry about it because I can sit back and listen to music or play the piano or something like that. I listen to a lot of classical music – I was brought up as a classical trumpeter – and quite modern stuff. I really love Messiaen.
In fact, Messiaen has a lot of mathematics running through his music. He was quite obsessed by prime numbers, and maybe that’s what I am picking up on. And you can do maths anywhere: that is the other great thing about it. I don’t need a laboratory, I don’t have to go and look after my cultures in the fridge at three o’clock in the morning to keep them alive.
In the 1960s a guy called Steve Smale proved something called the Poincaré conjecture for five dimensions and higher. But he got hauled in front of his funding authorities for wasting money, because he had been seen sitting on the beach in Rio. He told them: “Look, I’ve solved the Poincaré conjecture on the beach.” I approve of that kind of working day.
The conjecture you have spent a lot of time on has recently failed. How do you face up to that?
My student Luke Woodward came in and showed me this example that disproved my conjecture about palindromic symmetry. He was a bit worried that I was going to be angry or something, but he was obviously really pleased with his work and he did this lovely build-up. He showed me some of the things he had been doing, and then a few more, and he kept this example to the last minute and then said: “Oh, look, I’ve found this.” And you know, you just have to accept what nature gives you. There is always a silver lining to every discovery, so I’m not too disappointed: there is clearly something going on there. We can’t have calculated all these examples and seen all these beautiful symmetries without there being something going on.
Are there any ugly bits of maths?
There are messy bits of maths, and people get rather upset about them. Things like those proofs that use computers to check a huge number of cases. For many people, these aren’t good enough proofs because they don’t give us understanding, they just check the thing out. Then there’s something like the Goldbach conjecture. It might be just a strange coincidence that every even number greater than 2 is the sum of two prime numbers; maybe there isn’t a really good reason for this. That sort of conjecture might have a really messy proof.
Are there any applications of your work?
Well, not really. I used to veer towards saying “I don’t care if there are any applications,” but actually I’d really love it if somebody found that the work that I did was precisely what they needed. It was such a surprise that prime numbers turned out to be so essential to building security on the internet. At the moment I’m not motivated by finding practical applications. But the research councils are beginning to earmark grants more towards those who can prove that their work has a practical application. When it became clear that cancer could be treated by drugs, suddenly money poured into biochemistry. In a way, cryptography for the number theorists was a bit like our version of cancer.
Is controversy alive and well in maths the way it is in the rest of science?
In mathematics, when somebody has proved something, all the evidence is on the table. You’ve got a proof. OK, there’ll be a bit of controversy when someone will say they have proved something, but has made a mistake and that is pointed out.
For example, Louis de Branges said he had proved the Riemann hypothesis. Now we’ve been able to identify exactly where he made a mistake in his proof, though it was quite hard to go through all the details. And this is kind of interesting, because de Branges made the same claim many years ago about something called the Bieberbach conjecture, which concerns the geometry of mapping. His proof got thrown out because it was rather incoherent. But after a working seminar in Russia, mathematicians there said: “No, we’re sorry, this is proof and he’s done it.” And the mathematical community had to swallow its words.
Are there any areas of maths that are frowned upon, any mathematical equivalent of homeopathy?
There are certainly trends in mathematics. Some things are in, and others go out. It tends to be if the mathematics is important and connects to a lot of areas. I don’t think there is anything as terrible as homeopathy. You sometimes feel that someone is generating problems just to solve them, and that seems like treading water, while somebody else is making major breakthroughs. A bit of that is brought on by the need to pump out papers.
Is there anything that mathematicians don’t like to touch?
Yes. The Riemann hypothesis is one of them, but that is changing a little bit. If you said 20 years ago that you were working on the Riemann hypothesis, people would think that you were mad. But I think that it’s more acceptable now, partly because proving Fermat’s last theorem showed that we shouldn’t be too scared of these big theorems. I work by combining lots of different projects. If I hit a blank wall with one project, I’ll just move on to another one. Very often that will help me to find the way around the first problem.
Erasmus Darwin said you should always have one crazy project in your life. His was playing the trumpet to his tulips. He said, OK, it would be a wonderful discovery if tulips respond to music, and if they don’t, well, then I’ll just get some trumpet practice and no harm done.
My crazy project is on special prime numbers called Mersenne primes. I’m looking for the primes given by the function 2n – 1, where n is a prime and the function yields a prime. So for n = 3, 2n – 1 gives the prime 7. I’m working on this conjecture about whether there are infinitely many Mersenne primes. It looks so simple but it isn’t.
What keeps mathematicians going? Presumably it’s not just the idea of fame and a Fields medal [their Nobel equivalent]?
No, but they help. There are a lot of examples of longevity in mathematics. There is this myth that mathematicians will have done their best work by the age of 40. I certainly think that mathematicians do very innovative work very young, but they can still keep on going. There is something about wanting to be around to see whether these problems get solved.
Mathematics is something that will never come to an end. It is a bit like that Greek monster, the hydra. You chop off one head and two more appear. Each discovery suggests that more interesting, intriguing things lie beyond it. I think that is what keeps people going. You get so wound up in this world of numbers and you are just desperate to know what is on the other side of this mountain. That is what keeps you going.