Ian Stewart is Britain's best known populariser of mathematics, particularly in the fields of chaos theory and complexity. A professor at the University of Warwick, his books - including Does God Play Dice ? and Nature's Numbers - and his column in Scientific American recently won him the Michael Faraday Award for bringing mathematics to a wider audience. His new book From Here to Infinity (Oxford University Press) appeared this April in paperback.
Wired: You say that the desert is one of the most mathematical landscapes on earth.
Stewart: What you get in the desert are very interesting patterns in the sand dunes. Now, there are various fluid dynamics problems where you get interesting wave patterns of certain kinds: parallel waves, honeycomb waves and so on. If you take a snapshot of these waves and freeze it, then compare it to pictures of sand dunes in the desert, there's a very strong resemblance. Although the physical mechanisms are different, the mathematical features are the same.
And they're dynamic. Yes. Maths is very dynamic. It has a fundamentally dynamic structure. I think everything in mathematics is really a process. If you think of the number two, the way every child learns two and the way in which you think about it even as a logical structure within mathematics is as a process that goes one ... two. [Counts on fingers.] It's a counting process, and we label the process by the thing you reach at the end of it. But you couldn't get very far in maths if every time you did arithmetic you had to go all the way back to the counting process, so we use mathematical concepts as a way of catching your breath, saying, right, we've got this far, now we've got this thing. But the trick of turning a process into a thing is not an easy trick to grasp, which is maybe why many people find mathematics so difficult.
You stress in your books, though, that once you've turned the process into a thing, you can use that thing in many contexts.
One such crossover question that particularly fascinates me is to do with the wonderfully rich range of rhythmic patterns of motion in four- and six-legged animals. They seem to link up with other patterns in modern dynamics - images of attractors, patterns that change as you vary parameters, phenomena that persist up to some critical threshold and then suddenly disappear. And when you look from that perspective, you see that the phase transition from liquid to gas, or of an animal going from walking to trotting, are the same kind of process on some mathematical level. That's not just a pun and it's not just a metaphor. It is a genuine mathematical relationship. As a result of this you can be playing around with a problem about horses one day, and solve that problem with some bright new idea, and then that idea suddenly transports into some totally different area.
Do computers make it easier to shift ideas around ?
Computers are like any powerful technique or tool; there are problems if you misuse them. But the great thing about computers is that you can do esoteric calculations using stuff that you don't really understand and then apply it to some other problem. In other words, you can experiment on computers and see what seems to happen. So the theoretical people can produce a software package that the practical people can use without having to get tied up in all the nuts and bolts of how it works.
Does the mixture of a process-based approach and more computational ability give you more predictive power ?
Some of the new mathematical techniques let you predict things that were considered unpredictable before. The great thing about chaos is not that chaotic things are unpredictable - which we knew anyway - but that in a chaotic system you can make short-term predictions that are often pretty good. And the more accurate your short-term prediction of where a system is going is, the more accurately you can nudge it over to where you really want it to be instead. Modern computer control systems are to do with taking a system that wants to do something else and saying, no, you're going to do what I want you to do. Take the space shuttle: the space shuttle wants to plunge to earth and explode. It's completely unstable the whole time it's flying, but the computer is tweaking the control surfaces all of the time so that the instability never grows to the point where it matters.
So computer control systems are to do with making unstable things stable ?
That's right. If something's stable it's actually very hard to make it do anything different. That's what stable means. If you want something that's flexible and responsive it should be unstable. But it's still got to be controlled. There's a woman in Brussels called Agnessa Babloyantz who specialises in brain function, and she says brains have to be chaotic. They must be unstable in order to be able to switch very quickly from one task to a different one, from one thought to another thought, from one attractor to another attractor; on the other hand, they must have a kind of global stability so they continue to think sensible thoughts overall. This is the other characteristic of chaotic systems. They are stable as global objects but unstable on the small scale. So it's probably a good job chaos is around. Our brains wouldn't work terribly well without it.
All these applications must make it a very exciting time.
It is, it really is. I describe this as the golden age of mathematics. When was the golden age of mathematics ? Today. Any of the great mathematicians would have given their right arm to have been reincarnated today. Because there's so much going on and it's all knitting together.
James Flint (flint@wired.co.uk) is a section editor at Wired.