The scientist of Discworld
by Ian Simmons
[ people - august 02 ]
Ian Stewart is a mathematician, science writer and SF novelist. Nature's Numbers was shortlisted for the 1996 Rhone-Poulenc Prize. The Science of Discworld (co-authored with Terry Pratchett and Jack Cohen) was nominated for a Hugo Award and, like The Science of Discworld II: The Globe and Flatterland, topped bestseller charts in the UK and US. Wheelers, a SF novel written with Jack Cohen, recently appeared in the USA and UK.
Ian Simmons (Nth): Where's the exciting stuff happening in maths today?
Ian Stewart (IS): Exciting stuff? I'm tempted to say 'everything'. This is a very exciting time for mathematical research, because the subject is advancing on a large number of fronts. Several trends are apparent. Within 'pure' mathematics (I hate that word but it's a convenient way to label those parts of the subject that are developed to solve its own internal problems) the cross-connections between what used to be different areas are becoming so dense that the question 'what area do you work in?' is becoming impossible to answer succinctly. Also, the links between mathematics and the sciences are becoming stronger - and that is just as true of 'pure' mathematics as it is of 'applied'. (Pure mathematics builds tools: they can be used in many ways.)
However, that's a cop-out. So I'll mention a few topics that I happen to know about, where major new discoveries are being made.
NUMBER THEORY: A few years ago, Andrew Wiles proved Fermat's Last Theorem (two perfect nth powers cannot add up to a perfect nth power when n is 3 or higher). The theorem here is less important than the method of proof, which has opened up a very deep and powerful set of techniques. Expect more spectacular theorems here.
COMBINATORICS: This is the science (art?) of counting things without actually counting them. For instance, the number of ways to arrange a pack of cards in order is 8065817517094387857166063685640376 6975289505440883277824000000000000 (That took under a second on my computer). Combinatorics used to be a rather isolated area of mathematics, but in recent years it has acquired applications to many other areas. It's also important in the real world because digital computers basically work with whole numbers.
COMPUTATIONAL COMPLEXITY: how efficient is a computer algorithm? The big problem here is the P/NP problem (if an answer can be checked efficiently, can it be found efficiently?) where all the experts are sure the answer is 'no'. My prediction is that by 2050 we still won't have the foggiest idea.
KNOT THEORY: Big advances are being made in our understanding of knots and links - when they are the same up to some continuous motion, and when they are different. It may not sound important, but it opens up major new areas in topology. And it has direct applications to DNA structure and quantum field theory.
QUANTUM GROUPS: these are relatively new algebraic gadgets that relate quantum field theory and related areas of physics to what mathematicians call 'representation theory', which is a branch of abstract algebra. Big steps forward going on here, with ramifications all over mathematics and physics.
SYMPLECTIC GEOMETRY: a strange kind of geometry that is the 'natural' geometric language to describe mechanics. Applications range from wheeled robots to space-probes visiting comets. Watch out for symplectic topology. And possibly symplectic number theory.
NONLINEAR DYNAMICS: The true home of 'chaos theory', a fair amount of fractal geometry, and much else. This area has very direct contacts with the physical world. Beginning to get its theoretical tools effective enough to handle real applications - set to become very widely used indeed.
BIOMATHEMATICS: The application of mathematics to biology is going to be a very hot topic indeed in the 21st Century. And genuine new mathematics will come out of it, not just new applications of old techniques. One of the most exciting areas of them all, and the least predictable.
Nth: With films like 'Beautiful Mind', 'Good Will Hunting' and 'Pi', mathematics seems to be hot in Hollywood. Do these films really give even a vaguely realistic insight into the world of maths and is maths the new rock and roll?
IS: I'm from the generation that knows that the "new rock 'n' roll" is rock'n' roll. Or maybe just rock. As I once wrote somewhere, the total media attention to mathematics is insignificant compared to that devoted to Madonna's toenails. But... it's a lot bigger than it used to be, and the change is noticeable.
One reason for the change is that a lot more mathematicians are willing to talk to the press, or radio, or TV. So the world gets out. The media have discovered that mathematics can be really interesting - it's a new world to them and they didn't realise how extensive it was, or how many gripping stories there are. And to some extent they're getting better at telling them.
My only worry about 'A Beautiful Mind' is that a lot of people seem to have come away from it with the idea that you have to suffer from mental illness to be a great mathematician. It was really a film about mental illness, not about mathematics. So it ran the risk of reinforcing the stereotype: mathematicians are mad. That's why they can do mathematics, when ordinary mortals find it impossible.
That's reassuring for people who don't want to face reality, but the truth is that there are hundreds of thousands of mathematicians, and only a tiny proportion of them are mad. (Probably a lower proportion than you'd find in most professions.) Anyway, it was a great book, an excellent movie, and it deserved the plaudits. I hope we see more like it. But please don't assume that John Nash is a typical mathematician!
Nth: According to Stephen Wolfram, you've been wasting your time as a mathematician and should have been playing with cellular automata all along. What do you make of his claim that we have been looking at the universe through the wrong lens?
IS: The first thing to understand here is that cellular automata are not an alternative to mathematics: they are part of it. They were originally invented by John von Neumann to provide a mathematical proof of the existence of self-replicating systems. I agree with Stephen Wolfram that they are an alternative to 'conventional' mathematical modelling techniques, namely ordinary and partial differential equations. I just don't think they are as radical as he suggests. A few doors from me are ecologists who have been using cellular automata routinely to model ecosystems such as coral reefs, forests, and grouse moors - and have been doing this for a decade or more. See, for example, Note 4 page 262 in my book 'Life's Other Secret' of 1998, which was advocating the use of cellular automata in biology four years before 'A New Kind of Science' was published. And that was just following a long list of biologists. (Note 3 was a reference to Wolfram's 1986 book, by the way.) Wolfram makes an important point when he argues that algorithms, rather than equations, are a good way to model nature. However, it's not a new point. In the "complexity sciences" it long ago became a basic feature of the philosophy of modelling. On page 255 of 'Life's Other Secret' you will find a lengthy critique of partial differential equations as models of nature, and a call for new approaches to patterns. The book ends with a plea for the development of new mathematical techniques: "D'Arcy Thompson would have loved to be alive today, when his ideas are starting to bear fruit. He would have loved to learn about complexity, chaos, fractals, genetic algorithms, neural nets, and cellular automata..." However, it is quite wrong to suggest that algorithms are not mathematics, which Wolfram seem to me to do. The study of algorithms (basic to theoretical computer science) is an offshoot of mathematical logic. I agree that cellular automata are an interesting new computational tool. Sure, we don't have a theory that explains everything they do - but neither do we have such a theory for conventional differential equations. But for differential equations, we do have a theory that explains quite of lot of the phenomena observed in specific examples. The beginnings of such a theory for cellular automata also exists (and it is a part of mathematics, mainly ergodic theory). If cellular automata are ever to become a standard modelling technique in science, then a lot more of that theory will have to be worked out. So it's an exciting challenge for mathematicians - but it's not a radical alternative to mathematics itself.
Nth: The last few years has seen a huge upsurge in the number of hands-on science centres in the UK, do you feel this is important for the future of science and the public's involvement with it?
IS: I recently spent a marvellous weekend visiting (among other things) the Eden Project. Such centres help to raise public awareness of science. However, I suspect that very few people come away from them knowing more. Now, there are many scientists who think that "public understanding of science" means getting more basic scientific facts into the heads of the public. ("People don't even know that the Earth goes round the Sun...") I don't think that's the issue - that's education, not awareness. So even if my suspicions are correct, I'm not greatly bothered. The important things are to convince the public that scientists are actually doing new things, that those things are (or potentially are) useful, that science is not a belief system (as many in the arts fraternity think), that science has a strong social dimension (which many in the science fraternity deny), and so on.
These centres all help to make people aware that science is a normal human activity, done by normal people, and that behind the scenes it has a big effect on their lives. That's important background for any informed debate about the future of science and technology.
Moreover, they show that you can enjoy science; that is utterly fascinating. That's where the next generation of scientists will come from.
There's a 'Millennium Mathematics Centre' at Cambridge University, but it's not really a public centre like the Eden Project, with thousands of visitors every day (or even hundreds). I'd like to see a really good Mathematics Centre built - but there's a problem. Some years ago I got part way towards applying to the Millennium Commission for money for a national mathematics centre. That particular centre didn't get off the ground. One reason was that the Millennium funding did not cover running costs, only capital costs - and then only half, the rest being matched by industry or whatever. I didn't think the project would work without long-term support, and I still don't. (Another reason was that I objected to the tone of the discussion, which was about "what science can do for Great Britain plc". Great Britain is not a public limited company, and the orthodox business model is not appropriate for educational or social services. Indeed, post-Enron I think we should now seriously question whether it's appropriate for business.) Most of the recent science centres (Eden being a notable and glorious exception) are currently in big financial trouble precisely because of this feature of the funding. So I think my worries were justified.
Nth: You have written a variety of science fiction novels as well as a prodigious array of books communicating science and mathematics. In the field of science fiction, which authors do you find particularly interesting?
IS: I've been an SF fan since I was about 13, and still read the stuff avidly. I have about 6000 SF books and magazines. Early in the 1980s I started writing short stories, about 20 of which have been published. More recently Jack Cohen and I wrote a novel, 'Wheelers', and we have a sequel in the pipeline.
I like the kind of SF that's called "hard SF" in the community: basically, where you do your best to get the science right (except for that matter transmitter, or the aliens on Jupiter, or...). And I think that SF has a place in science - not in the journal papers, but in the thinking that goes into them. The essence of SF is the "what if?" question. (What if there were aliens on Jupiter? Would we know? What would they be like? How sure are we?). A lot of the current wrangling over genetic modification would make more sense if everyone involved considered the "What if?' questions.
Nth: You have now done two Discworld science books with Terry Pratchett and Jack Cohen. How difficult is it to get solid science and mathematics out of Pratchett's magical universe?
IS: That's a very interesting question because it contains an implicit misunderstanding. Resolving that makes it clear that 'The Science of Discworld' is not in the same genre as 'The Physics of Star Trek'.
When we were first discussing writing 'The Science of Discworld', Terry Pratchett pointed out that we had a problem: there is no science in Discworld. Discworld runs on magic, and even more crucially, it runs on narrative imperative, the power of story. The eighth son of an eighth son must become a wizard - even if the midwife made a mistake and she is a girl ('Equal Rites'). It took me and Jack six months to find a way round this: "Terry, if there's no science in Discworld, you've got to put some there." Which led us to the technique of fantasy/fact fusion, interweaving a Discworld story with Very Big Footnotes.
In the first 'The Science of Discworld', the wizards accidentally bring our own universe into being, in the form of the Roundworld Project, a Tardis-like sphere about the size of a football on the outside, but much bigger on the inside (because, there being no narrative imperative, it doesn't know what size it ought to be). Bringing solid science into the story is now straightforward (so the short answer to the question is "easy"). All we do is observe the wizards' actions, and their effects, and explain the associated science.
There's a bit more going on, though. This set-up gives us two distinct universes, which we can compare and contrast: Discworld and Roundworld. So we can tackle questions like the distinction between science and magic, and the extent to which technology comes to resemble magic. We felt that this was a new and fresh way to popularise various bits of science.
Nth: Elsewhere you have also trespassed into someone else's universe by writing 'Flatterland', updating Edwin A Abbots 'Flatland', the Victorian bestseller about one dimensional space. Why did you choose to do this and, looking back, how successful do you feel you were?
IS: "Trespassed"... I'm willing to plead guilty, but I get off on a technicality: thanks to the copyright laws, Edwin A Abbott's universe is public domain, hence fair game. And, unlike most of the modern sequels to 'Flatland', I make no attempt to mimic Abbott's style, or the social structure of Flatland itself. It's there in background, but I deliberately kept it out of the foreground, because Victorian attitudes don't fit well with superstrings.
Commercially, 'Flatterland' is my second most successful book (the first being 'Does God Play Dice?', but that's been around a lot longer). Not counting the Discworld collaborations, of course. So in that sense it was a success. An unavoidable fact of life for writers of any kind of 'trade' book - the sort you can buy in airport bookshops - is that the writer has signed a contract and the contract has a delivery date. So at some point you have to decide that the book is 'finished', and it goes into print. Six months after delivery, probably six months before the book goes into the shops, you may get second thoughts - but then all you can do is keep them on file for another book. In retrospect, I think I ought to have reduced the book's length by about 15%, and sometimes made more effort to explain exactly what's going on, to readers unfamiliar with the mathematics involved.
I deliberately chose to write the book with lots of wordplay. This polarises potential readers: either you like that kind of thing, or you don't. But I'd rather write a book that a lot of people love and a lot of people hate than one to which almost everyone is neutral. There aren't many humorous mathematics books (though I would recommend 'Mathematics Made Difficult' by Carl Linderholm), so I don't see it as being especially sinful to try to write one. People who dislike that kind of thing are free not to buy the book; they certainly can't complain that the jacket blurb disguised its nature.
A few people felt that it was sinful or at least frivolous to make use of Abbott's characters and setting. My response is: (a) I didn't, except as deep background: Vikki Line is four generations removed from A. Square. (b) They're public domain: tough. (c) Frivolity can be fun. (d) You think it can be done better: I'm sure you're right, so why not go ahead? (e) Get a life.
Nth: you have also ventured into other scientific fields such as biology, where you have looked critically at the way biologists use maths. What do you feel is the problem?
IS: I can get myself into real trouble here, but let's not worry too much about that. A lot of biologists are beginning (finally!) to appreciate what a lot of non-biologists and the occasional biologist has been telling them for half a century, which is: molecular biology alone is only part of the story. (Hence the sudden emphasis on 'proteomics', to take our minds off the question 'now that you've sequenced the human genome, where are all those wonder drugs you promised?)
My main concern about the way biologists use maths is that mostly they don't. There are a few traditional areas, like population genetics in the style of Ronald Aylmer Fisher, the Hodgkin-Huxley equations for nerve impulses, and of course lashings of statistics. But in many respects the kind of mathematics that you usually see in biology is very classical and outmoded, and its scope is very limited.
This is not the biologists' fault. There have been sociological influences: for a long time biology has been the one area of science where they would let you take a degree when you were hopeless at maths. (As an aside, the Chairman of the government's Science and Technology Committee recently advised universities to accept lower A-level grades from applicants wishing to study science, and to proved remedial classes to bring them up to speed. If you want to destroy British science, this is an excellent start. But it would open up physics, chemistry, and indeed mathematics to students who can't cope with physics, chemistry, or mathematics. Let's do the same for football, with a quota of at least four senior citizens per team, eh?)
But let's leave those aside. A big problem for biologists is that until recently very little mathematics was (or could be seen to be) useful in biology. And the mathematicians weren't making much effort to find out what biologists were trying to achieve, and to invent mathematical tools to help them achieve it.
All this is now changing very fast. Biomathematics has been selected for substantial development by national governments and by research funding bodies. A lot of the more recent mathematics has considerable potential to provide biological insights. A lot of the big problems of biology (such as how the cell works, how organisms develop, and how evolution leads to new species) have a substantial mathematical component. Even the human genome project, an icon of traditional molecular biology, would never have achieved its results without a lot of mathematics (to do efficient 'word-processing' on enormously long DNA sequences, for instance).
So I see huge potential here. I accept that any mathematician who wants to get involved had better listen very hard to the biologists. But I also think that every so often, the result of that listening will be an informed realisation that current biological thinking is missing the real point. What we need here is genuine interdisciplinary science, and that involves two-way traffic.
Nth: With Jack Cohen you have explored exobiology, so will aliens, if we encounter them look like T'Pau from Enterprise? Or are we in for a disappointment?
IS: My friend and regular co-author Jack Cohen is a biologist who, among other things, advises SF writers on the plausibility of their aliens. We both found a lot of science-writing about aliens to be much too unimaginative - basically, most writers seem to think that the only possible route to life is what happened on Earth, and the only kind of life possible is what we've got on Earth. We think that life is a general kind of behaviour that can in principle be much more varied, and use very different ingredients. What about creatures that live in a star, and are made from magnetic fields, for instance?
At any rate, we thought it would be fun to try to put these thoughts together in a book. The result, 'Evolving the Alien', is due out in September in the UK and November (under the title 'What Does a Martian Look Like?') in the USA. On the one hand, we try to assemble some solid science to show that life as it is in Earth could in principle be very different (such as novel bases in the genetic code, novel amino acids, non-oxygen metabolism, non-carbon chemistry). On the other, we advocate hard SF as a source of "what if?" ideas to illuminate just what might be out there in the galaxy. (With the caveat that any specific suggestion is a placeholder: "here's a possibility, but don't expect to find anything exactly like this".)
The thing about aliens is, they're alien. Which means that it is very difficult for us to imagine what they might really be like, or do. However, we can try. And we can say a lot about what they won't be like. Us.
Nth: What are you working on now, both in mathematics and in writing?
IS: I wrote six books in 15 months in 2001-2, so I've decided to take a rest from book writing for a few months, to concentrate on research and some DIY at home. However, there are two books in the pipeline: the book on aliens and the sequel to Wheelers. I'm also finishing editing a multi-author book for Cambridge University Press called 'Thinking Mathematically', in which experts explain their branch of mathematics in (moderately) accessible terms. And I'm putting together a new collection of Scientific American Mathematical Recreations columns.
Moreover, Jack and I are plotting a third SF novel, also set in the Wheelers universe, but around 13,000 BC. It will explain where Moses Odingo got his amazing powers to understand animals (he inherited it from Neanderthal forebears who were beast-masters to a pre-Egyptian civilisation). Where did they get it? Let's say that Neanderthal brains specialised in different ways from human ones.
Research: I'm concentrating on two (related) topics right now. One is to develop and analyse nonlinear dynamic models of the formation of new species in evolution. These models suggest that speciation can occur in populations that occupy the same location and are able to interbreed, unlike the traditional 'allopatric' theory. In this, it aligns itself with the alternative 'sympatric' theory, which is getting a lot of attention nowadays. (Both are probably correct: they just describe two different mechanisms.)
The other is to understand how networks of dynamical systems can synchronise (for example, nerve cells in the brain that fire together, or, indeed, organisms that belong to the same species and are thus synchronised in phenotypic space). I've just discovered that there is a beautiful formal setting for such questions, based on a branch of abstract algebra called Groupoid Theory. I've been tearing my hair out trying to get the formalism working, and finally I've sorted out the idea that was missing. There is already a lot of work on similar questions for symmetric networks, based on group theory: now we can go through the entire thing, adding 'oid' to every group, and seeing what happens.