Celebrated mathematician, Marcus du Sautoy speaks to Pritha Kejriwal. As he opens up new frontiers for mathematical thought, a new set of dialectics also emerges to frame the world with beautiful numbers and ideas…
So, tell me, how did it all start? When did you finally decide that numbers was what you wanted to do for your living?
Actually quite late. People usually assume that mathematicians are born and that somehow you have a brain to do maths or you don’t have a brain to do maths. And actually I wasn’t really interested in maths in the beginning…doing sums, multiplication, tables etc… I found them really boring. But at the age of fourteen, when a teacher at school, suddenly said, “du Sautoy, I want to see you at the end of the class” and then he took me around to the back of the Maths block, and I thought I was in real trouble, and then he said, “I think you should find out what mathematics is all about. It’s not really about what we are doing in the classroom, its not about long divisions, percentages, sines and cosines – it’s actually something much more exciting!” And I started thinking… first of all a teacher taking interest in you, that immediately kind of intrigues you – why did he pick me out! But then he recommended some books to me. He recommended Martin Gardner’s column in the Scientific American…he was one of the great popularisers of mathematics, and then I went to the book shop with my dad and got these books and as I started to read, I suddenly saw what he was talking about, mathematics was this subject full of beauty! One of the books was called – The Language of Mathematics and at that time I was more intrigued by languages. I had started to learn a lot of languages in school because I thought I wanted to join the foreign office and go and travel and become a spy…I used to love 007…but I found languages very frustrating, because they were full of strange spellings and words that you just had to learn, lots of irregular verbs and it was all so illogical, I couldn’t learn them; I had such an analytical mind, that I wanted some inner logic to things. Then I suddenly found this language of mathematics, where everything made sense, everything had a logic to it, you could reconstruct it, there was a reason for things being the way they were, and it was a language which seemed to describe the natural world around me as well… Fibonacci numbers seemed to be cropping up in flowers and fruits… symmetry, which is the subject that I study as a research mathematician that really fascinates me – the way that you can take the physical world around you and translate it into a world of algebra. I was also very lucky to go to something called the Royal Institution Christmas lectures which are lectures given for children during Christmas time. They were started in 1825 by Michael Faraday and it was only in 1978 that the first was given on mathematics… because, maths is such a hard thing to bring alive, compared to the other sciences. Take chemistry… you mix a couple of things together, and there’s this big explosion and the kids go wow! But, in mathematics, what do you do? You don’t have things to explode or animals to produce etc…so it’s much harder…but this guy in 1978, gave six lectures about mathematics and I went to one of them and I watched the rest on television. He was Erik Christopher Zeeman, a research mathematician but he also knew that mathematics was about getting the next generation excited and it was really important to do so. And I came away from those lectures, thinking, “Oh my god! That’s what I want to be when I grow up! I want to be him!” So, that’s what really got me interested in it and it was from there, that I kind of knew that that’s what I wanted to be – a mathematician.
You spoke about how you could analyse the physical world around you, through numbers. To what extent do numbers spill into your life? Can you explain human emotions through numbers as well? Can you explain love?
Yes! Love, you can actually! There’s a lot of game theory involved in any interaction between two human beings and that can be analysed – the give and take of a relationship can almost be quantified. But there is a warning here – which is – much of the natural world is actually described by equations, which have an essential element of something we call ‘chaos’ in them. It’s a technical chaos, which means that a small perturbation in your behaviour or the behaviour of the weather, for example, might cause the equations to take the dynamics in a completely different direction. You might have heard of something called the ‘Butterfly effect’, which says that the weather is actually controlled by quite simple equations but if you change the air speed by a very small amount – the amount that is perhaps changed by the butterfly flapping its wings, you can dramatically change the outcome of the equation.When I was making a programme for the BBC last year, we went to the met office and they showed us their weather simulations – there was the data for weather that day, and then they ran the equations to predict the weather over the next ten days and we saw what happened. And then they made a very small perturbation of that data – you know, you would hardly notice that it was different – and then they ran the equations again, and after five days it was pretty close to the earlier predictions but by the tenth day, it was so different, it was a spaghetti mess of prediction!
So, the same thing is probably also true of human interaction; that on a certain level, I can make certain predictions about cause and effect, about – if I put my hand on your knee, you would probably give me a slap, but on the other hand, there are a lot of small moves and things which are quite hard to interpret, partly because the smallest change can cause completely different reactions – and you know that in your relationships! It’s like the weather. In some parts of the world, you just know what the weather is going to be like, and in some parts, you just don’t know!
Well, Slavoj Zizek, at some point had said, “Love is just a cosmic imbalance”…
Oh Zizek! I like him…we’ve done some stuff together…yes that’s an interesting way of putting it. I think it probably is quite consistent with saying that it’s something that has a lot of instability in it.
You know, you come away from an evening feeling, that why didn’t I make a move? What would have happened – the world might have turned out sooo (sic) different if you had done something… or if you did do something, you might come back asking yourself, why did I do that? So, although I am a big reductionist at heart, I do believe that the whole of the universe, everything can be reduced to mathematics. Sometimes, it’s not very useful.
But mathematics is sometimes useful to be able to tell you, like the weather, that we are in a region where we cannot make predictions beyond five days, and actually knowing when you cannot know something is sometimes as important as knowing when you can control some things.
So, as a mathematician, how do you see wars? If love would be instability or arising out of cosmic imbalance… what are wars fundamentally fought over? Can you see one equation that would link or sum up all the wars in history?
Perhaps mathematics has more relevance to war than it does to love. I’m not sure on reflection that I agree with Zizek that love is a cosmic imbalance. Imbalance yes, but cosmic? It is a delicate micro-system. It may be part of a cosmic order as everything is but I think it is in the hands of the lovers and the loved. War on the other hand is a macro-structure and I think perhaps that’s where mathematics is at its most effective. It is brutal to admit but the way war is waged is a numbers game. I don’t think there is one equation that applies to all wars but I do think that any war necessarily involves counting casualties, applying the mathematics of game theory to assess the possible outcomes of particularly strategies. Hiroshima and Nagasaki were horrific events but some research suggests that more people would have died in the long-run had the bombs not been dropped. But this is a dangerous discourse. People are not numbers. Is mathematics a dangerous tool because it allows one to de-humanise war? Or is that just sentimental because ultimately that mathematical perspective can end up saving lives? How does one synthesise this dialectic tension between numbers and people, the personal and the political, I’m not sure. It’s interesting that one of my mathematical heroes, GH Hardy, writes about war at the end of his beautiful book A Mathematician’s Apology. Hardy of course has a big connection to India because of the work he did with Ramanujan. In the book he describes what it means to be a mathematician. Reading that book at school contributed to me wanting to become a mathematician. Although the book is primarily about the creative act of doing mathematics, he does address the question of war. Partly it was a response to the horror of the deaths inflicted on his generation in the First World War. He raises the interesting question of whether modern warfare facilitated by the advances of science is ultimately less horrific than the warfare of pre-scientific times. But he tries to persuade the reader that “real” mathematics, the mathematics that he and I do, has no effect on war. He writes that the mathematics of number theory and relativity will have no warlike purpose. How wrong he was! Number theory is at the heart of every modern code used to protect the secrets of war and relativity unlocked the pandora’s box of nuclear warfare.
For Hardy and I think for many mathematicians, that “real” mathematics is a place to escape the horrors of war. He wrote “when the world is mad, a mathematician may find in mathematics an incurable anodyne.”
But I think it is too easy to bury your head in the sand and enjoy maths for its own sake. We have a responsibility to engage because our language is so powerful at learning not just how the universe works but also to pick out the patterns in history.
What you just said, reminds me of Nietzsche’s idea of eternal recurrence…the idea that all the events in the universe can only attain meaning if they were to be repeated again and again…
It’s strange because I often call mathematics the science of pattern searching. Nietzsche’s comment really plays to my hand in that it is only using mathematics, spotting patterns, one can give meaning to the universe. But also I think the comment is relevant to the birth of ideas. Ideas only attain meaning if they are repeated.I think that’s really true and in a way it really resonates with the fact that I have spent half my life in communicating science to people, because, if somebody has a scientific idea and they don’t tell anybody about it, they don’t communicate it, then that idea might as well have not been had! So it’s important to communicate and repeat. I discovered some new symmetrical objects that had never been seen before, they are in high dimensions, and I can’t show them to you, but that night when I discovered them, I was in Germany, I contacted my colleague in Germany, we went out for a drink and I put this idea into the mind of another person. That repeat was when the thing first existed and had meaning, not when I had it in my own head! So as you said, one event is meaningless, two is significant.
Bertrand Russell had said once that the point where science ends, philosophy begins. Do you think Mathematics can collapse this space between the two, can speak in a more universal language?
It’s really interesting, and you are really helping me finish a point that I had started, which is that although I am a reductionist at heart and believe that everything can be reduced to maths but as I had already said, sometimes it’s not helpful – for example, the migration of a flock of birds, that can be reduced to a Schrodinger wave equation for every single particle inside that flock of birds, and if ultimately, I could solve that system of equations, it would tell me, that the birds are going to migrate from one place to another. But this isn’t the right language to explain that… sure… in reductionism, Schrodinger wave equations control that, but that’s the point about all of these different disciplines, actually, sometimes there is a better language to explain things, which are not fundamental physics or mathematics and that’s what all of these hierarchies of knowledge are there for… sometimes you need to not have complete information… complete information can often just overwhelm you, and that’s where, as you said, maths stops and philosophy takes over and sometimes, yes, the language of philosophy is the right one to apply to a particular context, not the language of science or mathematics.What do you mean by language, what does it connote here? Do you mean different streams of knowledge or different modes of expressing the same knowledge?
Ok, I have observed in my own subject, that there are moments when I have seen a new structure for the first time and I found it very hard to explain that and I remember when I was describing it in long sentences to my PhD supervisor, he said, “Name it!” And I didn’t realise that I had the power to name some new structure. But once I had named it, and actually defined what that name meant, it was an incredibly empowering moment. Because, basically knowledge is built in a pyramid like fashion and a new structure cannot really appear other than out of combining previous structures but that combination can create something new. But then when you’re naming it and articulating it, it’s almost like that moment, when you see the world in black and white and suddenly you start to see colour and you give names to them and you group them etc and again, it’s about communication, by articulating what that word is. Science is very good at being able to define almost outside of human interaction, what a word means… so I could define for you, what a topological space is, out of the language of mathematics.
Whereas philosophy is very frustrating, because the way a word is used, it’s often deliberately ambiguous – it starts to shift its meaning, so you can read a philosophical paper and a word that’s used by one philosopher can mean something different in the context of another, and that’s part of what philosophy is. That can be very frustrating for a scientist whose subject is very much built in a kind of hierarchical way.
So, is it like trying to say, that all these different streams of knowledge are fundamentally like different languages – say French, English, Hindi or Urdu etc – trying to express the same ideas, in different sounding words – and some are able to express some ideas better, depending upon their particular idioms and contexts?
It’s interesting… I think it’s more fundamental than that. Although of course, in the English language we have more words for rain, than probably in French, so I can articulate different sorts of rain, so may be English is a richer language to maybe talk about rain, and maybe French is a language to talk about love, but that’s a minor kind of difference of translation, of course, there are words that can’t be translated into other languages…but no, it is more fundamental than that – the difference between, scientific and philosophical languages…
You were talking about spaces where philosophy takes over, and gives you all these ambiguities to deal with. If mathematics was to come into that space, would it be able to cancel out or negate these ambiguities?
Well, I quite enjoy reading Wittgenstein. His early work was very much the work of a mathematician, an attempt to reduce everything to formal languages, then he suddenly experiences somebody sticking two fingers up at him and then he realises, “Oh my god, how do I fit that into this sort of a language?” and then he shifts and goes into a completely different direction. You know, I do a lot of work with artists – and both of us are interested in trying to understand structure and I often find that we are interested in the same things, but the big difference is that, ambiguity is anathema for a mathematician. It’s very important that there is no ambiguity, when I take you a along a logical proof. Whereas for an artist, it’s absolutely an essential part of allowing individual, personal experiences in a piece of art. But I think, philosophy aspires to rid itself of ambiguities, but it’s a work in progress, and you see that any philosophical debate is constantly evolving.
What do you make of numerology?
Ah! That’s really interesting, because the knee jerk response to numerology by every scientist would be, ‘Bull shit’! However, actually, at its heart, it is what drives me as a mathematician, which is looking at the patterns, the structure, to test everything to see whether it has meaning.
So you look at somebody like Johannes Kepler. He was obsessed with looking for patterns and he discovered the way the planets move – they make ellipses and that the area per second that a line joining the planet to the sun sweep out is constant so that’s why planets speed up past the sun and go slowly, further away from the sun. But he also came up with the most ridiculous theories, driven by the same desire for pattern and this is basically numerology; He saw that there were six observable planets, one being Earth, and he analysed the distances between the planets and he realised that they had a very close connection to the platonic solids. There were five platonic solids and he realised that if you put the platonic solids, one inside another, so the octahedron inside the icosohedron inside the dodecahedron inside the tetrahedron inside the cube and then place the planets on the circle of the spheres that are half way between, it seemed to describe exactly the solar system! And so he proposed that was why there were six planets, and why they were arranged like they were – it was complete nonsense! The discovery of a seventh planet blew his theory out of the water. But the theory arose from his desire to try and find reason and so he was playing around with things… and that was like numerology, it was matching of two different patterns, which did not have any genuine connection. So, most of the times, I am quite dismissive about these things. Often after my lectures, people come up to me asking, “I was born on this day, do you believe it is significant for my future?” and things like that, and I have to say, bull shit! But, on the other hand I have some respect for that desire to try and find meaning in everything and I think that the best scientific mind is open to anything, open to the possibility of weird connections and that’s when you make breakthroughs, but you also need to be extremely self critical. To be a good scientist, you need to combine those two elements, such as your good cop and bad cop – the good cop is just saying, wow, maybe there is a connection between the platonic solids and the planets and the bad cop is saying, why on earth should there be?
I often find in my mathematical collaborations – one of us is a good cop and the other the bad. I have a collaboration with an Israeli mathematician – he is the good cop – he keeps on coming up with these mad theories and I am the bad cop who is critical.
With another mathematician in Germany, I’m the good cop coming up with lots of different things and he has got the critical faculty to knock it down or show why it’s true. I think things break down when, for example in homeopathy – is it just that the good cop wants too much for something to be true and doesn’t allow the self critical science to come in.
Why this fascination with prime numbers?
Partly, because prime numbers represent the greatest challenge for a mathematician,a pattern searcher at heart.
I learnt this from my teacher that mathematics is a subject about patterns and not about arithmetic. And the primes are like the atoms of my subject – they are the hydrogen and oxygen of my world because multiplying primes together gets you every other number – you take a telephone number, if it’s not prime, you can divide it by something, you keep dividing and then suddenly you get to the indivisible numbers, the primes – so your telephone number is like a numerical molecule made up by these atoms which are the prime numbers. In chemistry, when you look at the periodic table, you understand there’s a structure in that, Mendeleev’s extraordinary realisation that there’s a pattern inside was an explosion in chemistry. But we are missing that moment in mathematics, that’s why the primes are so fascinating. Because we don’t understand the structure of these numbers, they surely can’t be random.
On the other hand, when you look at them, they look very random – there is no obvious pattern there, and so for me, they are like the ultimate challenge. There are infinitely many of them, so we have to use mathematical techniques, we can’t just use computers or something empirical… so that’s why I continue to remain fascinated. The other fascination I have is for symmetry. I try and understand what are the possible symmetrical objects and those are my two passions and that’s captured in my two books, Music of the Primes and on symmetry, Finding Moonshine (UK Title). Music of the Primes is a classic non-fiction narrative book whereas Finding Moonshine is a little bit more of a personal book, written in a diary form. So it takes a year in my life as a working mathematician and it tries to reveal what I do as a mathematician, and then embedded in there is the historical narrative of the human attempt to understand symmetry.
I wrote Music of the Primes partly because Fermat’s last theorem got proved in the middle of the 1990s and this was, in the public imagination, the great unsolved problem of mathematics, the one that everyone knew about and when it was solved, people thought, it was the last theorem. You know, as if maths was finished! So, partly, why I wrote that book was, now it was up to us to make people realise that maths was far from finished and even the most basic numbers in the subject were still deeply mysterious. So I always try and explain to people that maths is still a living subject, because, there are so many things we don’t know…
But being in India, I need to ask you, how important is zero to mathematics?
I think that’s a wonderful act of the imagination, and a really key moment.Whenever new numbers are admitted into the canon, it is a very exciting moment. It seems so obvious to us, now of course!
First of all, it facilitates computation – in that sense you were not the first to come up with the zero. The Babylonians had a mark for zero, the Greeks and the Romans didn’t get it, but the Mayans had a symbol for zero, but what you had was your abstract idea of creating something to denote nothingness. The Indians were also the first to come up with the idea of negative numbers. I made a programme for BBC, about the history of mathematics called The Story of Maths and we came to India and explored the development of Brahmagupta’s negative numbers. It is interesting that it is all related to a more philosophical view of the world. In Europe, I think they were frightened of nothing or the void. But in the Indian cultural landscape, it was very acceptable to talk about the zero, to talk about the infinite… in 13th century Florence, the use of zero was banned, it was illegal!
That’s very interesting! So, would that mean that the history of mathematics was also somewhere more sociological than we think it could be?
I think that sociology, history and culture are very integral to the discovery of new mathematics, but I wouldn’t say you could do anything! You can’t do anything, because there are limitations to it, it has to have logical consistency. So once negative numbers were introduced, there had to be an understanding of how do these numbers interact with the numbers that we already know about, and would it produce some irrational, illogical conclusions? So, for example, Brahmagupta struggles with the question – what’s a negative number times a positive number, say, -3 times 2, and he realises that it should be -6. By the logic of the way these numbers work, but he struggled when he considered what a negative number multiplied by another negative number would be. Eventually he realised that the way numbers work, -2 times -3 is +6. But you couldn’t just make up, and say you would like it to be -6, because there is a logical reason, if you admitted that into your axiom system, it would collapse the whole theory.
There’s an even more interesting example – negative numbers were also an act of the imagination because a number was meant to count things, so what’s minus 2 oranges? So it was a very abstract idea, but then you move to the16th century where suddenly mathematicians are coming up against the questions of what the square roots of a negative number would be? What’s the square root of -1?
At first, you can’t think of any numbers, which when you multiply by themselves could give you -1, because minus times minus is a plus. So for two centuries, mathematicians said, ok there are no such numbers, so this doesn’t have a solution, x2=-1, there is no x! But at some point they thought, what if we imagined a number, what if we added a new number to our world, which did square to give you -1. And this got accepted – here again, there is a connection to the social, cultural, historical side. This had to do with the French revolution, and you could read Hobsbawm and his description of the French revolution, he actually picks out the way the French helped to expand the mathematical horizon – there was a change of everything, the change of the whole regime allowed almost the acceptance of this new number, the square root of -1. But if this was going to be admitted, it had to be consistent with everything else, so although it was an act of imagination, and we call them imaginary numbers, the more technical term is probably complex numbers, but they were admitted only because they were logically consistent.
I heard this standup comic act, where the performer was making fun of learning mathematics in school and he says he didn’t know why he was ever taught trigonometry and sines and cosines…that he learnt more about triangles and their angles when he got into a love triangle! So, what would you have to say to that?
Hmm, that’s a very good point! I think this is the problem with our education system, and I’m having this battle with the British govt. and we are trying to redefine curriculum and I think that trying to justify the learning of sines and cosines because of utility is futile! You ask anybody here, whether they’ve used a sine or a cosine in their life, other than if they are engineers or some such thing, they would say no! So, does that mean, we shouldn’t teach it? Or let’s ask, why should we teach it? My first retort is that, in no other subject, the demand for utility of learning is so strong – my son learns English – sure he has to learn how to do grammar and spelling. But he also reads Othello, is that useful? Is he ever going to use it… maybe…maybe not… but we are still going to teach it, because it is a part of our cultural heritage and people should know about Othello. In physics, he learns that quarks are the fundamental particles, which make up the neutrons and protons; he is never going to need that! But it’s part of our collective knowledge about the universe. There is another reason why, actually, this guy who has learnt about sines and cosines, although he is not going to be using them directly in his life, being able to logically manipulate ideas is a skill he is using– I don’t think in any other subject, you are required to follow through a logical chain of arguments like in mathematics – if you have to prove that cos2q + sin2q = 1, why is that always true? By being able to prove that and understand that, you arrive at new knowledge. Only through things you know, you arrive there. That is a skill, which is almost unique to mathematics, which he will be using, even when he is doing his comedy, especially when he is doing his comedy! Because, what is a joke – a joke is setting up hypotheses, taking people through to an unexpected conclusion. The logic of that joke works because you are able to navigate an internal logic within that language.
So we must actually celebrate mathematics for its own sake, as a piece of art, a cultural heritage and also as a powerful analytical language that can be applied if you are a lawyer, a businessperson, even if you are a cook!
The cook at this hotel has this extremely varied selection of food and guests, and he has to manage the interaction of all these things to come out in the end, wasting as little food as possible, and wasting as little money as possible while satisfying everyone. That’s actually quite a mathematical skill, it’s a variable situation and if somebody can’t navigate logical steps, somebody who can’t see the advantages of using one system rather than another, is sort of dis-enfranchised.
Whenever there is any discussion or debate between faith and reason, as we saw in this one session as well (at the Thinkfest), the guy who stood defending reason was asked by the moderator, what should be her set of ideas or ideals, where should she posit her faith – what should be her system of belief to live a life? And he answered, how about common sense? What do you have to say to that?
I think common sense is the most dangerous! It leads you astray so often and we all know that. It is being hypercritical and analytical, that should direct us. The reason people don’t understand probability and so much of life is down to assessing risk and evaluating why do this or that, what might the consequences be, because probability is very counter-intuitive, it goes against the common sense and you can see this in the legal system – the legal system gets it wrong so many times because we use our common sense. There was a case of this woman who was convicted of killing her children because she had two children who she claimed had died of cot death. Now, the probability that a child dies of cot death was 1 in 8500. So the prosecution claimed that this meant that the chances of 2 children dying of cot death was 1 in 8500X8500, i.e. 1 in 72 million. But then how many mothers kill their children? What’s the probability of that? Probably 1 in 72 million! So, it was equally likely that she killed them both or they actually died of cot death. And anyway given that one child had died of cot death could dramatically change the chances of a second child dying, so you can’t simply multiply the probabilities. But common sense won’t let you see that and actually it was a mathematician who testified in court to explain how to do the mathematics. The judge misunderstood the evidence and dismissed it and the woman was sent to jail. When the re-trial came and they actually got the mathematical evidence admitted and understood by the jury, she was let free.
So then what would be your views on Marxism – it is perhaps one of the most scientific theories out there, and continues to be relevant in historically altered contexts, where numbers are the crux – the entire superstructure of human interactions hinged on to the base of solid numbers, defining the class structure.
Yes, I am a big fan! I read Marx as a student, and I really enjoyed Gramsci – and what was most appealing to me was the idea of social justice. I am a big believer in social justice – equal opportunities are very important, but also, the almost mathematical analysis of politics and economics was very appealing.
You keep referring to structure, and that’s perhaps also the beauty of Marxism, that it tends to be structural – hinged on to a certain framework – of fundamental ideas. What do you have to then, say, about post modernism?
Post-modernism might be good to a certain extent, because, often to find something new, you need to break the existing, decentralize and try and look at things in a new way. But, if you break everything, then you are lost. That’s why Schoenberg’s twelve tone system looks like you are throwing everything away, but he replaces it with something different. It’s actually quite mathematical – he takes a twelve tone row and he does symmetrical operations on these things – translations, reflections, rotations, to create out of one, twelve tone row, forty seven others. And he has these forty eight things which have some connection – so it’s a new structure that appeared. So, I think for any system that breaks, you very quickly need to find a new structure and that’s the point. So, post-modernism is the breaking, but then the point is to find new constructs. You talk to most artists, and they find themselves most creative when they are put in certain boundaries…That’s true; the frameworks shouldn’t ever disappear…
Yes! I have done a lot of theatre, writing and performing in a play, and one of the reasons I really hit it off with this theatre company in London called Complicite, was because they work a lot with theatrical improvisation – this wasn’t taking Shakespeare or something that was already written, this was creating something out of nothing – and how do you create new work? Well, if you tell an actor to go on stage and start acting by improvising, it would be rubbish! They wouldn’t know what to do…They’ll be very self conscious and they will try to come up with some stupid story. So what you do is give an actor, extremely rigid, simple instructions, but you try and find those rules which create something really organic and exciting and unexpected. And that’s what Schoenberg tries to do in his music – it’s very simple rules which generate something new and interesting. A very simple example is one of the workshops we did, where each actor had to choose two other actors, and when you say go, the actor has to try and create an equilateral triangle with the other two actors. Now the other two actors had probably chosen two completely different people. So what ensues is just beautiful to watch, because there is this kind of huge energy and surge because people are trying to rearrange themselves and after a while things seem to stabilize a little bit, because people have somehow come to a position, according to the rules, where they are in equilibrium forming equilateral triangles with the other actors. But then someone shifts a little trying to get a little more accuracy and that small shift, that flap of the butterfly’s wings again, causes a ricochet through the group and suddenly it collapses into utter chaos again… if you ever try to choreograph what you see on the stage, it would seem impossible. But if you have just a simple rule that you have given to the actors, you get this amazing effect coming out there… it looks like a wild crowd which then reaches a slight stability and then, as Zizek says, it’s unstable again… so, this is why mathematics and art has a lot of connection…
Don’t you think, that’s also the beauty of Marxism, that you have these simple set of beautiful rules to explain pretty much everything…
Yes, but which often can give rise to such huge complexities, that is often difficult to invert. So if you’ve watched that particular sequence, you would never know why it happens, you would never know what the rule was, you wouldn’t be able to reverse engineer it, which is kind of interesting. And of course, that’s what happens in the natural world. There is so much of the natural world which we don’t get but we haven’t been able to reverse engineer what the simple rules are, which generates them. But we are getting better. I think that’s why Marx was, in some sense, a mathematician at heart because he was very good at spotting the simple rules that give rise to the complexities of history.ma
Marx’s favourite pastime used to be algebra.
Oh…that’s interesting!
So this thing that you said, that even a simple set of rules gives rise to such complexities that often it becomes difficult to reverse engineer back to those rules… so this whole idea of Laissez faire seems like a simplified, convenient manifestation of that thought, that things would proceed in their own complex ways and that there could be no real rules… but then again, we have seen that many a times, as much as in the very recent past, that if we let things be, they would eventually collapse… like the recent meltdown of global economy in 2008…
The exciting thing about science is that you can realise that changing some parameters can alter the outcome, and that’s why we’ve got some hope with climate change… if it was all just laissez faire, and we would all just let it run the way it is…no, no that’s not how it can be…
Many people call the idea of climate change to be pure myth…
Because it is such a delicate system which is sensitive to small changes… people can use that argument… but no, it’s not about chaos theory… the graph is very very clear of what we are doing. It’s interesting that people would try and say, that it’s just numerology – the fact that you are putting in more carbon in the environment means that temperature’s going up – but that’s a false co-relation. Ha! Could be! But, let me give you an example – during my talk here, I gave some patterns where people had to pick up the pattern and make a prediction about what was going to happen next – I gave some pretty easy ones to start with, 1, 3, 6, 10, 15 – so, you’re just adding 1+2+3+4…so people spotted that… Fibonacci numbers – 1+1=2, 1+2=3 and so on… then I gave them something which was pretty obvious – 1, 2, 4, 8, 16 and then I asked people what next, and everyone shouted out 32, and then I said, no, the answer to this is 31! People went, “What?” And then I explained why 31 is as good an answer to that, because there is a sequence called the circular division numbers – you take a circle and you put one point on the circle, there is just one region but soon as I’ve got two points I can join those up to get two regions, 3 points, you get a triangle in the middle and three regions round the triangle to give you a total of 4 regions, now put 4 dots, and when I join all the dots, you get a little envelope in the middle, so there are in total 8 regions, now put 5 dots, you get a nice 5 pointed star with a pentagon around and you count all the regions and it’s 16, and then if you put 6 dots, what you get are 31 regions! What’s interesting is of course that 32 is the most obvious answer – if you ask anybody, it should be 32, but that sequence of numbers might be describing something where 31 is the right answer and actually you can prove that any answer is possible… there is a logical reason why any answer for a finite sequence is possible… the next one could be anything… AlanTuring actually proved that… so it makes you wonder…
You know all this talk about numbers takes me to a poem which I had written some time back, and it said,
“Can we build a calculator, which knows all about numbers?
So when we put, 10 divided by 5, it sometimes shows, 1 gets 8 and the rest 4 get a half each?
So when we put 10 multiplied by 5, it sometimes shows 500, depending on the weather?
And other such things?”
So, that’s how numbers elude us often politically and sociologically…
Ha ha, I like that! That’s the capitalist form of division! But don’t worry, real mathematics is very Marxist!
Yes, go on and finish that point about climate change.
Yes, so I come back to the issue of climate change and there may not actually be a co-relation between putting carbon into the environment and increase in temperature, but what you have to do is to choose – out of all of these infinite possibilities, what’s the most likely? In the case of climate change, I think the evidence is pretty convincing. In any situation you have to sometimes make assessments if you are going to make predictions about what’s the most likely outcome, but you have to be open to the fact that it might be an illusion…You know, when they did the first polls for the elections in America, they phoned people to ask who they were going to vote for, somehow it came out Republican and then when actually the vote came, the Democrats had won… and everyone asked what’s going on here, but they didn’t realise that anyone who could afford a telephone at that time was probably a Republican! So that co-relation between telephonic answers and results could be sometimes false… so one must always, always be very very critical.
So as you negotiate all these difficult and slippery terrains, what do you think is the biggest crisis of mathematics – and where do you think lies its true essence as someone said, “Even though it’s true essence seems to lie outside of mathematics, it must always use its own methods to justify its concepts…..”
That’s Hegel…right? It’s interesting that I am wrestling with precisely the tension that Hegel talked about inThe Science of Logic, in the play I am currently writing. It features two characters, one who navigates reality through the perspective of the scientist, but the other who views the world through the prism of mathematics. And although not a crisis, I would say that for me there is a fascinating tension between the two world views. In my mathematical world of logic, I can play with the concept of infinity. I can infinitely divide numbers. 1/2, 1/4,1/8… Nothing stops me dividing. But the physical reality does not permit you to put that into practice. There are limits.The world is quantised. I could never infinitely divide an orange. Is there anything truly infinite in the physical domain? Mathematics evolved as a language to describe the physical world. Indeed Hegel would argue that reality is nothing other than what we can describe in the language of logic.
And yet that language has created worlds that transcend our physical reality. But both perspectives need each other. It is out of this dialectic that we can begin to navigate our place in the universe. The play is evolving into something of a Becketian love story.
Your quote is actually very relevant to an event that happened at the beginning of the 20th century which many regard as the biggest crisis that hit mathematics with the work of the great Austrian logician Kurt Godel. I think ever since the Ancient Greeks began the analytic art of mathematics, we’ve had a belief that any true statement about numbers has a logical proof to confirm that truth. It may take humans centuries to discover but no one doubted that a proof should exist for every true statement. But by an extraordinary act of turning the subject on itself, Godel proved that within any axiomatic system for mathematics, there are in fact true statements about numbers that cannot be proved. It’s almost like a Heisenberg uncertainty principle for mathematics. It’s called Godel’s Incompleteness Theorem. And it shocked the mathematical community. What if you’d spent your life trying to prove the Goldbach Conjecture that every even number is the sum of two prime numbers only to find that it might be true but there is no proof? The amazing thing for me is that we have used our own methods, the methods of mathematical logic, to show the limitations of our subject.
