Is the world simple or complicated?
- Extra Reading
Stop some particle physicists in the street and they will probably try to persuade you that the world is simple and governed by a single Theory of Everything. But stop a biologist, an economist or a social scientist, and they will tell you quite the opposite: the world is a higgledy-piggledy collection of complexity and chaos. So who is right: is the world really complicated or is it simple? And what does the answer tell us about the nature of art and science?
Is the world simple or complicated?
Professor John Barrow
Good evening, ladies and gentlemen. My name is Jonathan Brayne. I am a partner of Allen & Overy, which is the firm that occupies these offices. My welcome to you all this evening, to what is going to be the first in a series of four lectures by Gresham College professors. We are absolutely delighted to be hosting them all. It is my pleasure to introduce to you the Provost of Gresham College, Lord Sutherland, who will be introducing our speaker and I hope also will tell us a little bit about the fascinating institution of which he is the Provost. Lord Sutherland…
Lord Sutherland of Houndwood KT FBA, Provost of Gresham College
Thank you very much indeed. Gresham College some of you know I think very well, looking round, but for some of you this may be a new introduction. I am delighted to welcome you here on behalf of Gresham College also.
Gresham College was created more than 400 years ago through the will of Sir Thomas Gresham, then Lord Mayor of London and whiz-kid financier. I think he broke every rule in the book as far as I know, but he accumulated a vast fortune, and some of it was dedicated, through his will, to the education of those who live and work in the City of London. That is marvellous – there was someone, more than 400 years ago, saying education is what we need. It is not first said by more recent politicians. In making that great commitment, he set up a college and the intention was that free lectures should be provided for those who live and work in the City of London. Our job in Gresham now is to interpret that for today. It is a very different world. There is free education for those up to the age of 16, compulsory indeed, but two things: one, we realise that the City of London has moved geographically in part, down here to the Docklands area, which is why we are especially delighted that this series has been inaugurated; and secondly, there is a vast thirst as we see it for the kinds of lectures we put on, and that is for good reason. We go for the highest quality, and as you will discover tonight, we attract people to be our professors of the highest quality.
But before introducing our lecturer tonight, I want to give a particular thanks in fact to one of our former professors, Honorary Professor of Gresham College, Richard Susskind, who gave the introduction to Allen & Overy, from which followed a very useful and worthwhile conversation. It became clear that Allen & Overy had something of Sir Thomas Gresham’s vision of the importance of education, and the outcome is that they are hosting this series of lectures in Docklands, whereas most of our productions are up near Holburn and near what others recognise as the City of London. It is a delight to be here, and I am sure you will find the lectures varied and fascinating. We went for a varied and rich mix, and credit and thanks to Allen & Overy for wanting to work with us in this way.
Our lecturer tonight, Professor John Barrow, Gresham Professor of Astronomy – I think he holds another chair in one of these other educational institutions - Cambridge, isn’t it?! Yes, that’s right! He holds another chair in Cambridge, but we are very proud to have him as our Gresham Professor of Astronomy (and other Physical Sciences), and he will be lecturing tonight on a very important topic, but I will leave that to him. His distinction is considerable. Apart from the chair he holds in Cambridge, he most recently won the Templeton Prize, which is the equivalent of the Nobel Prize for those who work on the interface between, in this case, science and religion, and we congratulate him on that. But as I say, we commit ourselves to the highest quality of content and of presentation, and it is a great delight to ask John Barrow to deliver this evening’s Gresham lecture – John Barrow.
Professor John D Barrow FRS, Gresham Professor of Astronomy
Well thank you. It is very nice to be here. Our lecture this evening is about a question, and if you want to place it somewhere within that historical title of Gresham Professor, it is in that little bracketed comment “(and other Physical Sciences)”, although I am going to try and talk about something that cuts across pretty much everything that might go all around in this complex of London as well as in science laboratories and the world of biology, the world of chemistry, the world of physics, and mathematics. It is the question, “Is the world really simple or complicated?”
If you went outside, down one of the streets here, stopping people at random and asking them this question, you would get a number of quite different answers. If by some misfortune you were to stop a particle physicist or an astrophysicist, like myself, then they would be very soon regaling you with wonderful stories about how simple the world really is if only you look at it in the right way – don’t worry about laws from the legal perspective, but there are laws of nature, there may be just one of them, and come back next year and you can put it on the front of a T-shirt and that’s all you need to know! So, for that person, the world really is simple if you look at it in the right way.
But if you stop just about anybody else, somebody trying to run an economic system, like the Chancellor perhaps, someone involved in financial management, someone involved in engineering, social science, or just trying to control a collection of school children in class, for them the world is not simple at all; it is a great higgledy-piggledy mixture of competing effects.
You would never hear the term “law of nature” being talked about by a biologist. What you see in the living world is not the result of symmetry or mathematical beauty, it is something more stable than other things. It tends to survive better and to multiply better than other things, even though they may be prettier.
You get completely different answers: for one group of people, the world is simple and symmetrical; for others, the world is complicated and a great higgledy-piggledy muddle. So is the world really simple or is it really complicated, like the second answer given to you? I hope by the end of this lecture if someone does stop you outside, you will be able to answer this question in 59 minutes.
Well, let’s take a look at the simple option. So this idea that the world is really simple is matched with a concept of what is called a law of nature. This concept has peculiar origins, which are interesting, but that is another lecture. There are origins in religious beliefs about monotheistic religious traditions in which the deity controls the whole universe and his or her decrees control what happens. In a rather small step, you move from those decrees of the omnipotent deity to the idea of laws of nature. But jumping up to the present, the most simple and illuminating way to think about laws of nature takes a lot from modern computer science.
Suppose that you just have a sequence of symbols, and they are going to be zeros and ones but they don’t need to be – they could be heads or tails, or diamonds or spades, square or round, whatever symbol you like – and I put two sequences of them, which you might imagine running on perhaps for ever, or even just a very long way, and we posed the question, “Which of these two sequences is random and which is ordered?” Well, I hope even at this late hour you might suspect that the first sequence is the one perhaps that is ordered and the other one perhaps not necessarily so. But what are you doing when you decide one way or the other?
The ordered sequence perhaps has a pattern to it, and that means if we wanted to tell our friends in Australia what is in that sequence, we would not have to print it out on a tape and mail it or fax it over to them. We could send them a rule or a formula for generating the sequence that was a good deal shorter than the sequence itself. The second sequence, suppose it is 50 digits long, there is no discernable pattern, and if you want to tell your Australian cousins what is in the sequence, you would have to send them the whole sequence. There is no abbreviation, there is no short formula and handy rule for replacing it. So this creates the notion of compressibility. If you can compress the information in the sequence into a rule, a formula, a programme, that is shorter than the sequence itself, then we will say that the sequence is not random, it is orderly; but if there is non-compression, if the shortest statement of the sequence, of the information, is the sequence itself, then we will say its random and not orderly.
My definition of science is that it is simply the search for compression. If we just go out collecting butterflies, or information about the positions of the planets in the sky, we are not really doing science, we are just like book-keepers, we are just cosmic civil servants, filling up filing cabinets full of pieces of paper about the positions of the planets at different times. We start to do science when we replace those pieces of data by a compression, where we have a rule, or, as we call it, a law, which enables us to see a few of the observations and then to predict what the future position, say, of the planets would be. The nice thing about this is that you can test those future predictions by looking out the window with your telescope and seeing whether your compression is correct. If it is not, you throw it away. So science is the search for compressions in this sense.
One of the oddities about modern science is the way in which we can see that there are four ways of looking at law and order in the universe, which are really all the same, although at first they might appear rather different. With those sequences I showed you, you could if you wished characterise the order by saying there is a rule, there is a law, which says whenever you see a zero, the next symbol will be a one, and whenever you see a one, the next symbol will be a zero. So it is like a causal law – “If this, then something else.” On the other hand, you might associate that law with a compression in the sense that I have just indicated, that there is a rule which makes the information content equivalent to something much briefer than all the outcomes of the laws. But that rule of going from one symbol to the next is rather like a pattern or a symmetry, so the sequence where zero follows one and one follows zero, you could also replace it by saying that there is an invariance, there is a pattern so that every other symbol is the same, or if you moved every symbol one step to the right, everything would change from zero to one and one to zero. So gradually scientists have recognised that rules that tell you how things change are the same as patterns, the recognition of patterns, which do not change.
Three hundred years ago, if you were Isaac Newton or one of his followers, your search for laws of nature would be focused upon aspects of laws – if I do this, then this follows. As we will see in a moment, the modern focus is very much upon identifying the things that do not change, regardless of what you do.
The curious thing about this way of looking at the world is the fact you probably notice, and began to notice when you were at school, that somehow mathematics is irritatingly ubiquitous; somehow everything about the world that you want to know involves learning about mathematics, that the world seems to be mathematical in some mysterious and persistent way. If you go to Departments of Philosophy around universities, you will find people perhaps making rather heavy weather of this, and trying to decide whether mathematics is something that we invent, or whether it is something that we discover, or something completely different, but I think our way of looking at the world, patterns and so forth, helps us to make this complicated story a little simpler.
I like to define mathematics to be simply the collection of all possible patterns. So some of those patterns are on the ceiling, some are on the palm of your hand, some are patterns of movement in time, some are patterns of computer operation, so mathematics is just a catalogue of all the patterns that there can be, some geometrical, some arithmetical, some logical. If you look at mathematics in this sense, then there is no longer any mystery about why mathematics is so useful and so ubiquitous. You see, the world must be mathematical in order that beings like ourselves can exist. If there was no order of any sort, no pattern of any sort in the universe, it would not be possible for living or conscious intelligences to exist. There would be nothing for natural selection to act upon. And so we should not be surprise to find that patterns exist in nature – we could not exist otherwise – and mathematics is just the description of those patterns.
But there is still a mystery, and the mystery is not that the world is described by mathematics, because patterns are inevitable, the mystery is that such simple mathematics, such simple patterns are so far-reaching and can tell you so much about the structure of the universe. The sort of mathematics you learn in school for five years of serious study, two or three more years perhaps at university, is virtually sufficient to understand everything we know about the world of elementary particles and certainly the world of astronomy and galaxies and the structure of the universe. It did not have to be like that. You can imagine universes that are much more complicated, that make use of forms of mathematics which we would not be able to compress in the way that I have just mentioned. So our universe is extraordinarily compressible in the sense that I introduced, that all the things that we see around us can be reduced to a very small number of rules and forms of law and order.
What happened in the 20th Century that is rather unusual is that there is a type of law of nature, which physicists have come to like very much, and it seems to be rather ubiquitous in the way the world works. It has a rather strange name, and it does not really matter any more – the name is a historical accident – they are called gauge theorems. We know there are four mathematical laws of nature, four forces of nature, that describe everything that we see around us and everything that is going on in all scales in this room: one is the force of gravity, which keeps our feet firmly on the ground; another is the force of radioactivity – we sometimes call that weak force; the other is the strong or nuclear force that binds the nuclei of atoms together; and the last is the force of electricity and magnetism, which involves the structure of atoms as well as moving electrons, electric current, computers, everything that involves electromagnetic interactions. Those four forces of nature are described by the rules of that if you do this, then something else will happen.
But starting perhaps about 100 years ago, 150 years ago, physicists began to notice that those four forces were each equivalent to the statement that some pattern, some abstract pattern in nature, never changes, so that whatever you do to the world, something about it remains the same. In the 20th Century, physicists became so impressed by this way in which rules of change are equivalent to patterns which don’t change that instead of finding the “if, then” rules, they started with the patterns. They looked at the catalogue of patterns and tried to discover which patterns could describe the way that electricity or magnetism or radioactivity operated, and they were extremely successful in doing that.
The situation today is that one has reached a state where you have four patterns of this sort which do not change, which tell you how these four forces of nature work. These patterns have a remarkable property. They do not just tell you how electrons behave or how electricity and magnetism behave. They have a sort of deep internal binding property that even tells you why electrons exist, why the particles on which these forces exist, and why they have the properties they do. This is not like the type of law that Newton first devised. It is as though the traffic laws in the Highway Code told you what the structure of the internal combustion engine was, and what sorts of cars could exist, so there is a deeper internal structure. How can that be?
Well, suppose that one of the patterns that you do not want to changesuch as the shape of my hand here, and you are fairly demanding and say, right, the symmetry that I want to keep preserved is that if I change the position of every point of my hand, then my hand has got to stay the same. Well, that is very easy – my hand just moves bodily from one place to the other and it looks the same. But that is a very strange symmetry to require to act everywhere in the universe at the same moment, for everything to keep in step. How do things on the other side of the universe know that such a symmetry is being actioned at this precise moment? So physicists require something that at first sounds completely impossible, that they want things to stay the same even though every point in my hand is subject to a completely different movement. First, it sounds as if what is going to happen is that my crazy mixed-up hand is going to fly off in all directions and it will not remain looking the same. But suppose a force exists, suppose I have an elastic band, or some sort of flex, which I wrap tightly around my figures, which has the properties that however my hand moves, it counters that motion and leaves my hand remaining the same? You can see that under that requirement, a force must exist in order that things remain the same. Gauge theories have that property; they require the forces of nature to exist in order that the underlying pattern of the laws remains the same, and they require particular particles to exist. These are very powerful types of theory. All the fore-known forces of nature are described by these sorts of symmetrical cause.
But physicists are not really content with having four different laws of natures governing the universe, as though you were living in a country with four different legislations in force at the same time – like modern Iraq or Liverpool or somewhere. There has always been a great desire, that began certainly with Einstein and has carried on to the present, to find a way of uniting these forces of nature into a single, so-called “Theory of Everything”. You can see what the idea would be, if you think of the different patterns as pieces of a jigsaw puzzle. You want to join them together in some way to make a single larger picture. You hope that the process of joining any two pieces together will place some constraint on the shapes of those pieces, and tell you that there has to be something new about them that you could go out and look at and check. This is the process of the search for unification, or a Grand Unification of the forces or the laws of nature.
I have a picture which shows what progress has been made over a long period of time in this sort of quest. So this is a picture of progress in discovering these patterns in laws of nature from about 300 years ago until the present. At the bottom, you have, just before Newton, people believed that there were two forces that acted in the way that gravity does, so there is something that keeps our feet firmly on the ground, and something else that is responsible for the orderly motions of the planets that Kepler had earlier discovered. But it was Newton’s genius to recognise that you did not need two forces of celestial and terrestrial gravitation; one force would do, one pattern describes both of those activities. And the movement in the progress of physics is very much progressed from more to less: why must you think that static electricity that makes balloons stick on the wall might be something completely different to dynamic electricity that makes currents flow in wires? They are just different aspects of the same underlying force. Likewise with electricity and magnetism: if you move magnets, you can make currents flow; by using electrical flow, you can create electromagnetism. Electricity and magnetism are just different aspects of a single force of electromagnetism. So again, you are going from more to less. In the 20th Century, the weak force of radioactivity, that is called nuclear force, were both discovered and the root of those forces are known as the deep interaction and the strong interaction.
In 1967, it was first proposed that the force of radioactivity, electromagnetism, may not be different, but again just different aspects of a single force, and it was proposed that there might be a bigger pattern in which they could both sit. That was confirmed by experiment at the end of the 1980s, and the telltale particles that had to exist if the pieces of the puzzle were to fit together were found at CERN in exactly the masses predicted. So the black part of the picture is the part of this unification story that has been accomplished. What remains, the top of the picture, exists in theory, proposals about how you might go on, and how you might join the strong nuclear force to the actual weak force to make a so-called Grand Unified Theory, and then finally, how you might add gravity to the story to create a complete Theory of Everything. The favoured candidate nowadays is the theory that was formerly known as Super-Strings..
The situation is that there appear to be at first five different possible patterns which could accommodate a Theory of Everything. People may be puzzled by that, you know, did our world just pick one of these possibilities and there are other universes where the others are on show? But then you must recognise that these five theories were not the final story. They were, as it were, shadows cast on the wall, where there was a solid object in the middle of the room, and if you shone light on it in that direction, you got one shadow, and if you shone light in this direction, you got another shadow, and in that direction, yet another one… So we have not identified the final theory. We were just seeing it in different projections, in different simple, limiting situations.
The final theory has become known as the M Theory and it has not been mapped out in full. That is the great challenge in modern physics, to find that theory whose shadows we can see in different limiting situations.
I do not want to say any more about that, but if we go back to our first question, is the world really simple or is it complicated, the person who tries to persuade you that the world is simple, the particle physicist who you stop at a tube station, has this picture in mind. If you look at the world in this way, at the level of the laws of nature, and the forces of nature, you can unify them into fewer and fewer patterns, the world does indeed appear simple, and there may ultimately be just one pattern, one overarching symmetry of pattern from which all the laws of nature that we discern around us in the world flood. This is the reason why you might believe that the world is really simple. It is where you look at the world at the level of the laws of nature.
It is not only particle physicists that like symmetry. There are interesting side issues that ask perhaps is there some other psychological or evolutionary reason why we might like symmetry. We all seem to like symmetry in different ways. If I could see the wallpaper in your house at home, I could tell what kind of symmetry you might like. Biologists have always suspected that there is a deep, evolutionary reason why we like symmetry, and why we search for it even if we are a particle physicist.
If you were around half a million years ago, or two million years ago, and you are seeking to survive, you are living in a jungle on the edge of the savannah, a crucial thing for you to be able to do is to tell the difference between things that are alive and things that are not. You see, things that are alive are going to eat you for lunch, or you might want to eat them for lunch, or you might want them to be a potential mate. If you want a rough and ready way of looking into a crowded or distant scene and telling whether something is alive or not, a very good rule of thumb is to have an ability to recognise lateral symmetry. Living things tend to have left/right symmetry, like you and me. They do not have up/down symmetry, because we all live in a gravitational field, and so we have a different structure far from the ground compared to close to the ground, and we tend not to have front/back symmetry because it seems as if the biology to engineer the ability to turn round would duplicate everything, back and front. But living things have remarkably accurate lateral symmetry, and we have become suspicious about our ability to recognise it because we know that most evaluations of superficial human beauty and good looks rely on respect for lateral symmetry. People in California and elsewhere will spend huge amounts of money enhancing or restoring lateral symmetry. Animals of course reject and choose mates very reliably on the basis of recognising bodily symmetry or rejecting it. So there is a good evolutionary reason why we like symmetry. We have a survival incentive to be good at recognising it.
In fact, we are over-sensitive to it, and if you look at aerial photographs or ancient pictures of the surface of Mars, of course you know people are very good at seeing symmetrical features where none really exists – canals on Mars, or runways in aerial photographs – that they are rather too good at seeing patterns and symmetries. You can perhaps understand that. You see, if you are not so good at seeing patterns where patterns really exist, then you are in trouble. If you do not see tigers in the bushes when there really are tigers in the bushes, you are going to be dead, but if you sometimes see tigers in the bushes when there are no tigers in the bushes, the worst that might happen is your family will think you are a bit paranoid! So this liking for symmetry, we can understand in psychological and evolutionary reasons as being something that is ingrained into our development.
You might like symmetry, but we also like symmetry breaking. We like the violation of symmetry. Most humour, at least on this side of the Atlantic, is based to some extent on symmetry breaking. You know, “There was a young man of Milan, whose rhymes they never would scan. When asked why it was, he said it’s because, I always try to cram as many words into the last line as ever I possibly can!” So we like this breaking of symmetry intentionally, in all sorts of humorous situations.
Physicists like it too. The most interesting feature of the world perhaps is to recognise that although I have been talking a lot about rules of nature, and symmetries, no one has ever seen a law of nature. You cannot look in your laboratory or out of your window and see a law of nature. What we see are the outcomes of the laws of nature – throwing this up in the air, dropping it on the floor – but the outcomes of the laws of nature are much more complicated and far less symmetrical than the laws. They are a completely different story.
Suppose I balance this pointer vertically on my hand. Then it is perfectly symmetrical. Let’s do it in the way that physicists like, at zero temperature in a perfect vacuum. If I let go of the pointer, it becomes subject to the law of gravitation. Gravity is perfectly democratic. It does not make all pointers fall towards Canary Wharf Tower, or the Tube Station. If we did it in a perfect vacuum, allow quantum fluctuations to act on it, it could fall in any direction with equal probability. That is the symmetry. But when it fell, it would always break the symmetry, so the outcome of this law of gravity does not have the same symmetry as the law of gravity.
You and I are at this moment located at particular positions in the universe. The laws of electricity and magnetism and gravity of which we are sophisticated, complicated outcomes, have no preference for any particular position in the universe, but if that were the case for the outcomes of those laws, nothing could exist. We are outcomes of those laws of gravity and electromagnetism in which those symmetries are broken, and so we can be situated at particular positions.
So this, if you like melodrama, is the nearest you get to what I call the secret of the universe: the fact that the outcomes of the laws of nature do not possess the same symmetries, the same patterns, the same simplicities, as the laws themselves. This is how it can be that you can have a universe like our own that is governed by four, or perhaps we may eventually find just the one, simple, symmetrical form of law, and yet give rise to an unlimited number of highly complex, completely asymmetrical outcomes, even including you and me, because those outcomes do not have to carry the symmetries and simplicities of the laws.
Now you are beginning to see a bit more of the answer to our first question. If your focus on the world is upon the symmetries and the laws, then you are in the realm perhaps of particle physics, permutation physics, and you are looking at the world at the level where it is simplest and most symmetrical. But most other people are interested in the complexities of the outcomes of the laws – solid state physics, chemistry, economics, social sciences. These are all complicated messy, pattern-free outcomes of those laws, and if you are studying these real world sciences, you are mostly impressed by the complexity of the world. So you begin to see why you have got the two answers. If you talk to somebody who spends their time looking at the world at the level of its laws, they will tend to think the world is simple and symmetrical, but if you start talking to people who are most interested in the complicated outcomes of the laws of nature, then they will be most impressed by the complexities of the world.
Until the late 1970s, most scientists, and certainly most scientific education, a rather platonic thread. There is a good reason for that. It is easier. This is the world of 30-minute examination questions that can be done with pencil and paper, it is the world of problems about frictionless surfaces and perfect spheres, and idealised, simple situation, things that can be solved by hand with pencil and paper. But in 1978, something dramatic happened in science, and it is something that came from the world of business and technology. It was the creation of the personal computer. The first personal computers revolutionised the study of science, because they made it possible for the first time to investigate what went on in a simple, inexpensive way. Computers had existed before, but they cost millions of pounds or dollars and they tended to be controlled by ferocious research groups who would not let other people use them and dedicated their use for blockbusting problems, like atom bombs, predicting the weather, predicting the future of the economy, studying how stars form or explode. But all of a sudden, it was possible for single individuals or just small research teams to have on their desks a reasonably priced, powerful machine, with very good interactive graphics, which you could use to investigate what went on in this world of complex outcomes. You could do it experimentally. You did not have to solve the problem exactly using pencil and paper; you could create it as a movie and watch what happened. You create an economic system that is governed by certain rules and constraints; you cannot envisage what is going to happen, but you can watch, over time, what ensues, try and spot patterns and simplifications, and then set about modelling it precisely with pencil and paper.
I want to spend the last bit of time talking about just a few of the curious things that emerged out of this world of the complex outcomes. I like to divide complexity into two varieties. It is rather like crime: it is either organised or it is disorganised.
Disorganised complexity has been around a long time, and I will talk about that first, because it generally goes by the name of chaos. This term was invented in about 1978. It is no accident you notice that serious, systematic study of chaos began pretty much at the same time as small computers became available. My definition of chaos, in the sense that you have seen in Jurassic Park and elsewhere, is simply that it is dramatic sensitivity to ignorance. So a chaotic situation is one where if you have a little bit of imperfection in your knowledge about situations, then very soon that imperfection is going to magnify and grow enormously and you will know essentially nothing of any practical use about the situation. The danger of course is that you are not aware that the situation you are studying has this property, and you are oblivious to the fact that you are calculating and predicting the course of the economy and it is all nonsense because it has this chaotic property.
Let me give you a simple example. Back in 1871, the first person to notice that the world had such a propensity was James Clark Maxwell, the person who joined together those forces of electricity and magnetism, the greatest physicist after Newton – and he was Scottish. He gave a beautiful example when he first introduced this idea, very Victorian – it is the railway point, of how you can have a situation where a very, very small change in the location of the point lever can produce an enormous divergence in the future course of the track. So a very small change in one place can produce a very big effect later on.
Here is a more sophisticated example, but still very simple. Suppose that we are in the simplest imaginable universe. This universe is governed by one law only. The universe is a bit like a clock face. All that happens in this universe is that a pointer moves around the circumference of a circle, like the minute hand on a clock. So we can locate it just by the angle, away from the vertical. This universe is governed by one law, which keeps on acting again and again and again, and what the law does is just to double the angle at each step. So it starts at 10 degrees, then it goes to 20, to 40, to 80, to 160, to 320, and then it comes round again and goes to 640. Just like the clock hand, it wraps around each time. Now this is where it is pretty simple: it is deterministic in principle – that is, if you know the initial position of the pointer, then after any number of applications of the doubling rule, you will know precisely where the point is, with 100% accuracy. So if this is the value of your investment, as it were, today, and you carry out all sorts of transactions and trades, 50 of them, then after 50 applications of the rule, you will know exactly the value of your investment. But unfortunately, the real world is never like that, that in this universe, there must always be some uncertainty in the location initially of your point. So in this example, it may just be the radius of an atomic nucleus or a single atom on a plastic film, but what happens when we apply the doubling rule is rather alarming. You see, the uncertainty doubles in each step as well, and gradually it gets bigger and bigger, and eventually it will become bigger than 360 degrees, and then you won’t know anything about the location of the point on the circle, even though you are in possession of the exact rule that moves it around. All you can say is that, in the long run, it is equally likely to be found anywhere! But you can see if you were not aware of that, you would go on calculating to ever-greater precision, kidding yourself that you had a perfectly predictable situation, but increasingly what you were predicting would be of no value at all. In fact, in this particular example, I think if you were to locate the point with an accuracy equal to the size of the atomic nucleus, it would take only about 40 applications of this rule before that uncertainty was bigger than the circumference of the circle. So these uncertainties grow very fast indeed.
There is an important lesson to learn about this type of example. If you read the wrong books or newspaper articles, you might get the view that because the world has this chaotic feature, that you cannot know anything about it, that all I said before about laws of nature is all of no value, because if this unpredictability is around, then how could you ever rely on any of the dictums you know? An important applicational worry about this type of feature is of course predicting the weather. The reason we cannot predict the weather is not because we are not good at predicting, that we do not know how the weather changes; it is because we do not know the state of the weather everywhere with perfect accuracy now. We know the state of the weather with high accuracy every 50 kilometres, or maybe every 200 kilometres, over the ocean, but the uncertainty that can exist in between is sufficient that when you run the computer programme, you will get a very different forecast for what might happen tomorrow. But nonetheless, you are sometimes quite right in certain respects, and the reason that is so is something that I think it was Florence Nightingale long ago first recognised – Florence Nightingale was quite a successful pioneer in the realm of statistics, one of the pioneering members of the Royal Statistical Society. That is why she was effective at nursing, because she carried out statistical analyses of what procedures worked and had beneficial effects and which did not. What she noticed was that one of the fascinating things about the world was not that at this moment things existed in some sort of harmony, but as time when on, that harmony was maintained – the number of males and females roughly balance over a long period of time.
Well, if everything was chaotic and unpredictable, think about the air molecules in this room – nitrogen, oxygen. Each one of the collisions that they undergo with others has this chaotic property. If you have an uncertainty in how they are moving before they hit something else, it will magnify massively by about a factor of a thousand in the course of the collision. So it is not just doubling, it is growing a thousand-fold, your ignorance, with each collision.
Nonetheless, when we were in school, we remember learning something called Boyle’s Law, that if you have a big volume like this room, and you measure the pressure of the gas in it, the volume of the room, multiply them together and divide by temperature, then even if you switch things around in the room, you turn the heater on so the temperature goes up, if you work out this quantity, pressure times volume divided by temperature, it will always be the same. So there is a simple rule, or law, which governs what happens to these air molecules. The reason that is possible is because although the individual motions are chaotically unpredictable, their average behaviour is entirely predictable. You see, the temperature is just the measure of the average speed of the molecules. The pressure is a measure of the average force that they exert per unit area of the walls. So in many chaotic systems, the small scale unpredictability is washed away by nice predictability on the average over the large scale.
Well, so much for disorganised complexity. It is pretty well understood. There are not many surprises I think to come in that area, but in a sense, a lot of scientists move on to something that was much more complicated and tricky to understand, and it is the problem of organised complexity. Organised complexity is a bit like having a beard – there is no definition of what constitutes a beard, but somehow, when you see one, you recognise it. Physicists’ approach to organised complexity is a bit like that.
I have put examples on a picture, characterised by just two properties, but these examples have many more properties than these, many examples of organised complexity. The graph is in computer-speak language. Here is plotted, numerically, the size of these objects, the memory capacity, bytes, so all repeat is the number of zeros and ones that you would need to encode the amount of information in the system. It shows how many zeros and ones would we need to encode all the information in those categories. This is just a measure of the storage capacity. The things along the bottom, from the abacus scoring one number on its lines all the way up to more complicated forms of DNA, up to these hybrids. These are things that are storing more and more information, but they are pretty boring. They do not do anything with the information.
Computational power – this is what your computer has and Microsoft tell you you need more of. All any computer does is to take one list of numbers and turn them into another list, and its power is just the speed that it does it. So anything that changes one list of numbers into another list, like your grandmother changing her shopping list, has a computational power. These objects, the radio channel, the television channel, these are things which can transform information very quickly from one form into another – digital into pictorial, say – but they do not store lots of information.
But if we march up the diagonal on this picture, we find all the most interesting and impressive things that we have encountered in our local universe, including ourselves, and as you march up the diagonal, the things on it combine the ability to store more and more information with an ability to transform it into new forms, faster and faster. That in practice is what we mean by organised complexity. The things on the picture are all made of atoms. If we had a Theory of Everything of the particle physicists’ sort, it would not help us understand the structure of the things in the picture by one iota more. A new theory of elementary particles would not help us understand how the national telephone system works any better, or how a piece of accounting software works, because the objects in the picture are what they are and they do what they do because of the way that they are hard-wired together, the way in which the components are organised. For the most part, it does not matter what the components are made of. This is one feature of organised complexity: it relies on the way things are organised and joined together. Also, most of the things in the picture have a remarkable property that sometimes people call emersion, that they are more than the sum of their parts, so that by taking them to pieces, you learn some vital aspect of their structure.
If we take something like a liquid, like water, which is rather complicated, like all liquids, if you try to pour it, it has a certain resistance to being poured. If it was treacle, it would have more resistance. That property we call viscosity. Now, if we look at an individual atom, we will not find a little bit of viscosity on a single atom. Viscosity is a property that arises collectively when you bring together a large number of atoms and molecules, so it emerges from that situation. By taking the molecules to pieces, you lose that collective emergent property. Something like consciousness perhaps is something that emerges when we have sufficient level of complexity in the system.
Also, every example is very far from thermal equilibrium. We are familiar that you can have order very far from thermal equilibrium so, if you like, a candle flame is much, much hotter than its surroundings, but it enjoys a stable equilibrium, consuming oxygen from its environment, cooling as the flame gets bigger. All these examples are far from equilibrium, and they store information in the same order.
So much for organised complexity. The last thing I want to describe was a discovery of something that exists on the interface between organised and disorganised complexity, where it turns out perhaps the most significant types of complexity exist. I just want to show you three examples: the first is a sand pile, the second is a river, and the last is a rather reluctant dog. Well, all these sound rather unlikely examples.
The first, the sand pile, I like because it shows that it is still possible to make fundamental discoveries in science without the need for billions of pounds of funding, or equipment or facilities, satellites, but something that could just exist on your dinner table and cost about 10 pence could be something which, if you look at it in the right way, enables you to understand something very profound about the world. Suppose you take some sugar, or some sand, some salt, and you start to pour it down on to a tablecloth into a pile, and you watch what happens. Well, you guess, at first, the grains fall around some area of the table and they start to form a pile that gets steeper and steeper. The grains come in and they tumble down the pile one way or the other. This is a chaotic process, in the sense that if you make a little change in how you drop the grains in, there will be a big difference in their future career – they will either go down this side or another side. So there is chaotic process here, which gives rise to an increasingly ordered pile. Keep pouring the sand, the pile gets steeper and steeper, avalanches occur all the time, but eventually, something very strange happens. The pile does not get any steeper. It reaches a particular critical slope, and any future sand that comes in just produces more tumbling avalanches which act to keep the whole slope at that special inclination. So what happens is that locally you get the little build up of sand for a tiny little hill-let, it gets steeper to a critical slope, and then it tumbles. Same thing happens again, build up, build up, avalanche, build up, avalanche.
What is happening here is that the overall organisation of this pile is a combination of chaos plus a force, and the force is gravity, that is making pieces of sand or sugar or rice tumble down. If there was an edge to the table, eventually the sand will be falling over the edge at the same rate that it was pouring in from the top. So an ordered pile would be transient in a sense that different grains would form it at different times, but its structure would remain invariant.
This example is something that is called self-organised criticality. Self-organised, well, you can see why you would call it self-organised: it organises itself into this orderly pile. Critical because, although it is stable, at every moment, it is on the verge of instability: there is an avalanche, then there is another avalanche, so it is instability that keeps it orderly. All you need for this to occur is one particular feature of the problem, that’s special but not unusual, and it is that there must not be a special size of avalanche. So if you have slightly sticky, gluey sand, which tended to form little globs of 50 grains which then rolled down the hill, this wouldn’t happen. If you have that incredibly fine silica sand, that is almost like dust, where surface forces make pieces join together, you would get a different type of behaviour, but for ordinary rice, and to some extent, sand, you get this behaviour, because the frequency, the chance of getting an avalanche of a particular size, just depends on the mathematical power of the size. The power can be negative so that the bigger avalanche will narrow the probability. What is remarkable about this activity here, you see it is on the interface between chaos and organised complexity.
Although weare talking about sand grains and sand piles, the suspicion for a long time was that this might be a paradigm for how all sorts of complexity develop in the world, that instead of avalanches of sand, suppose you had economic collapses, bankruptcies, that you had sort of build-up of capital investment companies, then there would be a collapse, build-up, collapse… The insight you get is that the collapses do not have entirely negative effect. People who work for those companies disperse into a workforce, the cash flows around somewhere else and comes back in. It is possible to have equilibrium even with local catastrophes.
Suppose you have a forest where there is a fire. At first, you think this is a disaster, everything is wiped out, all the trees are razed to the ground. Come back a few years later, you will find that the removal of the trees allowed light to reach the ground, and all sorts of new small plants and animals have sprung up, and gradually a new equilibrium has been found.
If the pile of sand and the avalanches is really traffic flow, so you’re looking at the M25, and the avalanches are like traffic jams – they stop, they start again – the size of the avalanche would be the waiting time in the jam. One of the curious things about this critical state is that it is unstable to little changes, and once an avalanche occurs somewhere, you cannot put your finger on what was the cause; you know, what was the sand grain that came in and caused the little avalanche, because it is a collective effect of many things interacting with another. So when you are in that traffic jam on the M25 and you have been stuck for five minutes, you think, oh there has been an accident down the road, the traffic starts to move, and there is no accident, there is no sign of any cause for your delay, all that is happening in the traffic flow is that it is organising itself in this way. If you stop and start and change lanes, you start a little perturbation that runs through the traffic system. It creates waiting lines of all lengths, and every so often, you will have a rather long one, and sometimes you will have a very short one, but you should not necessarily expect that you can identify the specific cause.
A second example that is a bit like that, to show how sometimes you can step into a completely different situation and see the same thing happening: suppose you have on a flat plain a meandering river, and meandering rivers have an interesting systematic type of shape. They are not random. You might begin to ask, well, is there a way of understanding something about the shape?
If you were at a school like mine, there is one fact about geography that everyone in the United Kingdom knows, and it is the oxbow lake. Like the one thing you know in history is the day and the month but not the year of the Gunpowder Plot! If you have a river that has a little bit of curvature in it, what tends to happen is that on the curves, you get some erosion, and gradual steepening of the curvature of the river, until eventually, there is too little flow and the oxbow lake breaks off. The result is that the river is then a bit straighter than it was before, so the formation flows at the places where the curvature is greatest leads to an organisation of the shape, the flow of the river. It is just like the sand pile: increased flow of the river, steepening of the curve, build-up of sand, build-up of sand, avalanche, break off. As a result, the sand pile is a little straighter than before and so is the river. So it is exactly analogous to the sand pile situation.
We might try and ask, well, what is the common factor between these sorts of examples? Here is our dog, which shows us why we see this type of behaviour in so many superficially different situations. If you are in a problem where there is a force acting – such as the sand falling - it was gravity; with the river, it was the centrifugal force of the water flow around the bend – and you have a number of different equilibrium states in which you can pool different slopes of the different configurations of the sand pile and so on. You can see what happens: the application of the force makes the dog begin to move uphill, rather slowly, rather gradually, eventually it jumps across…. then it walks up slowly, jumps across. So this slow climbing up the hill, sudden jump, slow climbing up the hill, sudden jump, is a very characteristic way for complicated systems to change. You can understand why combining the idea of chaos generates many possibilities, many equilibrium states.
To try and draw things together, if someone stops you outside and asks what did you learn, what could you say? Here is a picture that tries to summarise some of the things that we have looked at. A graph of a sort: along here, is a measure, superficially, of the complexity of some things that we see in the world, and up here is a measure of our ignorance and our uncertainty about the laws and equations that govern them. Along here, we have examples of things where we have very little ignorance about the laws – how the planets move in the solar system, or the rings of Saturn and so forth, the moons of Neptune, the weather, turbulence, what happens when you turn your tap on in your bath full on – highly complex situation, very poorly understood. Again, there is no problem with the laws that govern these things. We think we understand those, but we are in the realm of the complex outcomes, where the chaotic uncertainties and organised forms of complexity and criticality dominate what goes on. If we walk up here, we are in the realm of particle physics and the beginning of the universe. What happens in particle physics in a big accelerator at CERN is very, very simple. Two particles collide and come out at different speeds and energies. So there is very little complexity of phenomena, but we have considerable uncertainty often about the form of the laws, the fundamental laws that govern what goes on there, what is this string theory that ultimately governs gravity.
So we have two sorts of ignorance about the world, and the simple world of laboratory chemistry and thermal dynamics and simple engineering will work in this room from here, where we have simple phenomena and considerable certainty about the rules. As we move up here, we are entering dangerous ground, where we start to increase the complexity and the uncertainty of our understanding, the dotted line is somewhat randomly drawn – it lies where we sort of have a good grip on what is going on. Once we start to get up here – what is life, what is consciousness, how does it work, what is the complexity of terrestrial climate, what are the factors that govern it – we are on very dangerous ground, we do not understand fully what is going on.
If we go way up here, to the world of social science and finance and so forth, we enter a completely different sort of problem, things which are unpredictable in principle, not just in practice. Why do I say that? Well, predicting the weather does not actually change the weather, but predicting the course of the future of the economy does change the economy. If the Chancellor goes on television and predicts what the economy is going to do in the next three months, it will change what the economy does in the next three months in ways that it is logically impossible for him to incorporate fully in his prediction. There is no way that I can have a mathematical prediction of your future behaviour that you cannot choose to falsify if I make it known to you. I can know completely what you are going to do if I keep it secret and do not tell you, but if I tell you, you can always falsify it.
So this picture tries to combine something of what I have told you, and I hope you have been able to understand about why there are two ways of looking at the world. If you look at the world at the level of laws, then it is indeed simple, much simpler than we might have imagined, much more compressible than we could perhaps have hoped. But if we look at the world at the level of outcomes of those laws, it is fantastically and richly complex, and that is one of the things that makes it so fascinating. So I hope if someone stops you on the street outside, you will be able to spend an enjoyable hour explaining to them why the world is both simple and complicated!
© Professor John Barrow, Gresham College, 27 September 2006
This event was on Wed, 27 Sep 2006
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