If nature were not beautiful it would not be worth knowing, and if nature were not worth knowing, life would not be worth living.
Henri Poincaré
I wasted a lot of time at university playing billiards in our student common room. I could have pretended that it was all part of my research into angles and stuff, but the truth is that I was procrastinating. It was a good way of putting off having to cope with not being able to answer that week’s set of problems. But in fact the billiard table hides a lot of interesting mathematics in its contours. Mathematics that is highly relevant to my desire to understand my dice.
If I shoot a ball round a billiard table and mark its path, then follow that by shooting another ball off in very nearly the same direction, the second ball will trace out a very similar path to the first ball. Poincaré had believed that the same principle applied to the solar system. Fire a planet off in a slightly different direction then the solar system will evolve in a very similar pattern. This is most people’s intuition: if I make a small change in the initial conditions of the planet’s trajectory it won’t alter the course of the planet much. But the solar system seems to be playing a slightly more interesting game of billiards than the one I played as a student.
Rather surprisingly, if I change the shape of the billiard table this intuition turns out to be wrong. For example, fire balls round a billiard table shaped like a stadium with semicircular ends but straight sides and the paths can diverge dramatically even though they started in almost the same direction. This is the signature of chaos: sensitivity to very small changes in the initial conditions.
Two quickly diverging paths taken by a billiard ball round a stadium-shaped billiard table.
So the challenge for me is to determine whether the fall of my dice is predictable, like a conventional game of billiards, or whether the dice is playing a game of chaotic billiards.
Despite Poincaré being credited as the father of chaos, it is striking that this sensitivity of many dynamical systems to small changes was not very well known for decades into the twentieth century. Indeed, it really took the rediscovery of the phenomenon by scientist Edward Lorenz, when he, like Poincaré, thought he’d made some mistake, before the ideas of chaos became more widely known.
While working as a meteorologist at MIT in 1963, Lorenz had been running equations for the change of temperature in a dynamic fluid on his computer when he decided he needed to rerun one of his models for longer. So he took some of the data that had been output earlier in the run and re-entered it, expecting to be able to restart the model from that point.
When he returned from coffee, he discovered to his dismay that the computer hadn’t reproduced the previous data but had generated very quickly a wildly divergent prediction for the change in temperature. At first he couldn’t understand what was happening. If you input the same numbers into an equation, you don’t expect to get a different answer at the other end. It took him a while to realize what was going on: he hadn’t input the same numbers. The computer printout of the data he’d used had only printed the numbers to three decimal places, while it had been calculating using the numbers to six decimal places.
Even though the numbers were actually different, they differed only in the fourth decimal place. You wouldn’t expect it to make that big a difference, but Lorenz was struck by the impact such a small difference in the numbers had on the resulting data. Here are two graphs created using the same equation but where the data that is put into the equations differ very slightly. One graph uses the input data 0.506127. The second graph approximates this to 0.506. Although the graphs start out following similar paths, they very quickly behave completely differently.
The model that Lorenz was running was a simplification of models for the weather that analysed how the flow of air behaves when subjected to differences in temperature. His rediscovery of how small changes in the way you start a system can have such a big impact on the outcome would have huge implications for our attempts to use mathematical equations to make predictions into the future. As Lorenz wrote:
Two states that were imperceptibly different could evolve to two considerably different states. Any error in the observation of the present state – and in a real system, this appears to be inevitable – may render an acceptable prediction of the state in the distant future impossible.
When Lorenz explained his findings to a colleague, he received the reply: ‘Edward, if your theory is correct, one flap of a seagull’s wings could alter the course of history forever.’
The seagull would eventually be replaced by the now famous butterfly when Lorenz presented his findings in 1972 at the American Association for the Advancement of Science in a paper entitled: ‘Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?’
Curiously, both the seagull and the butterfly might have been pre-empted by the grasshopper. It seems that already in 1898 Professor W. S. Franklin had realized the devastating effect that the insect community could have on the weather. Writing in a book review, he believed:
An infinitesimal cause may produce a finite effect. Long-range detailed weather prediction is therefore impossible, and the only detailed prediction which is possible is the inference of the ultimate trend and character of a storm from observations of its early stages; and the accuracy of this prediction is subject to the condition that the flight of a grasshopper in Montana may turn a storm aside from Philadelphia to New York!
This is an extraordinary position to be in. The equations that science has discovered give me a completely deterministic description of the evolution of many dynamical systems like the weather. And yet in many cases I am denied access to the predictions that they might make because any measurement of the location or wind speed of a particle is inevitably going to be an approximation to the true conditions.
This is why the MET office, when it is making weather predictions, takes the data recorded by the weather stations dotted across the country and then, instead of running the equations on this data, the meteorologists do several thousand runs, varying the data over a range of values. The predictions stay close for a while, but by about five days into the future the results have often diverged so wildly that one set of data predicts a heat wave to hit the UK while a few changes in the decimal places of the data result in rain drenching the country.
Starting from nearly the same conditions, forecast A predicts strong wind and rain over the British Isles in 4 days’ time, while forecast B predicts incoming high pressure from the Atlantic.
The great Scottish scientist James Clerk Maxwell articulated the important difference between a system being deterministic yet unknowable in his book Matter and Motion, published in 1877: ‘There is a maxim which is often quoted, that “The same causes will always produce the same effects.”’ This is certainly true of a mathematical equation describing a dynamical system. Feed the same numbers into the equation and you won’t get any surprises. But Maxwell continues: ‘There is another maxim which must not be confounded with this, which asserts that “Like causes produce like effects.” This is only true when small variations in the initial circumstances produce only small variations in the final state of the system.’ It is this maxim that the discovery of chaos theory in the twentieth century revealed as false.
This sensitivity to small changes in initial conditions has the potential to sabotage my attempts to use the equations I’ve written down to predict the outcome of my dice. I’ve got the equations, but can I really be sure that I’ve accurately recorded the angle at which the cube leaves my hand, the speed at which it is spinning, the distance to the table?
Of course, everything isn’t completely hopeless. There are times when small changes don’t alter the course of the equations dramatically, like the paths in the classical billiard table. What is important is to know when you cannot know. A beautiful example of knowing the point when you can’t know what is going to happen next was discovered by mathematician Robert May when he analysed the equations for population growth.
Born in Australia in 1938, May had originally trained as a physicist working on superconductivity. But his academic work took a dramatic turn when he was exposed in the late 1960s to the newly formed movement in social responsibility in science. His attention shifted from the behaviour of collections of electrons to the more pressing questions of the behaviour of population dynamics in animals. Biology at the time was not a natural environment for the mathematically minded, but following May’s work that would all change. It was this fusion of the hardcore mathematical training he’d received as a physicist combined with a new sensibility to biological issues that led to his great breakthrough.
In a paper in Nature called ‘Simple Mathematical Models with Very Complicated Dynamics’, published in 1976, May explored the dynamics of a mathematical equation describing population growth from one season to the next. He revealed how even a quite innocent equation can produce extraordinarily complex behaviour in the numbers. His equation for population dynamics wasn’t some complicated differential equation but a simple discrete feedback equation that anyone with a calculator can explore.
Suppose I consider an animal population whose numbers can vary between 0 and some hypothetical maximum value that I will call N. Given some fraction Y (lying between 0 and 1) of that maximum, the equation determines what in the next season is the revised fraction of the population that survives after reproduction and competition for food. Let’s suppose that each season the reproduction rate is given by a number r. So that if the fraction of the maximum population that survived to the end of the season was Y, the next generation swells to r × Y × N.
But not all of these new animals will survive. The equation determines that the fraction that will not survive is also given by Y. So out of the r × Y × N animals that start the season, Y × (r × Y × N) die. So the total left at the end of the season is (r × Y × N) – (r × Y2 × N) = [r × Y × (1 – Y)] × N, which means that the fraction of the maximum population that exists in the current season is r × Y × (1 – Y).
Essentially the model assumes that at the end of each season the surviving population gets multiplied by a constant factor, called r, the reproduction rate, to produce the number of animals at the beginning of the next season. But there aren’t enough resources for them all to survive. The equation then calculates how many of these animals will make it till the end of the season. The resulting number of animals that survive then gets multiplied by the factor r again for the next generation. The fascinating property of this equation is that its behaviour really depends only on the choice of r, the reproduction rate. Some choices of r lead to extremely predictable behaviours. I can know exactly how the numbers will evolve. But there is a threshold beyond which I lose control. Knowledge is no longer within reach because the addition of one extra animal into the mix can result in dramatically different population dynamics.
For example, May discovered that if r lies between 1 and 3 then the population eventually stabilizes. In this case it doesn’t matter what the initial conditions are, the numbers will gradually tend to a fixed value depending on r. It’s like playing billiards on a table where there is a sinkhole in the middle. However I shoot the ball off, it eventually finds its way to the bottom of the sinkhole.
For r above 3, I still see a region of predictable behaviour but of a slightly different character. With r between 3 and 1 +√6 (which is approximately 3.44949), the population dynamics eventually ping-pong between two values that depend on r. As r passes 1 +√6, we see the population dynamics changing character again. For r between 1 +√6 and 3.54409 (or more precisely the solution of a polynomial equation of degree 12), there are 4 values that the population periodically cycles through. As r gets bigger, I get 8 values, then 16, and so on. As r climbs, the number of different values doubles each time until I hit a threshold moment when the character of the dynamics flips from being periodic to chaotic.
When May first explored this equation, he admitted that he frankly hadn’t a clue what was going on beyond this point – he had a blackboard outside his office in Sydney on which he offered a prize of 10 Australian dollars to anyone who could explain the behaviour. As he wrote on the blackboard: ‘It looks like a mess.’
It was on a visit to Maryland that he got his answer and where the term ‘chaos’ was actually coined. In the seminar he gave, he explained the region in which the period doubles but admitted he’d hit a point beyond which he didn’t know what the hell was happening. In the audience was a mathematician who did know. Jim Yorke had never seen the doubling behaviour but he knew exactly what was going on in this higher region. And it was what he called chaos.
Beyond r = 3.56995 (or more precisely the limit point of the solutions of a system of equations of increasing degree), the behaviour becomes very sensitive to what the initial population looks like. Change the initial number of animals by a minute amount and a totally different result can ensue.
Two populations with r = 4 that start off with a difference of just one animal in a thousand. Although they start behaving similarly, by year 15 they are demonstrating very different behaviours.
But as I turn up the dial on r, there can still be pockets of regular behaviour, as Jim Yorke had discovered. For example, take r = 3.627 and the population becomes periodic again, bouncing around between 6 different values. Keep dialling r up and the 6 changes to 12 which becomes 24, doubling each time until chaos strikes again.
Bob May recognized just what a warning shot such a simple system was to anyone who thought they knew it all: ‘Not only in research, but in the everyday world of politics and economics, we would be better off if more people realized that simple systems do not necessarily possess simple dynamic properties.’
Bob May is currently practising what he preaches. Or perhaps I should say Lord May of Oxford, as I was corrected by a man in a top hat who greeted me at the door to the peers’ entrance of the House of Lords. May has in recent years combined his scientific endeavours with energetic political activism. He now sits as a cross-party member of the House of Lords, which is where I popped in for lunch to find out how he was doing in his mission to alert politicians to the impact of chaotic systems on society.
Ushered through the peers’ entrance to the Lords by the man in the top hat and policemen with machine guns, I found May waiting for me on the other side of metal detectors and X-ray machines. May has no truck with all these formal titles and in his earthy Australian manner still insists on being called Bob. ‘I’m afraid I messed up and already ate lunch but I’ll come and eat cake while you get some lunch.’ As I ate fish he consumed an enormous piece of House of Lords chocolate cake. At 79, May is as energetic and engaged as ever and was rushing off after his second lunch to a select committee discussing the impact of a new rail link between London and the northwest of England.
Before joining the Lords, May was chief scientific adviser both to John Major’s Conservative government and then subsequently to Tony Blair’s Labour government. I wondered how tricky a balancing act such a political position is for a man who generally is not scared to tell it like it is.
‘At the interview I was told that there would be occasions where I would be called upon to defend the decisions of a minister and how would I feel about that? I said that I would never under any circumstances deny a fact. On the other hand, I’m fairly good at the kind of debating competition where you’re given a topic and according to a flip of a coin you’ve got to argue for either side of the debate. So I said I’d be happy explaining why the minister’s choice was arrived at. I simply wouldn’t agree to endorse it if it wasn’t right.’
A typical mathematical response. Set up the minister’s axioms and then demonstrate the proof that led to the conclusion. A judgement-free approach. That’s not to say that May isn’t opinionated and prepared to give his own views on the subject at hand.
I was curious how governments deal with the problems that chaos theory creates for anyone trying to make policy decisions. How do politicians cope with the challenges of predicting or manipulating the future, given that we can have only partial knowledge of the systems being analysed?
‘I think that’s rather a flattering account of what goes on here. With some notable exceptions it’s mostly a bunch of very egotistical people, very ambitious people, who are primarily interested in their own careers.’
What about May personally? What impact did the discoveries he’d made have on his view of science’s role in society?
‘It was weird. It was the end of the Newtonian dream. When I was a graduate student it was thought that with better and better computer power we would get better and better weather predictions because we knew the equations and we could make more realistic models of the Earth.’
But May is cautious not to let the climate change deniers use chaos theory as a way to undermine the debate.
‘Not believing in climate change because you can’t trust weather reports is a bit like saying that because you can’t tell when the next wave is going to break on Bondi beach you don’t believe in tides.’
May likes to quote a passage from Tom Stoppard’s play Arcadia to illustrate the strange tension that exists between the power of science to know some things with extraordinary accuracy and chaos theory, which denies us knowledge of many parts of the natural world. One of the protagonists, Valentine, declares:
We’re better at predicting events at the edge of the galaxy or inside the nucleus of an atom than whether it’ll rain on auntie’s garden party three Sundays from now.
May jokes that his most-cited works are not the high-profile academic papers he’s published in prestigious scientific journals like Nature, but the programme notes he wrote for Stoppard’s play when it was first staged at the National Theatre in London. ‘It makes a mockery of all these citation indexes as a way of measuring the impact of scientific research.’
So what are the big open questions of science that May would like to know the answer to? Consciousness? An infinite universe?
‘I think I’d look at it in a less grand way, so I’d look at it more in terms of the things I am working on at the moment. Largely by accident I’ve been drawn into questions about banking.’
That was a surprise. The question of creating a stable banking system seemed very parochial, but May has recently been applying his models of the spread of infectious diseases and the dynamics of ecological food webs to understanding the banking crisis of 2008. Working with Andrew Haldane at the Bank of England, he has been considering the financial network as if it were an ecosystem. Their research has revealed how financial instruments intended to optimize returns to individual institutions with seemingly minimal risk can nonetheless cause instability in the system as a whole.
May believes that the problem isn’t necessarily the mechanics of the market itself. It’s the way small things in the market are amplified and perverted by the way humans interact with them. For him the most worrying thing about the banking mess is getting a better handle on this contagious spreading of worry.
‘The challenge is: how do you put human behaviour into the model? I don’t think human psychology is mathematizable. Here we are throwing dice with our future. But if you’re trying to predict the throw of the dice then you want to know the circumstance of who owns the dice.’
That was something I hadn’t taken into account in my attempts to predict the outcome of my casino dice. Perhaps I need to factor in who sold me my dice in the first place.
‘I think many of the major problems facing society are outside the realm of science and mathematics. It’s the behavioural sciences that are the ones we are going to have to depend on to save us.’
Looking round the canteen at the House of Lords, you could see the sheer range and complexity of human behaviour at work. It makes the challenge of mathematizing even the interactions in this tiny microcosm of the human population nigh impossible. As the French historian Fernand Braudel explained in a lecture on history he gave to his fellow inmates in a German prison camp near Lübeck during the Second World War: ‘An incredible number of dice, always rolling, dominate and determine each individual existence.’ Although each individual dice is unpredictable, there are still patterns that emerge in the long-range behaviour of many throws of the dice. In Braudel’s view this is what makes the study of history possible. ‘History is indeed “a poor little conjectural science” when it selects individuals as its objects … but much more rational in its procedure and results when it examines groups and repetitions.’
But May believes that understanding the history and origins of the collection of dice that make up the whole human race is not as straightforward as Braudel makes out. For example, it’s not at all clear that we can unpick how we got to this point in our evolutionary journey.
‘I’ll tell you one of the questions that I think is a particularly interesting one: trying to understand our evolutionary trajectory as humans on our planet. Is the trajectory we seem to be on what happens on all or most planets, or is it the result of earlier fluctuations in the chaos which took us on this trajectory rather than another. Will we ever know enough to be able to ask whether the disaster we seem to be heading for is inevitable or whether there are lots of other planets where people are more like Mr Spock, less emotional, less colourful, but much more detached and analytical.’
Until we discover other inhabited planets and can study their trajectories, it’s difficult to assess whether evolution inevitably leads to mismanaged ecosystems based on just one dataset called Earth.
‘The question of whether where we’re heading is something that happens to all inhabited planets or whether there are other planets where it doesn’t happen is something I think we’ll never know.’
And with that May polished off the last few crumbs of his chocolate cake and plunged back into the chaos of the select committees and petty politics of Westminster.
May’s last point relates to the challenge that chaos theory poses for knowing something about the past as much as the future. At least with the future we can wait and see what the outcome of chaotic equations produces. But trying to work backwards and understand what state our planet was in to produce the present is equally if not more challenging. The past even more than the future is probably something we can never truly know.
May’s pioneering research explored the dynamics of a population as it went from season to season. But what determines which animals survive and which die before reproducing? According to Darwin, this is simply down to a lucky roll of the evolutionary dice.
The model of the evolution of life on Earth is based on the idea that once you have organisms with DNA, then the offspring of these organisms share the DNA of their parent organisms. But parts of the genetic code in the DNA can undergo random mutations. These are essentially down to the chance throw of the evolutionary dice. But there is a second important strand to Darwin’s proposal, which is the idea of natural selection.
Some of those random changes will give the offspring an increased chance of survival, while other changes will result in a disadvantage. The point of evolution by natural selection is that it is more likely that the advantageous change will survive long enough to reproduce.
Suppose, for example, that I start with a population of giraffes that have short necks. The environment of the giraffes changes such that there is more food in the trees, so that any giraffe born with a longer neck is going to have a better chance of survival. Let’s suppose that I throw my Vegas dice to determine the chance of a mutation for each giraffe born in the next generation following this environmental change. A roll of a 1, 2, 3, 4 or 5 condemns the giraffe to a neck of the same size or shorter, while a throw of a 6 corresponds to a chance mutation which causes a longer neck. The lucky longer-necked giraffes get the food and the shorter-necked giraffes don’t survive to reproduce. So it is just the longer-necked giraffes that get the chance to pass on their DNA.
In the next generation the same thing happens. Roll a 1, 2, 3, 4 or 5 on the dice and the giraffe doesn’t grow any taller than its parents. But another 6 grows the giraffe a bit more. The taller giraffes survive again. The environment favours the giraffes that have thrown a 6. Each generation ends up a bit taller than the last generation until there comes a point where it is no longer an advantage to grow any further.
It’s the combination of chance and natural selection that results in us seeing more giraffes with ancestors that all threw 6s. In retrospect it looks like amazing chance that you see so many 6s in a row. But the point is that you don’t see any of the other rolls of the dice because they don’t survive. What looks like a rigged game is just the result of the combination of chance and natural selection. There is no design or fixing at work. The run of consecutive 6s isn’t a lucky streak but is actually the only thing we would expect to see from such a model.
It’s a beautifully simple model, but, given the complexity of the changes in the environment and the range of mutations that can occur, this simple model can produce extraordinary complexity, which is borne out by the sheer variety of species that exist on Earth. One of the reasons I never really fell in love with biology is that there seemed to be no way to explain why we got cats and zebras out of this evolutionary model and not some other strange selection of animals. It all seemed so arbitrary. So random. But is that really fair?
There is an interesting debate going on in evolutionary biology about how much chance there is in the outcomes we are seeing. If we rewound the story of life on Earth to some point in the past and threw the dice again, would we see very similar animals appearing or could we get something completely different? It is the question that May raised at the end of our lunch.
It does appear that some parts of evolution seem inevitable. It is striking that throughout evolutionary history the eye evolved independently 50 to 100 times. This is strong evidence for the fact that the different rolls of the dice that have occurred across different species seem to have produced species with eyes regardless of what is going on around them. Lots of other examples illustrate how some features, if they are advantageous, seem to rise to the top of the evolutionary swamp. This is illustrated every time you see the same feature appearing more than once in different parts of the animal kingdom. Echolocation, for example, is used by dolphins and bats, but they evolved this trait independently at very different points on the evolutionary tree.
But it isn’t clear how far these outcomes are guaranteed by the model. If there is life on another planet, will it look anything like the life that has evolved here on Earth? This is one of the big open questions in evolutionary biology. As difficult as it may be to answer, I don’t believe it qualifies as something we can never know. It may remain something we will never know, but there is nothing by its nature that makes it unanswerable.
Are there other great unsolved questions of evolutionary biology that might be contenders for things we can never know? For example, why, 542 million years ago, at the beginning of the Cambrian period, was there an explosion of diversity of life on Earth? Before this moment life consisted of single cells that collected into colonies. But over the next 25 million years, a relatively short period on the scale of evolution, there is a rapid diversification of multicellular life that ends up resembling the diversity that we see today. An explanation for this exceptionally fast pace of evolution is still missing. This is in part due to lack of data from that period. Can we ever recover that data, or could this always remain a mystery?
Chaos theory is usually a limiting factor in what we can know about the future. But it can also imply limits on what we can know about the past. We see the results, but deducing the cause means running the equations backwards. Without complete data the same principle applies backwards as forwards. We might find ourselves at two very divergent starting points which can explain very similar outcomes. But we’ll never know which of those origins was ours.
One of the big mysteries in evolutionary biology is how life got going in the first place. The game of life may favour runs of 6s on the roll of the evolutionary dice, but how did the game itself evolve? Estimates have been made for the chances of everything lining up to produce molecules that replicate themselves. In some models of the origins of life it is equivalent to nature having to throw 36 dice and get them all to land on 6. For some this is proof of needing a designer to rig the game. But this is to misunderstand the huge timescale that we are working on.
Miracles do happen … given enough time. Indeed, it would be more striking if we didn’t get these strange anomalies happening. The point is that the anamolies often stick out. They get noticed, while the less exciting rolls of the dice get ignored.
The lottery is a perfect test bed for the occurrence of miracles in a random process. On 6 September 2009 the following six numbers were the winning numbers in the Bulgarian state lottery:
4, 15, 23, 24, 35, 42
Four days later the same six numbers came up again. Incredible, you might think. The government in Bulgaria certainly thought so and ordered an immediate investigation into the possibility of corruption. But what the Bulgarian government failed to take into account is that each week, across the planet, different lotteries are being run. They have been running for decades. If you do the mathematics, it would be more surprising not to see such a seemingly anomalous result.
The same principle applies to the conditions for producing self-replicating molecules in the primeval soup that made up the Earth before life emerged. Mix together plenty of hydrogen, water, carbon dioxide and some other organic gases and subject them to lightning strikes and electromagnetic radiation and already experiments in the lab show the emergence of organic material found only in living things. No one has managed to spontaneously generate anything as extraordinary as DNA in the lab. The chances of that are very small.
But that’s the point, because given the billion billion or so possible planets available in the universe on which to try out this experiment, together with the billion or so years to let the experiment run, it would be more striking if that outside chance of creating something like DNA didn’t happen. Keep rolling 36 dice on a billion billion different planets for a billion years and you’d probably get one roll with all 36 dice showing 6. Once you have a self-replicating molecule it has the means to propagate itself, so you only need to get lucky once to kick off evolution.
Our problem as humans, when it comes to appreciating the chance of a miracle such as life occurring, is that we have not evolved minds able to navigate very large numbers. Probability is therefore something we have little intuition for.
But it’s not only the mathematics of probability that is at work in evolution. The evolutionary tree itself has an interesting quality that is similar to the shapes that appear in chaos theory, a quality known as fractal.
The fractal evolutionary tree.
The evolutionary tree is a picture of the evolution of life on Earth. Making your way through the tree corresponds to a movement through time. Each time the tree branches, this represents the evolution of a new species. If a branch terminates, this means the extinction of that species. The nature of the tree is such that the overall shape seems to be repeated on smaller and smaller scales. This is the characteristic feature of a shape mathematicians call a fractal. If you zoom in on a small part of the tree it looks remarkably like the large-scale structure of the tree. This self-similarity means that it is very difficult to tell at what scale we are looking at the tree. This is the classic characteristic of a fractal.
Fractals are generally the geometric signature of a chaotic system, so it is suggestive of chaotic dynamics at work in evolution: the small changes in the genetic code that can result in huge changes in the outcome of evolution. This model isn’t necessarily a challenge to the idea of convergence, as there can still be points in chaotic systems towards which the model tends to evolve. Such points are called attractors. But it certainly questions whether if you reran evolution it would look anything like what we’ve got on Earth today. The evolutionary biologist Stephen Jay Gould has contended that if you were to rerun the tape of life that you would get very different results. This is what you would expect from a chaotic system. Just as with the weather, very small changes in the initial conditions can result in a dramatically different outcome.
Gould also introduced the idea of punctuated equilibria, which captures the fact that species seem to remain stable for long periods and then undergo what appears to be quite rapid evolutionary change. This has also been shown to be a feature of chaotic systems. The implications of chaos at work in evolution are that many of the questions of evolutionary biology could well fall under the umbrella of things we cannot know because of their connections to the mathematics of chaos.
For example, will we ever know whether humans were destined to evolve from the current model of evolution? An analysis of DNA in different animals has given us exceptional insights into the way animals have evolved in the past. The fossil record, although incomplete in places, has also given us a way to know our origins. But given the time scales involved in evolution it is impossible to experiment and rerun the tape of life evolving on Earth and see if something different could have happened. As soon as we find life on other planets (if we do), this will give us new sample sets to analyse. But until then all is not lost. Just as the MET office doesn’t have to run real weather to make predictions, computer models can illustrate different possible outcomes of the mechanism of evolution, speeding up time. But the model will only be as good as the hypotheses we have made on the model. If we’ve got the model wrong, it won’t tell us what is really happening in nature.
It’s such computer models that are at the heart of trying to answer the question Poincaré first tackled when he discovered chaos: will there even be a stable Earth orbiting the Sun for evolution to continue playing its game of dice? How safe is our planet from the vagaries of chaos? Is our solar system stable and periodic, or do I have to worry about a grasshopper disrupting our orbit around the Sun?
Poincaré wasn’t able to answer the King of Sweden’s question about the solar system: whether it would remain in a stable equilibrium or might fly apart in a catastrophic exhibition of chaotic motion. His discovery that some dynamical systems can be sensitive to small changes in data opened up the possibility that we may never know the precise fate of the solar system much in advance of any potentially devastating scenario unfolding.
It is possible that, like population dynamics with a low reproductive rate, the solar system is in a safe predictable region of activity. Unfortunately, the evidence suggests that we can’t console ourselves with this comforting mathematical hope. Recent computer modelling has provided new insights which reveal that the solar system is indeed within a region dominated by the mathematics of chaos.
I can measure how big an effect a small change will have on the outcome using something called the Lyapunov exponent. For example, in the case of billiards played on differently shaped tables, I can give a measure of how catastrophic a small change will be on the evolution of a ball’s trajectory. If the Lyapunov exponent of a system is positive, it means that if I make a small change in the initial conditions then the distance between the paths diverges exponentially. This can be used as a definition of chaos.
With this measure several groups of scientists have confirmed that our solar system is indeed chaotic. They have calculated that the distance between two initially close orbital solutions increases by a factor of ten every 10 million years. This is certainly on a different timescale to our inability to predict the weather. Nevertheless, it means that I can have no definite knowledge of what will happen to the solar system over the next 5 billion years.
If you’re wondering in despair whether we can know anything about the future, then take heart in the fact that mathematics isn’t completely hopeless at making predictions. There is an event that the equations guarantee will occur if we make it to 5 billion years from now, but it’s not good news: the mathematics implies that at this point the Sun will run out of fuel and evolve into a red giant engulfing planet Earth and the other planets in our solar system in the process. But until this solar blowout engulfs the solar system, I am faced with trying to solve chaotic equations if I want to know which planets will still be around to see that red giant.
This means that, like the predictions of the weather, if I want to know what is going to happen, I am reduced to running simulations in which I vary the precise locations and speeds of the planets. The forecast is in some cases rather frightening. In 2009 French astronomers Jacques Laskar and Mickael Gastineau ran several thousand models of the future evolution of our solar system. And their experiments have identified a potential butterfly: Mercury.
The simulations start by feeding in the records we have of the positions and velocities of the planets to date. But it is difficult to know these with 100% accuracy. So each time they run the simulation they make small changes to the data. Because of the effects of chaos theory, just a small change could result in a large deviation in the outcomes.
For example, astronomers know the dimensions of the ellipse of Mercury’s orbit to an accuracy of several metres. Laskar and Gastineau ran 2501 simulations where they varied these dimensions over a range of less than a centimetre. Even this small perturbation resulted in startlingly different outcomes for our solar system.
You might expect that if the solar system was going to be ripped apart it would have to be one of the big planets like Jupiter or Saturn that would be the culprit. But the orbits of the gas giants are extremely stable. It’s the rocky terrestrial planets that are the troublemakers. In 1% of simulations that they ran, it was tiny Mercury that posed the biggest risk. The models show that Mercury’s orbit could start to extend due to a certain resonance with Jupiter, with the possibility that Mercury could collide with its closest neighbour, Venus. In one simulation, a close miss was enough to throw Venus out of kilter, with the result that Venus collides with Earth. Even close encounters with the other planets would be enough to cause such tidal disruption that the effect would be disastrous for life on our planet.
This isn’t simply a case of abstract mathematical speculation. Evidence of such collisions has been observed in the planets orbiting the binary star Upsilon Andromedae. Their current strange orbits can be explained only by the ejection of an unlucky planet sometime in the star’s past. But before we head for the hills, the simulations reveal that it will take several billion years before Mercury might start to misbehave.
What of my chances to predict the throw of the dice that sits next to me? Laplace would have said that, provided I can know the dimensions of the dice, the distribution of the atoms, the speed at which it is launched, its relationship to its surrounding environment, theoretically the calculation of its resting point is possible.
The discoveries of Poincaré and those who followed have revealed that just a few decimal places could be the difference between the dice landing on a 6 or a 2. The dice is designed to have only six different outcomes, yet the input data ranges over a potentially continuous spectrum of values. So there are clearly going to be points where a very small change will flip the dice from landing on a 6 to a 2. But what is the nature of those transitions?
Computer models can produce very good visual representations that give me a handle on the sensitivity of various systems to the starting conditions. Next to my Vegas dice I’ve got a classic desktop toy that I can play with for hours. It consists of a metal pendulum that is attracted to three magnets, coloured white, black and grey. Analysis of the dynamics of this toy has led to a picture that captures the ultimate outcome of the pendulum as it starts over each point in the square base of the toy. Colour a point white if starting the pendulum at this point results in it ending at the white magnet. Similarly, colour the point grey or black if the ultimate destination is grey or black. This is the picture you get:
As in the case of population dynamics, there are regions which are entirely predictable. Start close to a magnet and the pendulum will just be attracted to that magnet. But towards the edges of the picture I find myself in far less predictable terrain. Indeed, the picture is now an example of a fractal.
There are regions where there isn’t a simple transition from black to white. If I keep zooming in, the picture never becomes just a region filled with one colour. There is complexity at all scales.
A one-dimensional example of such a picture can be cooked up as follows. Draw a line of unit length and begin by colouring one half black and the other white. Then take half the line from the point 0.25 to 0.75 and flip it over. Now take the half in between that and flip it over again. If we keep doing this to infinity then the predicted behaviour around the point at 0.5 is extremely sensitive to small changes. There is no region containing the point 0.5 which has a single colour.
There is a more elaborate version of this picture. Start again with a line of unit length. Now rub out the middle third of the line. You are left with two black lines with a white space in between. Now rub out the middle third of each of the two black lines. Now we have a black line of length 1⁄9, a white line of length 1⁄9, a black line of length 1⁄9, then the white line of length 1⁄3 that was rubbed out on the first round, and then a repeat of black–white–black.
You may have guessed what I am going to do next. Each time rub out the middle third of all the black lines that you see. Do this to infinity. The resulting picture is called the Cantor set, after the German mathematician Georg Cantor, whom we will encounter in the last Edge, when I explore what we can know about infinity. Suppose this Cantor set was actually controlling the outcome of the pendulum in my desktop toy. If I move the pendulum along this line, I find that this picture predicts some very complicated behaviour in some regions.
A rather strange calculation shows that the total length of the line that has been rubbed out is 1. But there are still black points left inside: 1⁄4 is a point that is never rubbed out, as is 3⁄10. These black points, however, are not isolated. Take any region round a black point and you will always have infinitely many black and white points inside the region.
What do the dynamics of my dice look like? Are they fractal and hence beyond my knowledge? My initial guess was that the dice would be chaotic. However, recent research has turned up a surprise.
A Polish research team has recently analysed the throw of a dice mathematically, and by combining this with the use of high-speed cameras they have revealed that my dice may not be as chaotic and unpredictable as I first feared. The research group consists of father-and-son team Tomasz and Marcin Kapitaniak together with Jaroslaw Strzalko and Juliusz Grabski, and they are based in Lódź. In their paper in the journal Chaos, published in 2012, the team draw similar pictures to those for the magnetic pendulum, but the starting positions are more involved than just two coordinates because they have to give a description of the angle at which the cube is launched and also the speed. The dice will be predictable if for most points in this picture when I alter the starting conditions a little the dice ends up falling on the same side. I can think of the picture being coloured by six colours corresponding to the six sides of the dice. The picture is fractal if however much I zoom in on the shape I still see regions containing at least two colours. The dice is predictable if I don’t see this fractal quality.
The model the Polish team considered assumes the dice is perfectly balanced like the dice I brought back from Vegas. Air resistance, it turns out, can be ignored as it has very little influence on the dice as it tumbles through the air. When the dice hits the table a certain proportion of the dice’s energy is dissipated, so that after sufficiently many bounces the dice has lost all kinetic energy and comes to rest.
Friction on the table is also key, as the dice is likely to slide in the first few bounces but won’t slide in later bounces. However, the model explored by the Polish team assumed a frictionless surface as the dynamics get too complicated to handle when friction is present. So imagine throwing the dice onto an ice rink.
I’d already written down equations based on Newton’s laws of motion for the dynamics of the dice as it flies through the air. In the hands of the Polish team they turn out not to be too complicated. It is the equations for the change in dynamics after the impact with the table that are pretty frightening, taking up ten lines of the paper they wrote.
They discovered that if the amount of energy dissipated on impact with the table is quite high, the picture of the outcome of the dice does not have a fractal quality. This means that if one can settle the initial conditions with appropriate accuracy, the outcome of the throw of my dice is predictable and repeatable. This predictability implies that, more often than not, the dice will land on the face that was lowest when the dice is launched. A dice that is fair when static may actually be biased when one adds in its dynamics.
But as the table becomes more rigid, resulting in less energy being dissipated and hence the dice bouncing more, I start to see a fractal quality emerging.
Moving from (a) to (d), the table dissipates less energy, resulting in a more fractal quality for the outcome of the dice.
This picture looks at varying two parameters: the height from which the dice is launched and variations in the angular velocity around one of the axes. The less energy that is dissipated on impact with the table, the more chaotic its resulting behaviour and the more it seems that the outcome of my dice recedes back into the hands of the gods.
What of the challenge to define God as the things we cannot know? Chaos theory asserts that I cannot know the future of certain systems of equations because they are too sensitive to small inaccuracies. In the past gods weren’t supernatural intelligences living outside the system but were the rivers, the wind, the fire, the lava – things that could not be predicted or controlled. Things where chaos lay. Twentieth-century mathematics has revealed that these ancient gods are still with us. There are natural phenomena that will never be tamed and known. Chaos theory implies that our futures are often beyond knowledge because of their dependence on the fine-tuning of how things are set up in the present. Because we can never have complete knowledge of the present, chaos theory denies us access to the future. At least until that future becomes the present.
That’s not to say that all futures are unknowable. Very often we are in regions which aren’t chaotic and small fluctuations have little effect on outcomes. This is why mathematics has been so powerful in helping us to predict and plan. Here we have knowledge of the future. But at other times we cannot have such control, and yet this unknown future will certainly impact on our lives at some point.
Some religious commentators who know their science and who try to articulate a scientific explanation for how a supernatural intelligence could act in the world have intriguingly tried to use the gap that chaos provides as a space for this intelligence to affect the future.
One of these religious scientists is the quantum physicist John Polkinghorne. Based at the University of Cambridge, Polkinghorne is a rare mind who combines both the rigours of a scientific education with years of training to be a Christian priest. I will be meeting Polkinghorne in person in the Third Edge when I explore the unknowability inherent in his own scientific field of quantum physics. But he has also been interested in the gap in knowledge that the mathematics of chaos theory provides as an opportunity for his God to influence the future course of humanity.
Polkinghorne has proposed that it is via the indeterminacies implicit in chaos theory that a supernatural intelligence can still act without violating the laws of physics. Chaos theory says that we can never know the set-up precisely enough to be able to run deterministic equations, and hence there is room in Polkinghorne’s view for divine intervention, to tweak things to remain consistent with our partial knowledge but still influence outcomes.
Polkinghorne is careful to stress that to use infinitesimal data to effect change requires a complete holistic top-down intervention. This is not a God in the detail but by necessity an all-knowing God. Given that chaos theory means that even the location of an electron on the other side of the universe could influence the whole system, we need to have complete, holistic knowledge of the whole system – the whole universe – to be able to steer things. We cannot successfully isolate a part of the universe and hope to make predictions based on that part. So it would require knowledge of the whole to act via this chink in that which is unknown to us.
Chaos theory is deterministic, so this isn’t an attempt to use the randomness of something like quantum physics as a way to have influence. Polkinghorne’s take on how to square the circle of determinism and influence the system is to use the gap between epistemology and ontology, between what we know and what is true. Since we cannot know a complete description of the state of the universe at this moment in time, this implies that from our perspective there is no determinacy. There are many different scenarios that coincide with our impartial description of what we currently know about how the universe is set up. Polkinghorne’s contention is that at any point in time this gives a God the chance to intervene and shift the system between any of these scenarios without us being aware of the shift. But, as we have seen, chaos theory means that these small shifts can still have hugely different outcomes. Polkinghorne is careful to assert that you allow shifts between systems where there is change only in information, not energy. The rule here is not to violate any rules of physics. As Polkinghorne says: ‘The succession of the seasons and the alternations of day and night will not be set aside.’
Even if you think this is rather fanciful (which I certainly do), a similar principle is probably key to our own feeling of agency in the world. The question of free will is related ultimately to questions of a reductionist philosophy. Free will describes the inability to make any meaningful reduction in most cases to an atomistic view of the world. So it makes sense to create a narrative in which we have free will because that is what it looks like on the level of human involvement in the universe. If things were so obviously deterministic, with little variation to small undetectable changes, we wouldn’t think that we had free will.
It is striking that Newton, the person who led us to believe in a clockwork deterministic universe, also felt that there was room in the equations for God’s intervention. He wrote of his belief that God would sometimes have to reset the universe when things looked like they were going off course. He got into a big fight with his German mathematical rival Gottfried Leibniz, who couldn’t see why God wouldn’t have set it up perfectly from the outset:
Sir Isaac Newton and his followers have also a very odd opinion concerning the work of God. According to their doctrine, God Almighty wants to wind up his watch from time to time: otherwise it would cease to move. He had not, it seems, sufficient foresight to make it a perpetual motion.
Newton and his mathematics gave me a feeling that I could know the future, that I could shortcut the wait for it to become the present. The number of times I have heard Laplace’s quote about ultimately being able to know everything thanks to the equations of motion is testament to a general feeling among scientists that the universe is theoretically knowable.
The mathematics of the twentieth century revealed that theory doesn’t necessarily translate into practice. Even if Laplace is correct in his statement that complete knowledge of the current state of the universe together with the equations of mathematics should lead to complete knowledge of the future, I will never have access to that complete knowledge. The shocking revelation of twentieth-century chaos theory is that even an approximation to that knowledge won’t help. The divergent paths of the chaotic billiard table mean that since we can never know which path we are on, our future is not predictable.
Chaos theory implies that there are things we can never know. The mathematics in which I had placed so much faith to give me complete knowledge has revealed the opposite. But it is not entirely hopeless. Many times the equations are not sensitive to small changes and hence give me access to predictions about the future. After all, this is how we landed a spaceship on a passing comet. Not only that: as Bob May’s work illustrates, the mathematics can even help me to know when I can’t know.
But a discovery at the end of the twentieth century even questions Laplace’s basic tenet of the theoretical predictability of the future. In the early 1990s a PhD student by the name of Zhihong Xia proved that there is a way to configure five planets such that when you let them go, the combined gravitational pull causes one of the planets to fly off and reach an infinite speed in a finite amount of time. No planets collide, but still the equations have built into them this catastrophic outcome for the residents of the unlucky planet. The equations are unable to make any prediction of what happens beyond this point in time.
Xia’s discovery is a fundamental challenge to Laplace’s view that Newton’s equations imply that we can know the future if we have complete knowledge of the present, because there is no prediction even within Newton’s equations for what happens next for that unlucky planet once it hits infinite speed. The theory hits a singularity at this point, beyond which prediction makes no sense. As we shall see in later Edges, considerations of relativity will limit the physical realization of this singularity since the unlucky planet will eventually hit the cosmic speed limit of the speed of light, at which point Newton’s theory is revealed to be an approximation of reality. But it nevertheless reveals that equations aren’t enough to know the future.
It is striking to listen to Laplace on his deathbed. As he sees his own singularity heading towards him with only a finite amount of time to go, he too admits: ‘What we know is little, and what we are ignorant of is immense.’ The twentieth century revealed that, even if we know a lot, our ignorance will remain immense.
But it turns out that it is isn’t just the outward behaviour of planets and dice that is unknowable. Probing deep inside my casino dice reveals another challenge to Laplace’s belief in a clockwork deterministic universe. When scientists started to look inside the dice to understand what it is made of, they discovered that knowing the position and the movement of the particles that make up the dice may not even be theoretically possible. As I shall discover in the next two Edges, there might indeed be a game of dice at work that controls the behaviour of the very particles that make up my red Las Vegas cube.
