Where does the complexity of the Universe come from? A simple computer program is generating it!
There is something fascinating about science. One gets such wholesale
returns of conjecture out of such a trifling investment of fact.
Mark Twain, Life on the Mississippi, 1884
God has chosen the world that is the most simple in hypotheses
and the most rich in phenomena.
Gottfried Leibniz, Discourse de métaphysique, 1686
It’s AD 2068 and the survey expedition from Earth is picking its way through the ruins of an alien civilisation, long departed from its home world for who knows where. Ahead, bathed in the sombre light of the twin red suns, is a great slab of a building – the planet’s central library, repository of the civilisation’s accumulated wisdom.
Struggling visibly in the strong gravity, the expedition members clamber up the giant steps and push open the creaking door. Their boots reverberating in the thick atmosphere, they hurry through an empty, echoing chamber – until, finally, they come to a single cabinet, displaying a lone tablet inscribed with arcane symbols. Everyone crowds around while someone scans it with a translator …
RECIPE FOR UNIVERSE:
REPEAT FOR 13.7 BILLION YEARS
One person laughs. Another gasps in disbelief. The cosmic computer program they are all staring at is only four lines long.
Could the recipe for making a universe really be as simple as this? One present-day physicist is convinced of it. His name is Stephen Wolfram and he claims to have stumbled on nature’s ‘big’ secret. The source of all its bewildering complexity – from spiral galaxies to rhododendrons to human beings – is the application of a few simple instructions, over and over again. ‘Our Universe is being generated by a simple computer program,’ says Wolfram.
Wolfram, a child prodigy from London, began publishing papers in professional physics journals at the age of fifteen. What led him to his extraordinary conclusion is a discovery he made around 1980. Contrary to all expectations, he found that simple computer programs have the ability to generate extraordinarily complex outputs.
Wolfram’s discovery came about when he became interested in problems such as how galaxies like our Milky Way form and how our brains work. ‘The trouble was that none of these “complex systems” seemed explicable by conventional science,’ he says.
Conventional science is synonymous with maths-based science. In the seventeenth century, Isaac Newton discovered that the laws which govern the motion of a cannon ball through the air and a planet round the Sun could be described by mathematical formulae, or ‘equations’. Following Newton’s lead, generations of physicists have found that mathematical equations exist that can perfectly describe everything from the character of the light given out by a hot furnace to the warping of space and time by the concentrated mass of a black hole.*
But, despite the tremendous successes of equation-based science in penetrating nature’s secrets, it has an Achilles’ heel: it cannot do ‘complexity’. It is utterly incapable of capturing the essence of what is going on in a whole range of complex phenomena, ranging from turbulence in fluids to biology itself.
Most scientists lose little sleep over this. Complex phenomena may be ‘hard’, they say, but this does not mean that science will not eventually get round to tackling them. Wolfram, however, emphatically disagrees. Controversially, he believes that mathematical science will never, ever penetrate the mystery of complex phenomena.
A streetlight illuminates merely what it can illuminate – the circle of ground immediately beneath. Similarly, Wolfram believes mathematical science illuminates merely what it is capable of illuminating – those phenomena whose essence can be captured by mathematical equations. But such phenomena, he contends, are rare and unusual. In the same way that a streetlight fails to reveal the subways and sports grounds and art galleries of its surrounding city, science as practised for the past three centuries is blind to the overwhelming majority of phenomena in the Universe – complex phenomena. ‘Since such phenomena include living things, the human brain and the biosphere, we are talking about all the truly interesting things that are going on in the Universe,’ says Wolfram.
This is radical stuff. For centuries, physicists have wondered why nature obeys mathematical laws, which can be distilled into neat mathematical equations and which can then be scrawled across blackboards. The Hungarian-American physicist Eugene Wigner famously drew attention to this when he talked of ‘the unreasonable effectiveness of mathematics in the physical sciences’.
According to Wolfram, however, Wigner was wrong to believe that the Universe is essentially mathematical. Mathematics is no more effective in revealing the inner workings of nature than a streetlight is in revealing the city that surrounds it. Nature may ‘appear’ to follow mathematical laws, he says. However, that is hardly surprising when scientists specifically seek out the rare natural phenomena that follow mathematical laws. ‘Wigner could equally well have remarked on the unreasonable effectiveness of streetlights in illuminating the ground beneath them,’ says Wolfram.
If Wolfram is right, science has a serious problem. After all, if mathematical equations are incapable of describing nature’s most interesting phenomena – complex phenomena – how can such phenomena be described? In the early 1980s, Wolfram gave this question a great deal of thought. It was clear to him that the Universe must obey rules of some kind. If it did not, after all, there would be no pattern or regularity in nature. The Universe would be a meaningless maelstrom of unpredictable randomness and chaos. But, if the rules are not embodied in mathematical equations, what are they embodied in? It was clear to Wolfram that it had to be something more general than a mathematical equation. After thinking about it, he could come up with only one thing that fitted the bill: a computer program.
Wolfram decided to find out what kind of science could be built starting with the more general kinds of rules embodied in computer programs. The first big question he needed to answer was: what are such rules capable of? Or, to put it in another way, typically what do simple programs do?
The simplest computer program Wolfram could think of is known as a ‘cellular automaton’. The most basic of these is simply a long line of squares, or ‘cells’, drawn across a page. A cell can be one of two colours – white or black. At regular intervals of time, a new line of cells is drawn on the page, immediately above the first. Whether a cell in this second line is black or white depends on a rule applied to its two nearest neighbours in the first line. The rule might, for instance, say: ‘If a particular cell in the first line has a black square on either side of it, it should turn black in the second line’. A third line of cells, immediately above the second, is then created by applying the cellular automata ‘rule’ to the second line, and so on.
What we are talking about here is the operation of a simple computer program embodying the cellular automaton rule. The program takes an input – the pattern of black cells and white cells on one line – and produces an output – the pattern of cells on the next line. The key thing is that the output is fed back in as the next input to the computer program, rather like a snake swallowing its own tail. Such tail-swallowing is commonly called ‘recursion’. And, as a wit once said: ‘To understand recursion, you must first understand recursion!’
For a one-dimensional, two-colour, adjacent-cell cellular automaton like this, it turns out there are 256 possible rules, 256 kinds of program.* The question is: what happens when the programs are run, starting, say, with a single black cell in the first line of cells? In true scientific fashion, Wolfram began experimenting to find out.
He soon discovered that some rules and some starting patterns led to nothing interesting. As new lines of cells were created, any pattern quickly fizzled out. Or a particular arrangement of black and white cells began repeating endlessly. However, in some cases, something very much more interesting happened.
The early 1980s was the time of the first cheap desktop computers so Wolfram was able to watch his cellular automata perform on a computer rather than on a piece of paper. Seeing the new lines of cells marching steadily up the screen was much like watching a movie. Occasionally, the patterns of black cells coalesced into discrete ‘objects’. These persisted – as unchanging and stable as a table or chair – despite the fact they were being continually destroyed and regenerated.
Wolfram played with his cellular automata for hours on end, mesmerised by the marching patterns. And then, one day, he stumbled on something extraordinary. ‘I found a pattern which appeared never to repeat, no matter how long I stared at it,’ says Wolfram.
If you see a complex thing like a car or a computer, you know it must have been made by a complex process. Even in biology, where natural selection is blind, the complexity of organisms is a result of a complex series of processes operating over billions of years of evolution. In the everyday world, simple things have simple causes and complex things have complex causes. What Wolfram had found, however, was something that bucked the trend – a complex thing that had a simple cause.
For Wolfram it was a life-changing moment. As he stared at his computer screen and the never-ending novelty scrolling down it, he wondered: Is this the origin of the Universe’s complexity? ‘When nature creates a rose or a galaxy or a human brain, is it merely applying simple rules – over and over again?’ he asks. ‘Is this its big secret?’
From that moment on Wolfram became obsessed with the origin of complexity. At the time of his epiphany he was at the California Institute of Technology in Pasadena. However, in the mid-1980s, he moved first to Princeton’s Institute for Advanced Study – Einstein’s old institute – then to the University of Illinois at Urbana-Champaign, where he founded the Center for Complex Systems Research. Around the same time, he started the first scientific journal on complexity. He even created his own computer language – ‘Mathematica’ – which helped him in his investigation of the origin of complexity.
Mathematica led him to start his own company, Wolfram Research, and attract scientists and mathematicians to help develop the software. The programming language turned out to be not only a tool but an inspiration to his work. Although Wolfram assembled it from simple program ‘modules’, it was nevertheless capable of carrying out enormously complex tasks. ‘It hammered home to me once again my central discovery – that simple programs can have hugely complex outcomes,’ says Wolfram.
Wolfram’s hope was that others would pile into the research area he had created and that this would lead to rapid progress in understanding complexity. To his disappointment and frustration, however, few joined in and progress was slow. He became increasingly impatient. By early 1991, he decided there was only one thing to do – carry out the work himself.
With several million users worldwide, Mathematica had made Wolfram a multimillionaire. He did not need to be employed by a university and he did not need to fight constantly for research money. He was free to concentrate all of his time on creating a science of complexity.
Wolfram had set himself a gargantuan task but even he did not realise it would take him a decade. During that time, he published not a single research paper. Although he certainly talked and corresponded with other scientists, he pretty much vanished off the edge of the scientific radar screen.
Month after month, year after year, while the rest of the world slept, Wolfram laboured through the night, painstakingly laying the foundations of a new way of doing science. In essence, he was carrying out a systematic computer search for simple rules with very complicated consequences. ‘He set out to survey all possible worlds – at least all the ones generated by simple rules,’ says the mathematician Gregory Chaitin of IBM in Yorktown Heights, New York. ‘The result was a treasure trove of small computer programs that, when repeated again and again, yield.’* extremely rich, complicated and interesting behaviour.
Among the many things Wolfram discovered is the remarkable property of cellular automaton rule 110. Starting with a single black cell, this simple rule turns out to be capable of generating infinite complexity, infinite novelty, infinite surprise. Not only that but a cellular automaton following rule 110 is a ‘universal Turing machine’.† Despite being amazingly simple, it is like a modern-day computer that can carry out any imaginable computation, simulate any other conceivable machine.
The remarkable ability of cellular automaton rule 110 is highly suggestive. After all, if even a simple one-dimensional cellular automaton can create never-ending complexity, it shows the kind of power that nature potentially has at its disposal. And Wolfram is convinced that nature avails itself of that potential. ‘I believe that physical systems subject to simple rules applied recursively – with the output fed back in as the input – can have created everything from the tip of your nose to the most distant cluster of galaxies,’ he says.
So is the Universe a giant cellular automaton – a three-dimensional version of the one-dimensional ones Wolfram has been playing with on his computer? Surprisingly, Wolfram thinks not. ‘I think the truth is actually much more strange and interesting,’ he says.
A serious shortcoming of a cellular automaton as a model of the Universe is that all the cells update themselves together. This kind of coordinated behaviour requires a built-in ‘clock’, whose ticks provide the all-important cue for the cells to ‘all change’. Unfortunately, this kind of clock is impossible to implement in the real Universe, the reason being the existence of a cosmic speed limit, as discovered by Einstein.
Nothing, it turns out, can travel faster than the speed of light. This constraint means that, wherever the cellular automaton clock happens to be located in the Universe, the signal carrying news of its tick will take longer to travel to a cell that is far away from it than to one that is nearby. A possible way round this might be to have lots of clocks distributed throughout the Universe. However, this does not overcome the fundamental problem because there is no way to make sure the clocks are all telling the same time. If a ‘reference clock’ is used, inevitably the signal carrying news of its time will take longer to reach some clocks than it does to reach others.
The impossibility of implementing a ‘global’ clock in our Universe means that at the very least the Universe cannot be a ‘standard’ cellular automaton. However, that does not rule out its being a cellular automaton of some non-standard type – one that somehow gets by without a global clock. This seems a bit of a tall order. But Wolfram can think of an ingenious way it can be achieved. ‘Say that, rather than updating all of its cells together, a cellular automaton updates just one cell at a time,’ he says.
At first sight this may seem crazy. But consider for a moment the advantage of such a scheme. If, at each step, only one cell is updated, the sticky problem of getting all the cells to update at the same time clearly goes away.
Of course there remains the small matter of how such a cellular automaton could possibly mimic our reality. After all, we have the very strong impression that everything in the Universe is travelling forward through time together, not that one detail of reality is being updated in turn while everything else remains doggedly rooted to the spot.
Say you are playing in a football game. You see all the other players running about the pitch simultaneously. You do not see first one player take a step, as their thought processes click on one notch, while everyone else remains paralysed in mid-stride; then another player take a step, and so on. ‘But, just because you do not see this happening, does not mean that this is not exactly what is going on,’ says Wolfram.
But surely you would notice? No, says Wolfram. The only time you notice the world about you is when it is your turn to be updated. And, when this happens, all you see is that all the other players have moved on a fraction. Because your awareness is frozen between your own updatings, it is impossible for you to notice when any of the other players are updated. Despite the fact that only one player on the pitch is moving at any one time, your perception is of everyone on the pitch running about simultaneously.
Between any two successive moments of time as perceived by you, there are very many updating events, none of which you have any awareness of. In fact, all you can ever really know about, says Wolfram, is what updating event influences what other updating event. For instance, the updating event that moved the football one step closer to the opposing team’s goal influenced the opposing team’s defenders and goal keeper, who altered their positions to intercept the ball. This, of course, is the familiar story of a football game. ‘But that’s all it is – a story,’ says Wolfram. ‘A network of cause and effect we impose on the underlying reality to make some kind of sense of it.’
Contrary to common-sense expectations, then, it appears that it is possible to mimic our Universe with a cellular automaton in which only one cell at a time is updated. The passage of the ‘time’ in the Universe is marked by the regular ticking of the cellular automaton’s clock. That only leaves ‘space’ to worry about. Unfortunately, it is here, according to Wolfram, that the idea of the Universe-as-a-cellular-automaton comes to grief.
Wolfram is convinced that the computer program generating our Universe is a simple one. Every scrap of evidence he has accumulated since his key discovery that simple programs can produce unexpectedly complex outputs bolsters this belief. But, if the program generating the Universe is simple, it stands to reason there will not be room in it for much ‘stuff’. In other words, very few of the features of our Universe – from gravity to space and time to koala bears – will be visible in the program. Instead, they will ‘emerge’ – like an inflatable raft unfolding from a canister – only after the program has been running for a long while.
But Wolfram does not simply think the Universe-creating program is simple. He goes further than this. He believes the program may be among the simplest possible programs capable of generating the Universe. This is a leap of faith. All Wolfram knows for sure is that the rule for the Universe is not really complicated. If it was, he argues, there would be no perceptible pattern to nature, which there clearly is. Wolfram thinks it is possible our Universe is the very simplest universe that is not obviously a silly one – for instance, a universe with no notion of space or of time. Consequently, he thinks it is worth first trying the simple rules for size because our Universe might be among them.
If the Universe-generating program is indeed among the simplest programs capable of generating the Universe, it will contain the absolute bare minimum of stuff. And it is this that persuades Wolfram that the Universe cannot possibly be a cellular automaton. A cellular automaton, after all, is a rigid array of cells laid out in ‘space’. In other words, the very notion of ‘space’is built into its very foundations. To Wolfram, this is already too much stuff.
Wolfram believes the Universe-generating program will be so simple, so pared down, that even something as apparently fundamental as space will not be built into it. Instead, it will emerge along with everything else only as the program runs, conjured out of something even more basic than space.
Wolfram believes space is not a smooth, featureless backcloth to the drama of the Universe. Instead, it has an underlying structure. The analogy he uses is water. Although water looks smooth and continuous, in fact it is made up of tiny motes of matter called molecules. Wolfram thinks space is similar. If it were possible to examine it with some kind of super-microscope, we would see that it is made of a huge number of discrete points. The points, or ‘nodes’, are connected together in a vast extended network.
But how can a mere network of points have the properties of familiar space? ‘Surprisingly easily,’ says Wolfram. ‘It simply depends on the way the nodes are connected to each other.’
Imagine being at one particular node, then going to all the nodes that are one connection away, then two connections, then three, and so on. After going, say, r connections, simply count how many nodes you have visited. If there are roughly pi × r2 nodes – the area of a circle – then the space is two-dimensional like the surface of a piece of paper. If there are roughly 4⁄3 pi × r3 – the volume of a sphere – then the space is three dimensional, like the space we live in. It turns out that a simple network of nodes can mimic the essential properties of absolutely any space imaginable, be it one-dimensional, two-dimensional or 279-dimensional.
According to Wolfram, space is nothing more than a bunch of nodes connected together. Of course, there is a little bit more to it than that.
Wolfram envisages a space network being updated in a similar way to a cellular automaton. After all, a constantly updated cellular automaton has a proven ability to generate complexity reminiscent of our Universe. Recall how it was possible to get over the synchronisation problem of a cellular automaton by updating just one cell at a time. Well, Wolfram thinks that this elegant solution can be carried right over to a space network. Instead of having a rule which says, if a cell is surrounded by a certain pattern of coloured cells – change its colour, Wolfram imagines a rule saying, if there is a piece of network with a particular form, replace it with a piece of network with another form. Remarkably, Wolfram claims that everything in our world can emerge from such a space network.
Take particles of matter. In a cellular automaton – for instance, the one subject to rule 110 – the system may quickly organise itself into a few localised structures which are persistent and appear to move through space just like fundamental particles – quarks and electrons and so on. What is actually happening is that, as fast as the structures are destroyed, they are refreshed again. It is just like a TV image of a football game. We may perceive that a football is in flight. But, in reality, what is happening is that a picture of the ball is being refreshed thirty times a second and giving us the illusion of the ball moving through the air.*
Sometimes, in a cellular automaton subject to rule 110, there is a collision between ‘particles’. They slam into each other and a whole bunch of other particles come out. This is just the kind of thing physicists observe at atom smashers like the one at the European centre for particle physics at CERN in Geneva. And what happens in a cellular automaton subject to rule 110 can also happen in a space network. Instead of being stubbornly persistent patterns of cells, however, the ‘particles’ are stubbornly persistent tangles of connections.
Remarkably, Wolfram has found that, with a constantly updated network of nodes, it is possible to create both the space we live in and the matter we are made of. ‘Reality,’ as Einstein remarked, ‘is merely an illusion, albeit a very persistent one.’
A problem arises, however, if a rule applies to a particular pattern of nodes and there are several places in the network with the same pattern. Which place should be updated first? Updating the places in a different order will in general lead to different networks of cause and effect. Rather than having a unique history, the Universe will have several possible histories. We will not know why we are following the history we are and not another, which is a highly unsatisfactory state of affairs.
Fortunately, there is a way out of this difficulty, says Wolfram. By a stroke of luck it turns out that there are certain rules with the property that it in fact does not matter in which order they are applied. Wolfram calls them ‘causally invariant’ rules. ‘Whenever they are used, there is always just a single thread of time in the Universe,’ he says.
Wolfram’s progression from the Universe-as-a-cellular-automaton to the Universe-as-a-constantly-updated-space-network is a good illustration of the way in which physicists grope their way towards a true picture of nature. They start with a crude model which mimics an aspect of reality which they consider to be important. In this case, the model is a cellular automaton which can generate complexity tantalisingly like the complexity we see in the world around us. Inevitably, the model falls short in some way. In the case of cellular automata, it contains too much ready-made stuff such as ‘space’. Nevertheless, physicists use the crude model as a bridge to reach a better model that mimics more reality more faithfully. Lastly, they throw away the bridge.
Wolfram’s talk of space and matter ‘emerging’ from a network may seem rather woolly. However, he maintains that it can explain concrete things too, such as the general theory of relativity, Einstein’s theory of gravity. In a nutshell, the theory says that matter distorts, or warps, space-time, and that warped space-time is what matter reacts to when it moves. In fact, warped space-time is all that gravity is. We think that the Earth pulls on the Moon with invisible fingers of force which somehow reach out across 400,000 kilometres of empty space. But, according to Einstein, this is an illusion. In reality, the Earth’s mass warps space-time, creating a sort of valley in its vicinity. We cannot see it because space-time is four-dimensional and we can experience only three dimensions. But the Moon ‘sees’ it. It skitters around the rim of the valley in space-time like a roulette ball round a roulette wheel.
Wolfram claims that his perpetually updated space network behaves exactly like Einstein’s warped space-time. For simplicity, imagine things in two dimensions. Also, imagine that the network is a network of hexagons which can be laid out flat like a fishing net spread out on a beach. What happens if some of the connections are changed so that some heptagons and pentagons are mixed in with the hexagons? The answer is that the network bulges out or in. ‘This is warped space,’ says Wolfram.
In ordinary, flat, two-dimensional space, as mentioned above, the number of nodes we get by going out r steps through the network goes up as r2. Well, in a warped network, it is not quite the same. There is what mathematicians call a ‘correction term’. And it turns out that the correction term is basically the ‘Ricci tensor’. It is not necessary to know exactly what the Ricci tensor is, but it crops up in Einstein’s equations, which in general relativity specify the warpage of space-time.
The story of how is quite complicated. But Wolfram maintains that, with just a few assumptions, he can work out the conditions which the Ricci tensor must obey. ‘And, guess what?’ he says. ‘They seem to be exactly Einstein’s equations of gravity.’
Wolfram believes the computer program that nature is using to generate the Universe is very short. We are certainly not talking about the ten million or so lines of a program like Microsoft Windows. Far from it. ‘Nature’s program may be expressible in as few as four lines of Mathematica,’ he says.
If he is right, those lines are responsible for creating everything from chocolate doughnuts to TV game shows to the very thought processes that have led Wolfram to the audacious claim that a mere four lines of computer code are generating reality.
Wolfram admits that his decade of investigation has not yet furnished him with the elusive cosmic computer program – the ‘one rule to bind them all’. But he is hopeful that he will one day find it.
One of the most important discoveries to have come out of Wolfram’s decade of toil is the recognition that a cellular automaton following rule 110 is far from unique. Wolfram has been surprised to find that many other real systems in the Universe – from turbulent fluids to colliding subatomic particles – also behave as universal computers. In other words, they too have the capacity to simulate any other machine, carry out any conceivable computation.
Because a universal computer can compute, or simulate, absolutely anything, it is trivial to deduce from this that all systems that behave as universal computers can compute as much as each other. In other words, they are equivalent. ‘Since universal computers are so widespread in nature, this has far-reaching implications,’ says Wolfram. ‘It means that everything from the behaviour of a cell to turbulence in a hydrogen cloud drifting in the depths of space to rain pattering on the pavement is equivalent in terms of the computational complexity required to generate it.’
Until now, scientists have assumed that the kind of complexity which is seen in living things – from single cells to human brains – can arise only in a system of large molecules based on carbon atoms. This, after all, is what we observe on Earth. But if, as Wolfram firmly believes, a large range of systems in nature have equivalent computational complexity, it means that the complexity we associate with life is not the unique preserve of planet-bound, water-soluble, carbon-based chemistry. Many of the things we thought were special about life and intelligence can be present in numerous other kinds of physical systems. ‘The Universe may contain life forms – including intelligent life forms – the like of which we cannot begin to imagine,’ says Wolfram.
He elevates his discovery that large numbers of natural systems have the same computational complexity to an over-arching natural principle. He calls it ‘The Principle of Computational Equivalence’. Put crudely, it says that systems of similar complexity are equivalent. Take, for instance, the Earth’s atmosphere. According to Wolfram’s Principle, because the atmosphere’s circulation is as complex as any living thing, it has exactly the same right to be classed as a living thing as you or me! ‘People say “The weather has a mind of its own” and think they’re just using a metaphor,’ says Wolfram. ‘I think there’s something much more literally true about it.’
Wolfram believes his Principle of Computational Equivalence is a revolutionary and fertile new idea in science. Moreover, he sees it as the next logical step along a road that science first embarked on more than four centuries ago.
In the sixteenth century, the Polish astronomer Nicolaus Copernicus realised that the Sun and planets did not turn about the Earth, as had generally been believed, but that the Earth occupied no special place in the Universe. Later, in the nineteenth century, Charles Darwin deduced that humans were just another product of evolution by the process of natural selection and so they occupied no special place in Creation. Wolfram sees himself as completing the revolution begun by Copernicus and Darwin. There is nothing special, he maintains, about the kind of computation that leads to living things and the thought processes of human brains. Life and intelligence could be implemented in a myriad different physical systems. One consequence of this is that there is no barrier preventing us from creating artificial intelligence – a machine that thinks and behaves like a human being.
All this spells trouble for a kind of reasoning currently favoured by some cosmologists. According to the ‘anthropic principle’, the reason the Universe has many of the features it has – for instance, laws of physics which permit the formation of galaxies, stars and planets – is because, if it did not, it would not have been possible for human beings to have arisen to notice those features. It is a curiously topsy-turvy logic. And an inevitable consequence is that biology is the ultimate determinant of the physics that we observe around us.
However, the anthropic principle is fatally undermined if, as Wolfram believes, life can be implemented in any number of different physical systems, some as far away from carbon-based chemistry as it is possible to imagine. ‘Cosmologists have no right to use the conditions necessary for our existence on Earth to deduce anything about the laws of physics that govern our Universe,’ says Wolfram.
Cosmologists wonder why the Universe appears so hospitable for life. The answer, Wolfram believes, is because almost any physical system, almost any set of parameters, can exhibit the complexity of a living thing.
Everything Wolfram discovered during his decade of toil – the equivalent, he maintains, of hundreds, maybe even thousands, of scientific papers – he eventually distilled into an enormous, epic book. A New Kind of Science was finished in January 2002. It was almost 1,200 pages long with about 1,000 black-and-white pictures and half a million words. On the first day of publication it sold 50,000 copies. And it annoyed the hell out of the scientific community.
Absolutely everything about the self-published book seemed to make other scientists see red. Wolfram was accused of not crediting the contributions of others. Wolfram was accused of breathtaking arrogance. After all, he was saying, ‘Here in my book is an entirely new way of doing science.’ And nobody had dared say that since Isaac Newton.
A striking feature of the venom directed at Wolfram was its swiftness. Within days of the book’s publication, some scientists had posted damning reviews on Amazon’s website. Yet the book’s 1,200 picture-filled pages were crammed with examples that had to be worked through by the reader. It was hard to believe that anyone could have digested enough to have dismissed it in just a few days.
Chaitin is philosophical about the knee-jerk reaction of the scientific community. ‘If you write a book that offends no one and make sure everything you write is absolutely, 100 per cent, correct, then you end up writing nothing,’ he says.
One specific criticism is that, although Wolfram has produced a 1,200-page book of pretty pictures of what simple computer programs can do, he has deduced very few universal laws of the kind first discovered by Newton. This, however, is to misunderstand Wolfram. His new kind of science is not at all like the old type – which, of course, is why he has called it ‘a new kind of science’. In the old, maths-based science, the motion of, say, a planet travelling around the Sun is distilled into an equation, which predicts its behaviour from now into the infinite past and future. In the new science, the only way to discover how something behaves is to run a computer program. There is no such shortcut. Or, rather, all the shortcuts have already been found – they are conventional, equation-based science.
It is Wolfram’s view that much of what is going on in the Universe cannot be distilled into neat equations. You have to run the program to find out what happens. Some of the programs can be run, and a result obtained, more quickly than the Universe. This is because, by some fortunate quirk, some of what the Universe is doing is ‘computationally reducible’. ‘Almost all of what traditional equation-based science has been doing is looking just at those computationally reducible parts,’ says Wolfram.
Wolfram suspects, however, that most of what is going on in the Universe is computationally irreducible. In other words, the only way to find out the outcome of the program the Universe is running is to run it for 13.7 billion years! This raises a spooky possibility. Is the program of the Universe being run by someone or something simply because there is no other way to discover the outcome? In The Hitchhiker’s Guide to the Galaxy, the Earth turns out to be a computer run by mice to discover the answer to the ultimate question. Might Douglas Adams’s jest, by some tremendous irony, actually be near to the truth?
The American physicist Ed Fredkin thinks so. He is convinced that the Universe is nothing more than a computer which is being used to solve a problem. As others have pointed out, this is both good news and bad news. The good news is that there really is a purpose to our lives. The bad news is that purpose may be to help someone or something work out pi to countless zillion decimal places!
The idea that the most fundamental stuff in the Universe – more fundamental even than matter or energy – is information, digital information, is certainly an idea which is taking hold among today’s physicists. Those who subscribe to this ‘digital philosophy’, such as Wolfram and Fredkin, are in absolutely no doubt that what the Universe is doing is computation, in the most general sense of the word. One consequence is unavoidable. Like the insects burrowing in the topsoil of Adams’s terrestrial computer, we are a part of the great cosmic computation. ‘We never perform a computation ourselves,’ says Tomasso Toffoli of Boston University.* ‘We just hitch a ride on the great Computation that is going on already.’
Of course, if you want to go one mystical step farther and talk about a computation not in its most general sense but in the sense of something directed to some end, like a human computation, then you come to the arena of religious speculation. ‘The Universe begins to look more like a great thought than a machine,’ wrote the British astronomer Sir James Jeans. And Jeans was really only echoing Bishop George Berkeley, the Irish philosopher who in the eighteenth century declared: ‘We exist only in the mind of God.’ Chaitin likes to put it in more modern terms: ‘Is God a programmer?’
If the idea of the Universe ‘computing’ something is not mind-blowing enough, consider what it really means if Wolfram is right and the complexity of the Universe is generated merely by applying a simple computer program – a simple rule – over and over again. Information cannot be created out of nothing. Common sense says that what comes out cannot be any more than what is put in. If Wolfram is right, it means that the Universe can contain no more complexity than the simple program responsible for generating it. Consequently, the complexity we see around us cannot be real complexity. It must be ‘pseudo complexity’. The Universe only looks complex because we are unaware of the simple underlying rule generating it.
Newton’s worldview was one in which the laws of physics orchestrate a predictable world. The planets, for instance, circle the Sun with the regularity of clockwork. However, in such a clockwork universe, where the future is always utterly predictable, scientists faced a conundrum: how can there be any free will?
Wolfram sidesteps this problem. In his clockwork universe, the future of the Universe is predictable – but only in principle. In practice, you can never finish the computations and discover the outcome faster than the Universe does. Free will survives – as pseudo free will!
Chaitin puts Wolfram’s worldview in purely mathematical terms. Pi, the ratio of a circle’s circumference to its diameter, is a number that appears to be extraordinarily complicated, its digits never repeating, but it can in fact be generated by a short computer program. Chaitin, however, has invented a number which is truly complex. ‘Omega’ requires an infinitely long computer program to generate it.* ‘Is the Universe like pi or like Omega?’ says Chaitin. ‘Most people think it’s like Omega. Wolfram thinks it’s like pi.’
The reason that most people think the Universe is like ‘Omega’ – in other words, that it has unadulterated, infinite complexity – is that most people believe in quantum theory. And quantum theory tells us that events in the microscopic world such as the disintegration of an atom or the absorption of a photon of light by a window pane are completely random, as unpredictable as a perfect coin toss. Such events generate an infinite amount of complexity – which is the same as randomness.* And this gets permanently imprinted on the Universe – for instance, when a high-energy photon strikes a strand of DNA and causes a mutation, which echoes down the generations, frozen into the fabric of life for all time.
As a consequence of quantum theory, then, much of what we see around us in the Universe is inherently unpredictable. It is the result of countless quantum coin tosses, which have been happening one after the other since the beginning of time. We will therefore never be able to comprehend the Universe in its entirety.
On this score, Wolfram is far more optimistic than the majority of physicists. Because he believes the Universe has finite complexity like pi, he believes that quantum theory as currently practised is wrong. All the randomness that quantum theory generates is therefore really only pseudo randomness, like that in the digits of pi. If he is right, then we may eventually be able to comprehend everything.
Who is right – Wolfram or the rest of the scientific community? Chaitin confesses to spending long hours at Wolfram’s house near Boston arguing with him about his ideas. ‘In A New Kind of Science, Wolfram develops an extremely interesting and provocative vision,’ says Chaitin. ‘The question is: Does the physical Universe share Wolfram’s vision? Time alone will tell.’
* A black hole is a region of space where gravity is so strong that not even light, the fastest thing in the Universe, can escape.
* Why 256 possible rules? Well, for each successive line, the colour of a cell (black or white) depends only on its own previous colour and the colour of the cell on the left and the cell on the right. This means there are eight possible starting situations. For instance, a square can be black with a black square to its left, or black with a black square to its right, or black with black squares to the left and right, or black with white squares to the left and right. Another four possibilities arise if the central square is white.
Each rule maps all these eight input situations to an output (black or white). This means there are 2^8 = 256 possible rules for such a one-dimensional, two-colour, adjacent-cell cellular automaton.
* For more on Chaitin – in fact, for a whole chapter on the man, not to mention the amazing number he invented that contains the secret of life, the Universe and everything, see Chapter 6, ‘God’s Number’.
† See Chapter 6, ‘God’s Number’.
* Even the atoms and molecules that compose you are ‘refreshed’ at intervals. They are not the same ones that were a part of you last year. Most cells such as blood cells are replenished within a matter of weeks and even those that persist longer such as neurones have their component molecules changed at regular intervals.
* Toffoli is famous for inventing the Toffoli Gate, a logical computing circuit which can be implemented by transistors in a computer. Not only is the gate universal – which means that any conceivable calculation can be carried out solely by a collection of Toffoli Gates – but it is reversible – which means the gate produces the same result regardless of whether current flows through it forwards or backwards. This is important because, in physics, reversible processes use no energy. The Toffoli Gate is therefore extremely energy-efficient.
* See Chapter 6, ‘God’s Number’.