GALILEO once said that the book of nature is written in the language of mathematics. James Jeans made the point equally metaphorically when he said God is a mathematician. Although scientists often take for granted the effectiveness of mathematics in describing the physical world, there has long been a philosophical debate over why the Universe should be circumscribed by mathematical laws. Mathematics is, after all, a construct of the human mind; so there is no obvious reason for it to mirror physical reality.
Combine weights on a pair of scales and you see their effect accumulate precisely according to the laws of addition. Travel in a rocket into outer space and you feel the Earth’s pull weaken in proportion to the inverse square of your distance. Gently swing a simple pendulum and you witness the embodiment of one of the fundamental objects of mathematics, the sine wave. All of these are simple examples of the power of mathematics in physics. But need our Universe be like this? Could an alternative Universe harbour pendulums and rockets that were not described by mathematics?
The question is an old one, but in the past few years, some new clues have emerged from the study of complex systems. Complexity has become something of an ‘in’ word among physicists. Cynics might say that this is because what used to be the physicist’s best game in town – searching for the simplest components of matter, the elementary particles – is looking a little passe given the dearth of discoveries since the W and Z particles were found in 1983. So some physicists have turned their attention to alternative pursuits such as looking at nature’s most complex entities.
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But there is another good reason why complexity has become fashionable. The increasing power of computers has enabled scientists to construct progressively more elaborate simulations of complex phenomena. In particular, models called cellular automata have taken such simulations to a new plane of complexity.
Cellular automata are machines that simulate the behaviour of large collections of autonomous biological cells. Imagine a draughtboard with counters in only some of the squares. Then follow these rules. If an empty square has three occupied neighbours, it too acquires a counter. The cell comes alive, if you like, nurtured by its neighbours. If a square has two occupied neighbours then it remains unchanged. The cell is unperturbed. Finally, if an occupied square has any other number of occupied neighbours (0,1,4,5,6,7 or 8), then it loses its counter. In other words, the cell dies either because of a lack of neighbourly support or because of overcrowding. These rules form what is known as the Game of Life, invented by mathematician John Conway (‘The life and times of cellular automata’, ¿ìè¶ÌÊÓƵ, 8 October 1988).
You start out with some pattern, apply the rules to every square, and see the pattern change. Then you keep repeating the game and the shapes evolve. In practice, the Game of Life is played on a computer with the shapes displayed on a screen. What you obtain is a sort of toy universe in which the draughtboard rules play the role of the laws of physics (or life) and the patterns represent material objects. You can see these objects move about, interact, reproduce and even perform computations.
Recently, Tomaso Toffoli of the Massachusetts Institute of Technology built an advanced cellular automaton using some specially designed hardware that plugs into an IBM PC. The machine can perform very rapid and detailed animations, and it also lets him vary the draughtboard laws at will. Some people refer to the machine as ‘The God Game’ because on it you can, in effect, play God with the Universe. Certainly such toy universes can be fascinating to watch and have been used to simulate complex phenomena such as biological growth, fluid turbulence, chemical reactions and crystal formation.
The striking point that emerges from these simulations is that you can generate remarkably complex, even life-like, patterns from very simple laws. Of course, such patterns are a bit like cartoons – they are, after all, simply flat images that sometimes resemble features of the real world. Yet people such as Toffoli are convinced that cellular automata could, in principle, model any real-life process. If this is the case, we could envisage a simulation that encompassed the whole of the Universe. Given such a possibility, some scientists argue that the Universe is in some sense a gigantic computational system.
Not surprisingly, there are several objections to the view that the Universe is a computer. The first and most obvious one is that computers (and cellular automata) are far too limited to model anything other than a few simplified features of the Universe. However, in the 1930s, Alan Turing, one of the pioneers of the computer, showed that a general-purpose computing machine could, in principle, calculate anything computable by any other machine. In other words, all general-purpose computing machines are equivalent. In practice, people use different types of computers for different applications. Big companies, for example, tend not to use microcomputers to run their payrolls. But this is mainly because some computers are much faster than others and also because some have much larger memories than others. In deriving his result, Turing conceived of a computational machine with a potentially infinite memory. Imagine the power of a cellular automaton with a three-dimensional screen of infinite size. Here we could surely simulate any physical object in unlimited detail. Turing’s result shows that such a simulation could be accomplished on any computer provided it had a big enough memory and sufficient time.
The second objection concerns the reversibility of physical laws and the apparent irreversibility of computers. The microscopic laws of physics, described by classical mechanics and quantum theory, are fully reversible with respect to time. If you could turn time backwards, the planets would continue to revolve around the Sun without bumping into one another. Atoms, too, would appear to have the same properties as before. On the other hand, all known computers are irreversible in their operation. This can be traced to the irreversibility of the logic gates that make up the computer’s central processor. Whenever a gate is switched, some energy is irretrievably lost as heat.
So the question arises: how can the Universe be a computer if the laws of physics are reversible and computers are not? Remarkably enough in the mid-1970s, Charles Bennett of IBM’s Research Center at Yorktown Heights and Ed Fredkin at MIT independently discovered how it might be possible to construct reversible computers. They took their cue from another IBM researcher, Rolf Landauer, who had previously shown that the minimum energy needed to perform a computation was directly related to the amount of information thrown away. Bennett and Fredkin both devised computational schemes in which no information would be lost. Fredkin considered an AND logic gate, for example, in which there are normally two inputs and one output and asked how you could reverse the operation of the gate to regenerate the inputs from the output. The answer usually is that you cannot. If, however, you arrange for the gate to pass on the value of its inputs as well as the AND output (so the gate now has three outputs), the AND gate becomes reversible because no information is lost.
Fredkin and his colleague Toffoli at MIT went on to show that a computer built using such gates could perform anything that a normal one could. The idea of a reversible computer remains a theoretical one – no one has yet made such a machine but it demolishes the old notion that there is an irreducible quantum of energy required to execute any computation.
Two objections to the idea of the Universe as a computer are thus swept aside. But a third arises in the form of arguments over what many people regard as the most mysterious manifestation of organised complexity: human consciousness. If the Universe is a computer and if all computers are functionally equivalent (as Turing’s work would suggest), computers must be able to simulate every feature contained within the Universe. That means computers must ultimately have the power to simulate conscious rational thought. Here, of course, we enter a minefield of philosophical debate.
Roger Penrose, a theoretical physicist at the University of Oxford, has been one of the latest explorers to intrude into the battle zone. He has collected new ammunition for the ‘computers can’t think’ brigade in the shape of his book, The Emperor’s New Mind. His foray begins with Kurt Godel’s undecidability theorem which states that certain mathematical propositions are unprovable (‘The incompleteness of arithmetic’, ¿ìè¶ÌÊÓƵ, 5 November 1987). Turing used Godel’s theorem to show that certain mathematical functions are uncomputable (‘A random walk in arithmetic’, ¿ìè¶ÌÊÓƵ, 24 March 1990). An example is the so-called halting problem. This involves devising a computer program that can examine a second computer program and decide whether that second program would ever terminate or simply ‘loop’ forever. In a few simple cases, it might be easy to spot whether a program will stop or loop indefinitely. But to make the prediction for any program is much harder. In fact, Turing showed that the general problem is uncomputable and so cannot be tackled by a computer program, no matter how complex.
Can we outsmart computers?
Strangely, though, as Penrose argues, human beings are sometimes capable of circumventing such logical constraints. Simplifying somewhat, Penrose’s argument goes like this. Imagine a mathematical proposition, P, which says that proposition P cannot be proved. We now ask: ‘Can P be proved?’ If the answer is: ‘Yes, P can be proved’, we would have to accept P as true. But P states that the proposition P cannot be proved, so we are led to an unacceptable contradiction. Thus the answer to the question must be: ‘No, P cannot be proved’. P is, in fact, a good example of one of Godel’s unprovable propositions. And once recast in Turing form, P becomes an uncomputable function. In other words, no computer starting from a given set of mathematical axioms could prove that P was true. It is here, however, that we human beings seem to be able to outsmart the computers. To avoid contradiction, we know we cannot prove P. Yet that is precisely what P states: P cannot be proved. Thus P must be true even though we cannot prove it! We can see that but computers cannot.
Though this argument may make your head swim a little, Penrose believes that it confirms that ‘out there’ somewhere is an infinite reservoir of truths to which mathematicians have access but from which computers are excluded. All very well for the mathematicians but it raises some awkward new questions because, as Penrose admits, the brain is a machine that works within the laws of physics. What aspect of the brain’s functioning and hence the laws of physics is it that enables our (or mathematicians’) minds to overcome the laws of computation? Penrose believes that new physics is required to answer the question – physics related to unsolved questions in quantum theory.
Intriguingly, David Deutsch at the University of Oxford has shown that computers whose logic gates work at the quantum level, so-called quantum computers, can solve certain problems in ways that cannot be reproduced on conventional computers. Quantum computers seem to have modes of computation not available on orthodox machines. Indeed, Bennett has devised a method of encryption at Yorktown Heights using a limited form of quantum computation (¿ìè¶ÌÊÓƵ, Science, 9 December 1989). Unfortunately, no one has yet found any interesting examples of classically insoluble problems, such as the halting problem, that can be cracked by quantum machines. But even if they did, the question of whether the brain harnesses quantum phenomena remains highly speculative without more direct experimental evidence.
Despite the arguments about the biological phenomenon of consciousness, there remains a hard core of computer converts like Fredkin who argue that their machines, given enough memory and speed, have the power to replicate the Universe in unlimited detail. The notion raises a question about Penrose’s Godel-like argument. Given that it can be expressed in the form of mathematical and linguistic symbols, why could such reasoning not be translated into computational terms? In other words, why cannot the process of human insight in this problem be mechanised so that a computer could come to the same conclusions as us?
Further studies of the science of complexity may cast light on Fredkin’s claim. At least, it is in the arena of complex systems that we expect the computational view of nature to be put to the test. If Fredkin and others are right about the connection between physics and computation, the mathematical properties of the Universe might seem to be inevitable.
It may sound like a tempting solution to why the Universe is mathematical, but Deutsch thinks that it only creates new problems. If the Universe is a computer and the laws of physics are part of its software, we would be prevented from knowing anything about its hardware. This is because, as Turing showed, computers are universal machines and thus the behaviour of a program is essentially independent of the particular hardware on which it runs. The upshot is that there would be an underlying physics responsible for the cosmic computer. If so, we, the ghostly meta-entities of the cosmic program, would be forever prevented from elucidating either the nature of that physics or the origin of the physical laws.
Nevertheless, Fredkin remains undaunted. In fact, he anticipates a time when computers will be so powerful that we, ourselves, will be able to choose the laws. For he can envisage a simulation not just of an individual thinking person but of an entire society. He tells an amusing parable about a futuristic computer simulation called the Heaven Machine.
One day, Fredkin muses, you hear about an advertising campaign by the Heaven Machine Corporation and you become curious to find out what it is. Arriving at their offices, you are ushered into a room and shown some brochures. The sales people explain that what they have is huge computer simulation into which they can load an exact copy of your brain state. Unfortunately, if you decide to accept the company’s offer, the process of duplication will destroy your original brain at which point your life on Earth will have ended.
Instead, however, you are promised eternal life within the machine and, not only that, you are told life there is heavenly. Because you look pretty sceptical, they offer to let you talk to your neighbour who recently took up a deal with the Heaven Machine Corporation. A computer screen the size of a wall is unveiled. Initially, the picture turns milky but gradually the clouds separate and you see your neighbour. You say: ‘Hi there Joe. How are you doing?’. Joe replies: ‘Just fantastic. It’s really heavenly up here. There are all these amazing people; Einstein, Buddha, Confucius – you wouldn’t believe the conversations I’ve had. I do as much fishing as I want and, boy, you should see the fish! You remember the guy I play tennis with at the club and how he always used to beat me? Well, now I always beat him! By the way, there’s something that’s been bothering me. Before I came here, I borrowed your lawn mower. Well, it’s still in the garage so just go and take it.’
Hearing this, you think, that must be Joe. So, is this the machine that offers eternal heavenly life or just a nightmarish fantasy? Well, Fredkin, at least, is prepared to believe that one day science fiction ideas like that may come true.
Julian Brown is a producer at the BBC Radio Science Unit. This article is based on a recent documentary ‘From white noise to a symphony’, broadcast on Radio 3. His most recent book is Superstrings: A Theory of Everything?, Paul Davies and Julian Brown (eds), Cambridge University Press, 1989
Further reading: The Emperor’s New Mind, Roger Penrose, Oxford University Press, 1989.