ON THE closing page of his famous book A Brief History of Time, Stephen Hawking celebrated the idea of a theory of everything that would unify all the forces of nature. He argued that it would be the ultimate triumph of human reason: 鈥渇or then we would know the mind of God鈥�.
This controversial statement gained the author great notoriety. It read like a declaration of science鈥檚 supremacy over religion and philosophy, while many critics saw it as a sign of supreme arrogance. Perhaps they were right. Today, 15 years after Hawking鈥檚 book was published, it seems the Cambridge cosmologist has changed his mind.
鈥淯p to now, most people have implicitly assumed that there is an ultimate theory that we will eventually discover. Indeed, I myself have suggested we might find it quite soon,鈥� he told an audience this week in Davis, California. But now he has doubts. 鈥淢aybe it is not possible to formulate the theory of the Universe in a finite number of statements.鈥� If this is true, we can kiss goodbye to the idea of a theory of everything.
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Hawking鈥檚 turnaround has been prompted by his work on one of the most advanced ideas in physics: a would-be theory of everything called M theory. This monolith is not so much a single idea as a basket of string theories, all of which build on the idea that matter and energy arise from the vibrations of tiny subatomic strings.
String and M theories were created to tie together Einstein鈥檚 thinking on gravity with quantum theory, the description of how matter and energy interact on tiny scales. But although individual string theories are successful in limited conditions, they cannot deal with all eventualities. Even together, they do not truly describe reality.
The problem physicists face with M theory is one that we might have seen coming for decades, Hawking says. It comes from the work of the Austrian-born mathematician Kurt G枚del, who in 1931 proved that there exist true but unprovable mathematical statements. And, Hawking believes, the same may well be the case in physics.
G枚del鈥檚 work blew mathematicians off their feet. The prevailing notion at the time was that in formal mathematical systems 鈥� which are built up from a handful self-evident statements, or axioms 鈥� a mathematician could prove any theorem true or false simply by reasoning from the axioms.
In 1900, the renowned German mathematician David Hilbert had set out a list of 23 problems as a challenge for the new century. He argued that every mathematical problem had a solution: be clever enough, look hard enough and everything would be tamed. For thirty years, mathematicians celebrated the supremacy of their discipline. Then G枚del came along. He showed that not every theorem could be proved from the axioms: mathematics was 鈥渋ncomplete鈥�.
Although this result might sound depressing, it has stimulated mathematics ever since, spawning a wealth of new understanding about the limits to what we can ever know. The British mathematician Alan Turing used G枚del鈥檚 finding to show that there are things a computer can never do. And IBM mathematician Gregory Chaitin used it to show that there exists a number, called Omega, that is real but utterly incalculable. Now Hawking thinks that G枚del鈥檚 result, or at least its analogue in physics, signals that the mind of God may stay hidden forever.
G枚del鈥檚 master stroke was to create the arithmetical equivalent of a sentence that refers to itself, such as 鈥淭his statement cannot be proved true.鈥� If the statement is false, then it can be proved to be true, and there鈥檚 a contradiction. So it must be true, but then it cannot be proved. The statement, then, produces inconsistent results.
Hawking has a direct physical analogy for this problem. In days gone by, Newtonian reasoning told us that we could calculate the future, such as the position of a car racing along a road, simply by extrapolating from our understanding of the present. But these days of certainty are no more: modern notions of gravity and quantum theory show that this approach is inadequate. 鈥淲e and our models are both part of the Universe we are describing,鈥� says Hawking. 鈥淲e are not angels who view the Universe from outside.鈥�
This means that these physical theories are self-referential, as in G枚del鈥檚 theorem, so we shouldn鈥檛 be surprised if they are inconsistent or incomplete. 鈥淭he theories we have so far are both inconsistent and incomplete.鈥�
M theory is incomplete in a very real sense. It assumes that we can define the Universe鈥檚 鈥渨ave function鈥� 鈥� the full quantum description of its properties 鈥� at each and every point in space. In an infinite universe, this would require an infinite density of information, but there鈥檚 a fundamental problem with that idea.
In their work on black holes, Hawking and Jacob Bekenstein of the Hebrew University of Jerusalem have shown that the amount of information contained in a black hole is not proportional to its volume, as you might expect, but to the area of its boundary 鈥� the event horizon, inside which the black hole鈥檚 gravity is too strong for anything to escape. This fact rules out any possibility that M theory can utilise an infinite density of information. 鈥淲hat we need is a formulation of M theory that takes account of the black hole information limit,鈥� says Hawking.
But G枚del鈥檚 shadow will loom large over that model. The information the model itself contains has to be represented by something 鈥� the arrangement of particles on magnetic tape, for example. After all, as the IBM physicist Rolf Landauer famously remarked, 鈥渋nformation is physical鈥�. Arranging these particles costs energy, so the model will change the energy 鈥� and information 鈥� in the very system it is trying to represent. Just like G枚del鈥檚 arithmetic statement, it refers back on itself. So a theory of everything may be out of reach for ever. Once again, you might find this conclusion depressing. But Hawking is sanguine. 鈥淕枚del鈥檚 theory ensured there would always be a job for mathematicians,鈥� he says. 鈥淚 think M theory will do the same for physicists.鈥�