The world about us looks so bewilderingly complex, it seems impossible that human beings could ever understand it completely. But dig deeper, and the richness and variety of nature are found to stem from just a handful of underlying mathematical principles. So rapid has been the advance of science in elucidating this hidden subtext of nature that many scientists, especially theoretical physicists, believe we are on the verge of formulating a “theory of everything”.
When Stephen Hawking accepted the Lucasian Chair of Mathematics at Cambridge University in 1980 he chose as the title of his inaugural lecture “Is the end in sight for theoretical physics?”. What he meant was that physicists could glimpse the outlines of a final theory, in which all the laws of nature would be melded into a single, elegant mathematical scheme, perhaps so simple and compact it could be emblazoned on your T shirt. Now Hawking has done something of a U-turn by claiming in a lecture that we will never be able to grasp in totality how the universe is put together.
The quest for a final theory began two and a half thousand years ago. The Greek philosophers Leucippus and Democritus suggested that however complicated the world might seem to human eyes, it was fundamentally simple. If we could only look on a small enough scale of size, we would see that everything is made up of just a handful of basic building blocks, which the Greeks called atoms. It was then only a matter of identifying these elementary particles, and classifying them, for all to be explained.
Today we know that atoms are not the elementary particles the Greek philosophers supposed, but composite bodies with bits inside them. However, this hasn’t scuppered the essential idea that a bottom level of structure exists on a small enough scale. Physicists have been busy peering into the innards of atoms to expose what they hope is the definitive set of truly primitive entities from which everything in the universe is built. The best guess is that the ultimate building blocks of matter are not particles at all, but little loops of vibrating string about twenty powers of ten smaller than an atomic nucleus.
String theory has been enormously beguiling, and currently occupies the attention of an army of physicists and mathematicians. It promises to describe correctly not only the entire inventory of familiar particles – electrons, protons, neutrinos, and so on – but the forces that act between them, like electromagnetism and gravity. It could even explain the existence of space and time too.
Though string theorists are upbeat about achieving the much sought-after theory of everything, others remain sceptical about the entire enterprise. A bone of contention has always surrounded the word "everything". Understanding the basic building blocks of physical reality wouldn’t help explain how life originated, or why people fall in love. Only if these things are dismissed as insignificant embellishments on the basic scheme would the physicist’s version of a final theory amount to a true theory of everything.
Then there is a deeper question of whether a finite mind can ever fully grasp all of reality. By common consent, the most secure branch of human knowledge is mathematics. It rests on rational foundations, and its results flow seamlessly from sequences of precise definitions and logical deductions. Who could doubt that 1 + 1 = 2, for example? But in the 1930s the Austrian philosopher Kurt Gödel stunned mathematicians by proving beyond doubt that the grand and elaborate edifice of mathematics was built on sand. It turns out that mathematical systems rich enough to contain arithmetic are shot through with logical contradictions. Any given mathematical statement (e.g. 11 is a prime number) must either be true or false, right? Wrong! Gödel showed that however elaborate mathematics becomes, there will always exist some statements (not the above ones though) that can never be proved true or false. They are fundamentally undecidable. Hence mathematics will always be incomplete and in a sense uncertain.
Because physical theories are cast in the language of mathematics, they are also subject to the limitations of Gödel’s theorem. Many physicists have remarked over the years that this will preclude a truly complete theory of everything. Now it seems that Stephen Hawking has joined their ranks.
So does this mean physicists should give up string theory and other attempts at unifying the laws of nature, if their efforts are, in the final analysis, doomed to failure? Certainly not, for the same reason that we don’t give up teaching and researching mathematics because of Gödel’s theorem. What these logical conundrums tell us is that there exist ultimate limits to what can be known using the rational method of inquiry. It means that however heroic our efforts may be at understanding the world about us, there will always remain some element of mystery at the end of the universe.
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