Seeing Further (40 page)

This makes the point that mathematics also saves lives. Have you had a medical scan recently? How do you think the scanner works out what’s inside you? There’s a whole branch of mathematics devoted to such questions.
Are you concerned about crime? The FBI uses ‘wavelets’, a very recent piece of mathematics, to analyse and record fingerprint information to help catch criminals. Other police forces use similar techniques. Do you use oil or natural gas, for heating, cooking, or transport? The oil companies use powerful mathematical techniques to find out what the rocks miles underground look like, based on the echoes from explosions at the surface. Do you use anything with a spring in it – ballpoint pen, video recorder, mattress? The spring-making industry uses mathematics for quality control.

Another huge area that relies on mathematics is science, and science is our most successful method for understanding the natural world. The development of science, and that of mathematics, have gone hand in hand for about five hundred years. Newton invented calculus to understand the movements of the planets. Independently, Gottfried Leibniz developed much the same ideas for purely intellectual reasons. These two sources of mathematical inspiration can be roughly characterised as ‘applied’ and ‘pure’ mathematics. The main differences are motivation and attitude, rather than content. The same mathematical concept may appear in the solution of Fermat’s last theorem (pure mathematics) or in the construction of a secure code for Internet banking (applied mathematics). Some areas are traditionally considered as being ‘pure’, others as ‘applied’, but these are convenient distinctions, not impassable barriers. Today’s science is increasingly multi-disciplinary; so is mathematics.

Initially, the main beneficiaries of mathematical techniques were the physical sciences, and these are still the areas in which the use of mathematics is greatest. But the biological and medical sciences are catching up rapidly, and some of the most interesting new problems for research mathematicians are coming out of biology. A century or two from now we will look back at today’s Newtons and Booles, and understand how vital their work has been to the development of our society. Provided we do not lose sight of the hidden mathematics that rules our world – because if we do, those advances will never happen.

M
AKING SENSE OF THE WORLD SCIENTIFICALLY HAS OFTEN MEANT SEARCHING FOR SIMPLICITY UNDERLYING THE APPARENTLY COMPLEX. FINE, SAYS JOHN BARROW, EXCEPT WHEN THE COMPLEXITY TURNS OUT TO BE IRREDUCIBLE. OR DOES IT?

Is the world simple or complicated? As with many things, it depends on who you ask, when you ask, and how seriously they take you. If you should ask a particle physicist you would soon be hearing how wonderfully simple the universe appears to be. But, on returning to contemplate the everyday world, you just know ‘it ain’t necessarily so’: it’s far from simple. For the psychologist, the economist, or the botanist, the world is a higgledy-piggledy mess of complex events that just seemed to win out over other alternatives in the long run. It has no mysterious penchant for symmetry or simplicity.

So who is right? Is the world really simple, as the particle physicists claim,
or is it as complex as almost everyone else seems to think? Understanding the question, why you got two different answers, and what the difference is telling us about the world, is a key part of the story of science over the past 350 years from the inception of the Royal Society to the present day.

Our belief in the simplicity of Nature springs from the observation that there are regularities which we call ‘laws’ of Nature. The idea of laws of Nature has a long history rooted in monotheistic religious thinking, and in ancient practices of statute law and social government.
¹
The most significant advance in our understanding of their nature and consequences followed Isaac Newton’s identification of a law of gravitation in the late seventeenth century, and his creation of a battery of mathematical tools with which to unpick its consequences. Newton made his own tools: with them we have made our tools ever since. His work inspired the early Fellows of the Royal Society, and scientists all over Europe, who followed the advances reported at its meetings and in its published
Transactions
closely during the years of his long Presidency from 1703 to his death in 1727, to bring about a Newtonian revolution in the study of the mathematical description of motion, gravity and light. It gave rise to a style of mathematics applied to science that remains distinctively Newtonian.

Laws reflect the existence of patterns in Nature. We might even define science as the search for those patterns. We observe and document the world in all possible ways; but while this data-gathering is necessary for science, it is not sufficient. We are not content simply to acquire a record of everything that is, or has ever happened, like cosmic stamp collectors. Instead, we look for patterns in the facts, and some of these patterns we have come to call the laws of Nature, while others have achieved only the status of by-laws. Having found, or guessed (for there are no rules at all about how you might find them) possible patterns, we use them to predict what should happen if
the pattern is also followed at all times and in places where we have yet to look. Then we check if we are right (there are strict rules about how you do this!). In this way, we can update our candidate pattern and improve the likelihood that it explains what we see. Sometimes a likelihood gets so low that we say the proposal is ‘falsified’, or so high that it is ‘confirmed’ or ‘verified’, although strictly speaking this is always provisional, none is ever possible with complete certainty. This is called the ‘scientific method’.
²

For Newton and his contemporaries, the laws of motion were codifications into simple mathematical form of the habits and recurrences of Nature. They were idealistic: ‘bodies acted upon by no forces will …’ because there are
no
such bodies. They were laws of cause and effect: they told you what happened if a force was applied. The future is uniquely and completely determined by the present.

Later, these laws of change were found to be equivalent to statements that quantities did not change. The requirement that the laws were the same everywhere in the universe was equivalent to the conservation of momentum; the requirement that they be found to be the same at all times was equivalent to the conservation of energy; and the requirement that they be found the same in every direction in the universe was equivalent to the conservation of angular momentum. This way of looking at the world in terms of conserved quantities, or invariances and unchanging patterns, would prove to be extremely fruitful.

During the twentieth century, physicists became so enamoured of the seamless correspondence between laws dictating changes and invariances preserving abstract patterns when particular forces of Nature acted, that their methodology changed. Instead of identifying habitual patterns of cause and effect, codifying them into mathematical laws, and then showing them to be equivalent to the preservation of a particular symmetry in Nature, physicists did a U-turn. The presence of symmetry became such a
persuasive and powerful facet of laws of physics that physicists began with the mathematical catalogue of possible symmetries. They could pick out symmetries with the right scope to describe the behaviour of a particular force of Nature. Then, having identified the preserved pattern, they could deduce the laws of change that are permitted and test them by experiment.

Since 1973, this focus upon symmetry has taken centre stage in the study of elementary-particle physics and the laws governing the fundamental interactions of Nature. Symmetry is the primary guide into the legislative structure of the elementary-particle world, and its laws are derived from the requirement that particular symmetries, often of a highly abstract character, are preserved when things change. Such theories are called ‘gauge theories’. All the currently successful theories of four known forces of Nature – the electromagnetic, weak, strong and gravitational forces – are gauge theories. These theories prescribe as well as describe: preserving the invariances upon which they are based requires the existence of the forces they govern. They are also able to dictate the character of the elementary particles of matter that they govern. In these respects, gauge theories differ from the classical laws of Newton, which, since they governed the motions of all bodies, could say nothing about the properties of those bodies. The reason for this added power of explanation is that the elementary-particle world, in contrast to the macroscopic world, is populated by collections of identical particles (‘once you’ve seen one electron, you’ve seen ’em all,’ as Richard Feynman remarked). Particular gauge theories govern the behaviour of particular subsets of all the elementary particles, according to their shared attributes. Each theory is based upon the preservation of a pattern.

This generation of preserved patterns for each of the separate interactions of Nature has motivated the search for a unification of those theories into more comprehensive editions based upon larger symmetries. Within those larger patterns, smaller patterns respected by the individual forces of Nature might be accommodated, like jigsaw pieces, in an interlocking fashion that places some new constraint upon their allowed forms. So far, this strategy has resulted in a successful, experimentally tested, unification of the electromagnetic and weak interactions, and a number of purely theoretical proposals for a further unification with the strong interaction (‘grand unification’), and candidates for a four-fold unification with the gravitational force to produce
³
a so-called ‘theory of everything’, or ‘TOE’. It is this general pattern of explanation by which forces and their underlying patterns are linked and reduced in number by unifications, culminating in a single unified law, that lies at the heart of the physicist’s perception of the world as ‘simple’. The success of this promising path of progress is the reason that led our hypothetical particle physicist to tell us that the world is simple. The laws of Nature are few in number and getting fewer.

The first candidate for a TOE was a ‘superstring’ theory, first developed by Michael Green and John Schwarz in 1984. After the initial excitement that followed their proof that string theories are finite and well-defined theories of fundamental physics, hundreds of young mathematicians and physicists flocked to join this research area at the world’s leading physics departments. It soon became clear that there were five varieties of string theory available to consider as a TOE: all finite and logically self-consistent,
but all different. This was a little disconcerting. You wait nearly a century for a theory of everything then, suddenly, five come along all at once. They had exotic-sounding names that described aspects of the mathematical patterns they contained – type I, type IIA and type IIB superstring theories, SO(32) and E8 heterotic string theories, and eleven-dimensional super-gravity. These theories are all unusual in that they have ten dimensions of space and time, with the exception of the last one, which has eleven. Although it is not demanded for the finiteness of the theory, it is generally assumed that only one of these ten or eleven dimensions is a ‘time’ and the others are spatial. Of course, we do not live in a nine- or ten-dimensional space so in order to reconcile such a world with what we see it must be assumed that only three of the dimensions of space in these theories became large and the others remain ‘trapped’ with (so far) unobservably small sizes. It is remarkable that in order to achieve a finite theory we seem to need more dimensions of space than those that we experience. This might be regarded as a prediction of the theory. It is a consequence of the amount of ‘room’ that is needed to accommodate the patterns governing the four known forces of Nature inside a single bigger pattern without hiving themselves off into sub-patterns that each ‘talk’ only to themselves rather than to everything else. Nobody knows why three dimensions (rather than one or four or eight, say) became large, or what is the force responsible. Nor do we know if the number of large dimensions is something that arises at random and so could be different – and may be different – elsewhere in the universe, or is an inevitable consequence of the laws of physics that could not be otherwise without destroying the logical self-consistency of physical reality.

One thing that we do know is that only in spaces with three large dimensions can things bind together to form structures like atoms, molecules, planets and stars. No complexity and no life is possible except in spaces with three large dimensions. So, even if the number of large dimensions
is different in different parts of the universe, or separate universes are possible with different numbers of large dimensions, we would have to find ourselves living where there are
three
large dimensions, no matter how improbable that might be, because life could exist in no other type of space.