Seeing Further

by Bill Bryson

SIMPLE REALLY:FROM SIMPLICITY TO COMPLEXITY – AND BACK AGAIN

  John D. Barrow FRS is a cosmologist, Professor of Mathematical Sciences, Director of the Millennium Mathematics Project, University of Cambridge, and Gresham Professor of Geometry at Gresham College, London. His many books include The Anthropic Cosmological Principle, The World Within the World, Pi in the Sky, Theories of Everything, The Origin of the Universe, The Left Hand of Creation, The Artful Universe, Impossibility: The Limits of Science, The Science of Limits, Between Inner Space, Outer Space, The Constants of Nature: From Alpha to Omega and Cosmic Imagery: Key Images in the History of Science. His latest is 100 Essential Things You Didn’t Know You Didn’t Know.

  MAKING 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?

  Symmetry calms me down, lack of symmetry makes me crazy.

  – Yves Saint Laurent

  WHAT IS THE WORLD LIKE?

  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.

  THE QUEST FOR SIMPLICITY

  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 produce3 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.

  At first, it was hoped that one of these theories would turn out to be special and attention would then narrow down to reveal it to be the true theory of everything. Unfortunately, things were not so simple. Progress was slow and unremarkable until Edward Witten, at Princeton, discovered that these different string theories are not really different. They are linked to one another by mathematical transformations that amount to exchanging large distances for small ones, and vice versa in a particular way. Nor were these string theories fundamental. Instead, they were each limiting situations of another deeper, as yet unfound, TOE which lives in eleven dimensions of space and time. That theory became known as ‘M-theory’, where M has been said to be an abbreviation for Mystery, Matrix, or Millennium, just as you like.4

  Do these ‘extra’ dimensions of space really exist? This is a key question for all these new theories of everything. In most versions, the other dimensions are so small (10–33 cm) that no direct experiment will ever see them. But, in some variants, they can be much bigger. The interesting feature is that only the force of gravity will ‘feel’ these extra dimensions and be modified by their presence. In these cases the extra dimensions could be up to one hundredth of a millimetre in extent and they would alter the form of the law of gravity over these and smaller distances. This gives experimental physicists a wonderful challenge: test the law of gravity on submillimetre scales. More sobering still is the fact that all the observed constants of Nature, in our three dimensions, are not truly fundamental, and need not be constant in time or space:5 they are just shadows of the true constants that live in the full complement of dimensions. Sometimes simplicity can be complex too.

  ELEMENTARY PARTICLES?

  The fact that Nature displays populations of identical elementary particles is its most remarkable property. It is the ‘fine tuning’ that surpasses all others. In the nineteenth century another of the Royal Society’s greatest Fellows, James Clerk Maxwell, first stressed that the physical world was composed of identical atoms which were not subject to evolution. Today, we look for some deeper explanation of the sub-atomic particles of Nature from our TOE. One of the most perplexing discoveries by experimentalists has been that such ‘elementary’ particles appear to be extremely numerous. They were supposed to be an exclusive club, but they have ended up with an embarrassingly large clientele.

  String theories offered another route to solving this problem. Instead of a TOE containing a population of elementary point-like particles, string theories introduce basic entities that are loops (or lines) of energy which have a tension. As the temperature rises the tension falls and the loops vibrate in an increasingly stringy fashion, but as the temperature falls the tension increases and the loops contract to become more and more point-like. So, at low energies the strings behave like points and allow the theory to make the successful predictions about what we should see there as the intrinsically point-like theories do. However, at high energies, things are different. The hope is that it will be possible to determine the principal energies of vibration of the superstrings. All strings, even guitar strings, have a collection of special vibrational energies that they naturally take up when disturbed. If we could calculate these special energies for super-strings, then they would (by virtue of Einstein’s famous mass-energy equivalence – E = mc2) correspond to the masses of the ‘particles’ that we call elementary. So far, these energies have proved too hard to calculate. However, one of them has been found: it corresponds to a particle with zero mass and two units of a quantum attribute called ‘spin’. This spin value ensures that it mediates attractions between all masses. It is the particle we call the ‘graviton’ and it is responsible for mediating the force of gravity. Its appearance shows that string theory necessarily includes gravity and, moreover, its behaviour is described by the equations of general relativity at low energies – a remarkable and compelling feature since earlier candidates for a TOE all failed miserably to include gravity in the unification story at all.

  WHY IS THE WORLD MATHEMATICAL?

  This reflection on the symmetries behind the laws of Nature also tells us why mathematics is so useful in practice. Mathematics is simply the catalogue of all possible patterns. Some of those patterns are especially attractive and are studied or used for decoration, others are patterns in time or in chains of logic. Some are described solely in abstract terms, while others can be drawn on paper or carved in stone. Viewed in this way, it is inevitable that the world is described by mathematics. We could not exist in a universe in which there was neither pattern nor order. The description of that order, and all the other sorts that we can imagine, is what we call mathematics. Yet, although the fact that mathematics describes the world is not a mystery, the exceptional utility of mathematics is. It could have been that the patterns behind the world were of such complexity that no simple algorithms could approximate them. Such a universe would ‘be’ mathematical, but we would not find mathematics terribly useful. We
could prove ‘existence’ theorems about what structures exist but we would be unable to predict the future using mathematics in the way that NASA’s mission control does.

  Seen in this light, we recognise that the great mystery about mathematics and the world is that such simple mathematics is so far reaching. Very simple patterns, described by mathematics that is easily within our grasp, allow us to explain and understand a huge part of the universe and the happenings within it.

  THE COPERNICAN PRINCIPLE APPLIED TO LAWS

  It is often said with hindsight that Nicholas Copernicus taught us not to assume that our position in the universe is special in every way. Of course, this does not mean that it cannot be special in any way, simply because life is only possible in certain places.6 Once we start distinguishing between the laws of Nature and their outcomes we should also bring this Copernican view to bear upon the laws of Nature as well as their outcomes.

  Universal laws of Nature should be just that – universal – they should not just exist in special forms for some privileged observers at special locations, or who are moving in particular ways, in the universe. Alas, Newton’s laws do not have this democratic property. They only have simple forms for privileged observers who are moving in a special way, neither rotating nor accelerating with respect to the distant ‘fixed’ stars. So there were privileged observers in Newton’s universe for whom all the laws of motion look simple.

  Newton’s first law of motion demands that bodies acted upon by no forces do not accelerate: they remain at rest or move with constant speed. However, this law of motion will only be observed by a special class of observers who are neither accelerating nor rotating relative to the fixed stars. The appearance of these special observers for whom all the laws of motion look simpler violates the Copernican principle.