John D Barrow Quotes

History is full of people who thought they were right -- absolutely right, completely right, without a shadow of a doubt. And because history never seems like history when you are living through it, it is tempting for us to think the same.

Any universe simple enough to be understood is too simple to produce a mind able to understand it -Barrow's Uncertainty Principle

There are only certain intervals of time when life of any sort is possible in an expanding universe and we can practise astronomy only during that habitable time interval in cosmic history.

to paraphrase science writer John D. Barrow … we know they are impossible and yet we can imagine them anyway. Our brains, it turns out, are not prisoners of the world we live in; we can fly free! We can, any time we like, create the impossible.

No non-poetic account of reality can be complete.

Prior to then it was believed that black holes were just cosmic cookie monsters, swallowing everything that came within their gravitational clutches.

If the deep logic of what determines the value of the fine-structure constant also played a significant role in our understanding of all the physical processes in which the fine-structure constant enters, then we would be stymied. Fortunately, we do not need to know everything before we can know something.

I love cosmology: there’s something uplifting about viewing the entire universe as a single object with a certain shape. What entity, short of God, could be nobler or worthier of man’s attention than the cosmos itself? Forget about interest rates, forget about war and murder, let’s talk about space.” Rudy Rucker21

Turing attended Wittgenstein's lectures on the philosophy of mathematics in Cambridge in 1939 and disagreed strongly with a line of argument that Wittgenstein was pursuing which wanted to allow contradictions to exist in mathematical systems. Wittgenstein argues that he can see why people don't like contradictions outside of mathematics but cannot see what harm they do inside mathematics. Turing is exasperated and points out that such contradictions inside mathematics will lead to disasters outside mathematics: bridges will fall down. Only if there are no applications will the consequences of contradictions be innocuous. Turing eventually gave up attending these lectures. His despair is understandable. The inclusion of just one contradiction (like 0 = 1) in an axiomatic system allows any statement about the objects in the system to be proved true (and also proved false). When Bertrand Russel pointed this out in a lecture he was once challenged by a heckler demanding that he show how the questioner could be proved to be the Pope if 2 + 2 = 5. Russel replied immediately that 'if twice 2 is 5, then 4 is 5, subtract 3; then 1 = 2. But you and the Pope are 2; therefore you and the Pope are 1'! A contradictory statement is the ultimate Trojan horse.

We can measure the fine structure constant with very great precision, but so far none of our theories has provided an explanation of its measured value. One of the aims of superstring theory is to predict this quantity precisely. Any theory that could do that would be taken very seriously indeed as a potential 'Theory of Everything'.

nothing is higher than heaven; nothing is beyond the walls of the world; nothing is lower than hell, or more glorious than virtue.”48

universal laws prescribe how things will behave not, like human laws, how they ought to behave.

There are about one hundred billion galaxies within this visible universe and the average density of material within a galaxy is about one million times greater than that in the visible universe as a whole, and corresponds to about one atom in every cubic centimetre.

Einstein argued that the laws of Nature should appear to be the same for all observers in the Universe, no matter where they were or how they were moving. If they were not then there would exist privileged observers for whom the laws of Nature looked simpler than they did for other observers.

The Indian system of counting has been the most successful intellectual innovation ever made on our planet. It has spread and been adopted almost universally, far more extensively, even than the letters of the Phoenician alphabet which we now employ. It constitutes the nearest thing we have to a universal language…

Gravity acts on all forms of mass and energy, but energy comes in a host of very different forms that behave in peculiar ways that were not known in Newton's day. Wotst of all, gravity gravitates. Those waves of gravity that spread out, rippling the curvature of space, carry energy too and that energy acts as a source for its own gravity field. Gravity interacts with itself in a way that light does not.

THE MEETING" "Scant rain had fallen and the summer sun Had scorched with waves of heat the ripening corn, That August nightfall, as I crossed the down Work-weary, half in dream. Beside a fence Skirting a penning’s edge, an old man waited Motionless in the mist, with downcast head And clothing weather-worn. I asked his name And why he lingered at so lonely a place. “I was a shepherd here. Two hundred seasons I roamed these windswept downlands with my flock. No fences barred our progress and we’d travel Wherever the bite grew deep. In summer drought I’d climb from flower-banked combe to barrow’d hill-top To find a missing straggler or set snares By wood or turmon-patch. In gales of March I’d crouch nightlong tending my suckling lambs. “I was a ploughman, too. Year upon year I trudged half-doubled, hands clenched to my shafts, Guiding my turning furrow. Overhead, Cloud-patterns built and faded, many a song Of lark and pewit melodied my toil. I durst not pause to heed them, rising at dawn To groom and dress my team: by daylight’s end My boots hung heavy, clodded with chalk and flint. “And then I was a carter. With my skill I built the reeded dew-pond, sliced out hay From the dense-matted rick. At harvest time, My wain piled high with sheaves, I urged the horses Back to the master’s barn with shouts and curses Before the scurrying storm. Through sunlit days On this same slope where you now stand, my friend, I stood till dusk scything the poppied fields. “My cob-built home has crumbled. Hereabouts Few folk remember me: and though you stare Till time’s conclusion you’ll not glimpse me striding The broad, bare down with flock or toiling team. Yet in this landscape still my spirit lingers: Down the long bottom where the tractors rumble, On the steep hanging where wild grasses murmur, In the sparse covert where the dog-fox patters.” My comrade turned aside. From the damp sward Drifted a scent of melilot and thyme; From far across the down a barn owl shouted, Circling the silence of that summer evening: But in an instant, as I stepped towards him Striving to view his face, his contour altered. Before me, in the vaporous gloaming, stood Nothing of flesh, only a post of wood.

Navy: Please divert your course 15 degrees to the North to avoid a collision. Civilian: Recommend you divert your course 15 degrees to South to avoid a collision. Navy: This is the Captain of a US Navy ship. I say again, divert your course. Civilian: No, I say again, divert your course. Navy: This is the aircraft carrier Enterprise. We are a large warship of the US Navy. Divert your course now!! Civilian: This is a lighthouse. Your call.” Canadian naval radio conversation38

This picture of matter curving space and curvaceous space dictating how matter and light will move has several striking features. It brings the non-Euclidean geometries that we talked about in the last chapter out from the library of pure mathematics into the arena of science. The vast collection of geometries describing spaces that are not simply the flat space of Euclid are the ones that Einstein used to capture the possible structures of space distorted by the presence of mass and energy.

Three laws governing black hole changes were thus found, but it was soon noticed that something unusual was going on. If one merely replaced the words 'surface area' by 'entropy' and 'gravitational field' by 'temperature', then the laws of black hole changes became merely statements of the laws of thermodynamics. The rule that the horizon surface areas can never decrease in physical processes becomes the second law of thermodynamics that the entropy can never decrease; the constancy of the gravitational field around the horizon is the so-called zeroth law of thermodynamics that the temperature must be the same everywhere in a state of thermal equilibrium. The rule linking allowed changes in the defining quantities of the black hole just becomes the first law of thermodynamics, which is more commonly known as the conservation of energy.

The universe appears to be a system of very low density wherever we look. This is no accident. The expansion of the Universe weds its size and age to the gravitational pull of the material that it contains. In order that a universe expands for long enough to allow the building blocks of life to form in the interiors of stars, by a sequence of nuclear reactions, it must be billions of years old. This means that it must be billions of light years in extent and possess a very small average density of matter and a very low temperature. The low temperature and energy of its material ensures that the sky is dark at night. Turn off our local Sun and there is just too little light around in the Universe to brighten the sky. The night is dark, interspersed only by pinpricks of starlight. Universes that contain life must be big and old, dark and cold. If our Universe was less of a vacuum it could not be an abode for living complexity.

The success of discovering a thermodynamic principle associated with the gravitational field of a black hole has led to a speculation that there might exist some thermodynamic aspect to the gravitational field of the whole Universe. The simplest assumption to make, following the black hole case, would be that it is the surface area of the boundary of the visible universe. As the Universe expands, this boundary increases and the information available to us about the Universe increases. But this does not seem promising. It would appear to tell us only that the Universe must continue expanding forever, for if it were ever to begin to recollapse the entropy would fall and violate the second law of thermodynamics. The universe can expand in all sorts of different ways and still have the increasing area. What we really want is some principle that tells us why the organization of the Universe changes in the way that it does: why it now expands so uniformally and isotropically.

All our puzzles about whether or not lambda exists and, if so, what is responsible for giving it such a strange value, are like questions about the inflationary scalar field's potential landscape. Why is its final vacuum state so fantastically close to the zero line? How does it 'know' where to end up when the scalar field starts rolling downhill in its landscape? Nobody knows the answer to these questions. They are the greatest unsolved problems in gravitation physics and astronomy. The nature of their answers could take many forms. There could exist some deep new principle that links together all the different forces of Nature in a way that dictates the vacuum levels of all the fields of energy that feel their effects. This principle would be unlike any that we know because it would need to control all the possible contributions to lambda that arise at symmetry breakings during the expansion of the Universe. It would need to control physics over a vast range of energies.

If water is bombarded with intense sound waves, under the right conditions, then air bubbles can form which quickly contract and then suddenly disappear in a flash of light. The conventional explanation of what is being seen here is that a shock wave, a little sonic boom, is created inside the bubble, which dumps its energy, causing the interior to be quickly heated to flash point. But a more dramatic possibility, first suggested by the Nobel prize-winner Julian Schwinger, has been entertained. Suppose the surface of the bubble is acting like a Casimir plate so that, as the bubble shrinks, it excludes more and more wavelengths of the zero point fluctuations from existing within it. They can't simply disappear into nothing; energy must be conserved, so they deposit their energy into light. At present, experimenters are still unconvinced that this is what is really happening, but it is remarkable that so fundamental a question about a highly visible phenomena is still unresolved.

Then, in 1974, Stephen Hawking made a dramatic discovery. He decided to examine for the forst time what occurs when one applies the notions of quantum mechanics to black holes. What he discovered was that black holes are not completely black. When quantum mechanics is included in the discussion of their properties, it is possible for energy to escape from the surface of the black hole and be recorded by an outside observer. The variation in the strength of the gravitational field near the horizon surface is strong enough to create pairs of particles and antiparticles spontaneously. The energy necessary to do this is extracted from the source of the gravitational field, and as the process continues, so the mass of the black hole ebbs away. If one waits long enough, it should disappear completely unless some unknown physics intervenes in the final stages. Such a discovery was exciting enough, but its most satisfying aspect was the fact that the particles radiated away from the surface of the black hole were found to have all the characteristics of heat radiation, with a temperature precisely equal to the gravitational field at the horizon and an entropy given by its surface area, just as the analogy had suggested. Black holes did possess a non-zero temperature and obeyed the laws of thermodynamics, but only when quantum mechanics was included in their description.

We have seen that our numerical zero derives originally from the Hindu sunya, meaning void or emptiness, deriving from the Sanskrit name for the mark denoting emptiness, or sunya-bindu, meaning an empty dot. These developed between the sixth and eighth centuries. By the ninth century, the assimilation of Indian mathematics by the Arab world led to the literal translation of sunya into Arabic as as-sifr, which also means 'empty' or the 'absence of anything'. We still see a residue of this because it is the origin of the English word 'cipher'. Originally, it meant 'Nothing', or if used insultingly of a person it would mean that they were a nonentity-a nobody-as in King Lear where the fool says to the King "Now thou art an 0 without a figure. I am better than thou art now. I am a fool, thou art nothing.

Non-Euclidean' became a byword for non-absolute knowledge. It also served to illustrate most vividly the gap between mathematics and the natural world. Mathematics was much bigger than physical reality. There were mathematical systems that described aspects of Nature, but there were others that did not. Later, mathematicians would use these discoveries about geometry to discover that there were other logics as well. Aristotle's system was, like Euclid's, just one of many possibilities. Even the concept of truth was not absolute. What is false in one logical system can be true in another. In Euclid's geometry of flat surfaces, parallel lines never meet, but on curved surfaces they can. These discoveries revealed the difference between mathematics and science. Mathematics was something bigger than science, requiring only self-consistency to be valid. It contained all possible patterns of logic. Some of those patterns were followed by parts of Nature; others were not. Mathematics was open-ended, uncompleteable, infinite; the physical universe was smaller.

Fitzgerald had noticed that if this sqrt(I-v^2/c^2) correction factor was applied to the analysis of Michelson's apparatus fixed on the earth's surface as it moved around the Sun, it could explain why Michelson measured no effect from the ether. The arm of the interferometer contracts by a factor sqrt(I-v^2/c^2) in the direction of its motion through the ether at a speed v. At an orbital speed of 29 kilometers per second this results in a contraction of only one part in 200,000,000 in the direction of the Earth's orbital motion. The length of the arm perpendicular to the ether's motion is unaffected. This small contraction effect exactly counterbalances the time delay expected from the presence of a stationary ether. If the Fitzgerald-Lorentz contraction occurred then it allowed the existence of a stationary ether to be reconciled with the null result of the Michelson-Morley experiment. Space need not be empty after all.

When an important change takes place in science, in which a new theory takes the stagw, the incoming theory is generally an extension of the old theory which has the property of becoming more and more like the old theory in some limiting situation. In effect, it reveals that the old theory was an approximation (usually a very good one) to the new one that holds under a particular range of conditions. Thus, Einstein's special theory of relativity becomes Newton's theory of motion when speeds are far less than that of light, Einstein's general theory of relativity becomes Newton's theory of gravity when gravitational fields are weak and bodies move at speeds less than that of light. In recent years we have even begun to map out what the successor to Einstein's theory may look like. It appears that Einstein's theory of general relativity is a limiting, low-energy case of a far deeper and wider theory, which has been dubbed M theory.

Bindu is used to describe the most insignificant geometrical object, a single point or a circle shrunk down to its centre where it has no finite extent. Literally, it signifies just a 'point', but it symbolises the essence of the Universe before it materialized into the solid world of appearances that we experience. It represents the uncreated Universe from which all things can be created. This creative potential was revealed by means of a simple analogy. For, by its motion, a single dot can generate lines, by whose motion can be generated planes, by whose motion can be generated all of three-dimensional space around us. The bindu was the Nothing from which everything could flow.

The pulsar is like a lighthouse beam spinning at high speed. Every time it comes around to face us we see a flash. Its rotation can be very accurately monitored by timing observations of its periodic pulses. Twenty years of observations have shown that the pulsing of the binary pulsar is slowing at exactly the rate predicted if the system is losing energy by radiating gravitational waves at the rate predicted by Einstein's theory.

The phenomenon of symmetry breaking reveals something deeply significant about the workings of the universe. The laws of Nature are unerringly symmetrical. They do not have preferences for particular times, places and directions. Indeed, we have found that one of the most powerful guides to their forms is precisely such a requirement. Einstein was the first to recognise how this principle had been used only partially by Galileo and Newton. He elevated it to a central requirement for the laws of Nature to satisfy: that they appear the same to all observers in the Universe, no matter how they are moving or where they are located. There cannot be privileged observers for whom everything looks simpler than it does for others. To countenance such observers would be the ultimate anti-Copernican perspective on the Universe. This democratic principle is a powerful guide to arriving at the most general expression of Nature's laws.

Yet, despite the symmetry of the laws of Nature, we observe the outcomes of those symmetrical laws to be assymetrical states and structures.

Each of us is a complicated assymetrical outcome of the laws of electromagnetism and gravity.

One of Nature's deep secrets is the fact that the outcomes of the laws of Nature do not have to possess the same symmetries as the laws themselves. The outcomes are far more complicated, and far less symmetrical, than the laws. Consequently, they are far more difficult to understand. In this way it is possible to have a Universe governed by a very small number of simple symmetrical laws (perhaps just a single law) yet manifesting a stupendous array of complex, asymmetrical states and structures that might even be able to think about themselves.

Thus we see that the quantisation of atomic energies into a ladder of seperate values, rather than allowing them to take on the entire continuum of possible values, lies at the heart of the life-supporting stability and uniformity of the world around us.

First, energy is quantised: in atoms it does not take on all possible values but only a ladder of specific values whose separation is fixed by the value of a new constant of Nature, dubbed Planck's constant and represented by the letter h. An intuitive picture of how the wavelike character of the orbital behaviour leads to quantisation can be seen in Figure 7.1, where we can see how only a whole number of wave cycles can fit into an orbit. Second, all particles possess a wavelike aspect. They behave as waves with a wavelength that is inversely proportional to their mass and velocity. When that quantum wavelength is much smaller than the physical size of the particle it will behave like a simple particle, but when its quantum wavelength becomes at least as large as the particle's size then wavelike quantum aspects will start to be significant and dominate the particle's behaviour, producing novel behaviour. Typically, as objects increase in mass, their quantum wavelengths shrink to become far smaller than their physical size, and they behave in a non-quantum or 'classical' way, like simple particles.

Curiously, Schrodinger's equation describes the change in the probability that we will obtain a particular result if we conduct an experiment. It is telling us something about what we can know about the world. Thus, when we say that a particle is behaving like a wave, we should not think of this wave as if it were a water wave or a sound wave. It is more appropriate to regard it as a wave of information or probability, like a crime wave or a wave of hysteria. For, if a wave of hysteria passes through a population, it means that we are more likely to find hysterical behaviour there. Likewise, if an electron wave passes through your laboratory it means that you are more likely to detect an electron there. There is complete determinism in quantum theory, but not at the level of appearances or the things that are measured. Schrodinger's amazing equation gives a completely deterministic description of the change of the quantity (called the wave function') which captures the wavelike aspect of a given situation. But the wave function is not observable. It allows you only to calculate the result of a measurement in terms of the probabilities of different outcomes. It might tell you that fifty percent of the time you will find the atom to have one state, and fifty percent of the time, another. And, remarkably, in the microscopic realm, this is exactly what the results of successive measurements tell you: not the same result every time but a pattern of outcomes in which some are more likely than others.

When one looks at the numbers, the situation becomes even more perplexing. The effect of lambda grows steadily with respect to the familiar Newtonian force of gravity as the Universe gets bigger. If it is only recently becoming the dominant force, after billions of years of expansion of the Universe, it must have started out enormously smaller than the Newtonian force. The distance of that final minimum energy level in Figure 8.14 from the zero line in order to explain the value of lambda inferred from the supernova observations is bizarre: roughly 10^-120 - that is, 1 divided by 10 followed by 119 zeros! This is the smallest number ever encountered in science. Why is it not zero? How can the minimum level be tuned so precisely? If it were 10 followed by just 117 zeros, then the galaxies could not form. Extraordinary fine tuning is needed to explain such extreme numbers. Extraordinary fine tuning is needed to explain such extreme numbers. And, if this were not bad enough, the vacuum seems to have its own defence mechanism to prevent us finding easy answers to this problem. Even if inflation does have some magical property which we have so far missed that would set the vacuum energy exactly to zero when inflation ends, it would not stay like that. As the Universe keeps on expanding and cooling it passes through several temperatures at which the breaking of a symmetry occurs in a potential landscape, rather like that which occurs in the example of the magnet that we saw at the beginning of the chapter. Every time this happens, a new contribution to the vaccum energy is liberated and contributes to a new lambda term that is always vastly bigger than our observation allows. And, by 'vastly bigger' here, we don't just mean that it is a few times bigger than the value inferred from observations, so that in the future some small correction to the calculations, or change in the trend of the observations, might make theory and observation fit hand in glove. We are talking about an overestimate by a factor of about 10 followed by 120 zeros! You can't get much more wrong than that.

We know what we are, but know not what we may be.

Anselm conceives of God as something than which nothing greater or more perfect can be conceived. Since this idea arises in our minds it certainly has an intellectual existence. But does it have an existence outside of our minds? Anselm argued that it must, for otherwise we fall into a contradiction. For we could imagine something greater than that which nothing greater can be conceived; that is the mental conception we have together, plus the added attribute of real existence.

For even if we expunged all the matter in the Universe the lambda force could still exist, causing the universe to expand or contract. It was always there, acting on everything but unaffected by anything. It began to look like an omnipresent form of energy that remained when everything that could be removed from a universe had been removed, and that sounds very much like somebody's definition of a vacuum.

There is, as we shall see, a real and precise difference between the number zero and the concept of a set that posesses no members - the null, or empty set. Indeed, the second idea, pointless as it sounds, turns out to be by far the most fruitful of the two. From it, all of the rest of mathematics can be created step by step.

Universes that expand too slowly will collapse back to a big crunch before galaxies can form; universes that expand too quickly do not allow islands of matter to condense out into galaxies and form stars.

It is the cosmological vacuum energy that contributes a repulsive lambda force to the gravitational force of Newton.

The inclusion of just one contradiction (like 0 = 1) in an axiomatic system allows any statement about the objects in the system to be proved true (and also proved false). When Bertrand Russell pointed this out in a lecture he was once challenged by a heckler demanding that he show how the questioner could be proved to be the Pope if 2 + 2 = 5. Russell replied immediately that 'if twice 2 is 5, then 4 is 5, subtract 3; then 1 = 2. But you and the Pope are 2; therefore you and the Pope are 1'! A contradictory statement is the ultimate Trojan horse.

The ubiquity of chaotic phenomena raises a further problem for our dreams of omniscience through the medium of a Theory of Everything. Even if we can overcome the problem of initial conditions to determine the most natural or uniquely consistent starting state, we may have to face the reality that there is inevitable uncertainty surrounding the prescription of the initial state which makes the prediction of the exact future state of the Universe impossible. Only statistical statements will be possible.

Science is predicated upon the belief that the Universe is algorithmically compressible and the modern search for a Theory of Everything is the ultimate expression of that belief, a belief that there is an abbreviated representation of the logic behind the Universe's properties that can be written down in finite form by human beings.

The brain is the most effective algorithmic compressor of information that we have so far encountered in Nature. It reduces complex sequences of sense data to simple abbreviated forms which permit the existence of thought and memory. The natural limits that nature imposes upon the sensitivity of our eyes and ears prevents us from being overloaded with information about the world. They ensure that the brain receives a manageable amoubt of information when we look at a picture. If we could see everything down to sub-atomic scales then the information-processing capacity of our brains would need to be prohibitively large. The processing speed would need to be far larger than it now is in order for bodily responses to occur quickly enough to evade dangerous natural processes. This we shall have more to say about in the final chapter of our story, when we come to discuss the mathematical aspects of our mental processing.

The inverse-square law singles out no particular scale of length as special and has an infinite range. To explain objects of particular sizes requires a balance between gravity and some other force. Boscovich proposed a grand unified force law which included all known physical effects. This was his 'Theory', as he called it. It approached the inverse-square law of Newtonian gravitation at large distances as required by observations of the lunar motions. But on smaller length scales, it is alternately attractive and repulsive and so gives rise to equilibrium structures whose sizes are dictated by the characteristic length scales built into the force law.

There are many other innovations in Boscovich's detailed treatise, but we are interested here in drawing attention to just this one point: that he was the first to envisage, seek, and propose a unified mathematical theory of all the forces of Nature. His continuous force law was the first scientific Theory of Everything. Perhaps, in the eighteenth century, only a generalist like Boscovich, who successfully unified intellectual and administrative activities in every area of thought and practice would have the presumption that Nature herself was no less multicultural.

A 'group' is simply a collection of changes which possess three simple properties: there must be the possibility of no change, there must exist the possibility of undoing or reversing each change to restore its original state, and any two consecutive changes must give a result that could equally well be attained by another single change.

Each of the most basic physical laws that we know of corresponds to some invariance, which in turn is equivalent to a collection of changes which form a symmetry group. The symmetry group describes all the variations that can be formed from an initial seed pattern whilst still leaving some underlying theme unchanged. Thus, for example, the conservation of energy is equivalent to the invariance of the laws of motion with respect to translations backwards or forwards in time (that is, the result of an experiment should not depend on the time at which it was carried out, all other factors being identical); the conservation of linear momentum is equivalent to the invariance of the laws of motion with respect to the position of your laboratory in space, and the conservation of angular momentum to an invariance with respect to the directional orientation of your laboratory in space.

Any string of symbols that can be given an abbreviated representation is called algorithmically compressible.

The inflationary period of expansion does not smooth out irregularity by entropy-producing processes like those explored by the cosmologies of the seventies. Rather it sweeps the irregularity out beyond the Horizon of our visible Universe, where we cannot see it . The entire universe of stars and galaxies on view to us. , on this hypothesis, is but the reflection of a minute, perhaps infinitesimal, portion of the universe's initial conditions, whose ultimate extent and structure must remain forever unknowable to us. A theory of everything does not help here. The information contained in the observable part of the universe derives from the evolution of a tiny part of the initial conditions for the entire universe. The sum total of all the observations we could possibly make can only tell us about a minuscule portion of the whole.

The first scientist to contemplate the significance of places where things apparently cease to exist or become infinite ('singularities' that we would Now call them) in Newtonian Theory where the 18th century scientist leonhard Euler and Roger boscovich.

The effect of the 'time becomes space' proposal is that there is no definite moment or point of creation. In more conventional quantum mechanical terms we would say that the universe is the result of a quantum mechanical tunneling process, where it must be interpreted as having tunneled from nothing at all. Quantum tunneling processes, which are familiar to physicists and routinely observed, correspond to Transitions which do not have a classical path.

It is generally agreed by modern cosmologist that we have established the general framework of how the universe behaved from when it was just one second old until the present, some fifteen billion years later. This is not to claim that we understand everything that occurred. We do not understand the detailed process by which galaxies formed, but such processes actually exert a negligible influence upon the course of the overall expansion. Prior to one second after the apparent beginning, we are on all together shakier ground. We no longer have direct fossil remnants from the early Universe against which to check the accuracy of our reconstruction of its history. In order to reconstruct the history of the universe in these first instants we require knowledge of the behavior of matter at far higher energies than are accessible to us by terrestrial experiments.

Carbon originates in the Universe via a two-step process from nuclei of helium, or alpha particles as we usually call them. Two alpha particles combine under stellar conditions to make a nucleus of the element beryllium. The addition of a further alpha particle is necessary to transform this into a carbon nucleus. One would have expected this two-step process to be extremely improbable, but remarkably the last step happens to possess a rare property called 'resonance' which enables it to proceed at a rate far in excess of our naive expectation. In effect, the energies of the participating particles plus the ambient heat energy in the star add to a value that lies just above a natural energy level of the carbon nucleus and so the product of the nuclear reaction finds a natural state to drop into. It amounts to something akin to the astronomical equivalent of a hole-in-one. But this is not all. While it is doubly striking enough for there to exist not only a carbon resonance level but one positioned just above the incoming energy total within the interior of the star, it is well-nigh miraculous to discover that there exists a further resonance level in the oxygen nucleus that would be made in the next step of the nuclear reaction chain when a carbon nucleus interacts with a further alpha particle. But this resonance level lies just above the total energy of the alpha particle, the carbon nucleus, and the ambient environment of the star. Hence, the precious carbon fails to be totally destroyed by a further resonant nuclear reaction. This multiple coincidence of the resonance levels is a necessary condition for our existence. The carbon atoms in our bodies which are responsible for the marvellous flexibility of the DNA molecules at the heart of our complexity have all originated in the stars as a result of these coincidences. The positioning of the resonance levels are determined in a complicated way by the precise numerical values of the constants of physics.

All we can assert with confidence is a negative: if the constants of Nature were not within one percent or so of their observed values, then the basic buildong blocks of life would not exist in sufficient profusion in the Universe. Moreover, changes like this would affect the very stability of the elements and prevent the existence of the required elements rather than merely suppress their abundance.

A true standard requires us to find some way of defining what we mean by one unit of time which is the same for everyone no matter where or when they are observing. This desire for universality naturally moves us to seek some time standard that is determined by the constants of Nature alone. And this is indeed how modern absolute time standards are defined. They avoid the use of any characterisitic of the Earth or its gravitational field and focus instead upon the natural oscillation frequencies of certain atomic transitions between states of different energy. The time for one of these transitions to occur in an atom of caesium is determined by the velocity of light in a vacuum, the masses of the electron and proton, Planck's constant, and the charge on a single electron. All these quantities are taken to be constants of Nature. A time interval of one second is then defined to be a certain number of these oscillations. Despite the esoteric nature of this definition of time, it is a powerful one. It should allow us to communicate precisely what length of time we were talking about to the inhabitants of a distant galaxy.

This 'Planck length' is the only quantity with the dimensions of a length that can be built from the three mist fundamental constants of Nature: the velocity of light c, Planck's constant h, and Newton's gravitational conatant G. It is given by Lp = (Gh/c^3)^1/2 = 4 X 10 ^ -33 cm. This tiny dimension encapsulates the attributes of a world that is at once relativistic (c), quantum mechanical (h), and gravitational (G). It is a standard of length that makes no reference to any artefact of man or even of the chemical and nuclear forces of Nature. Relative to this unit of length, the size of the entire visible universe today extends roughly 10^60 Planck lengths, but the cosmological constant must be less than 10^-118 when referred to these Planck units of length rather than centimetres. To have to consider such a degree of smallness is unprecedented in the entire history of science. Any quantity that is required to be so close to zero by observation must surely in reality be precisely zero. This is what many cosmologists believe. But why?

For many years, cosmologists have sought, with little success, some fundamental principle which would reveal why the cosmological constant mist be zero. The elementary-particle physicists have searched as well, but far from finding an answer to the problem they have merely compounded it by showing that, even if such a principle were to exist which started the universe on its way at the Big Bang with a zero value of the cosmological constant, there arise complicated elementary-particle processes which produce stresses that mimic the presence of a cosmological constant with an unacceptably large value, billions and billions of times larger than observation allows.

Moreover it is assumed that wormholes only join universes to baby universes, or universes to themselves; there are no wormholes joining different baby universes in this approximation, nor are there allowed to be wormholes which split up into two or more other wormholes.

Newton, as we have said, was the promoter of a Design Argument based upon the precision and universality of the laws of motion and gravitation that he had discovered.

Topology is that branch of mathematics which is interested in the forms of things aside from their size and shape, Two things are said to be topologically equivalent if one can be deformed smoothly into the other without sticking, cutting, or puncturing it in any way. Thus an egg is equivalent to a sphere. The first application of topology to an analogous problem-the interaction of atoms rather than elementary particles-was made in the mid-nineteenth century by Lord Kelvin. It has many striking parallels with the aims and attractions of modern string theory.

Aristotle's later view of the relationship between mathematics and Nature could not have been more different. He wanted to rescue physical science from the mathematical stranglehold that Plato had placed upon it. He believed there to exist three completely autonomous realms of purely theoretical knowledge-metaphysics, mathematics, and physics-each possessing its own methods of explanation and accordant subject matter. But over-arching these divisions there existed a more general principle of 'homogeneity'-that like follows like-which mush always be obeyed:

Aristotle draws a sharp dividing-line between the activities of the physicist and those of the mathematician. The mathematician limits his enquiry to the quantifiable aspects of the world and so dramatically restricts what is describable in mathematical terms. Physics, for Aristotle, was far wider in scope and encompassed the earthly reality of sensible things. Whereas Plato had maintained that mathematics was the true and deep reality of which the physical world was but a pale reflection, Aristotle claimed mathematics to be but a superficial representation of a piece of physical reality. Such is the contrast between idealism and realism in the ancient world.

In the Middle Ages, this conflict between the Platonic and Aristotelian views of the relationship between mathematics and the world began to re-emerge after the sleep of centuries. The question became intricately entwined with the labyrinthine syntheses of Aristotelian and Platonic ideas within early Christian theology. Influential thinkers like Augustine and Boethius implicitly supported the Platonic emphasis upon the primary character of mathematics. Both of them pointed to the fact that things were created in the beginning 'according to measure, number, and weight' or 'according to the pattern of numbers'. This they took to exhibit an intrinsic feature of the mind of God and thus mathematics took its place as an essential part of the medieval quadrivum without which the search for all knowledge was impaired. Yet Boethius later veered towards the Aristotelian viewpoint that some act of mental abstraction occurs en route from physics to mathematics which renders these two subjects qualitatively distinct.

Grosseteste influenced Roger Bacon's ideas about mathematics and Nature. Bacon wrote hundreds of pages on the subject and, indeed, no historical figure has ever appeared more preoccupied with the question than he. He believed that mathematical knowledge was innate to the human mind and mathematics was a unique form of thought known both by ourselves and by Nature. Its uniqueness is characterized by the fact that it allows complete certainty to be achieved and hence our knowledge of Nature can be secure only in so far as we found it upon mathematical principles.

But, as powerful mathematicians like Minkowski and Hilbert found striking harmony between their pure mathematical results and the workings of the physical world, many found the claims for such a harmony hard to resist. Thus, in the early years of the twentieth century, we begin to see why Minkowski's application of complex numbers to the description of space and time was hailed by one physicist as 'one of the greatest revolutions in our accepted views'.

Moreover, there is no known reason why the geometry of space and time should be described by the particular types of curved geometry defined by Riemann. There exist other more complicated varieties that could in principle have been employed by Nature. Only observation can at present tell us which mathematics is chosen by Nature for employment in particular situations. This may of course merely be a transient manifestation of our relative ignorance of the bigger picture in which everything that is not excluded is demanded.

Impressed by the success of high-level mathematics in the formulation of the general theory of relativity in 1915, we find that Einstein's life-long quest for a unified field theory was dominated by the search for more general mathematical formalisms that could bring together the existing descriptions of gravity and electromagnetism. We find none of Einstein's compelling thought experiments and beautifully simple physical reasoning that lay at the heart of his early success. As the last quotation tells, he had become convinced that by pursuing mathematical formalisms alone, the compelling simplicity of a unified description of the world would become inescapable.

Besides the traditional questions of where or what the Platonic world of perfect mathematical blueprints actually is, this view moves us towards a number of deep and fascinating questions. It elevates mathematics pretty close to the status of God in traditional theology. Just take any work of medieval theology and alter the word 'God' to 'mathematics' wherever it appears and it makes pretty good sense. Mathematics is part of the world, and yet transcends it. It must exist before and after the Universe. In this respect, it is reminiscent of our discussion of the nature of time in earlier chapters. In the Newtonian world-view, both space and time were absolute and independent of the events played out upon them. Then the Einsteinian transformation of our concepts of space and time (whose radicalness is obscured by the fact that the concepts retained the same names) linked space and time to events going on within the Universe. Maybe a similar evolution of this interpretation of mathematics will emerge? Although at present mathematics seems to transcend the Universe because cosmologists think they can actually describe the Universe as a whole in terms of mathematics and use mathematics to study the process of creation and annihilation of universes, perhaps the nature of mathematics will become more closely associated with physically realizable processes like counting or computing?

Most scientists and mathematicians operate as if Platonism is true regardless of whether they believe that it is. That is, they work as though there were an unknown realm of truth to be discovered.

Particle physicists are the most deeply Platonic because their entire subject is built upon a belief that the deepest workings of the world are based upon symmetries. They examine symmetry after symmetry, confident in the expectation that the biggest and the best will have found employ in the grand scheme of things.

The great unanswered question is whether there exists some undiscovered organizing principle which complements the known laws of Nature and dictates the overall evolution of the Universe. To be a true addition to what we know of Nature's laws, this principle would need to differ from any laws of gravitation and particle physics that might emerge in final form from some Theory of Everything. It would not be specific to Universes but would govern the evolution of any complex system. True, its general notions ought to be tailored in some way to the notions which characterize the specific things that go on in an evolving universe-the clustering of matter into stars and galaxies, the conversion of matter into radiation-but it would also need to govern the invisible ways in which the gravitational field of the Universe can change. Any such discovery would be profoundly interesting because the Universe appears to be far more orderly than we have any right to expect. It has a tiny entropy level compared with the largest value that we could conceive of it possessing if we were to reorganize the observed matter into other configurations. This implies that the entropy level at the beginning of the expansion of the Universe must have been staggeringly small, which implies that the initial conditions were very special indeed.

The fact that simultaneous discovery occurs in mathematics, as well as the sciences, points toward some objective element within their subject matter that is independent of the psyche of the investigator.

The ease with which collaboration occurs in mathematical research and the essential similarity of the fruits of such collaboration to that of individual work points suggestively towards a powerful objective element behind the scenes that is discovered rather than invented.

Since so much of the physical universe, from brain waves to quantum waves, relies upon travelling waves we appreciate the key role played by the dimensionality of our space in rendering its contents intelligible to us.

As we look way back into the first instants of the Big Bang, we find the quantum world that we described in Chapter 3. From that state, where like effects do not follow from like causes, there must somehow emerge a world resembling our own, where the results of most observations are definite. This is by no means inevitable and may require the Universe to have emerged from a rather special primeval state.

All the best physical theories are associated with equations which allow the continuation of data defined at present into the future, and hence allow prediction. But this situation requires space and time to possess a rather particular type of mathematical property which we shall call 'natural structure'. Other theories, like those describing statistical or probabilistic outcomes, which attempt to use mathematics for prediction, often fail to possess a mathematical substratum with a 'natural structure' of this sort, and so there is no guarantee that its future states are smooth continuations of its present ones.

There is no formula that can deliver all truth, all harmony, all simplicity. No Theory of Everything can ever provide total insight. For, to see through everything, would leave us seeing nothing at all.

In a Newtonian world, all physical quantities, like energy and spin, can take on any values whatsoever. They range over the entire continuum of numbers. Hence, if one were to form a 'Newtonian hydrogen atom' by setting an electron in circular orbit around a single proton then the electron could move in a closed orbit of any radius because it could possess any orbital speed. As a result, every pair of electrons and protons that came together would be different. The electrons would find themselves in some randomly different orbit. The chemical properties of each of the atoms would be different and their sizes would be different. Even if one were to create an initial population in which the electrons' speeds were the same and the radii of their orbits identical, they would each drift away from their starting state in differing ways as they suffered the buffetings of radiation and other particles. There could not exist a well-defined element called hydrogen with universal properties, even if there existed universal populations of identical electrons and protons. Quantum mechanics shows us why there are identical collective structures. The quantization of energy allows it to come only in discrete packets, and so when an electron and a proton come together there is a single state for them to reside in. The same configuration arises for every pair of electrons and protons that you care to choose. This universal state is what we call the hydrogen atom. Moreover, once it exists, its properties do not drift because of the plethora of tiny perturbations from other particles. In order to change the orbit of the electron around the proton, it has to be hit by a sizeable perturbation that is sufficient to change its energy by a whole quantum packet. Thus the quantization of energy lies at the root of the repeatability of structure in the physical world and the high fidelity of all identical phenomena in the atomic world. With the quantum ambiguity of the microscopic world the macroscopic world would not be intelligible, nor indeed would there be intelligences to take cognisance of any such a totally heterodox non-quantum reality.

If the Universe possesses intrinsically random elements in their make up, inherited from its quantum origins or from random symmetry-breakings during its early evolution, then we must take our own existence into account when evaluating the correspondence between reality and the cosmological predictions of any Theory of Everything. Moreover, if these random cosmological elements lead to a Universe which differs significantly from place to place over the very large distances, then our local observations of a possibly infinite Universe will inevitably leave our knowledge of its global structure seriously incomplete.

Plato argued that the material world of visible things was but a shadow of the true reality of eternal forms. He proceeds to explain the nether world of eternal blueprints most completely in the case of the elements of matter: earth, air, fire, and water. These he represents by geometrical solids: the earth by a cube, water by an icosahedron, air by an octahedron, and fire by a tetrahedron. His position is that ultimately the elements are just these solid geometrical shapes not simply that they possess geometrical shapes as one of their properties. The transmutation of elements one into the other is then explained by the merger and dissolution of triangles. This strictly mathematical description characterizes Plato's discussion of many other physical problems, For him, mathematics is a pointer to the ultimate reality of the world of forms that overshadows the visible world of sense data. The better we can grasp it, the closer we can come to true knowledge. Thus, for Plato, mathematics is more fundamental, truer, closer to the eternal forms of which the visible world is an imperfect reflection, than the objects of physical science. Because the world is mathematical at its deepest level, all visible phenomena will have mathematical aspects and be describable by mathematics to a greater or lesser extent, determined by their closeness to their underlying forms.

Strings possess one defining property which is their tension. This quantity plays a crucial role in the overall picture of how strings can be reconciled with the miraculous experimental success of the point-like Quantum field theories in explaining the observed features of the world at lower Energies. For the strings possess a tension that varies with the energy of the environment, so that at low energies, the tension is high and pulls the strings taut into points and we recover the favorable features of a world of Point like Elementary particles. At high energies where the string tension is low, their essential stringiness becomes evident and creates Behavior that is qualitatively different from that of the point particle theories. Unfortunately at present the mathematical expertise required to reveal these properties is somewhat Beyond us. For the first time modern physicists have found that off-the-shelf mathematics is insufficient to extract the physical content of their theories. But, in time, suitable techniques will no doubt emerge, or perhaps a better way to look at the theory will be found: one that is conceptually and technically simpler.

The presence of the particles of matter , and their motion, determine the local topography of the space in which they sit.

String theory promises to take a further step beyond that taken by Einstein's picture of force subsumed within curved space and time geometry. Indeed, string theory contains Einstein's theory of gravitation within itself. Loops of string behave like the exchange particles of the gravitational forces, or 'gravitons' as they are called in the point-particle picture of things. But it has been argued that it must be possible to extract even the geometry of space and time from the characteristics of the strings and their topological properties. At present, it is not known how to do this and we merely content ourselves with understanding how strings behave when they sit in a background universe of space and time.

One of the most striking properties of the visible universe is the preponderance of matter over antimatter. Although particle accelerators produce matter and antimatter in equal abundances quite routinely and there is a democratic relationship between the two, we see no antiplanets, no antistars, no antigalaxies, and there is no evidence of any antimatter in the cosmic rays that come from outside our solar system. Nor do we see any evidence of the wholesale annihilation of matter and antimatter which would erupt anywhere in the Universe where the two came into contact. Thus, for some mysterious reason, there exists a form of cosmic favouritism. The observable universe is made of matter rather than antimatter. The other thing that it most obviously consists of is radiation. Indeed, on a straight count the photons have it; for there are on the average about two billion photons of light to be found for every proton in the Universe. Since every time a proton meets an antiproton and annihilates, two photons of light are produced, we can see that a universe such as ours, possessing about two billion photons for every proton, needs to have arisen from a hot dense state in which there were on average a billion and one protons for every billion antiprotons. A billion antiprotons knock out a billion protons producing two billion photons for every left-over proton. But why should the early Universe possess such a weird skewness of matter over antimatter to start off with?

Our brains are the most complicated objects that we have so far encountered in the Universe. We are far from simple. Indeed, were our brains significantly simpler, we would be too simple to know it.

Life as we know, and partially understand it, is a classical example of what can occur when a sufficient level of complexity is attained. Consciousness appears to be a manifestation of an even more elaborate level of organization.

Complexity is a delicate business. Chemical and molecular bonds require a particular range of temperature in which to operate. Liquid water exists over a mere one hundred degree range on the centigrade scale. Even Earth-based life is concentrated towards particular climatic zones. The temperature at the Earth's surface keeps it tantalizingly balanced between recurrent ice ages and the roasting that results from a runaway greenhouse effect. Very slight differences in the size of our planet or its distance from the Sun would have tipped the scales irretrievably towards one or other of these fates. That such a delicate balance, which is essentially the outcome of those random symmetry-breakings that we discussed in Chapter 6, should be so crucial suggests that natural complexity may be a rather rare thing in the Universe.

Humans are distinguished further by the highly effective way in which they have pooled the individual intelligence of single individuals to produce a collective intelligence that greatly outweighs the capability of any single individual.

Today, a science fiction writer looking for a futuristic tale of silicon dominance would not pick upon the chemistry of silicon so much as the physics of silicon for his prognostications. But this form of silicon life could not have evolved spontaneously : it requires a carbon-based life-form to act as a catalyst. We are that catalyst.

A future world of computer circuits, getting smaller and smaller, yet faster and faster, is a plausible future "life- form" more technically competent than our own. The smaller a circuit can be made, the smaller are the regions over which voltages appear, and hence the smaller these voltages can be. Tiny layers of material just a few atoms thick allow the electronic properties of a material to be finely tuned and rendered far more effective. The first transistors were made of germanium but were far from reliable and failed at high temperatures. When high-quality silicon crystals could be grown they were used in a generation of faster and more reliable silicon transistors and integrated circuitry. Newer materials like gallium arsenide allow electrons to travel through them even faster than through silicon and has given rise to the line of cray supercomputers. The evolution of computer power is represented in figure 7.3. Undoubtedly other materials will eventually take over. The story may even come full circle back to carbon again. Pure carbon in the form of diamond is about the best conductor of heat, a property that is a premium in a densely packed array of circuits.

The deep significance of this discovery appears to be that we have found a physical situation where two different natural principles, of quantum mechanics and general relativity, come together, which admits of a simple thermodynamic description. We expected all the rules governing how things behave in such a quantum gravitational situation to be complicated and novel. Many undoubtedly are; yet we find that the tried and tested principles of thermodynamics encompass them within their dominion. Besides giving physicists confidence that they might be able to elucidate still more complicated problems of basic science by appeal to simple thermodynamic principles, this case history bolsters our faith in thermodynamics as a paradigm for a 'law' governing the organization of complex systems.

At first, one might think that something like thermodynamics is a rather restrictive concept because it concerns itself with temperature and heat. But its application is not just restricted to all things thermal. It is possible to relate the notion of entropy, which is a measure of disorder, to the more general and fruitful notions of 'information', of which we have already made use in discussing the richness of certain systems of axioms and rules of reasoning. We can think of the entropy of a large object like a black hole as being equal to the number of different ways in which its most elementary constituents can be rearranged in order to give the same large-scale state. This tells us the number of binary digits ('bits') that are needed to specify in every detail the internal configuration of the constituents out of which the black hole is composed. Moreover, we can also appreciate that, when a black hole horizon forms, a certain amount of information is forever lost to the outside observer when a horizon forms around a region of the universe to create a black hole.

Particles like protons or electrons occupy microscopic niches into which only one particle is allowed to sit. Any attempt to compress matter so that more than one particle would be squeezed into each niche is met by a resisting force. The balance between this force and the inward push of gravity results in the large, stable, cold bodies we see in the solar system.

A star is a body that is massive enough for the gravitational comptession at its centre to produce a temperature great enough for nuclear reactions to occur spontaneously. When this ignition temperature is achieved, nuclear reactions in the central region will produce an energy outflow that is eventually radiated away from its surface in the form of heat and light. The star is kept in equilibrium by the balance between its internal pressure and gravity. This is a stable balance because if the gravitational force were made slightly greater then the star's centre would be squeezed a little more, so producing faster nuclear reactions and hence a corresponding extra pressure pushing outwards. Thus a balance is quickly restored.