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The conception of 'nothing' has played a crucial role in the development of logic, mathematics, physics, philosophy and theology. This lecture looked at how these currents have led to the modern picture of the 'vacuum' in quantum physics and how its non-uniqueness and changeability lie at the root of our modern picture of the universe.

Professor John D Barrow

Let’s start with the mathematical side of things. If you were to write down 3 strokes – 111 – and you were to talk to a Roman centurion, they would imagine that this was the number 3, but if you talk to somebody here, 111 means a hundred and eleven. But if you talk to the wrong sort of computer scientist, who works in binary, 111 means something different altogether, so it means one, plus two to the one, plus two squared times one. So if you use a base two arithmetic, 111 means something different to what it means if you use base 10. This idea of the position of a numeral having a meaning was something that was introduced originally by the Sumerians and the Babylonians. It’s a rather sophisticated idea. It wasn’t used by the Egyptians, it wasn’t used by the Ancient Greeks, it wasn’t used by the Romans. But once you start to record numbers, like the Babylonians did, you have a stylus of a particular shape; you can represent it vertically, horizontally, and the position of the marks you make carries some meaning. The Babylonians used a system which combined a base 10 with a base 60, so instead of having hundreds, they had multiples of 60 times 60; instead of having tens, they would have multiples of 60, and then they would have the rest.

For a long period of time, they simply left a space on the tablet, but that could create an ambiguity. It wasn’t too clear what the number was. So inclined marks of the stylus were used to indicate an empty slot in the register, what we would call a zero. All this may sound rather remote, but on the other hand, what we’ve derived from the Babylonian system is our measure of keeping time – 60 seconds to the minute, 60 minutes to the hour – and of angular measure – degrees, minutes and seconds of arc. Their way of representing numbers was almost like us thinking in terms of degrees, minutes and seconds of arc.

The Babylonians weren’t the only people to think of zero. The other culture that developed a system that needed to use a zero, a great culture, but very mysterious, was the Mayans. I had the great privilege to go the Mayan Riviera, as you might call it, into Mexico, in November for a couple of weeks, and was able to see some of the Mayan glyphs and old cities there – an extraordinary culture that developed mathematics to a relatively sophisticated level, but never invented the wheel. What the Mayans did was to produce images of numbers, in rows. In the first row, you would keep track of what we would call units, and then they had a base 20 system rather than 10, so they counted the number of digits feet and their fingers originally. So the number 400 would be denoted by the dot, two dots, and a strange shell-like symbol, which represented zero. At the top you would see lots of pictures from different glyphs. What they would then do was to add a number alongside a rather exotic picture of a creature, which was representing the number being shown. What’s curious – they have a place value system in a sense, so they sat in particular positions, so when they created the artistic work on the glyph, if they didn’t have a symbol for zero, they had a gap in the ornamentation. And so zero was in effect invented for aesthetic reasons to complete the picture, as it were.

Babylonians finally got round to introducing their explicit zero around 500BC. The Mayans, as you know, just before William the Conqueror came along, were dying out. Nobody seems to quite know why they did; they weren’t necessarily conquered completely by the Spaniards in any way.

The other great culture that developed the zero in the form that we know it is the Indian tradition. The early Indians developed mathematics and geometry in a sophisticated way using very nice notation. They used the decimal system, they used the same numerals that we use today, and this entire way of doing mathematics came to Europe, and ultimately to us, by way of the Arab cultures in the early Middle Ages. What’s interesting about the Indian conception is it does something that really the Greeks were unable to do. The Greeks, because of their philosophical views about nothing, never had a zero symbol, and nor did the Romans. All these logical problems about whether it was an illegal move to regard nothing as something made them very reluctant to introduce the idea of zero as a formal element in logical argument, because if you introduce one false element, then everything collapses and you can prove any statement. The Indians of course, because of their philosophical views, were much more mystical, much more conversant with the concept of nothingness and non-being, very comfortable with these ideas, and so what you find in Indian culture is not just a zero symbol, but the whole panoply of interconnections of the words for zero with nothingness and the void and the vacuum that we’re familiar with. For the Babylonians, a nation of accountants and astronomers, the zero symbol was just a gap in the register, but for the Indians had basic words which were used for the void, or for the zero, or for that red dot on the woman’s forehead. In Indian poetry, you’ll see women’s beauty is extolled in the same language, rather like mathematics of generating dots and zeros. From this, you generate all the concepts that we’re now familiar with about non-existence and the void and things being worthless and having no value, going all the way down to the more abstract notions about emptiness, nothing, and the mathematical ideas that if you multiply something by zero, you get zero, and even if you divide by zero, you get infinity. All these ideas were present in early Indian arithmetic and mathematics.

We’re familiar today with the words zero and cipher. In slightly older English, there’s a usage of the word cipher rather different to how we use it today. We tend to use it as meaning a code of some sort, but a cipher was just a nonentity, so if you said that a gentleman was a cipher in his own household, you meant that he was a zero presence. So these two terms have come to have slightly different meanings, and we still see them in language today. The interesting message is the way that the philosophical climate in India was conducive to the development of the mathematical idea of zero.

Shakespeare really did it better than anybody, and if you look through his plays, you’ll find constant word play and use of this idea of nothing. If you look through Lear and other plays, you’ll find that he’s constantly using this idea of nothing being something.

The other ingredient of it, which had a sort of theological element, was that there was something that we might now in the press regard as the moral vacuum. So the world of nothingness and the vacuum, was something without god, it was something where there were no morals, where humanity wouldn’t go. This whole period of thinking that you could learn something about nothingness and the vacuum and the void just by playing games with linguistic paradoxes really went on right up until Galileo came along. Galileo did away with this whole idea of thinking that just by studying what people said long ago, by creating linguistic paradoxes, you might learn something about the vacuum, and he replaced it by real experiments.

While all sorts of the arts are fascinated by nothingness and the void, there are countless works of art around the world that look like blank canvases. There’s something terribly unoriginal about these sort of works; for some reason they all have a rectangular frame – you never see pictures that have an elliptical or asymmetrically shaped frame. And the musician John Cage, with his 4 minutes and 33 seconds of silence, that’s 273 seconds of silence, the absolute zero of sound. And the other matter of silence in music, which is more serious, experts can talk to us at great length, is the whole matter of timing in music, the silences between the notes, which are absolutely crucial for determining the structure and the effect.

I want to complete what one might say about mathematics by telling you about something that developed after the ordinary idea of zero and counting. By the time we got to the 19 th Century, mathematicians realised that there wasn’t one mathematics – counting, geometry and so on – and these were the thoughts of god and this was the way the world worked, as even Galileo believed, but mathematics was made of all sorts of mathematical systems, collections of axioms. There were different geometries – non-Euclidian ones, Euclidian ones - different arithmetics, lots of different mathematical systems. You specify what the objects are – whether they’re lines or points or symbols - and you specify the rules of the game, rather like chess or drafts, and all the possible games of chess are like all the truths and theorems that can be proved using that system. What this means is that each system has its own zero, and although the word might be the same for each system, the idea is completely different. So if your zero is something that you do to the objects in your system which doesn’t change them, then if you have a collection of symbols and just the operation of addition, then you have what we denote by our zero symbol, but if you had a system where only multiplication is defined, then the symbol that doesn’t change your A, which is what we normally call one. Every system of logic and mathematics has an operation, if you wish, that corresponds to zero, but they’re different, so there is an infinite number of zeros or versions of that concept.

The nearest to the original idea of zero in mathematics is something that’s known to mathematicians as the empty set. A set’s just a collection of things, a collection of cars in your garage, in the toy box, and the empty set is the set that doesn’t have any members. This is quite different from the numerical zero, which is defined as part of arithmetic. So this is the set that has no members, and out of that set, you can define all the numbers of arithmetic. Out of nothing, you can generate everything. How do you do that? Well, you define what we normally call zero just to be the empty set, and you’ll define by the number one the set which has got a single member, and you’ll define two to be the set that contains the object zero, and one as members. So the number two is defined to be the empty set and the set which contains the empty set, and so on with three. The number three is defined to be the set that has these members; the empty set, the set containing the empty set, and the set containing the empty set, and the set that contains the empty set, and so on forever. So you can define all the numbers just in terms of the empty set.

The other great paradigm that began really with Lucretius was the idea of just having two plates, maybe of metal, later of glass. You put them very close together, and then you ask when you separate them, does a vacuum form instantaneously or not? Why is that interesting? Well, ancient philosophers had all sorts of great ideas about the vacuum on which an awful lot rested because they were pivotal ideas in great philosophical systems. For Plato, the whole idea of a vacuum, of having nothing, was inconceivable, because for Plato, what we saw around us were just expressions of the abstract forms in some other world of ideas, so even if there was nothing here, there still had to be a form corresponding to that, so there is no possibility of there being nothing at all. Aristotle believed it was impossible to create a physical vacuum, so it was not possible to make a vacuum in the world. And this became bound up with the whole idea that it was not possible for nothing to be something, for nothing to be part of the whole story and argumentation about the world. What Aristotle and his followers for thousands of years would have argued is that you cannot create nor have what they called intra-cosmic void. You can’t make a void, a complete vacuum, anywhere in the world, or in the visible universe, but it might be possible to have what they called extra-cosmic void. So if you imagine that the world is some great finite ball, then beyond it, there might be a void, a complete vacuum. But that’s another story.

All these little experiments with the water catcher and the two strips all revolved around the question of whether it was really possible to make an intra-cosmic void. Some people were arguing that if you put two slabs together perfectly, then there’s nothing between. When you pull them apart, air rushes in, but there must be a fleeting moment before the air gets in when there’s a vacuum. And if you look at the whole medieval argument about this, it’s really quite sophisticated and very interesting. The next stage, people say, no, you can never have perfectly smooth surfaces, so actually there isn’t a perfect vacuum to start with, and so you don’t have to wait for the air to rush in. And then Blasius of Parma pointed out, well, it’s worse than that; if they were perfectly smooth, you wouldn’t be able to separate them. And then other people pointed out, you don’t need two slabs at all; just take one slab, drop it on the floor, and at the moment when it’s in contact before it bounces, it’s in perfect contact, and when it bounces off, that’s the same process of a vacuum being replaced after a finite time by air rushing in. Other people said, the air rushes in at exactly the right rate never to allow the vacuum to form, because there’s a sort of celestial agent, as William Burley called it, rather like Roger Penrose’s cosmic sensor that wants to stop naked singularities forming in the universe – exactly the same sort of philosophy. The celestial agent stops a vacuum ever forming in the world. And so there was a great argument then about whether the laws of nature really could have negative aspects, whether there could be a law of nature that says no vacuum ever forms. The trouble with that was, when you looked at your water catcher, the celestial agent could stop a vacuum forming in all sorts of ways, in that when you took your finger off, it could fill it up with water, but it could also have made the metal collapse and squash everything out. Why did it choose to do one thing rather than another? The celestial agent didn’t really tell you enough.

All these arguments about whether it was possible to make a vacuum or not went on pretty much unabated until the famous event in 1277, the Paris condemnations, Albert the Great, when there were theological statements for the first time trying to place no restrictions on the actions of God. It wasn’t possible for you to say, for example, that there couldn’t be infinities in the world because this was somehow a challenge to the operation of God’s actions. So this whole story about the physical vacuum had an interesting experimental, philosophical background.

This was all changed in the early 1600s by Galileo and Torrichelly, his student. As he wandered the country, Galileo had noticed rather interesting things when farmers were pumping water in their fields. He noticed that nobody could pump water higher than a particular height, about 10.5 metres in our units, and he obviously had a strong idea why this was so. He talked to his student and secretary, Torrichelly, who became a great scientist and mathematician himself, and Torrichelly picked a different, much denser, material than water, the densest liquid there is, mercury, to demonstrate this effect and then understand what was going on. He created the manometer, or the barometer that we know today. You take a long tube, longer than 76 centimetres in length, you fill it up with mercury, and then you invert it in a little jug of mercury and watch what happens. The level of the mercury in the tube drops, and you get a height of 76 centimetres, what we call atmospheric pressure. This was very mysterious, philosophically, theologically. You see, when the tube started, it was completely full, there was nothing in it, and nothing’s been allowed to get in it, but when it was stood up vertically, what happened was that a vacuum appeared. It doesn’t matter what shape the tube is, the mercury will always rise to the same level, and so the issue was, what was that? Torrichelly maintained this is a perfect vacuum – I have created a vacuum here before you in the laboratory. If you’re a physicist, the way you work out the height, you work out the weight of the mercury that’s pressing down, which is its density times its volume times acceleration due to gravity, and that’s equal to the force exerted by the pressure of air, which is just the pressure times the area over which the pressure is acting. You notice the areas cancel out. It doesn’t matter what the cross-sectional shape of your tube is. The height of the mercury is given by air pressure, divided by the density of mercury, and the acceleration due to gravity. As Galileo and Torrichelly appreciated, if you did this experiment in different places, at different heights above the Earth’s surface, you get a slightly different rise. But this created a whole sea change in attitude, so you see what I meant by saying Galileo did away with the old way of thinking about nothing and vacuum and the void. Suddenly, it’s an experimental issue, and you don’t discuss the vacuum by linguistic paradoxes, but you make a vacuum and you try to do things to it.

In this country, Robert Boyle, not very far away, carried out all sorts of exotic experiments – putting canaries in tanks, and evacuating the air and watching the canary become unconscious, then putting the air back in again and it comes back to life; seeing if magnetic fields were propagated, as it were, across a vacuum, and noting that chemical reactions came to a halt if you took all the air out of a chamber. So there began an early experimental phase of investigating the vacuum.

The most spectacular experiment of all was carried out by Otto von Guericke, who was the Mayor of Magdeburg. They don’t do experiments like this anymore. It might be a great Gresham experiment on the day of the Lord Mayor’s processionI What what Guericke did was to have two enormous bronze hemispheres made, which he joined together, and then pumped out the air, and then of course they were locked together, they couldn’t be separated. He then got two teams of eight horses attached to each side of the hemispheres, and had the horses revved up, pull in opposite directions, to try and separate the hemispheres, and they couldn’t do it. Then he just went along and turned the switch and let the air in, and the hemispheres just fell apart. You can still see the hemispheres in the city museum in Munich today. What was remarkable about this experiment, great showman that he was, he wanted to demonstrate that the vacuum really was something, that it wasn’t just a vague idea. Worse than that, Aristotle and early Judeo-Christian philosophy told you that the vacuum was something that always tried to go away; if you made a vacuum it was always filled, so it was unstable. But this vacuum didn’t want you to destroy it, so if you created a vacuum inside the hemispheres, pull as you might, you couldn’t separate them and destroy it.

Pascal was another to really complete the story. He played the game of taking Torrichelly’s type of barometers up to the top of Notre Dame Cathedral and up into the French countryside, to the highest mountains that he could find, and of course what he discovered was that the height of mercury that was raised at the top of the cathedral, on the mountains, was different, because air pressure varies with altitude, and even the acceleration due to gravity.

This is an era of demonstration that the vacuum is not just a vague idea, that it’s part of science, you can manipulate it. In many ways, it was the mathematicians who, by introducing the idea of zero into Europe, along with eventually the Indo-Arab system of mathematics, really smoothed the way for the physicists and other scientists and engineers to use the idea of the vacuum without huge amounts of persecution and oppression; so the zero in mathematics was relatively uncontroversial. It eventually assumed a significant role in accountancy and mathematical science in Europe, and this gave credence and acceptance to the idea of nothing as being something.

Moving to the 19 th century, following Newton, and at the time of Maxwell and others, in physics and astronomy, there grew up a belief that the whole of the universe was filled with a mysterious fluid, the so-called ether, and you should imagine us just sitting inside a great celestial fluid of ether. Physicists took this very seriously. As the Earth moved in orbit around the Sun, it was moving through this ether. The challenge was could you identify the ether? Could you measure its existence?

A famous experiment took place in 1881 by two Americans, Michelson and Morley, and what they did was to test whether there exists such an ether by examining what happens to the motion of light in the universe. The idea, to begin with, is rather like going swimming in a river, so you think of the ether as flowing past us, rather like a great stream. Suppose you swim across to the other bank; suppose this is a hundred metres, and you swim a hundred metres against the flow and a hundred metres back, and you time yourself on these two swims. Well, if you do everything exactly the same each time, there’ll be a time difference in how long it takes you to swim 200 metres by the route where you never go against the flow, and where you have to against the flow and then with the flow. You would be able to detect whether there was a flow of the stream by comparing the round swim times. If there was no current, you should have exactly the same time to do the two trips. What Michelson and Morley did was to set up an experiment which timed the travel of light on two different paths at right angles to each other. This requires rather fancy technology and very, very high precision measurement using what’s now become known as interferometer. You can shine a beam which gets split, you can send light up and back, and you can allow light to go through and come back. You have the distances equal, and if the light returning is not coming back at the same moment and is slightly out of phase, you get interference fringes, which you can measure with fantastic precision. This great experiment which they performed, probably the most famous null experiment in modern physics, discerned no time travel difference for the light in the two paths, and this, as Einstein predicted, is what you should see if there is no ether. If there had been ether, you would find a time travel difference. This experiment pretty much did away with the idea that there was this extraordinary ethereal fluid that we were moving through. Einstein then developed the general theory of relativity which contains the idea that there can be so-called vacuum universes, universes which just contain waves of gravity; they don’t contain any material at all.

After that, the vacuum in physics changed its character quite extraordinarily. Maxwell was the first person to make a rather precise and clever statement about what is meant by the vacuum in physics. So if you have a region or a container, then he defined the vacuum to be what is left after everything that can be removed is removed. What is left after everything that can be removed is removed is not necessarily anything at all in our ordinary understanding of the word. What the idea of the vacuum has been become in modern physics is that it’s just the lowest energy state that a system can be in. Quantum mechanics doesn’t allow us to say what the position in motion is with perfect accuracy of every particle and every state of matter in the world, and so it is not possible in quantum mechanics to say that a region contains nothing at all. There must always be some residual zero point energy, just like there’s always something sitting at the bottom of your bag that somehow you can never get rid of.

But this idea of defining the vacuum simply to be the lowest energy state of a system has all sorts of rather unusual consequences. If you think of the state of some physical situation being measured by the level of its energy as you change some parameter of its configuration, then the stable situations are where the energy forms are minimum locally. If you disturb it a little bit, it’s like a marble at the bottom of a goldfish bowl, it’ll wobble a bit and come back to rest, but if you try to put it at the peaks of one of the hills, and you nudge it a little bit, it’ll roll one side or the other. There can exist many vacuum states, so there can be many vacuum states of a system. They can all have the same energy or some can be of lower energy than others. In the second case, they’re all so-called local vacuums. It’s possible to do things to physical systems that make them change from one vacuum state to another, so if you hit a system very hard, it could go over the top and come to rest. Similarly, you might go from one quite high energy minima to one that’s much lower, and when you do that, there’s some energy which can be given out. It’s situations like that that are much studied in physics. The inflationary universe theory, which I’ve talked about on several occasions, our picture of what happens very close to the beginning of the universe, is based on just such an idea, that as the universe cools, first of all, the elementary particles are in a landscape, as it were, where there’s a single valley and all the matter sits there, but as the world cools a little more, other minima open up and turn out to have much lower energy values than the original one. So what happens is that all the matter after it gets disturbed a bit and can shift and roll down into the new minimum, giving out lots of energy in the process, and that can inflate and produce a surge in the expansion of the universe.

One of the most exciting things about these different minima in particle physics, is that there are theories of high energy physics, which may well describe what happens near the beginning of our universe, where there are many different minima with different levels. As a result of that inflationary process, our universe maybe goes from an unstable maximum down to a minimum, but in different parts of the universe, matter may have rolled in the opposite direction and may end up in a different minimum. What does different minimum mean? The minima defined the values of the constants of nature, whether things like quarks have mass or whether they’re massless like the photon, and also how many different forces of nature there are. So having a universe where there is a non-uniqueness of the vacuum state of the universe, the lowest energy state that it could find itself rolled into, is tantamount to the idea that there can be different values of the constants of nature; different logically consistent scenarios in different places in the universe.

One of the most spectacular experiments I think that’s been done in the last 50 or so years is something which is called the Casimir effect, and it relates to this idea of the quantum vacuum. As Maxwell taught us, the vacuum is not nothing at all, in fact it’s seething with energy and particles coming and going. Hendrik Casimir, a Dutch physicist, in 1948, proposed a beautiful experiment, a beautiful effect, whereby the vacuum would reveal itself. The idea is very simple. Imagine that the vacuum is just a sea of waves of energy of all possible wavelengths, large and small, and then you introduce into that vacuum two parallel metal plates, and you think what happens to all the waves. Outside the plates, there are still waves of every possible wavelength, but in between the plates, only waves whose wavelengths exactly fit between the plates will be present. So you can appreciate the first order of what this means is that there’s more waves outside the plates than there are in between, and so this means there’s more pressure pushing the plates together than pushing them apart, and so the plates should attract one another by this so-called Casimir force. It’s really tiny. If the plates are just a thousandth of a millimetre apart, then the force is about a thousandth of a gram, something like that, so it’s like a fly’s wing sitting on your finger, but easily measurable in physics, as long as you can remove all the other sources of noise and interference and stickiness that might make the plates come together. This experiment has been done and the effect has been measured and it’s seen, so the plates do indeed attract, just as Casimir predicted. The formula depends on planks, constant, the speed of light and the separation “D” of the plates, so it varies inversely as the fourth power of the separation, so you can change the separation, make sure that the force there is like that.

I discovered a few years ago a wonderful nautical version of this force. Back in the 1820s, the French Navy produced a handbook of advice to their sailors, and one of the pieces of advice to captains is rather extraordinary. There’s no reason for the advice, it’s just based on experience, and the experience was that if you had two large ships and you were sailing in a strong swell but there were not high winds, then if the boats were to come quite close together, 30 or 40 yards close to one another, then the captain should set down small boats full of 20 or 30 men, and they whould attach lines and pull the boats apart, because experience showed that once the boats which were gently swaying together got closer than about 30 or 40 yards, they would be pulled together and their riggings would become entwined and there would be disaster. If you calculate this, you find that it’s exactly the same effect in the large, that you get an out of phase generation of waves between the boats, so the crests of one wave match the troughs of the other, so it’s in effect there is no wave motion between the two boats, but the waves coming in and rebounding off the outside push the boats together. Mathematically, it’s exactly the same type of effect, but not with quantum waves, just with classical waves. In fact, a little mathematics shows you, if you have two ships that weigh about 700 tons or so, then a couple of small boats with 20 or 30 men will be able to overcome the force that pulls them together, so it’s a nautical Casimir effect.

In particle physics, we’re familiar with the role of the vacuum in very subtle and sophisticated ways. If you take an electron and you put it down in a vacuum, that vacuum is full of particles of negative and positive electric charge, appearing and disappearing all the time on immeasurably short time intervals, once you put the electron down, the oppositely charged, positively charged virtual electrons in the vacuum all get slightly attracted to this electron and surround it with a shield of positive charge. So if you fire in another electron, it doesn’t interact and scatter as strongly as you might have thought. If it comes in with a rather low energy, it doesn’t get a look at the negative charge at the centre at all; it just sees the positive shield, and deflects rather weakly. If you throw it in at high energy, then it goes right through the penetrating shield and gets a look at the full negative charge in the middle and deflects strongly. This can all be measured, checked, rather precisely. It’s like having two hard billiard balls, and you surround one with a woolly cloth cover. If you throw the other ball in rather gently, it will tend just to interact with the cloth cover, but if you throw it in rather hard, then it will really bang into the ball and it will rebound strongly. This is known as asymptotic freedom: as the energy becomes greater, the electromagnetic interaction becomes stronger. The Nobel Prize was given in physics this year for that discovery, to Wilczek, Gross and Politzer.

In modern cosmology, nobody knows why the world was created out of nothing, if it was, but there are rather precise ways to make clear what you mean by the beginning of the universe, or the origin of the universe, the notion of a singularity in space in time, an edge to space and time. You have no description of what’s beyond the edge, but you have a very precise way of characterising the edge. Current astronomical observations can’t really tell us whether the world really did have an infinite past or just a finite one. The favourite picture of the universe, which once was just rather like an expanding bubble of some sort, you should really imagine as being replaced by not a single bubble, but it’s like an expanding foam of bubbles, possibly an infinite foam, and each of the bubbles in the foam behaves perhaps rather differently, and eventually gives rise to other bubbles in the foam, and we are just living in one of the bubbles in the foam, and each bubble may have just a finite age. It may have had a finite past and only live for a finite time in the future, but the whole process, the whole foam, may have gone on forever for the future; it may have gone on forever in the past.

How do these ideas of mathematics, philosophy, physics really come together? I think there’s an important way in which they don’t, and thinking that they do can lead you to some type of confusion. Mathematical existence is a rather weak thing. All it means when you read a mathematician saying that they’ve proved an existence theorem or that a triangle of this sort exists or a counter-example to some conjecture exists, all they mean is that that thing is not logically inconsistent, so existence in mathematics just means logical self-consistency, so four-sided triangles do not exist, given the definition of triangles four and sided, but even numbers bigger than two certainly do exist. But this is quite different to physical existence. So there is nothing illogical about unicorns, and you could write down a mathematical equation which described their genetic make-up, but just because you can describe them, doesn’t mean that they must exist. It’s important not to confuse physical existence with mathematical existence. Mathematical non-existence means that things are logically inconsistent, and that would be certainly enough to guarantee physical non-existence.

These ideas about the non-uniqueness of the vacuum lead to rather worrying considerations by cosmologists. If the world is like a series of floors, and we’re at the basement energy level, we like to think that the universe has done all the dropping downstairs that it is ever going to do, but that may not be the case. We may not yet be really in the basement energy level in the universe’s true vacuum, and there may be another transition still to come, and the fact that the universe began accelerating just a few billion years again may indicate that in some sense we may be on the way to another type of vacuum state over a timescale of many billions of years. After all, if the universe did appear out of nothing and the laws of physics are time symmetric, as we know, it may be that one day it’ll just pop back into nothing from whence it came.

© John D Barrow, Gresham College, 1 February 2005

Professor John D Barrow FRS

Professor of Astronomy

Professor John D Barrow FRS has been a Professor of Mathematical Sciences at the University of Cambridge since 1999, carrying out research in mathematical physics, with special interest in cosmology, gravitation, particle physics and associated applied mathematics.

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