The Book of Nothing: Vacuums, Voids, and the Latest Ideas about the Origins of the Universe - John D. Barrow (2002)

Chapter 2. Much Ado About Nothing

“Among the great things which are found among us the existence of Nothing is the greatest.”

Leonardo da Vinci1

“Nothing really matters.”

Queen

WELCOME TO THE HOTEL INFINITY

“… the library contains … Everything: the minutely detailed history of the future, the archangels’ autobiographies, the faithful catalogue of the Library, thousands and thousands of false catalogues, the demonstration of the fallacy of the true catalogue, the Gnostic gospel of Basilides, the commentary on that gospel, the commentary on the commentary on that gospel, the true story of your death, the translation of every book in all languages, the interpolations of every book in all books.”

Umberto Eco2

“Nothin’ ain’t worth nothin’, but it’s free.”

Kris Kristofferson & Fred Foster3

The development of European thinking about the puzzles created by Nothing is a story about grasping two horns of a dilemma. Five hundred years ago, if you were a philosopher you might have had to get a grip on the slippery abstract concept of Nothing and persuade your peers that Nothing could be something after all – not least, something worth studying. But if you were a practising scientist, a ‘natural philosopher’, you faced the deeper paradox of whether there could exist a physical Nothing: a perfect vacuum of empty space. Worst of all, both of them risked serious disapproval from the religious status quo for letting their thoughts stray into such potentially heretical territory. Nothing was an ultimate issue, what nowadays we might call a ‘meaning-of-life question’: a question whose answer has the potential to unsettle the foundations of entire edifices of thought, carefully arranged to withstand the perturbations of new ideas. Any theology that had doctrines about the beginning of the world, and from whence the world had sprung, had to have a view about Nothing. Nor is any answer quite as simple as it seems. Say ‘Nothing at all’ to the question of what was before the beginning of the world and trouble could be in store.

It does not immediately occur to us that Nothing might be an impossible state. But there was a time when it was hard for many to think otherwise. Plato’s influential philosophy taught that the things that are seen around us are just imperfect manifestations of a collection of perfect ideal forms – blueprints from which all material things take their character. These forms are eternal, indestructible and invariant. Remove every material thing in the physical universe and the Platonist would still hold that these eternal forms exist. They are the ‘mind of God’ in modern parlance.4 If we were to assume that Nothingness is one of these forms then it is impossible to conceive of an imperfect manifestation of it that would still merit the title Nothingness. A vacuum that contains a single thing is no sort of vacuum at all.

The problems facing anyone thinking about Nothing are not unlike those that face anyone contemplating what we call ‘infinity’. They are problems because we stand firmly and finitely between the two extremes marked by zero and infinity. At first they appear intimately linked. Divide any number by zero and we get infinity. Divide any number by infinity and we get zero. But just like the ski resort full of girl-chasing husbands and husbandchasing girls, the situation is not as symmetrical as it might first appear. For a mathematician, the idea of zero is straightforward and uncontroversial: we see concrete examples of it when the quantity of any commodity is exactly exhausted. It obeys simple rules of addition and multiplication.5 But infinity is quite another matter. Some currents of mathematical opinion have, in the past, argued that mathematics should only be allowed to deal with finite collections of things that can be enumerated in a step-by-step fashion. The more conventional view is that formal infinities are all right in mathematics but you must be very careful how you handle them. They do not obey the usual laws of arithmetic for finite quantities. Take an infinity away from an infinity and you can still be left with infinity: for example, the list of all whole numbers (1, 2, 3, 4, 5, …) contains an infinite number of odd numbers (1, 3, 5, 7, …) and an infinite number of even numbers6 (2, 4, 6, 8, …). Take the infinite number of odd numbers away from the infinity of all numbers and you are left with an infinity of even numbers!

The problem of infinity is beautifully captured by the story of Hilbert’s Hotel.7 In a conventional hotel there are a finite number of guest rooms. If they are all taken then there is no way you can be accommodated at the hotel without evicting one of the existing guests from their room. But with an infinite hotel things are different. Suppose that one person turns up at the check-in counter of the Hotel Infinity with its infinite number of rooms (numbered 1, 2, 3, 4, … and so on, for ever), all of which are occupied. No problem: the manager asks the guest in room 1 to move to room 2, the guest in room 2 to move to room 3, and so on, for ever. This leaves room 1 vacant for you to take and everyone still has a room.

You are so pleased with this service that you return to the Hotel Infinity on the next occasion that you are in town, this time with an infinite number of friends. Again, this popular hotel is full. But again, the manager is unperturbed. He moves the guest in room 1 to room 2, the guest in room 2 to room 4, the guest in room 3 to room 6, and so on, for ever. This leaves all the odd-numbered rooms empty. There are an infinite number of them free to accommodate you and your infinitely numerous companions without difficulty. Needless to say room service was a little slow.

The contrast between zero and infinity is most marked when it comes to the physical realisation of these ‘numbers’. Zeros are no problem – there are no wheels on my wagon – but no one knows whether infinities are physically manifested. Most scientists believe that they are not: their appearance in a calculation merely signals that the theory being employed has reached the limits of its validity and must be superseded by a new and improved version which should replace the mathematical infinity by a finite measurable quantity. In controllable situations, like the flow of a fluid, we can observe the physical situation in which the spurious infinity was predicted to occur, see that no physical infinity arises, and so be certain that more accurate mathematical modelling of the situation will exorcise the predicted infinity. However, there are more exotic situations, like that of the apparent beginning to the expansion of the Universe, where we can assure ourselves by observation that everything is physically finite. The situation being considered there is so singular in many respects that it is not clear why a physical infinity could not be present. Nevertheless, a large part of cosmologists’ studies of this situation is directed towards trying to find a superior theory in which any beginning to the Universe is not accompanied by physical infinities.

Another contrast between zero and infinity is the psychological effect that each produces on human minds. In modern times there is little fear of zero – except when it appears too often in your bank balance – but many find the concept of the infinite to be awesome, mind-boggling, even terrifying, echoing Blaise Pascal’s famous confession that ‘The silence of infinite space terrifies me’. Nor are such sentiments confined to the seventeenth century. The famous Jewish philosopher Martin Buber, who died in 1965, wrote of how the mere thought of the infinite led him to contemplate suicide:

“A necessity I could not imagine swept over me: I had to try again and again to imagine the edge of space, or its edgelessness, time with a beginning and an end or time without a beginning or end, and both were equally impossible, equally hopeless … Under an irresistible compulsion I reeled from one to the other, at times so closely threatened with the danger of madness that I seriously thought of avoiding it by suicide.”8

Existentialist philosophers have struggled to extract some sense from the contrast between Being and non-Being from a vantage point that sees all existence as deriving from human existence. The most well-known work of this sort is Jean-Paul Sartre’s book Being and Nothingness, which contains tortuous ruminations over the meaning and significance of Nothingness. Here are some typical extracts:

Nothingness haunts being. That means that being has no need of nothingness in order to be conceived and that we can examine the idea of it exhaustively without finding there the least trace of nothingness. But on the other hand, nothingness, which is not, can have only a borrowed existence, and it gets its being from being. Its nothingness of being is encountered only within the limits of being, and the total disappearance of being would not be the advent of the reign of non-being, but on the contrary the concomitant disappearance of nothingness. Non-being exists only on the surface of being.”9

Here, Sartre is contesting the idea, argued by Hegel, that Being and Nothingness are merely equal and opposite. He does not believe they can logically be contemporaries at all. Nor are they merely both ‘empty abstractions, and the one is as empty as the other’ as Hegel claimed, for the key feature that creates the asymmetry between them ‘is that emptiness is emptiness of something’.10 They are quite different.

GREEKS, BEARING GIFTS

“‘I see nobody on the road,’ said Alice.

‘I only wish I had such eyes,’ the King remarked in a fretful tone, ‘to be able to see Nobody! And at that distance too! Why, it’s as much as I can do to see real people, by this light!’”

Lewis Carroll

Ever since the early Greeks grappled with these problems the contemplation of Nothing has been bedevilled by paradoxes like those that afflict the contemplation of the infinite. Philosophers like Parmenides and Zeno marshalled these paradoxes to attack the self-consistency of the concepts of Nothing and infinity.

For Parmenides the Universe must be a unity. It is limited but fills all of space. Symmetry demands that it must be spherical in shape. A vacuum is impossible because it constitutes non-Being and contradicts the assumption that the Universe fills all space. Parmenides went so far as to protect his Universe from any intercourse with a vacuum anywhere else. He argued that things could not appear from Nothing or disappear into Nothing; he asked why such a creation from Nothing should have occurred at a particular moment and not sooner. Later supporters of the idea of creation out of Nothing, like Simplicius, answered this charge by suggesting that there might exist an orderly sequence of events, with individual forms of matter appearing one after the other. By reference to this logical sequence we can date any particular appearance.

European Christianity tried to wed together two pictures of Divine activity. One was the Greek picture of God as an architect who fashions the world out of pre-existing eternal material. The other was the Jewish tradition of God as the Creator of the World and all its properties out of Nothing. The Greek tradition held on to the belief that there was always something there originally from which the World was moulded. In this way it avoided having to wrestle with the concept of nothingness and thus with all the philosophical problems it carried with it. Greek philosophers recoiled from the concept of emptiness. The word chaos originally meant Nothing and shows us the anarchy that was attached to the very idea of regarding Nothing as something that had Being.

Philosophers like Parmenides and his disciple Zeno tried to defend their belief in the static unchanging nature of Being by a variety of ingenious arguments. Zeno’s paradoxes of motion are amongst the gems of Greek thought and they were never refuted by other Greek thinkers, merely ignored. The Greek tradition focuses upon elements that do not change: points, lines, circles, curves and angles in geometry; numbers, ratios, sums and products in arithmetic. It is nervous of dealing with the limitless, and the opposition of zero and infinity attached a label saying ‘beware’ of both. Each dangled at the crumbling edge of thought. Aristotle saw them both as loose cannons in the logical structure of cause and effect. Nothing had no cause and no effects; no reason and no end. This presented a real quandary if one wanted to fit all concepts into a single harmonious logical structure, because as Brian Rotman pungently remarks:

“For Aristotle, engaged in classifying, ordering and analysing the world into its irreducible and final categories, objects, causes and attributes, the prospect of an unclassifiable emptiness, an attributeless hole in the natural fabric of being, isolated from cause and effect and detached from what was palpable to the senses, must have presented itself as a dangerous sickness, a God-denying madness that left him with an ineradicable horror vacui.”11

Greek philosophy and psychology could find no room in their indivisible Universe of unchanging Being for the sort of gap that the reality of Nothing would require. And so it simply could not be. One could not make something of Nothing. Aristotle defined the void to be a place where no body could be. This step would have allowed him to take off in many different philosophical explorations, moving East to contemplate the notions of non-Being and nothingness so beloved of the Indian thinkers. Instead, he concluded that the void could not exist. Eternal things occupy every place. There can be no state of perfect emptiness, devoid of Being.

Despite this antipathy to Nothing one does occasionally find some of the paradoxical wordplay that was to overtake English writers in the seventeenth century. The most striking is the encounter between Ulysses and the Cyclops, Polyphemos, created by Homer in The Odyssey.12 Ulysses sets about lowering the one-eyed monster’s guard by providing him with an abundance of wine. When asked by the Cyclops for his name he replies ‘my name is Noman;13this is what my father and mother have always called me’. But the Cyclops vows to devour him, so Ulysses seizes his opportunity to blind the Cyclops with a burning stake from the fire. The Cyclops screams out to his neighbours for help: ‘Noman is killing me by fraud! Noman is killing me by force!’ No help comes, merely the replies that ‘if no man is attacking you, you must be ill; when Jove makes people ill, there is no help for it’. Ulysses and his men slip by the blinded Cyclops, disguised by the fleeces of sheep, and make good their escape, but as they sail into the distance the Cyclops curse them never to return to their homes alive.

It is strange that this ancient epic bestseller did not stimulate any other Greek philosophers to take up the paradoxes of Nothing. They were ripe for the treatment that Zeno administered to the idea of infinity in memorable scenarios like those summarized in Figure 2.1.

Greek philosophy denied the concept of Nothingness right from its outset in the fifth and sixth centuries BC. Thales and his school in Miletus maintained first that ‘something’ can never emanate from Nothing or disappear into Nothing. He used this intuition to deny the possibility that the Universe could have appeared out of Nothing, a difficult idea to grasp and one that we in the Christian West have become comfortable with only because of two millennia of religious tradition. Parmenides was the first of the Greek philosophers to take the idea of ‘non-Being’ seriously and grapple with it in order to make sense of it. Thales had focused upon the attributes of Being and simply ignored the concept of non-Being. Parmenides maintained that non-Being did not exist but his exploration of these ideas never considered the practical questions of empty space and regions devoid of matter: of actually looking for a space that might potentially be empty. That more detailed step of speculative natural philosophy was taken by the Sicilian Empedocles, who later in life was to come to a grisly end by leaping into the active volcano on Mount Etna, perhaps ultimately coming to believe his delusions of divinity.

Figure 2.1 Zeno’s paradoxes of motion.

Empedocles imagined matter to contain pores of a mysterious light medium, called ‘ether’. This quintessential part of the world was devised in order to avoid having to introduce the concept of empty space when trying to account for the granular structure of many forms of matter. In places where there was no evidence of any matter at all, Empedocles could maintain that there was always some of this ethereal substance, lighter than all known materials (except possibly air), permeating tiny pores and guarding us against the horror of a perfect vacuum ever forming. To his credit, he was not content to let the ether be simply a spoiler for the vacuum; he envisaged emanations proceeding from the pores within bodies so that they could influence one another in different ways. In some respects this intuition has a rather modern ring to it. Empedocles does not have the idea (that Newton used about two thousand years later) that forces act instantaneously between different bodies. Rather, when a magnet pulls a piece of iron towards it, the attraction takes a finite time to occur:

“Why does a magnet attract iron? Empedocles says that the iron is drawn to the magnet, because both give off emanations and because the size of the pores in the magnet corresponds to the emanations of the iron … Thus, whenever the emanations of the iron approach the pores of the magnet and fit them in shape, the iron is drawn after the emanations and is attracted.”14

This was the beginning of a belief in an ether. We shall see that it was maintained in different forms until the start of the twentieth century. Its original purpose was simply to avoid having to admit the existence of empty space in the physical universe and to reconcile the picture of physical space and matter with the philosophical conceptions of Being and the inconceivability of non-Being.

Empedocles was not just a philosopher. He made an important experimental discovery in the course of his researches into human breathing and the nature of air. He explains what he has observed about the behaviour of a perforated vessel used to catch water,15 displayed in Figure 2.2. He notices that if the water-catcher is submerged before the air has been expelled from it then the water cannot flow into it,16

“As when a girl, playing with the water-catcher of shining brass – when, having placed the mouth of the pipe on her well-shaped hand she dips the vessel into the yielding substance of silvery water, still the volume of air pressing

Figure 2.2 The ancient water-catcher experiment. Immerse the perforated vessel and then seal the tube with a finger. On removing it from the water the water remains trapped in the vessel but when the finger is removed from the tube it escapes. An explanation for this behaviour was a challenge to scientists and philosophers for more than 2000 years.

from inside on the many holes keeps out the water, until she uncovers the condensed stream [of air]. Then at once when the air flows out, the water flows in in an equal quantity.”

He is on the verge of deducing something about the pressure exerted by the air in the Earth’s atmosphere. Two thousand years would pass before Torricelli provided the correct explanation for the behaviour of devices like this.

Anaxagoras, like Empedocles, lived in the middle of the fifth century BC, working first in Ionia and then in Athens. Like Empedocles, he also denied the existence of empty space and believed strongly in the conservation of the ‘essence’ of the world. This conservation principle meant that things could not appear out of Nothing, or disappear into it. It is an idea that is similar in spirit to our modern concept of the conservation of energy. Anaxagoras viewed ‘creation’ as the bringing of order into a state of primordial chaos rather than as an event from which the World came into being out of Nothing. He also used this conservation principle to understand how things change from one substance into another; for example, how fruit or other forms of food that we eat can turn into flesh and bones. He believed that something must be passed on in each of these changes, that there are ‘seeds’ within all forms of matter which are passed on but neither created nor destroyed. ‘For in everything there is a portion of everything.’ One might even view these seeds as being the molecules of modern chemistry. Yet these ingredients were held to be infinitely divisible, so that space could be continuously filled with matter. No need for Empedocles’ pores, and no room for empty space either.

Anaxagoras shared Empedocles’ fascination with the water-catcher and repeated that experiment, extending it by compressing air inside wineskins so as to demonstrate that the air offers a resistance when the skins are stretched. From this, he concluded that air is not the same thing as empty space and that we have no observational evidence for the existence of empty space. A subtle thinker, he was the first philosopher to recognise that our observations of the world are conditioned by the frailty of our senses. Our ability to decide whether one thing is really different from another (his favourite example was distinguishing very similar shades of colour) is just a reflection of our senses and because of ‘the weakness of the sense-perceptions, we cannot judge truth’. Our senses are sampling partial information about a deeper reality that they cannot fully apprehend. His ideas were ones that would be used by the Greek atomists who came after him as a fundamental feature of their picture of the world.

The atomists maintained that all matter was composed of atoms, tiny indivisible particles (the Greek word atomos means having no parts), which were eternal, indivisible and unchangeable. Atoms moved through empty space and their different degree of clustering from place to place was responsible for changes in density and the distinctive properties of different forms of matter. This powerful picture of the world was appealing because of its simplicity and wide applicability. It was proposed first by Leucippus of Miletus in the mid-fifth century BC, developed further by his student Democritus, and eventually upgraded into an entire philosophical system by Epicurus of Samos (341–270 BC), after which it became extensively known. Even so, today its most memorable articulation is to be found in the remarkable poem De Rerum Natura (On the Nature of Things) composed by the Roman poet Lucretius in honour of Epicurean atomism in about 60 BC.

Leucippus has the double distinction of introducing the concept of matter being composed of identical basic units and of taking seriously the idea that there does indeed exist something called empty space in which these atoms move. Here we see for the first time the concept of a true vacuum being rigorously employed as an axiomatic part of a natural philosophy. Because the world was differentiated into atoms and the void in which they moved, the vacuum was necessary for any movement or change to be possible, and Leucippus reminds us that17

“unless there is a void with a separate being of its own, ‘what is’ cannot be moved – nor again can it be ‘many’, since there is nothing to keep things apart.”

Atoms could differ in concentration, in shape and in position, but they could not appear and disappear from or into Nothing. This immutability of atoms rules out any possibility that they contain regions of vacuum. They must be solid and finite in size. It may be significant that Leucippus spent some time as a pupil in the philosophical school of Elea where Zeno had worked, and where his paradoxes of the infinite were much studied. Zeno had demonstrated some of the bizarre paradoxes that could occur if you considered a process of halving things indefinitely. For example, one of his paradoxes of motion invites us to contemplate how it is possible to walk, say, to the door of our room one metre away. First, we must cross half a metre, then half of half a metre, then half of half of half a metre, and so on, ad infinitum. It appears that we will never be able to reach the door because we have to cover an infinite number of distances! It is possible that these awkward problems of dealing with things that were allowed to become arbitrarily small convinced Leucippus of the importance of having a smallest possible size for his atomic units of matter to avoid such paradoxes. There were physical reasons as well. Epicurus argued that allowing matter to be infinitely divisible would result in the irreversible destruction of its identity, slipping ultimately into non-existence, or give rise to aggregates of matter that were too fragile to persist. This was a far-reaching step because it drew a sharp distinction between mathematical and physical reality: in the former, infinite division of any quantity was possible; in the latter, it was not. You had to choose which mathematical structure to apply to physical existence.

According to Epicurus,18 atoms must also have a maximum possible size in order to explain why they are not seen with the naked eye. Democritus is silent19 on this point, but he agrees with all the other atomists that the number of atoms in the Universe, like its size and age, is infinite. Thus, their conception of the Universe is as a vacuum of infinite size filled with moving, solid, indivisible particles of different shapes and sizes.20 Lucretius poetically describes how it can be that the random motion of these imperceptible atoms can give rise to everyday objects that seem to be steady and unchanging:21

“Although all the atoms are in motion, their totality appears to stand totally motionless … This is because the atoms all lie far below the range of our senses. Since they are themselves invisible, their movements also must elude observation. Indeed, even visible objects, when set at a distance, often disguise their movements. Often on a hillside fleecy sheep, as they crop their lush pasture, creep slowly on-ward, lured this way or that by grass that sparkles with fresh dew, while the full-fed lambs gaily frisk and butt. And yet, when we gaze from a distance, we see only a blur – a white patch stationary on the green hillside.”

There is a curious parallel between the atomists’ picture of atoms separated by the void and Pythagoras’ picture of numbers. Pythagoras and his followers believed that everything could be expressed by numbers and these numbers possessed intrinsic meanings, they were not merely ways of expressing relationship between things. If two quite different things possessed an element of threeness, or fiveness, then they were deeply related by a fundamental harmony. Like the atomists, the Pythagoreans required a void to exist in order to maintain the identities of things. For the atomists, it was empty space that separated atoms and allowed them to move. For the Pythagoreans, everything was number: the void existed between numbers. Aristotle reports that the Pythagoreans maintained that22

“the void exists … It is the void which keeps things distinct, being a kind of separation and division of things. This is true first and foremost of numbers; for the void keeps them distinct.”

The atomists were not the only ancient philosophers to have strong views about the vacuum. From the third century BC, there emerged a completely different theory of the nature of things. It became known as Stoicism, after its first adherents were dubbed Stoics because they chose to meet under a painted corridor (stoa) on the north side of the market place in Athens. Its founders were Zeno of Cition (not to be confused with Zeno of the paradoxes), Chrysippus of Soli in Cilicia, and Poseidonius of Apamea in Syria.

In complete contrast to the atomists’ dogma, the Stoics believed that all things were a continuum, bound together by a spirit – an elastic mixture of fire and air – or pneuma, that permeated everything. No empty space could exist within or between the component pieces of the world, but this did not mean that there couldn’t exist any empty space at all. Quite the contrary, the Stoics’ Universe was a finite continuous island of material diffused by pneuma, but sitting in an infinite empty space.23 The void was the great beyond and the pneuma bound the constituents of the world together so as to prevent them diffusing out into the formless void.

The Stoic conception is of interest to us now because the pneuma was a forerunner of the long-lasting idea that space is filled with a ubiquitous fluid, an ether, which can be acted upon and which responds to the actions of other material. The Stoics envisaged their ether as a medium through which the effects of sound or other forces could propagate, just as when we disturb the surface of water in one place we can see the waves emanating outwards over the surface to create effects elsewhere, causing a nearby floating leaf to oscillate up and down.

Remarkably, neither the views of the atomists nor those of the Stoics proved influential over the next fifteen hundred years. The dominant picture of the natural world that emerged from Greek civilisation and wedded itself to the Judaeo-Christian world view was that of Aristotle. Aristotle’s approach to natural phenomena was dominated by a search for purpose in motion and change. While this teleological perspective could be of help in understanding what was going on in the natural world, or in the study of human psychology, it was a real obstacle to the study of problems of physics and astronomy. Aristotle’s picture of Nature was extremely influential and his views about the vacuum fashioned the consensus view about it until the Renaissance.24 He rejected the possibility that a vacuum could exist, either in the world as the atomists maintained, or beyond it as the Stoics believed. The Aristotelian universe was finite in volume; it contained everything that exists; it was a continuum filled with matter; space was defined by the bodies it contained. But unlike the dynamical ether suggested by the Stoics, Aristotle’s continuous ether was static and passive, eternally at rest.

ISLAMIC ART

“Humility collects the soul into a single point by the power of silence. A truly humble man has no desire to be known or admired by others, but wishes to form himself into himself, to become nothing, as if he had never been born. When he is completely hidden to himself in himself, he is completely with God.”

Isaac of Ninevah (AD 600)25

When compared with ancient Greek or later Western representational art, the intricate mosaics and tessellations of Islamic art seem like an ancient form of mathematical art: computer art before there were computers. We can picture their teleported ancient creators manipulating fractals and modern tiling patterns to continue a tradition that vetoed the representation of living things. Their patterns are extremely revealing of their religious views. God alone was infinite. God alone was perfect. But by creating finite parts of patterns that were evidently infinite it was possible to capture a little piece of the Divine in a humble yet inspiring manner. The partial character of the design served to reinforce the frailty and finiteness of humanity in contrast to God’s infinity.

Islamic art directed the mind towards the infinite by creating regular patterns that could be infinitely repeated. These designs have become familiar to us through the work of the Dutch artist Maurits Escher and the mathematical designers he inspired. Born in Leeuwarden, Holland, in 1898, Escher began his artistic career as a landscape artist, painting little Mediterranean towns and villages. But his life’s work was changed in the summer of 1936 by a visit to see the fabulous designs of the Alhambra, in Granada, Spain (Figure 2.3).

Escher was deeply impressed by the intricate patterns he saw and the fabulous geometric precision of the creators of this fourteenth-century Moorish palace. He spent many days studying the detailed patterns and periodicities, and went away to develop his own synthesis of symmetry and impossibility. Unlike the patterns of the Alhambra, Escher animated his designs with living creatures: fish, birds, winged horses and people. This expression of the abstract by means of recognisable images was, he remarked, the reason for his ‘never-ceasing interest’ in these patterns.

In Islamic art we see how the Moslems celebrated infinity where the Greeks feared it. They made it the hidden engine of their artistic creations. While not quite on central stage, it was never far away in the wings. The treatment of zero and Nothingness is just as confident. Rather than sweep Nothing away under the carpet as a philosophical embarrassment, the Islamic artists simply saw the void as a challenging emptiness to be filled. No blank space could be left alone. They filled friezes and surfaces with intricate patterns.26 This urge seems to be shared by human cultures the world over. Wherever anthropologists look they find elaborate decorations.

Figure 2.3 Islamic decorations from (top) Badra in Azerbaijan, and (bottom) the Alhambra Palace in Granada, Spain.

We just do not like an empty space. As we saw with the Mayan need to fill their mathematical pictograms with an image for Nothing, the human mind longs for pattern and for something to fill any void. The great art historian Ernst Gombrich termed this impulse to decorate the horror vacui. It inspires a wealth of persistent procedures, sometimes linking different parts, sometimes filling in space, allowing a network of growing intricacy to emerge and develop.

ST AUGUSTINE

“Miracles are explainable; it is the explanations that are miraculous.”

Tim Robinson

In medieval and Renaissance thought the paradoxical aspects of the something that is Nothing became interwoven with the doctrines and traditions of Christian theology. These doctrines were founded upon the Jewish tradition of turning away from Nothing because it was the antithesis of God. God’s defining act was to create the world out of Nothing. What stronger evidence could there be that Nothing was something undesirable: a state without God, a state which He had acted to do away with. Nothing was the state of oblivion to which the opponents and enemies of God were dispatched. Any desire to produce a state of nothingness or empty space was tantamount to attempting what only God could do, or to remove oneself from God’s domain. A single Divine creation of everything out of Nothing was a basic tenet of faith. To speak seriously of the void or of empty space was atheistic. It countenanced parts of the Universe where God was not present.

The most innovative thinker to grapple with the problem of synthesising the Greek horror of the void with the Christian doctrine of creation out of Nothing was Augustine of Hippo (354–430), pictured in Figure 2.4. He had a broader and deeper view of what creation should mean. It needed to be something more than the mere refashioning of primitive pre-existent materials into an ordered cosmos, something more than the unfurling of the cosmic scene at some moment in the distant past. Rather, it must provide the ground for the continued existence of the world and an explanation for time and space itself. It was for him a total bringing into being. Nothingness was therefore an immediate precursor state to the one that God sustains. This makes it more negative in its attributes than merely not being what is now in the Universe. It was characteristic of being apart from God.

Figure 2.4 St Augustine.

Augustine equated Nothing with the Devil: it represented complete separation from God, loss and deprivation from all that was a part of God, an ultimate state of sin, the very antithesis of a state of grace and the presence of God. Nothing represented the greatest evil. This was the ‘something’ that he believed non-Being to be. This formula led Augustine into dangerous waters because by introducing Nothing into the realm of Being he admitted that there was something that God lacked before he created the world. This difficulty he sidestepped, along with other problems about the beginning of time, by arguing that when God created the world he created time as well. There was no ‘before’ the first moment of time and so no time when God needed to change an unsatisfactory state of affairs.

These pieces of theological legerdemain were never entirely persuasive and centuries later they led Thomas Aquinas to create a fuller negative theology in which the attributes of God were only to be spoken of negatively: He was not finite, not temporal, unchangeable, and so forth. Aquinas supported the Aristotelian abhorrence of Nothing by viewing the creation of the world as an annihilation of Nothing in an act of Divine creative transformation. Yet, despite this careful circumscription, the Church was wary of Nothing and its mathematical representations during the tenth to thirteenth centuries. It tried to keep Nothing confined to the realm of arithmetic symbols where zero could be relegated to a harmless place holder on a counting board, far from the philosophical implications that the Indians had embraced but from which the Greek-Christian synthesis had recoiled.

There were two threads to the theological writings: one which drew out the nature of the nothingness from which creation had sprung; another which emphasised the nothingness and ephemerality of all temporal things. Both were directed at refuting the dualist heresy that the world was created out of pre-existing matter, rather than out of Nothing. The first thread was the preserve of serious theological treatises whereas the second was the substance of metaphysical poets trying to prove the nothingness of life when viewed in the cosmic scheme of things.27

It is important to recognise that although Christian doctrine included the notion of creation out of nothing (creatio ex nihilo), it did not include the idea that the creation was caused by nothing. The cause28 of the creation is God, not some latent property of the void. God always exists but the Universe just lacks a material cause to initiate its structure. Aquinas argued that if there is absolutely nothing – no Universe, no God, no Being at all – then nothing can appear. For to cause itself, a being would have to exist in order to give itself existence and this, he claims, is absurd. So if absolutely nothing ever existed in the past, nothing exists now.29

THE MEDIEVAL LABYRINTH

“But if there is a void above and a void below, a void within and a void without, he who is intent on escaping void has need of a certain imaginative mobility.”

Robert M. Adams30

It is easy to skip over the Middle Ages as though they were a time of darkness and delusion, an antechamber in the history of scientific ideas that awaits the arrival of Copernicus, Galileo and Newton. But in order to understand why scientific ideas about space and the vacuum developed in the way they did, when they did, it is important to take some snapshots of the way in which human thinking about the concept of Nothing developed from the time when Aristotle’s ideas held sway until the early eighteenth-century arguments between Newton and Leibniz. Scholars of all complexions struggled for more than five hundred years to harmonise subjects like the nature of space, infinity and the vacuum. Their task was made more difficult by their need to relate all these concepts to the nature and capacity of God. The synthesis of Aristotle’s philosophy and Christianity created a complex web of philosophical ideas whose theological consistency was more important than the mere assimilation of experimental facts; not because those facts were regarded as of little relevance, but because their significance was often ambiguous and they could be incorporated into the World model in a variety of ways consistent with their overall World view.

As a result of Aristotle’s rejection of the idea that a separate vacuum could exist, on the grounds that it was logically incoherent,31 it was believed almost universally in the early Middle Ages that Nature abhorred the creation or persistence of any vacuum state. Almost all scholars believed that it was not possible to create a vacuum within the universe of space that we experience and see around us, a so-called intracosmic void. Things became more complicated when attention was turned to consider the possibility that there might exist an infinite extracosmic void beyond our finite, spherical, Aristotelian universe. This idea began to have credence in the fourteenth century and became very widely accepted over the next three hundred years.

Medieval philosophers inherited a strong Aristotelian opposition to the vacuum. In order to leave no hiding place from his arguments for its non-existence, Aristotle defined the vacuum carefully. He characterised it as that in which the presence of a body is possible, although not actual. Aristotle attempted to show that admitting the idea of a vacuum would paralyse the Universe. Motion would be impossible because there was no reason to move one way or the other in a vacuum because it was necessarily the same everywhere and in every direction. There was neither ‘up’ nor ‘down’ and so no way for things to adopt their ‘natural’ motion. In any case, if motion did occur it would continue for ever because there would be no medium offering any resistance to its motion.32 Perpetual motion was a reductio ad absurdum. Nor would it make sense for a moving body to stop anywhere in this perfectly homogeneous void: for why should it stop at one place rather than another?

Considerable attention was devoted by scholars in the thirteenth and fourteenth centuries to the idea that Nature disliked the presence of a vacuum so acted always so as to remove it or resist its creation. As ever, there were shades of opinion. Some – strict Aristotelians – maintained that it was impossible to make a perfect vacuum for even a fleeting instant of time. Others were content to permit the ephemeral existence of a vacuum so long as events inevitably overcame it and refilled it quickly with air or other material. They did not believe in the existence of a stable vacuum.

Some scholars like Roger Bacon were unhappy with a law of Nature that was negative. A rule like ‘no vacuums allowed’ could not be primary; it needed to be a consequence of a deeper positive principle about what Nature did. Negative principles were very powerful vetoes but they allowed too many things to happen that were not seen. As a specific example, the workings of Empedocles’ water-catcher, or clepsydra, were much debated. Bacon argued that the veto on the formation of a vacuum was insufficient to explain what is seen. The formation of a vacuum inside its walls could equally well be avoided by the walls imploding. Why does Nature choose to hang on to the water rather than cave in the walls of the vessel? What principle decides?

Another favourite puzzle that taxed medieval scholars was a simple example noticed first by Lucretius.33 It is the problem of separating two smooth surfaces; for example, two flat sheets of glass or metal, as shown in Figure 2.5. The concern was that if they begin in perfect contact but are then suddenly pulled apart then a vacuum must be briefly formed when they separate: there must be a change from a state in which there is nothing between the sheets and one in which there is air between them. Here is Lucretius’ ancient version of the problem:34

“If two bodies suddenly spring apart from contact on a broad surface, all the intervening space must be void until it is occupied by air. However quickly the air rushes in all round, the entire space cannot be filled instantaneously. The air must occupy one spot after another until it has taken possession of the whole space.”

The way this problem was investigated gives a wonderful insight into the ingenuity and serious medieval interest in these problems.35 The theological stakes were surprisingly high, as we shall see.

Figure 2.5 Two parallel sheets sliding across their surface of contact.

The Scholastics tried hard to show that this ancient puzzle, rediscovered by Bacon and others, did not allow even a short-lived vacuum to arise. Some claimed that while a vacuum could indeed form in principle if the surfaces were slid across each other so that they remained perfectly parallel, this could never happen in practice. A slight angle would develop between the surfaces and air would enter and gradually fill the gap between the surfaces bit by bit. Bacon turned the discussion around and argued that two smooth plane parallel surfaces could not be separated if they were in perfect contact (which would have a ring of truth about it for those who had tried it) unless they were first inclined. This was Nature displaying her resistance to the creation of a vacuum.

One of Bacon’s English contemporaries, Walter Burley, saw through this suggestion, pointing out that the inclining of the surfaces makes no difference of principle at all. The ephemeral vacuum must still form, it merely lasts for a shorter time than if the plates are parallel when separated. Probing deeper still, he pointed out that perfectly parallel surfaces don’t exist; but no matter, even though real surfaces always exhibit microscopic undulations which restrict their points of contact to occasional protrusions, all that the argument for an ephemeral vacuum needs is for there to be a single point of contact. When that point of contact is broken a vacuum must arise momentarily.

A notable opponent of the direction of this reasoning was Blasius of Parma, a student of motion and fluid flow. He argued that it was possible for plates to be separated by parallel motion yet not form a vacuum. This requires the particles of air to move at just the right speeds at just the right times to fill the void space immediately it forms.36 But then he resorts to a very interesting argument, reminiscent of Zeno, that the vacuum can never form because there is no first moment when it exists – if you think there is, then halve it, and so on, ad infinitum. By denying the logical possibility of a first instant of separation for the plates, Blasius tried to eliminate the possibility of an instantaneous separation of the surfaces which would have allowed a vacuum to make even the briefest appearance.

Despite the ingenuity of these suggestions, the most widely accepted resolution of the dilemma stayed close to Aristotle’s own treatment of a very similar problem. Aristotle had argued that there would always be some air trapped between two surfaces in contact, just as two surfaces which touch under water always get wet at their interface. Most people had regarded this as a conclusive and simple resolution of the surface contact problem until Bacon raised a simple objection. Forget about the two solid surfaces, he said. They are just a device to allow some air to be trapped in between the surfaces and avoid a vacuum forming. Suppose, instead, that you have only one surface and consider its interface with the water around it. Nothing lies between the surface and the water and so whenever they are separated a momentary vacuum must form!37

Burley responded by claiming that there was still a thin film of air at the interface between a liquid and a solid. When you began to separate them it would quickly expand to fill up any potential void space. But what if the air was as rarefied as it could be and there was no scope for further expansion? Burley responded again by claiming that the surfaces would be inseparable in this situation. The only way of parting them would be to bend one of them and so create the inclined-surface problem that he and Bacon both believed to be the only way to effect a separation without producing a vacuum.

Despite his appeal to a specific physical process to avoid the vacuum’s appearance, Burley felt he needed more protection from uncontrollable natural events than this. What if a heavy rock should fall to the ground and expel all the air in between its surface and the ground at the point where they first made contact? Would there not be an instantaneous vacuum there? To stop all the air being expelled, he appeals to a celestial agent38 which prevents the air yielding ‘to the stone because [the stone] is held back by the superior agent, which powerfully seeks to prevent a vacuum’. If natural processes were inadequate to overcome the threat of a real vacuum forming then one needed to fall back on the cosmic censorial power of this supernatural agency to stave off the creation of Nothing from something. In the much-debated example of the water-catcher, the celestial agent could be invoked to explain why the water behaved ‘unnaturally’ by not falling downwards, rather than imploding the walls of the vessel to prevent a vacuum forming. This type of explanation, reminiscent of the Just So Stories, was not very persuasive. Unfortunately, the celestial agent was annoyingly inconsistent in policing the formation of vacua. Others were quick to point out that on other occasions the universal agent did choose to stave off the vacuum by distorting the walls of the container, as occurs if the water freezes.

Alongside these detailed arguments about the ways in which Nature staves off the creation of an intracosmic void, there were centuries of debate about the existence of an extracosmic void – a vacuum beyond the physical Universe. Aristotle considered the idea briefly but rejected it along with the whole idea of the plurality of worlds. His definition of a vacuum as that in which ‘the presence of a body is possible but not actual’ demanded such a conclusion. For ‘outside’ the Universe there is no possibility of body and hence no vacuum. In this respect, as we have seen, he is diametrically opposed to the Stoic view that there exists an infinite extracosmic void.

The extracosmic void introduced further dilemmas for medieval philosophers. It was imagined to consist of ‘imaginary’ space; that is, space that could be imagined to exist even when bodies did not. We can imagine all sorts of things that seem to go on for ever, like the list of all numbers, and these may be imagined to ‘live’ in this infinite imaginary space.39 Although it couldn’t contain ordinary matter it had a property that proved crucial in the subsequent development of ideas. It was completely filled with the presence of God and was both the expression of God’s immensity and the means by which His omnipresence was achieved and maintained. This considerably narrowed the options when it came to pinpointing its properties. Try to make the extracosmic void finite, or endow it with dimensions, and you risk reaching a heretical conclusion about the nature of God. For in order that God remains omnipresent yet indivisible, He needs to be wholly located at every single point of the infinite space of the extracosmic void: One who ‘is an infinite sphere whose centre is everywhere and circumference nowhere’.40

The key moment in the early stages of these debates was the famous Paris condemnations of 1277 in which Bishop Etienne Tempier strove to reassert the doctrines of God’s power to do whatever He chooses. Before Tempier’s intervention, there was a widespread belief amongst theologians that Aristotelian philosophy showed that God was constrained in various ways. For example, God could not make two and two make five; God could not create a plurality of worlds; and, inevitably, there was a veto that God could not cause a movement of things that would produce a local vacuum. By denying these restrictions on God’s power, Bishop Tempier made space for an extracosmic vacuum. For if many worlds exist what lies between them? And if God should choose to move our whole world in a straight line then what would remain in its former place? ‘A vacuum’ was the answer that the prompter was whispering from the wings. And if you didn’t hear, then beware of suggesting to the Bishop that God could not create a vacuum. After 1277, the vacuum became admissible because any attempt to exclude it on philosophical grounds was tantamount to limiting the power of God.

Another great medieval question was whether a vacuum had existed before the creation of the world. Aristotle had denied the possibility that the world (or anything else) could be created from Nothing. The original Aristotelian scenario of an eternal, uncreated Universe had the drawback of clashing with Christian doctrine and so the more appealing alternative was a version in which the world had been created from a pre-existing void containing nothing. Yet this was not entirely without problems of its own. It required the existence of something eternal that was seemingly independent of God. It was this stance that spawned infamous questions like ‘What was God doing before the creation of the world?’ and the development of Augustine’s response that entities like time and space were created along with the Universe so there was no ‘before’.

By the sixteenth century the tide had begun to turn. The rediscovery of lost texts by Lucretius and accounts of Hero’s ancient experiments on pressure inspired assaults on Aristotelian dogma. The horror at allowing a vacuum to form abated and there was a change of attitude to the existence of an infinite void space which would alter the relationship of God to that space, culminating in a complete decoupling of the scientific and theological debates about the nature of space and the vacuum.

In the sixteenth and early seventeenth centuries those who began to subscribe to the Stoic cosmology, with its finite cosmos surrounded by an infinite extended void, were all agreed on many of the attributes of the surrounding void: it was the same everywhere, immutable, continuous and indivisible, and offered no resistance to movement. But what was new was a growing disagreement as to God’s relationship to the infinite void space. Notable atomists like Pierre Gassendi denied that the infinite void had anything to do with the attributes of the Deity. A third way was provided by the philosopher Henry More who, while regarding space as an attribute of God, also regarded God as an infinite extended Being. More is interesting primarily because his views seem to have influenced Isaac Newton’s views about space. Newton introduced God as a three-dimensional presence everywhere and as the underpinning intelligence behind the mathematical laws of Nature. Indeed, he introduced a new form of the ancient Design Argument for the existence of God, appealing to the fortuitous structure of the laws of Nature rather than their outcomes as evidence for a Grand Designer behind the scenes.41 Newton clung to the Stoic picture of a finite world surrounded by an infinite void space. He could imagine an empty space but not the absence of space itself. Thus space was something that was quite independent of matter and motion. It was the cosmic arena in which matter could reside, move and gravitate. Newton writes that,42

“we are not to consider the world as the body of God, or the several parts thereof as the parts of God. He is a uniform Being, void of organs, members or parts, … being everywhere present to the things themselves. And since space is divisible in infinitum, and matter is not necessarily in all places, it may also be allowed that God is able to create particles of matter of several sizes and figures, and in several proportions to space, and perhaps of different densities and forces, and thereby to vary the laws of Nature, and make worlds of several sorts in several parts of the Universe. At least I see nothing of contradiction in this.”

For Newton, the extracosmic void space was entirely real and not in the least imaginary. When he was preparing the 1706 edition of his Opticks for the press he considered adding to his list of ‘queries’ – a series of far-reaching questions and speculations about the physical world – a final question asking43

“what the space that is empty of bodies is filled with.”

These Newtonian views about the reality of the extracosmic void space and its relationship to God were articulated by his champion, Samuel Clarke, in a famous debate with Leibniz. Leibniz disagreed fundamentally with Newton. He denied that the infinite void even existed and objected to Newton’s idea that we equate it with the immensity of God. He saw how difficult it was to sustain a relationship between God and space and opposed any such attempt. In the end, his view about the separation of God from space prevailed amongst philosophers and theologians, even though the infinite void space of Newton was retained by scientists.

Newton’s God was no longer located in the void beyond the material world. The great idea of the Scholastics, that God was inextricably linked to the nature of space and to its infinite extent, lived long enough to influence Newton’s great conception of the world and the laws of motion and gravity that governed it, but by the end of the eighteenth century the theological complexion of the problem of space had been eroded. The proposals for explaining God’s omnipresence in space had lost credibility and played no further role in understanding the things that were seen. The Almighty could then be removed without reverberations spreading into the theological domain. Gradually, it was God’s transcendence rather than His omnipresence that would become the centrepiece of the theologian’s discussion of God. Once this transformation was complete, God needed no place in the infinite void of space that the astronomers took as the backdrop for the finite world of matter and motion. It was an arena that finally allowed mathematical deductions to be made without the need for a theological conscience. The vacuum was at last safe for scientists to explore.

WRITERS AND READERS

“Now is the discount of our winter tents.”

Advertisement in Stratford-upon-Avon camping shop44

Not everyone spoke so seriously. In order to sidestep the risk of being accused of blasphemously toying with the demonic concept of empty space, writers and philosophers cloaked their thoughts in more playful deliberations, inventing and pursuing paradoxes and puns in a way that could always be defended as undermining the coherence of the concept of empty space regardless of the true intent. The paradox that would bring the argument to an end could always be defended as a reductio ad absurdum. The American commentator Rosalie Colie concludes her study of the poems and paradoxes of Nothing that were all the rage in the fifteenth and sixteenth centuries with the opinion that the writers of these paradoxes

“were engaged in an operation at once imitative and blasphemous, at once sacred and profane, since the formal paradox, conventionally regarded as low, parodies at the same time as it imitates the divine act of Creation. And yet, who can accuse the paradoxist of blasphemy, really? Since his subject is nothing, he cannot be said to be impious in taking the Creator’s prerogative as his own – for nothing, as all men know, can come of nothing. Nor indeed is he directing men to dangerous speculation, since at the very most he beguiles them into – nothing. And most important of all … if the paradoxist lies, he does not lie, since he lies about nothing.”45

The two most common trends of this sort are to be found in the ‘all or nothing’ paradoxes and the amusing penchant for double entendres about Nothing displayed by writers and playwrights. Poets joined in the game as well, with works like The Prayse of Nothing:46

“Nothing was first, and shall be last
for nothing holds for ever,
And nothing ever yet scap’t death
so can’t the longest liver:
Nothing’s so Immortall, nothing can,
From crosses ever keepe a man,
Nothing can live, when the world is gone,
for all shall come to nothing.”

and On the Letter O,

“But O enough, I have done my reader wrong
Mine O was round, and I have made it long.”47

or Jean Passerat’s Nihil, informing us that

“Nothing is richer than precious stones and than gold; nothing is finer than adamant, nothing nobler than the blood of kings; nothing is sacred in wars; nothing is greater than Socrates’ wisdom – indeed, by his own affirmation, nothing is Socrates’ wisdom. Nothing is the subject of the speculations of the great Zeno; nothing is higher than heaven; nothing is beyond the walls of the world; nothing is lower than hell, or more glorious than virtue.”48

and so on, and on, and on.

These word games soon become a little tedious to our ears. They had the goal of generating lots of words from nothing by means of talking about Nothing. For a time, the genre was a fashionable form of philosophical nonsense verse. Several paradoxical juxtapositions occur again and again. There is the picture of the circle representing, on one hand zero, and, on the other, the encompass of everything. There is the egg, shaped like zero but promising to become the generator of new life. It was pregnant with creativity just like the mathematicians’ zero, waiting to be added to other figures to create larger numbers. And in the background there is the sexual allusion to the circle which represents the female genitalia. This is a running joke in Elizabethan comedies although much of the humour is lost on us. Fortunately, there is a famous example of this genre that is widely known and appreciated. It is intriguing because it shows that the paradoxes and puns of Nothing attracted the interest of the greatest of all wielders of words.

SHAKESPEAREAN NOTHINGS

“Is this nothing?
Why, then the world and all that’s in’t is nothing;
My wife is nothing: nor nothing have these nothings,
If this be nothing.”

William Shakespeare, The Winter’s Tale49

Shakespeare was much taken with all the linguistic and logical paradoxes of Nothing. For good measure he entwined them with the double entendres of the day to add yet another dimension to the many-layered works that are his hallmark. The comedy Much Ado About Nothing is a wonderful example50 of the deftness with which games could be played with words that others had struggled to enliven. First appearing in print in 1600, and probably written during the preceding two years, the title of this play immediately illustrates the general fascination with the ambiguities of Nothing that were in vogue in Shakespeare’s time.51 In the fourth act, the prospective lovers Beatrice and Benedick use the ambiguities of Nothing as a subtle smokescreen so that each hearer can choose to interpret Nothing in a positive or a negative way:

“Benedick: I do love nothing in the world so well as you.

Is not that strange?

Beatrice: As strange as the thing I know not. It were as possible for me to say I loved nothing so well as you. But believe me not; and yet I lie not. I confess nothing, nor I deny nothing.”52

Shakespeare plays upon other dimensions of Nothing as well. In the tragedies Hamlet and Macbeth we find the philosophical and psychological paradoxes of Nothing deeply interwoven with human experience. Macbeth is repeatedly confronted with the paradoxes of Nothing and the horrors of non-Being: he despairs that

“Nothing is
But what is not.”53

Hamlet explores how Nothing can have paradoxical meaning and content. In contrast to Macbeth, who rails that

“Life’s …a tale
Told by an idiot, full of sound and fury,
Signifying nothing”,54

the Prince of Denmark finds consolation in death and convoluted speculation about Nothing. They stand in stark contrast about what it means to be and not to be, for

“where Macbeth discovers that death is oblivion, Hamlet discovers that it is not. Macbeth discovers that, when death is oblivion, life is insignificant. Hamlet discovers that when one does not fear death, life with all its painful responsibilities can be borne and even borne nobly. In the end Hamlet knows for himself the relation between ‘to be’ and ‘not to be’ by which even his own death can affirm life.”55

Yet even Hamlet makes full use of the double entendres associated with Nothing and the female form in this exchange with Ophelia:56

“Hamlet: Lady, shall I lie in your lap?
Ophelia: No, my lord.
Hamlet: I mean, my head upon your lap?
Ophelia: Ay, my lord.
Hamlet: Do you think I meant country matters?
Ophelia: I think nothing, my lord.
Hamlet: That’s a fair thought to lie between maids’ legs.
Ophelia: What is, my lord?
Hamlet: Nothing.”

In King Lear, Shakespeare tells of the destruction of Lear by all that emanates from Nothing. The play has a recurrent theme of quantification, numbering and reduction. Two of Lear’s daughters make pretentious statements of love and respect for him in return for parts of his kingdom, but the third, Cordelia, will not play this cynical game or just remain silent. Her encounter with her father introduces a typical play on Nothing:57

“Lear: … what can you say to draw
  A third more opulent than your sisters’? Speak!
Cordelia: Nothing, my lord.
Lear: Nothing?
Cordelia: Nothing.
Lear: Nothing will come of nothing. Speak again.”

From this ominous beginning many are reduced to Nothing. Cordelia is hanged. Lear’s Fool asks him ‘Can you make use of nothing?’ and Lear repeats his admonition to Cordelia, ‘Why, no, boy. Nothing can be made out of nothing.’ But the Fool responds by reducing Lear to Nothing:

“Thou art an O without a figure. I am better than thou art now; I am a fool, thou art nothing.”

Lear’s other daughters, Goneril and Regan, reduce Lear to zero in more practical ways, demanding that he reduce the size of his entourage, halving and halving it until there is only one left and then Regan asks, ‘What need one?’ Lear shows Shakespeare58 grappling with the double meanings of Nothing, the metaphysical void and the end result of taking away what one has, bit by bit, if one exports the arithmetic of buying and selling into the human realms of love, loyalty and duty. Things then don’t always add up. Madness is not far away. On that you can count.

Shakespeare explored all the meanings of Nothing: from the simplicity of zero, the nonentity of the cipher, the emptiness of the void, and the absence of everything it witnessed, to the contrast between the whole and the hole that was zero, the circle and the egg, hell, oblivion and the necromancer’s circle. His explorations can be roughly divided into those that pursue the negative aspect of Nothing and those that pursue the positive. On the negative side we see the focus on the absence of things, on denial, apathy and silence. These invariably bring bad consequences and reveal some of the awful results of meaninglessness. By contrast, the positive side of Nothing lays stress on the power of Nothing to generate something. Just as zero lay at the beginning of an ever-increasing sequence of numbers, so the sexual connotations of Nothing and the pregnant power of the egg symbolised fruitfulness and multiplication, the growing of something out of nothing. Indeed, it was just this multidimensional proliferation that Shakespeare’s own work displayed.59

One should not think that the linguistic gymnastics of nihil paradoxes are a thing of the past. While it is not common for these word games to be played by modern writers, they can still be found if you know where to look. Here is Jean-Paul Sartre trying to convey information about the origin of negation:

“Nothingness is not, Nothingness is ‘made-to-be’, Nothingness does not nihilate itself; Nothingness ‘is nihilated’ … It would be inconceivable that a Being which is full positivity should maintain and create outside itself a Nothingness or transcendent being, for there would be nothing in Being by which Being could surpass itself towards Non-Being. The Being by which Nothingness arrives in the world must nihilate Nothingness in its Being, and even so it still runs the risk of establishing Nothingness as a transcendent in the very heart of immanence unless it nihilates Nothingness in connection with its own being. The Being by which Nothingness arrives in the world is a being such that in its Being, the Nothingness of its Being is in question. The being by which Nothingness comes to the world must be its own Nothingness …”60

and so on, for more than 600 pages.

PARADOX LOST

“What did the mystic say to the hot-dog vendor?
Make me one with everything.”

Laurence Kushner

By the end of the seventeenth century the literary fascination with the paradoxes of Nothing had run its course.61 It ceased to be a mainspring of imaginative exploration in both literature and philosophy. Writers simply mined out the seam of possibilities and moved on to explore new ideas. Philosophers came to distrust these games with words and they were seen increasingly as mere puzzles to amuse. They were no longer considered to provide a route into deep truths about the nature of things. The increasing stress placed upon observation and experiment relegated the paradoxes of Nothing to a linguistic backwater from which they would not reappear until the beginning of the twentieth century. The sea change in attitudes is displayed in Galileo’s Dialogue Concerning Two World Systems,62 which includes a discussion of the dangers of treating the contemplation of ‘words’ as a superior route to truth than the study of ‘things’. Simplicio cautions that ‘everybody knows that you may prove whatever you will’ by means of linguistic paradoxes. Galileo equated ‘paradox’ with vague, unverifiable word games that had no place in the development of science, which was typified by the logic of testable chains of cause and effect. For example, the famous ‘Liar paradox’, credited to Epimenides, which St Paul repeats, that ‘all Cretans are liars, one of their own poets has said so’ was condemned as ‘nothing but a sophism … a forked argument … And thus, in such sophisms, a man may go round and round for ever and never come to any conclusion.’

Galileo had the highest regard for mathematical knowledge of the world. He recognised that our knowledge of most things was necessarily imperfect. We can only know as much as Nature reveals to us, but in the field of mathematics we have access to a part of the absolute truth at the heart of things. For

“the human intellect does understand some propositions perfectly, and thus in these it has as much absolute certainty as Nature itself has. Of such are the mathematical sciences alone; that is, geometry and arithmetic, in which the Divine intellect indeed knows infinitely more propositions, since it knows them all. But with regard to those few which the human intellect does understand, I believe that its knowledge equals the Divine in objective certainty.”63

This remarkable passage shows how mathematics and geometry came to support the belief that it is possible for humans to know some of the absolute truth of things. Because Euclid’s geometry was believed to be true – a precise description of reality – it provided important evidence that human thought could penetrate the nature of ultimate truth in at least one area. And, if it could do this in the realm of mathematics, then why not in theology too? Paradoxes were not part of this domain of ultimate reality. Ironically, in the twentieth century Kurt Gödel would turn these beliefs on their head in a striking way. Gödel showed that there are statements of arithmetic that can be made using the rules and symbols of arithmetic which it is impossible to show to be either true or false using those rules. The golden road to truth that Galileo loved must always give rise to statements that are unverifiable. Remarkably, Gödel established this extraordinary truth about the limits of mathematics by taking one of the linguistic paradoxes that Galileo rejected and transforming it into a statement about mathematics. But long before Gödel’s work, the absolute truth of mathematics had been undermined. Mathematicians of the nineteenth century had shown that Euclid’s classical geometry was but one amongst many. There were an infinite number of possible geometries, each obeying their own set of self-consistent axioms, different from Euclid’s. These new geometries described lines and figures drawn on curved surfaces rather than the flat ones that Euclid assumed. None of these systems was any ‘truer’ than any of the others. They were each logically consistent, but different, axiomatic systems. None of them had any special claim to be part of the absolute truth at the heart of things. Later, this ‘relativism’ would spread even to logic itself. The simple logic of Aristotle was revealed to be but one system of reasoning amongst an unlimited catalogue of possibilities.

The Galilean distinction between the quagmire of paradox and the sure path of science paved with conjectures and refutations was an important one. It moved science towards the modern era of experimental investigation. No longer were important questions solved by recourse to authorities like Aristotle.64 Human self-confidence was reawakened. It was possible to do better than the ancients. And one did not have to be more inspired to do so. A superior method was what was needed: look and see. If the question was whether or not there could be moons around the planet Jupiter the answer was not to be found by philosophical arguments about the appropriateness of this state of affairs or the natural places for moons to reside, it could be decided by just looking through a telescope.

In this chapter we have traced the fate of Nothing in the hands of philosophers and writers with very different aims. Medieval scholars inherited the world pictures of the Greeks and the mathematical systems of the Far East. Both had distinctive pictures of Nothingness etched into their fabrics. The need to handle the philosophical and theological implications of Nothing was in many ways fuelled by the acceptance of the idea in simple mathematics, where it proved uncontentious and useful. It replaced nothing and it could exist merely as a sign that signalled the divide between profit and loss, prosperity and ruin. It was a symbol with a prosaically positive message. The books balanced; nothing was missed out; all debts were repaid. These were the messages that the zero symbol sent throughout the world of business. Away from the world of numbers there were bigger issues at stake. Nothing was entwined with theological issues of the greatest consequence. Was it the realm from whence the world was made? And if it was, how could it not be something? We are content with the cogency of nothing at all, so long as we do not pursue the idea too closely. But there was an influential Greek view of the nature of things which made the whole concept of nothing at all quite incomprehensible. Plato’s explanation for things saw them as manifestations of the eternal forms behind the appearances. Even if there were no things, no expressions of those eternal forms, the blueprints themselves must always exist. If they didn’t then there would be no way in which the appearance of the world could be in-formed. The eternal forms were the source of the in-formation required to turn the potential into the actual. Nothing was no part of either.

One of things that we have seen in the struggle to make sense of the vacuum and its possible reality is a medieval willingness to conduct experiments, both thought-experiments which appeal to common experience and more contrived sequences of events which demand careful observation and interpretation. This appeal to the behaviour of the world as a source of reliable knowledge did not begin with Galileo, but with him it started to become the only trusted guide to the truth behind everyday things. This was not so much because other guides were mistrusted, merely that they were so hard to interpret clearly and reliably. The medieval philosophers like Bacon and Burley began a tradition of inquiry and a search for the vacuum that would be taken up by Galileo and his contemporaries with a brilliant acuteness. Nothing better displays the phase transition from natural philosophy to natural science.