The Book of Nothing
Page 6
Zero was like a genie. Once released it could not be restrained, let alone removed. Once words existed for the concept that the zero symbol represented, it was free to take on a life of its own, unconstrained by the strictures of mathematics, and even those of logic. The mathematicians had played a vital role in making legitimate the concept of Nothing in a place where it was easiest to define and control. In the centuries to follow, it would emerge elsewhere in different guises, with even deeper consequences, and more puzzling forms.
“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 madne
ss 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;13 this 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. I
n 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.