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

Chapter 5. Whatever Happened to Zero?

“The fool saith in his heart that there is no empty set. But if that were so, then the set of all such sets would be empty, and hence it would be the empty set.”

Wesley Salmon

ABSOLUTE TRUTH – WHERE IS IT TO BE FOUND?

“As lines, so loves oblique may well
Themselves in every angle greet
But ours, so truly parallel,
Though infinite can never meet.”

Andrew Marvell, ‘Definition of Love’1

Like the Grand Old Duke of York, who marched his men to the top of the hill and marched them down again, nineteenth-century physicists had been busily filling the ancient void with ether and emptying it out again. In the meantime, what had been happening to zero, that handy little circle that provided the final piece in the jigsaw of symbols that went to make our modern system of arithmetic?

During the nineteenth century, mathematics began to move in a new direction and its scope expanded beyond the paths mapped out by the ancients. For them, mathematics provided a way of making precise statements about quantities, lines, angles and points. It was divided into arithmetic, algebra and geometry, and formed a vital part of the ancient curriculum because it offered something that only theology would also dare to claim – a glimpse into the realm of absolute truth. The most important exemplar was geometry. It was the most impressive and powerful instrument wielded by mathematicians. Euclid created a beautiful framework of axioms and deductions that led to truths called ‘theorems’. These truths led to new knowledge of the motions of the planets, new techniques for engineering and art; Newton’s greatest insights were achieved by means of geometry.

Geometry was not seen as merely an approximation to the true nature of things, it was part of the absolute truth about the Universe. Like part of some holy writ, the great theorems of Euclid were studied in their original language for thousands of years. They were true, perfectly so, and they provided human beings with a glimpse of absolute truths. God was many things but he was undoubtedly also a geometer.

We begin to see why mathematics was of such importance to theologians and philosophers. With no knowledge of mathematics you might have been persuaded that the search for absolute truth was a hopeless quest. How could we fathom its bottomless complexity given the approximate and incomplete nature of our understanding of everything else in the world around us? How could a theologian claim to know anything about the nature of God or the nature of the Universe in the way that medieval philosophers seemed to do so confidently in their pronouncements about the vacuum and the void? Their justification was in the success of Euclid’s geometry. It was the prime example of our success in understanding a part of the ultimate truth of things. And if we could succeed there, why not elsewhere as well? Euclid’s geometry was not just a mathematician’s game, a rough approximation to things, or a piece of ‘pure’ mathematics devoid of contact with reality. It was the way the world was. A similar exalted status was afforded the system of logic that Aristotle introduced as the means by which the truth or falsity of deductions made from premises could be ascertained. Aristotle’s logic was accepted as being true and perfectly representative of the working of the human mind. It was the one and only way of reasoning infallibly.2

Euclid’s geometry is a logical system that defines a number of concepts, makes a number of initial assumptions, sets down what rules of reasoning are to be allowed, and then allows an edifice of geometrical truths to be deduced by applying the rules of reasoning to the concepts and axioms. It is rather like a game of chess. There are pieces and rules governing their movement together with a starting position for all the pieces on the board. Applying the rules to the pieces produces a sequence of positions for the pieces on the board. Each possible configuration of pieces that can be reached from the starting position could be regarded as a ‘theorem’ of chess. Sometimes one encounters inverse chess puzzles that challenge you to decide whether or not a given board position could have been the result of a real game or not.

Euclid’s geometry described points, lines and angles on flat surfaces. It is now sometimes called ‘plane geometry’. He set out definitions of twenty-three necessary concepts and five postulates. To get the flavour of how pedantically precise Euclid was, and how little he took for granted, here are a selection of his definitions:3

Definition 1: A point is that which has no part.

Definition 2: A line is a breadthless length.

Definition 4: A straight line is a line that lies evenly with the points on itself.

Definition 23: Parallel straight lines are straight lines that, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

Euclid’s aim was to avoid using pictures or practical experience. All truths of plane geometry must be deduced by using these definitions and five other axioms or ‘postulates’ from which everything follows by logical reasoning alone. This arena for plane geometry was circumscribed most potently by one of its axioms, the fifth, which stated that parallel lines never meet.4 Usually it is known as the ‘parallel postulate’. There had always been special interest in this axiom because some mathematicians suspected that it might be an unnecessary stipulation: they believed it could be deduced as a logical consequence of Euclid’s other axioms. Many claims were made at different times to have proved the parallel postulate from the other axioms, but all were found to have cheated in some way, subtly assuming precisely what was to be proved along the way.

The great success of Euclidean geometry had done more than merely help architects and astronomers. It had established a style of reasoning, wherein truths were deduced by the application of definite rules of reasoning from a collection of self-evident axioms. Theology and philosophy had used this ‘axiomatic method’, and most forms of philosophical argument followed its general pattern. In extreme cases, as in the works of the Dutch philosopher Spinoza, philosophical propositions were laid out like the definitions, axioms, theorems and proofs to be found in Euclid’s works.5

This confidence was suddenly undermined. Mathematicians discovered that Euclid’s geometry of flat surfaces was not the one and only logically consistent geometry. Carl Friedrich Gauss (1777–1855), Nikolai Lobachevski (1793–1856) and Janos Bolyai (1802–1860) all contributed to the revolutionary idea of giving up the quest to prove Euclid’s parallel postulate from his other axioms and, instead, see what happens if one assumes that it is false.6This revealed that the fifth axiom was by no means a consequence of the other axioms. In fact, it could be replaced by another axiom and the system would still be self-consistent.7 It would still describe a geometry but not one that exists on a flat surface.

There exist other, non-Euclidean, geometries that describe the logical interrelationships between points and lines on curved surfaces (see Figure 5.1). Such geometries are not merely of academic interest. Indeed, one of them describes the geometry on the Earth’s surface over large distances when we assume the Earth to be perfectly spherical. Euclid’s geometry of flat surfaces happens to be a very good approximation locally only because the Earth is so large that its curvature will not be noticed when surveying small distances. Thus, a stonemason can use Euclidean geometry, so can a tourist travelling about town, but an ocean-going yachtsman cannot.

This simple mathematical discovery revealed Euclidean geometry to be but one of many possible logically self-consistent systems of geometry. All but one of these possibilities was non-Euclidean. None had the status of absolute truth. Each was appropriate for describing measurements on a different type of surface, which may or may not exist in reality. With this, the philosophical status of Euclidean geometry was undermined. It could no longer be exhibited as an example of our grasp of absolute truth. Mathematical relativism was born.

Figure 5.1 A vase whose surface displays regions of positive, negative and zero curvature. These three geometries are defined by the sum of the interior angles of a triangle formed by the shortest distances between three points. The sum is 180 degrees for a flat ‘Euclidean’ space, less than 180 degrees for a negatively curved ‘hyperbolic’ space and more than 180 degrees for a positively curved ‘spherical’ space.

From this discovery would spring a variety of forms of relativism about our understanding of the world.8 There would be talk of non-Euclidean models of government, of economics, and of anthropology. ‘Non-Euclidean’ became a byword for non-absolute knowledge. It also served to illustrate most vividly the gap between mathematics and the natural world. Mathematics was much bigger than physical reality. There were mathematical systems that described aspects of Nature, but there were others that did not. Later, mathematicians would use these discoveries about geometry to discover that there were other logics as well. Aristotle’s system was, like Euclid’s, just one of many possibilities. Even the concept of truth was not absolute. What is false in one logical system can be true in another. In Euclid’s geometry of flat surfaces, parallel lines never meet, but on curved surfaces they can (see Figure 5.2).

These discoveries revealed the difference between mathematics and science. Mathematics was something bigger than science, requiring only self-consistency to be valid. It contained all possible patterns of logic. Some of those patterns were followed by parts of Nature; others were not. Mathematics was open-ended, uncompleteable, infinite; the physical Universe was smaller.

Figure 5.2 Lines on flat and curved surfaces, where ‘lines’ are always defined by the shortest distance between two points. On a flat surface only parallel lines never meet; on the spherical surface all lines meet whilst on a hyperbolic space many lines never meet.

MANY ZEROS

“The ultimate goal of mathematics is to eliminate all need for intelligent thought.”

Ronald Graham, Donald Knuth & Owen Patashnik9

The discovery that there can exist logically self-consistent geometries which are different from Euclid’s was a landmark.10 It showed that mathematics was an infinite subject. There was no end to the number of different logical systems that could be invented. Some of those logical systems would have direct counterparts in the natural world, but others would not. Only a fraction of the possible patterns of mathematics are used in Nature.11 From now on, some new choices would have to be made. What mathematical system is appropriate for the problem under study? If we wish to survey distances we need to use the right geometry. Euclid is no good for determining distances on the Earth’s surface which are great enough for its curvature to be important.

The proliferation of mathematical systems (see Figure 5.3) led to the notion of what is now called ‘mathematical modelling’. Particular pieces of mathematics help us describe aerodynamic motion but if we want to understand risk and chance we may have to use other mathematics. On the purer side of mathematics, it was recognised that there exist different mathematical structures, each defined by the objects (for example, numbers, angles or shapes) they contain and the rules for their manipulation (like addition or multiplication). These structures have different names according to the richness of the rules that are allowed.

One of the most important families of mathematical structures of this sort is that of a group. It is a precise prescription for a collection of objects that are related in some way. A group contains members, or ‘elements’, which can be combined by a transformation rule. This rule must possess three properties:

a.     closure: if two elements are combined by the transformation rule, it must produce another element of the group.

b.    identity: there must be an element (called the identity element)12 which leaves unchanged any transformation it is combined with.

c.     inversion: every transformation has an inverse transformation which undoes its effect on an element.

These three simple rules are based on properties that are possessed by many simple and interesting procedures. Let’s consider a couple of examples. First, suppose that the group elements are all the positive and negative numbers (… −3, −2, −1, 0, 1, 2, 3, …). The group transformation rule will be addition (+). We see that this defines a group because the closure condition is obeyed: the sum of any two numbers is always another number. The identity condition is obeyed. The identity element is zero, 0, and if we add it to any element it is left unchanged by +. The inversion property also holds: the inverse of the number N is −N so that if we combine any number with its inverse we always get the identity, zero; for example 2 + (−2) = 2 − 2 = 0.

Note that if we had taken our elements to be the same natural numbers but the transformation combining them to be multiplication rather than addition and the identity element to be 1, then the resulting structure is not a group. This is because the inversion property fails for all numbers other than +1 and −1. The quantity that we need to multiply, say, the number 3 by to give the identity, 1, is ⅓, which is not a whole number and so is not another element of the group. If we allow the elements to be fractions, then we do have a group with transformation defined by multiplication.

We notice that in these two examples the identity operation which leaves an element of the group unchanged is a null operation. In the first example of adding numbers it corresponds to the usual zero of arithmetic. Its status as the identity element of our group is guaranteed by the simple property that N + 0 = N for any number N. In the second example the identity element is not the usual zero at all. The null operation for multiplication is provided by the number 1 (or, as a fraction 1/1, which is the same thing). The usual zero is not a member of the second group.13

Figure 5.3 The structure of modern mathematics, showing the development of different types of structure, from arithmetics, geometries and algebras. The simple natural numbers can be found at the heart of the network.

The elements in the second group structure are quite different from those in the first. The zero in the first group is quite distinct from that in the second group. Similarly, in every mathematical structure in which an element producing no change appears we must regard this ‘zero’ or ‘identity’ element as logically distinct from that in other structures.

When mathematicians were interested only in Euclidean geometry and arithmetic it was reasonable to regard mathematical existence and physical existence as being the same things. The discovery of non-Euclidean geometries, other logics and a host of other possible mathematical structures defined only by specifying the rules for combining their elements to generate new elements, changed this presumption.14

Mathematical existence parted company with physical existence. If the structure being invented on paper was free from logical inconsistency, then it was said to have mathematical existence. Its properties could be studied by exploring all the consequences of the prescribed rules. If a bad choice had been made initially for the elements and rules of transformation of a mathematical structure so that they turned out to be inconsistent with each other, then the structure was said not to exist mathematically.15 Mathematical existence does not require that there be any part of physical reality that follows the same rules, but if we believe Nature to be rational then no part of physical reality could be described by a mathematically non-existent structure.

This explosion and fragmentation of mathematics (see Figure 5.3) has unusual consequences for the concept of zero. It creates a potentially infinite number of zeros. Each separate mathematical structure, fanned into mathematical existence by a judicious choice of a self-consistent set of axioms, may have its own zero element.16 That zero element is defined solely by its null effect on the members of the mathematical structure in which it lives.17

The distinct nature of these zeros that inhabit different mathematical structures is nicely illustrated by an amusing paper written by Frank Harary and Ronald Read for a mathematics conference in 1973, entitled ‘Is the null graph a pointless concept?’18

To a mathematician, a graph is a collection of points and lines joining some (or all) of the points. For example, a triangle made by joining up three points by straight lines is a simple ‘graph’ in this sense; so is the London Underground map. The null graph is the graph that possesses no points and no lines. It is shown in Figure 5.4.

There is a real difference between our old friend, the zero symbol, that the Indian mathematicians introduced long ago to fill the void in their arrays of numbers, and the zero or null operation that is needed to signify no change taking place in exotic mathematical structures. This zero operator is clearly something. It acts upon other mathematical objects; it follows rules; without it, the system is incomplete and less effective: it becomes a different structure.

This distinction between the traditional zero and other null mathematical entities is most spectacularly illustrated by the introduction of a definite notion of a collection, or a set, of things in mathematics. There is, as we shall see, a real and precise difference between the number zero and the concept of a set that possesses no members – the null, or empty, set.

Figure 5.4 The null graph!19

Indeed, the second idea, pointless as it sounds, turns out to be by far the most fruitful of the two. From it, all of the rest of mathematics can be created step by step.

CREATION OUT OF THE EMPTY SET

“A set is a set
(you bet; you bet!)
And nothing could not be a set,
you bet!
That is, my pet
Until you’ve met
My very special set.”

Bruce Reznick20

One of the most powerful ideas in logic and mathematics has proved to be that of a set, introduced by the British logician George Boole. Boole was born in East Anglia in 1815 and is immortalised by the naming of Boolean logic/algebra/systems after him. He was responsible for the first revolution in human understanding of logic since the days of Aristotle. Boole’s work appeared in a classic book, published in 1854, entitled The Laws of Thought.21 It was then developed in important ways to deal with infinite sets by Georg Cantor between 1874 and 1897.

A set is a collection. Its members could be numbers, vegetables or individual’s names. The set containing the three names Tom, Dick and Harry will be written as {Tom, Dick, Harry}. This set contains some simple subsets; for example, one containing only Tom and Dick {Tom, Dick}. In fact, it is easy to see that given any set we can always create a bigger set from it by forming the set which contains all the subsets of the first set.22 The sets in this example have a finite number of members, but others, like that containing all the positive even numbers {2, 4, 6, 8,… and so on}, can have an infinite number of members generated by some rule.

Boole defined two simple ways of creating new sets from old. Given two sets A and B, the union of A and B, written A∪B, consists of all members of A together with all members of B; the intersection of A and B, written A∩B, is the set containing all the members common to both A and B. If A and B have no members in common they are said to be disjoint: their union is empty. These combinations are displayed in Figure 5.5.

One further idea is needed in order to use these notions. It is the concept of the empty set (or null set): the set that contains no members and is denoted by the symbol ∅, to distinguish it from our zero symbol, 0, of arithmetic. The distinction is clear if we think of the set of married bachelors. This set is empty, ∅, but the number of married bachelors in existence is zero, 0. We can also form a set of symbols whose only member is the zero symbol {0}.

We need the concept of the empty set to deal with the situation that arises when we encounter the intersection of two disjoint sets; for example, the set of all the positive even numbers and the set of all the positive odd numbers. They have no members in common and the set that is defined by their intersection is the empty set, the set with no members. This is the closest that mathematicians can get to nothingness. It seems rather different to the mystic or philosophical idea of nothingness which demands total non-existence. The empty set may have no members but it does seem to possess a degree of existence of the sort that sets have. It also possesses some similarities with the physical vacuum that we have already met. Just as the vacuum of nineteenth-century physics had the potential to be a part of everything, and has nothing inside it, so the empty set is the only set that is a subset of every other set.

All this sounds rather trivial but it turns out to have a remarkable pay-off. It allows us to define what we mean by the natural numbers in a simple and precise way by generating them all from nothing, the empty set. The trick is as follows.

Define the number zero, 0, to be the empty set, ∅, because it has no members. Now define the number 1 to be the set containing 0; that is, simply the set {0} which contains only one member. And, since 0 is defined to be the empty set, this means that the number 1 is the set that contains the empty set as a member {∅}. It is important to see that this is by no means the same thing as the empty set. The empty set is a set with no members, whereas {∅} is a set containing one member.

Figure 5.5 Venn diagrams23 illustrating the union and intersection (C) of two sets A and B.

Carrying on in this way we define the number 2 to be the set {0, 1}, which is just the set {∅,{∅}}. Similarly the number 3 is defined to be the set {0, 1, 2} which reduces to {∅,{∅},{∅,{∅}}}. In general, the number N is defined to be the set containing 0 and all the numbers smaller than N, so N = {0, 1, 2, … N-1} is a set with N members. Every one of the numbers in this set can be replaced by their definition in terms of nested sets, like Russian dolls, involving only the concept of the empty set ∅. Despite the typographical nightmare this definition creates, it is beautifully simple in the way that it has enabled us to create all of the numbers from literally nothing, the set with no members.24 This curious foundation for sets and numbers on the emptiness of the null set is nicely captured in a verse by Richard Cleveland:25

“We can’t be assured of a full set
Or even a reasonably dull set.
It wouldn’t be clear
That there’s any set here,
Unless we assume there’s a null set.”

These strange sets within sets are mind-boggling at first. The incestuous way in which sets refer to themselves is not easy to get a feel for. But there is a more graphic way of visualising them26 if we think about the part of our experience where the same self-reference constantly occurs – the process of thinking. Let’s picture a set as a thought, floating in its thought balloon. Now just think about that thought. The empty set, ∅, is like an empty balloon but we can think about that empty thought balloon. This is like creating the set that contains the empty set {∅}. This is what we called the number 1. Now go one further and think about yourself thinking about the empty set. This situation is {∅,{∅}}, which we call the number 2. By setting up this never-ending sequence of thoughts about thoughts, we produce an analogy for the definitions of the numbers from the empty set, as shown by the cartoons in Figure 5.6.

SURREAL NUMBERS

“In the beginning everything was void and J.H.W.H. Conway began to create numbers. Conway said, ‘Let there be two rules which bring forth all numbers large and small. This shall be the first rule. Every number corresponds to two sets of previously created numbers, such that no member of the left set is greater than or equal to any member of the right set. And the second rule shall be this: One number is less than or equal to another number if and only if no member of the first number’s left set is greater than or equal to the second number, and no member of the second number’s right set is less than or equal to the first number.’ And Conway examined these two rules he had made, and behold! they were very good.”

Donald Knuth27

Figure 5.6 The mental analogy for the creation of the numbers from the empty set. A ‘set’ is represented by a thought and the empty set by an empty thought. Now think about that empty thought to generate the number 1, and so on.

The fascination with using the empty set to create structure out of nothing at all didn’t stop with the natural numbers. Quite recently, the ingenious English mathematician and maestro of logical games, John Conway, devised an imaginative new way of deriving not just the natural numbers, but the rational fractions, the unending decimals, and all other transfinite numbers as well, from an ingenious construction.28 This population of children of nothing have been called the ‘surreal’ numbers by the computer scientist Donald Knuth,29 who provided a novel exposition of the mathematical ideas involved by means of a fictional dialogue which traces his own exploration of Conway’s ideas. Knuth has another serious purpose in mind in this story, besides explaining the mysteries of surreal numbers. He wants to make a point about how he believes mathematics should be taught and presented. Typical teaching lectures and textbooks are almost always a form of sanitised mathematics in which the intuitions and false starts that are the essence of the discovery process have been expunged.30 The results are presented as a logical sequence of theorems, proofs and remarks. Knuth thinks that maths should be ‘taken out of the classroom and into life’, and he uses the surreal numbers as the prototype for this informal style of exposition. Here is something of the flavour of Conway’s creation.

There are only two basic rules. First, every number (call it x) is made from two sets (a ‘left set’ L and a ‘right set’ R) of previously constructed numbers, so we write it down as31

x = {L|R}. (*)

These sets have the property that no member of the left set is greater than or equal to any member of the right set. Second, one number is less than or equal to another number if and only if no member of the first number’s left set is greater than or equal to the second number, and no member of the second number’s right set is less than or equal to the first number. The number zero can be created by choosing both the right and the left set to be the empty set, ∅, so

0 = {Ø|Ø}.

This definition follows the rules: first, no member of the empty set on the left is equal to or greater than any member of the right-hand empty set because the empty set has no members; second, 0 is less than or equal to 0. With a little thought, the rule can be extended to make the other natural numbers. We have Ø and 0 to play with now and there are only two ways of combining them, which yield 1 and −1, respectively

1 = {Ø|0} and −1 = {0|Ø}.

Carrying on in the same vein, we just put 1 and −1 into the formula (*) and use it to generate all other natural numbers. Thus the positive number N allows us to generate N+1 by combining it with the empty set through

{N|Ø} = N + 1

and for the negative numbers we have

−N −1 = {Ø|−N}.

Operations like addition and multiplication can also be defined self-consistently.32 The empty set behaves in a simple way. The empty set plus anything is still just the empty set and the empty set multiplied by anything else is still the empty set.

Again, this is all very pretty but what does it enable us to do that we couldn’t do with the old scheme that we discussed above? The pay-off comes when Conway extends his scheme to include more exotic numbers in the L and R slots. For example, suppose one takes the set L to be an infinity of natural numbers (called a countable infinity) 0, 1, 2, 3,… and so on, for ever. Then we can define infinity to be33

inf ={0,1,2,3,… |Ø}

Now put inf in the right-hand slot and we have a peculiar definition for infinity minus 1, an infinite number less than infinity!,

inf −1 = {0,1,2,3,…| inf}

and also

1/inf = {0| ½, ¼, ⅛, 1/16,…}

and even, the square root of infinity:

None of these peculiar quantities had been defined by mathematicians previously. Starting from the empty set and two simple rules, Conway man-ages to construct all the different orders of infinity found by Cantor, as well as an unlimited number of strange beasts like √inf that had not been defined before. Every real decimal number that we know finds itself surrounded by a cloud of new ‘surreal’ numbers that lie closer to it than does any other real number. Thus the whole of known mathematics, from zero to infinity, along with unsuspected new numbers hiding in between the known numbers, can be created from that seeming nonentity, the empty set, Ø. Who said that only nothing can come of nothing?

GOD AND THE EMPTY SET

“You know the formula: m over nought equals infinity, m being any positive number? Well, why not reduce the equation to a simpler form by multiplying both sides by nought? In which case, you have m equals infinity times nought. That is to say that a positive number is the product of zero and infinity. Doesn’t that demonstrate the creation of the universe by an infinite power out of nothing?”

Aldous Huxley34

Our discussion of the unexpected richness of the empty set leads us to take a look at its relationship to the infamous ontological argument for the existence of God.35 This argument was first propounded by Anselm, who was Archbishop of Canterbury, in 1078. Anselm conceives36 of God as something than which nothing greater or more perfect can be conceived. Since this idea arises in our minds it certainly has an intellectual existence. But does it have an existence outside of our minds? Anselm argued that it must, for otherwise we fall into a contradiction. For we could imagine something greater than that which nothing greater can be conceived; that is the mental conception we have together, plus the added attribute of real existence.

This argument has vexed philosophers and theologians down the centuries and it is universally rejected by modern philosophers, with the exception of Charles Hartshorne.37 The doubters take their lead from Kant, who pointed out that the argument assumes that ‘existence’ is a property of things whereas it is really a precondition for something to have properties. For example, while we can say that ‘some white tigers exist’, it is conceptually meaningless to say that ‘some white tigers exist, and some do not’. This suggests that while whiteness can be a property of tigers, existence cannot. Existence does not allow us to distinguish (potentially) between different tigers in the way that colour does. Despite its grammatical correctness, it is not logically correct to assert that because something is a logical possibility, it must necessarily exist in actuality.

We see that there is an amusing counterpart to these attempts to prove that God, defined as the greatest and most perfect being, necessarily exists because otherwise He would not be as perfect as He could be. For suppose that the empty set, conceived as that set than which no emptier set can be conceived, did not exist. Then we could form a set that contained all these non-existent sets. This set would be empty and so it is necessarily the empty set! One can see that with a suitable definition of the Devil as something than which nothing less perfect can be conceived, we could use Anselm’s logic to deduce the non-existence of the Devil since a non-existent Devil has a lower status than one which possesses the attribute of existence.

LONG DIVISION

“Now: heaven knows, anything goes.”

Cole Porter

The mathematical developments we have charted in this chapter show how a great divide came between the old nexus of zero, nothingness and the void. Once, these ideas were part of a single intuition. The rigorous mathematical games that could be played with the Indian zero symbol had given credibility to the philosophical search for a meaningful concept of how nothing could be something. But in the end mathematics was too great an empire to remain intimately linked to physical reality. At first, mathematicians took their ideas of counting and geometry largely from the world around them. They believed there to be a single geometry and a single logic. In the nineteenth century they began to see further. These simple systems of mathematics they had abstracted from the natural world provided models from which new abstract structures, defined solely by the rules for combining their symbols, could be created. Mathematics was potentially infinite. The subset of mathematics which described parts of the physical universe was smaller, perhaps even finite. Each mathematical structure was logically independent of the others. Many contained ‘zeros’ or ‘identity’ elements. Yet, even though they might share the name of zero, they were quite distinct, having an existence only within the mathematical structure in which they were defined and logically underwritten by the rules they were assumed to obey. Their power lay in their generality, their generality in their lack of specificity. Bertrand Russell, writing in 1901, captured its new spirit better than anyone:

“Pure mathematics consists entirely of such asseverations as that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is of which it is supposed to be true … If our hypothesis is about anything and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.”38

Pure mathematics became the first of the ancient subjects to free itself of metaphysical shackles. Pure mathematics became free mathematics. It could invent ideas without recourse to correspondence with anything in the worlds of science, philosophy or theology. Ironically, this renaissance emerged most forcefully not with the plurality of zeros that it spawned, but with the plethora of infinities that Georg Cantor unleashed upon the unsuspecting community of mathematicians. The ancient prejudice that there could be potential infinities, but never actual infinities, was ignored. Cantor introduced infinities without end in the face of howls of protest by conservative elements in the world of mathematics. Cantor was eventually driven into the deep depression that overshadowed the end of his life, yet he vigorously maintained the freedom of mathematicians to invent what they will:

“Because of this extraordinary position which distinguishes mathematics from all other sciences, and which produces an explanation for the relatively free and easy way of pursuing it, it especially deserves the name of free mathematics, a designation which I, if I had the choice, would prefer to the customary ‘pure’ mathematics.”39

These free-spirited developments in mathematics marked the beginning of the end for metaphysical influences on the direction of the mathematical imagination. Nothingness was unshackled from zero, leaving the vagueness of the void and the vacuum behind. But there were more surprises to come. The exotic mathematical structures emerging from the world of pure mathematics may have been conceived free from application to Nature, but something wonderful and mysterious was about to happen. Some of those same flights of mathematical fancy, picked out for their symmetry, their neatness, or merely to satisfy some rationalist urge to generalise, were about to make an unscheduled appearance on the stage of science. The vacuum was about to discover what the application of the new mathematics had in store for time and space and all that’s gone before.