The Book of Nothing Page 16
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 pro
perty 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: