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vol 8: A theory of Peace
page 3: Symbol

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Lecture 3: Symbols

Introduction

1 In the first two lectures I outlined a program to produce the theory necessary to guarantee peace on earth. I summarised this task at the end of the last lecture:

Global peace requires global law. Ultimately there is only one law from which all other laws and structures derive, that of consistency. Conflict points to inconsistency.

The conflict between the United States and the Soviet Union indicates that the laws of at least one and probably both of them (and us as well) are inconsistent with the rest of the universe. To prevent nuclear war we must identify and correct this inconsistency.

To identify and correct this inconsistency, we need a theory of consistency, that is mathematics. In the first lecture I stated that my basic physical hypothesis is that the universe is mathematics incarnate. My task in the next few lectures is to explain and illustrate this hypothesis, and then use it to find a way to overcome war.

Symbol

2 In this lecture I wish to use a discussion of symbols to link mathematics and the visible universe.

3 There are two sides to every symbol. On the one hand, a symbol is a definite mark, any definite thing. On the other hand, every symbol points to other things. It has a set of meanings or interpretations.

4 Symbolism is part of life. To the anatomist, the breast is a thing constructed from numerous different tissues whose principal function is to synthesise, store and deliver milk to an infant.

5 On the other hand the breast in its role as a pointer has a range of meanings or interpretations which could easily fill a very large encyclopaedia. Many of these meanings are deeply established in our biological makeup. Many others are to be found in the thousands of religions and cultures that fill the world.

6 Everything in the universe is a symbol. This is clear, because first, every thing is a definite thing; and second, every thing can be used to point to something else.

7 The most versatile symbols in common use are the numbers. We can name anything by matching it with a number. Every car, washing machine, camera, television set and just about every other substantial product has a unique name, its serial number. We have registration numbers, social security numbers, income tax file numbers and telephone numbers. In principle there is no limit to the number of things that may be represented by each number.

8 The term symbol is itself a symbol and points to two things: the symbol as a thing, such as a number, and the symbol as a pointer to something else, such as a particular telephone.

9 There is an essential difference between the two aspects of a symbol. A symbol as a thing is finite. As a pointer it is infinite, since, as we have seen with numbers, a particular thing may be used to point to an infinite variety of other things.

10 We can look at the relationship between the finite and infinite aspects of a symbol from another point of view.

Dualism

11 Take a blank sheet of white paper and put a nice big black mark in the middle. You now have a symbol, or more precisely, you have two symbols. The black mark in the middle of the paper is a symbol. So is the white paper surrounding the black mark.

13 If you sit and think for a while, you can convince yourself that symbols always come in pairs, the mark and the not-mark. You simply cannot make a mark without simultaneously making a not-mark.

14 You will also see that the same information is contained in both. What defines the mark is its boundary, and the boundary is shared by both white and black.

15 The situation is like the relationship between a mould and a casting made from the mould. All the information about the shape of the casting is contained in the casting itself, and exactly the same information is contained in the mould.

Space

16 I have used the image of a mark on a piece of paper to lead you to the mathematical concept of dualism. This image is a symbol which points to the mathematical idea. The image is not yet perfect.

17 The dot is finite; it is the thing or mark aspect of the symbol. If the paper is to represent the pointer aspect of the symbol, it must become infinite. So let us curve the paper round and join its edges to form a ball. At the soe time we reduce the size of the dot until it becomes a point, infinitely small compared to the size of the ball.

18 Now imagine that you are a two dimensional creature living on the surface of the white ball. You have no knowledge of the third dimension, to that the idea of jumping off the ball is meaningless to you. The surface of the ball is your whole world.

19 If you travel around the ball in different directions you will occasionally come close to the black dot, but you will never come to the edge of the ball because in two dimensions the surface of a sphere has no edge.

20 We now have a finite black dot in an infinite white space, an image that recalls to mind the finite and infinite aspects of a symbol which I explained a few moments ago.

21 This image is a powerful one. We are four dimensional creatures in a closed four dimensional spacetime, and our experience is exactly the same as that of a two dimensional creature on the closed two dimensional surface of a sphere. We may go as far as we like in any direction in spacetime, and we will never come to the edge of the universe.

22 Our spacetime has no boundary. We are four dimensional creatures in a closed four dimensional spacetime, and our experience is exactly the same as that of a two dimensional creature on the closed two dimensional surface of a sphere. We may go as far as we like in any direction in spacetime and we will never come to the edge of the universe.

23 Back to the white ball with the black dot. The dot is symbol. The boundless white space around it represents the infinity of things which the dot may point to. This image captures the dualism of a symbol a little better, but we must take one more step to arrive at the full picture.

24 At the moment, we have a white ball with a black dot. Looking at this image in two dimensions, we consider the white surface of the ball to be infinite and the black dot to be finite. One is the dual of the other, and the same information is contained in both.

25 Now let the black dot grow, eating up the white surface until we have a black ball with a white dot. This is the exact reverse of the previous situation. Both dot and space still contain exactly the same information.

26 What I am trying to show you is that in reality, it is impossible to decide what is the symbol and what is the meaning. It all depends on your point of view.

27 The modern masters of symbolism are in the advertising industry. They take a small paper roll of shredded tobacco leaf and use it to point to wild horses, power, wealth, yachts and nights of debauchery in Rio. Or is it the other way around? It is the drug they are pushing.

28 For a mathematician, duals are exactly equivalent. Each throws light on the other. The same information is contained in the mark and the space. Which is mark and which is space depends entirely upon how you wish to look at the symbol.

29 I can show you the power of this idea by applying it to myself. I am a symbol. My dual is all the universe that is not me. Since the same information is contained in both sides of a duality, to study the universe is to study myself and vice versa. Myself and all that is not me are simply two aspects of the same reality.

Mathematics

30 The universe, I claim, is mathematics incarnate. Mathematics is the study of symbols as symbols. It therefore deals with all symbols equally, whether they be a point or a line or a galaxy. If it is true that everything is a symbols, mathematics deals as easily with sexuality as it does with counting or with war and peace. This is why I have enlisted mathematics in my assault on war.

31 We usually associate mathematics with the study of numbers. This definition is a little too narrow for my purposes, and a little out of touch with modern mathematical developments. Instead I define mathematics as the study of symbols. This definition has a long history which I do not have time to touch. I must simply ask you to accept it as another assumption.

32 Perhaps I should warn you that I am not particularly concerned here with mathematical finesse. I simply want to get some ideas across. These lectures are an attack on war. My only concern is to punch through a bridgehead to the possibility of peace, no matter how rough the hole. There will be plenty of time later for mopping up the messy bits.

33 Not only is mathematics about symbols, it uses symbols in its work. This makes it a very self-conscious science, deeply concerned with its own methods and foundations.

34 I say that the universe is mathematics incarnate, and wish to show that the universe is god, able to explain its own origins and its own creation. My aim, in other words, is to show that mathematics can explain mathematics, and that mathematics is a symbol which points to everything.

Cantor

35 We want to know how the universe creates itself so that we can learn how to create peace. Since I claim that everything is a symbol, what we need to know is how symbols come to be.

36 The first part of the answer lies in the work of Georg Cantor. Cantor's most important work, Contributions to the Founding the the Theory of Transfinite Aggregates .

37 Cantor began with a thing he called an aggregate. Modern mathematical terms which have similar meaning are set and class. He defined an aggregate as any collection to a whole of definite and separate objects of our intuition or our thought. The objects in an aggregate are called the elements of the aggregate.

38 For our purposes , an aggregate is a symbol formed by collecting a number of separate symbols together. We are quite familiar with the construction of symbols out of symbols. In written texts, the words are aggregates of letters, the sentence are aggregates of words, and so on.

39 Every aggregate has a definite cardinal number., The cardinal numbers of finite aggregates are the ordinary counting numbers. We in this room form an aggregate. I can count us, one two three ... . The cardinal number of our aggregate is the number of people in the room.

40 Cantor was able to prove that for any aggregate with a certain cardinal number there is another aggregate with a greater cardinal number. This result holds even for aggregates whose cardinal numbers are infinite. Cantor's proof points to the existence of an infinite hierarchy of numbers greater than the whole infinity of ordinary counting numbers. These numbers are called the transfinite numbers.

41 In these lectures I interpret Cantor's proof to mean that any symbol, even an infinite one, generates an infinity of other distinct symbols.

42 Cantor proved this, that is he showed that a symbolic system in which his theorem was not true would be inconsistent. Since we have assumed that the universe must be consistent, we are led by Cantor's theorem to the conclusion that the universe must generate new symbols.

43 More succinctly, Cantor's theorem seems to require that a consistent universe must be creative.

Cantor's paradox

44 Cantor's work caused a great stir in the mathematical world. The aggregate of counting numbers is infinite. This is a consequence of the way the numbers are generated. Each successive number arises by adding one to the number before it. There is no end to this process. No matter how big a number is, we can always add one to it and get a bigger number.

45 Some people saw a potential conflict between the infinity of the counting numbers and the infinity of god. This conflict was overcome by pointing out that although we can go on counting forever, each of the numbers we reach is itself finite, even if it is huge, The fact that there is no end to the counting numbers does not require that an actually infinite number exists.

46 Cantor claimed that actually infinite numbers do exist. The first transfinite number is the first number after all the infinite aggregate of counting numbers.

47 This claim involved Cantor in bitter debate with his teacher Leopold Kronecker. The debate was partly theological, but its important dimension, from our point of view, is its mathematical component.

48 Cantor proved the existence of the transfinite numbers, but he did not produce a transfinite number. He simply showed that the assertion that the transfinite numbers do not exist is inherently inconsistent with the existence of the infinite aggregate of counting numbers, while the assertion that the do exist is consistent with the existence of the counting numbers.

49 This type of proof is called non-constructive, and some claim that it is not good mathematics. I consider non-constructive proof acceptable. In addition, in a couple of lectures' time I will produce an interpretation of the transfinite numbers which allows us to point to them and see that they exist.

50 These problems are almost insignificant compared to Cantor's Paradox, which Cantor himself discovered in 1899. We can imagine an aggregate which is the aggregate of all aggregates. We would expect this to be at least as great as any other aggregates, since it contains all the others. Cantor's proof, however, tells us that for every aggregate there is a greater aggregate. In other words, there must be an aggregate greater than the aggregate of all aggregates, which is a contradiction.

51 This paradox, and other like it, caused great soul searching in the mathematical world. Since we have arrived at a contradiction by a valid line of reasoning, the trouble must be somewhere in the assumptions. If we assume that Cantor's proof is good, the trouble must lie in the aggregate of all aggregates.

52 Since our assumption of the existence of an aggregate of all aggregates leads to a contradiction, we must discard it. There is no aggregate of all aggregates. I interpret this paradox to mean that there is no greatest aggregate, that is there is no last symbol. There is no end to the creative power of the universe.

Interpretation

53 Mathematics is the study of symbols as symbols. It is a creative occupation. AS mathematician spends his or her life thinking up different possible structures and then trying to write the structure down in symbolic form.

54 Every symbol has two realities. On the one had it is a thing. On the other had it is an interpretation. We may consider the chemical or electrical impulses and patterns in a mathematician's brain as symbols which may be interpreted by the written expression of these ideas on paper.

55 The written expression, in turn, may point to physical symbols in the universe. So we have the mathematical theory of the counting numbers, one, two, three ... . These numbers may be interpreted as sheep, so we say one sheep, two sheep, three sheep ... .

56 As with the counting numbers, so with all other mathematical symbols. My claim that the universe is mathematics incarnate is tantamount to the claim that any mathematical structure may be found somewhere in the universe, and, because mathematical structures owe their existence only to the requirement of consistency, we can expect to find all mathematical structures realised in some way everywhere in the universe.

57 We have thus uncovered, through the concept of symbol, an intimate link between mathematics and the universe. Ultimately this link depends upon the fact that both mathematics and the universe are consistent systems.

Conclusion

58 Since human beings became conscious of themselves and began to record their thoughts in writing, we have seen our conception of the size of the universe gradually expand. From Aristotle to Copernicus to Newton to Einstein we have seen the universe grow to an unbounded spacetime at least 20 billion light years around, if that has any meaning.

59 What I now propose is not an expansion in physical size, but in something less tangible. For too long we have though of our universe, our planet and ourselves as second class or defective, in some way puppets who do not know who or what is pulling their strings. I claim that this is not so. This universe is it. In theological terms, it is god.

60 How do we understand god? The only constraint we put on god is that it be consistent. In time, we will see that this is no constraint at all.

61 If we allow non-constructive existence proofs, consistency is the only constraint acting on mathematics. This leads us to identify mathematics and the universe with the slogan The universe is mathematics incarnate.

62 In this lecture we have seen that Cantor's theorem must be true in a consistent universe. As I interpret it, Cantor's theorem tells us that there is no limit to the generation of new symbols from old. There is no limit to the creative power of a consistent universe.

63 This has been the third lecture in my series entitled A Theory of Peace. Printed copies of this and the first two lectures are available from the 2BOB studios for 50 cents per lecture if you collect them yourself, or a dollar posted.

64 I will be giving the next lecture here at the 2BOB studios next Thursdays evening at 8.00 pm. You are welcome to come and join the studio audience.

65 There will be a short music break now, and then we will have time for questions. Thankyou.

 

Originally broadcast on 2BOB Radio, Taree, NSW on 2 July 1987

Books

Cantor, Georg, Contributions to the Founding of the Theory of Transfinite Numbers (Translated, with Introduction and Notes by Philip E B Jourdain), Dover 1955 Jacket: 'One of the greatest mathematical classics of all time, this work established a new field of mathematics which was to be of incalculable importance in topology, number theory, analysis, theory of functions, etc, as well as the entire field of modern logic.'  Amazon  back
Damon, S Foster, A Blake Dictionary: The Ideas and Symbols of William Blake, Brown University Press 1998 Jacket: '... compiled by one of the world's best known Blake scholars; it assembles, synthesizes and interprets the clues to Blakes meaning that are scattered throughout the whole body of his work, both literary and graphic.'  Amazon  back
Hallett, Michael, Cantorian set theory and limitation of size, Oxford UP 1984 Jacket: 'This book will be of use to a wide audience, from beginning students of set theory (who can gain from it a sense of how the subject reached its present form), to mathematical set theorists (who will find an expert guide to the early literature), and for anyone concerned with the philosophy of mathematics (who will be interested by the extensive and perceptive discussion of the set concept).' Daniel Isaacson.  Amazon  back
Hughes, Patrick, Vicious Circles and Infinity: A Panoply of Paradoxes, 1975   Amazon  back
Jech, Thomas, Set Theory, Springer 1997 Jacket: 'This book covers major areas of modern set theory: cardinal arithmetic, constructible sets, forcing and Boolean-valued models, large cardinals and descriptive set theory. ... It can be used as a textbook for a graduate course in set theory and can serve as a reference book.'  Amazon  back
Joyce, James, Finnegans Wake, Faber and Faber 1982   Amazon  back
Kramer, Edna E , The Nature and Growth of Modern Mathematics, Princeton UP 1982 Preface: '... traces the development of the most important mathematical concepts from their inception to their present formulation. ... It provides a guide to what is still important in classical mathematics, as well as an introduction to many significant recent developments. (vii)  Amazon  back
Tymoczko, Thomas, New Directions in the Philosophy of Mathematics: An Anthology, Princeton University Press 1998 Jacket: 'The traditional debate among philosophers of mathematics is whether there is an external mathematical reality, something out there to be discovered, or whether mathematics is the product of the human mind. ... By bringing together essays of leading philosophers, mathematicians, logicians and computer scientists, TT reveals an evolving effort to account for the nature of mathematics in relation to other hman activities.'  Amazon  back

 

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