Lecture 6: Uncertainty

Introduction

1 I am doing my best to keep you up to date with what I think is happening in these lectures.

2 I am looking for a lost paradise. During my religious life I gained a feeling for a paradise, the catholic heaven. In time this turned out to be an impossible paradise.

3 The loss of this particular paradise has not shaken my faith that paradise, which I here call peace, is possible. My hope is to find peace. Some might call me a fool, claiming that peace is inherently impossible. My intention is to prove that opinion wrong.

4 In these lectures, I am on the track of a mathematical picture of peace that not only defines paradise, but suggests where to look for it and provides a means of estimating how close we are and how close we can get.

The fall

5 We find our archetypal story of paradise in the book of Genesis. This book and the rest of the Bible explains how we arrived in the Garden of Eden, it explains why we were thrown out, and it describes many efforts to recover the promised land. It is a powerful myth, and a real one. It has inspired our culture for thousands of years.

6 There are many schools of though about how to interpret the Bible. I consider it to be like any other literature, written by people trying to understand themselves and their environment. The Bible is special because it is the distillation of thousands of years of written and oral tradition, recorded by people who had a very deep and direct experience of life.

The tribe

7 My guess, from studying the Bible and other learned tomes, is that the lost paradise of the Bible is the tribe. Very few of us have ever lived in a tribe. I haven't, but I'll define a tribe as the human environment that takes care of me as a whole human being.

8 The tribe is an environment for all those things we have in common with the other animals. In a tribe I can be born and educated, eat, talk, smoke, fight, make love, have children and bring them up, lie in the sun, dream, and be understood for what I am.

9 Not only can I do these things, but I can learn from infancy how to do them well and pleasurably by direct contact with people who have experienced the same life. In a tribe the young and the old overlap and interact, mingling creativity and experience to survive.

10 In more abstract terms, the tribe is a human structure made out of human beings. The tribe was once a basic survival unit of the human species. It contained enough individuals and information to replace itself continually and so it was able to exist for very much longer than a human lifetime.

11 On the whole tribes have been destroyed by the march of civilisation. We have been left with the family and the nation. A family is a much smaller unit than a tribe, and is incapable of independent existence. There are many reasons for this, but the fundamental one is genetic. The children in a family cannot breed with one another without introducing an unacceptably high level of genetic error in their offspring.

12 A nation is a much larger unit than a tribe. Although people identify with their nationality, the members of a nation are predominantly strangers to one another and do not interact with the closeness and familiarity of people in a tribe. The nation has taken over many of the supporting roles of the tribe through social security and national health, but it tends to do these things in a remote inhuman way.

13 The result of this inhumanity, I suggest, is a very high level of stress which exhibits itself in the extremes of patriotism and war, and has ultimately come to a head with the threat of global nuclear war.

14 Is the structure we call the tribe gone forever? I say no, because I feel that tribe is a structure which arises naturally and consistently out of human beings of the environment is right. In the next lecture, on communication theory, I will tell you what I mean by 'if the environment is right'.

Recursion

15 In this lecture I wish to continue relating mathematical ideas about structure to the observed physical world. Last week I tried to explain how a more complex structure is formed out of a simpler one using as an example the special and general theories of relativity.

16 The essential idea is that the structure of the universe is recursive. Let us imagine that we have a machine we will call the creator. We feed it on a simple structure, and it puts out a more complex structure. We feed it on that more complex structure, and it puts out an even more complex structure, and so on.

17 The universe that this machine produces will be layered. Each layer will correspond to one cycle of the machine. The layers will be different. But they will all have something in common, which is the ingredient added by the creator.

18 In addition, each layer will contain all the features of the layer before it, since that is what it is made from. When atoms unite into molecules they are modified slightly, but their essential structure remains. When we wish to understand molecules, we learn about atoms first.

19 The old tribal structure is behind us, destroyed forever, but the same structure is also in front of us, transformed into something new, because the world is recursive. We cannot go back to the old paradise, but we can look forward to a new one.

The creator

20 Remember our slogan: the universe is mathematics in the flesh! The gist of these lectures is that the fundamental structure of the universe and everything in it, including human groups, is simply the structure of the number system, and that the number system is nothing other than an expression of the assumption that a viable structure is consistent.

21 A moment ago we introduced a machine called the creator to explain the recursive structure of this universe. What is this machine?

22 Everything must die. This seems in accord with experience and appears to be a consequence of Goedel's theory.

23 If everything must die, the only permanent structures are those which can reproduce themselves.

24 Because the universe is recursive, more complex structures are made out of simpler structures.

25 So we are led to ask: what is the simplest permanent structure; what is the simplest thing that can reproduce itself? This surely must be the machine, that thing which when fed with nothing can at least reproduce itself.

26 And what is this? My answer is that the machine is simply the shape imposed upon things by logical confinement, that is the logical requirement that what exists must be consistent. From this, via Cantor's theory, comes the numbers. I think that if we can understand the numbers in all their richness, from one, two, three to the transfinite numbers, we can understand creation.

27 I assume that at its very simplest that structure is to be found in the general theory of relativity, and that the general theory is a faithful interpretation of the theory of number, albeit a partial one.

28 The general theory is a local theory. It is the structure that arises when we assume that all the points in space are autonomous, so that they respect one another's independence when they communicate.

29 Local theory is inherently consistent, since points do not destroy one another when they find themselves unable to communicate. Instead they generate a new language which enables them to unite themselves into a new consistent whole.

30 In the physical theory of relativity, these points are interpreted as geometric objects. But if we adhere to the principle of local theory, these points may be anything, human beings for instance.

Quantum mechanics

31 This new consistent whole is not the last word, however. Remember Goedel's theorem: consistency implies incompleteness. We have mentioned Goedel's theorem often, but now we come to its physical realisation in quantum mechanics.

32 Quantum mechanics is a large and complex subject. I will introduce it at a great distance as it were, so that you don't see too much detail.

33 Imagine that each definite object in the universe is an island surrounded by a sea of uncertainty. The seas of uncertainty belonging to different objects coalesce, since there is nothing to differentiate them.

34 The sea is the communication medium. On the sea a ship (or a bottle) can go from any point on any coast to any other point. Because the sea of uncertainty is continuous and lies between all of the islands of certainty, it enables any two of them to communicate.

35 Out of the infinity of routes available on the sea, some have more traffic than others. The routes of heavy traffic give discernible structure to the communication between islands of certainty, but the fact remains that any two points whatever can communicate through the sea.

36 Theories like general relativity describe in a clear mathematical form the ideal communication links between objects in the same sea. The actual relationships, however, are rather more uncertain. Ships sink and go off course, they encounter storms and crews go mad.

37 Relativity describes certain structures in the universe as they would be with perfect communication. Quantum mechanics describes the actual act of communication, with its certainty and uncertainty.

The quantum of action

38 What is quantum mechanics? First, lets look at the name. A mechanic is someone who takes things apart and puts them together. Mechanics, then,. is the general science of how things fit together.

39 A quantum is a discrete thing, what we have been calling a definite and separate object. Continuous mechanics thinks of the world like water or clay, something smooth which can be moulded into any shape with no limitations whatever on fineness of detail.

40 Quantum mechanics thinks of the world as something made of discrete parts, like the parts of a car, each with definite properties. These parts are usually called particles.

41 Particles exist in time as well as space. They have limits in space analogous to the shape of a car part. They have limits in time which are their creation and annihilation. They have a lifetime. They are born and they die.

42 Each birth or death is an action, like the tick of a clock. It is an action that can be measured with a discrete unit, called the quantum of action, whose physical value is a number called Planck's constant.

43 In quantum mechanics, particles are not half born or half dead. They either exist or they don't. This reminds us of our definition of the elements of an aggregate and gives further meaning to the phrase definite and separate object.

Uncertainty in the theory of numbers

44 We can approach a slightly deeper understanding of quantum mechanics through the theory of numbers.

45 One of the simplest operations imaginable is the addition of one. We have already see that this is the operation by which we generate the natural numbers. It can work on any number.

46 Adding one is a definite action which leaves a definite result. It changes one into two, for instance.

47 In the scientific world, to know something is to measure it. There is a mathematical measure of action which enables us to learn a little more about the action of adding one.

48 One is not two. This is what we mean when we say symbols are definite and separate objects. It was the action of adding one that caused this definiteness and separateness.

49 We are talking about a specific action, adding one, but we want our measure to apply to all actions. We therefore define the unit of action as the distance between two definite and separate objects. If we call one of these objects yes and the other no, the measure of action becomes the distance between yes and no.

50 This measure was first used by Claude Shannon in his mathematical theory of communication. We'll meet that theory next week.

51 Imagine you are asking yourself a yes-no question. She loves me? She loves me not? You are in the classical state of human romantic uncertainty. The measure of this uncertainty is one bit.

52 I cannot tell you how to answer this question. You can find out by tossing coins, picking daisies or actual experience. When you know for sure, yes or no, you have lost one bit of uncertainty and acquired one bit of information. Information and uncertainty are duals of one another and the measure of both is therefore the same.

53 Goedel's theorem guarantees that there will always be one bit of uncertainty in every situation, for it says that every consistent system is incomplete, and so that it must generate questions that it cannot answer.

54 My program is to see the physical universe as the realisation of mathematics. I am therefore led to identify the bit which is the mathematical measure of uncertainty with Planck's constant, which is the physical measure of uncertainty.

Probability

55 How does quantum mechanics work? It is a theory, and like any theory, its aim is to make sense of the world that we find around us. Like relativity, it is a theory expressed in mathematical language.

56 It is a difficult theory to understand because it has to unite certainty and uncertainty. One the one hand the universe has incredible precision. Some times are calculated and can be measured to one second in a million years or better.

57 On the other hand, there is the radical uncertainty predicted by Goedel's theorem which says we cannot even decide if some things exist or not.

58 The link is the theory of probability. Take a coin. Idealise it so that it is perfectly balanced and has such a sharp edge that it will always fall either heads or tails.

59 This coin is a definite and separate object, yet every time we spin it we do not know whether it will come down heads or tails. While it spins, we have one bit of uncertainty. When it lands, we have one bit of information.

60 You might say OK, you don't know which way the coin will land, but it can be calculated, so the information is there in the universe. All you need to know is exactly how hard you flip it, how much friction there is in the air, and so on.

61 This is the old deterministic view that the universe works like clockwork, so that if we know exactly what is happening now we will be able to calculate all of the future.

62 This view is wrong. All the physical evidence and Goedel's theorem point to the same fact: uncertainty is built into the universe. There are gaps between the definite and separate objects that we can never know about because there is nothing there to be known.

63 Probability theory models uncertainty, but the probabilities have structure. There is only one bit of information in the motion of a coin because it will always come down either heads or tails.

64 The probability structure of the throw of two coins is different. There are now two bits of uncertainty, one for each coin. We might get two heads, two tails, or a head and a tail.

65 Probability theory tells us how to calculate the odds. If you play two-up, you know the answer. There is one chance in four of two heads, one change in four of two tails, and two chances in four of a head and a tail.

Superposition

66 Quantum mechanics proceeds in a similar manner. It uses the probabilities of some events to calculate the probabilities of other events. The way it does this is called the principle of superposition.

67 Everything in the universe exists in spacetime. The structure of uncertainty in spacetime is modelled by waves. You can get a feel for this by thinking about waves on water.

68 Imagine the top of a wave is heads and the bottom of it is tails. If you keep throwing a coin you will sometimes get heads and sometimes get tails, just as if you swim in the sea you will sometimes be on top of a wave and sometimes on the bottom.

69 When two waves meet , they go through one another and mix to produce new waves. You can watch this if you throw two stones into a pond some distance apart and watch the complex patters that are formed when the ripples mix.

70 This mixing is called superposition. It can be described mathematically, and it is just this mathematical description which is used in quantum mechanics to combine the probability waves of different events.

71 Superposition is local. The height of a wave on a pond can be calculated right there on the spot from the heights of all the incoming waves. None of the waves is changed by the others. The result contains them all.

72 You can hear superposition. The sound of a violin does not change the sound of a flute. They are both there in the finished effect and you can listen to them together or pick them out separately. For a quantum mechanic, the universe is something like an incredible orchestra with every distinct particle as an instrument.

73 I am trying to give you a picture of quantum mechanics, but my images are far from perfect because the waves of quantum mechanics are not physical objects like waves in water and air. They are part of our mathematical model, probability waves in a sea of uncertainty.

74 Because they dwell in the region of uncertainty, they cannot be observed any more than we can observe that heads or tails have equal probability. All we can observe is how the coin actually falls. By throwing a coin enough times we can guess that heads and tails have equal probability. By studying the mathematical theory of probability, we can convince ourselves that this is mathematically consistent.

75 On the pond we can observe the superposition of waves coming from two different particles. In quantum mechanics we cannot do this. The only probability waves we can superpose are those belonging to the same particle. In other words, a particle decides its own fate by interacting with itself.

76 This brings us back to the dualism we find in every symbol. A symbol is a definite and separate thing which may relate to an infinity of other things. The word food is a symbol. It is a thing. It points to the infinite cuisine of the universe.

77 The symbol and what it points to are distinct. One is not the other, and so the gap between them is one bit. On the other hand, they contain the same information. One is not the other and yet they are the same. Only uncertainty can resolve this dilemma.

78 In the formalism of quantum mechanics, the two distinct selves of each particle are represented by complex numbers. The natural numbers have a natural order, one, two, three. The complex numbers have no natural order, and are therefore specially suited for dealing with uncertainty.

79 These complex numbers interact by superposition to produce a real number, which is the probability of finding the in a given state.

80 I cannot take this story any further here. If you get a chance find a text about quantum mechanics and read at least as far as the description of the famous two slit experiment. This will introduce you to the mysterious relationship between certainty and uncertainty.

81 The mystery of quantum mechanics is reflected in the mystery of our own moods, decisions and actions. Goedel's theorem and the uncertainty principle hold for every definite and separate object, not just subatomic particles where quantum mechanics was invented. That includes me.

82 My decisions and actions arise from the superposition of unobservable influences within myself. If even I am uncertain about myself, how am I to be judged by others? How indeed. Local theory says I cannot. Only my observable actions can be judged.

Conclusion

83 The universe is mathematics incarnate. But mathematics in the flesh cannot be just the hard edged perfection that we find in rigorous mathematical monographs. Mathematics in its physical realisation must obey its own theorems. It must respect Goedel's theory, which says that there are some decisions that cannot be made on principle, and Turing's theory, which says that there are some decisions which can never be made in practice. From the very hardness of mathematics springs an equal softness.

84 The duality of uncertainty and information assures us that in the real world hardness and softness are also duals, and that you cannot be hard unless you are soft and vice versa.

85 The formalism of quantum theory demonstrates this dualism at work in the very heart of modern physics. So far quantum theory has never failed.

86 If we identify the hardness with law and the softness with love, we find that love and law are duals of one another. If we identify the law with mathematical theories like general relativity which describe with total precision the relationships of points, and love with the sea of uncertainty that lies between these certain laws, we see that love is the communication out of which law is born, and law is the structure that guarantees love.

87 Jesus Christ ushered in an era of human development by pointing out that there is but one law, and that is the law of love. We can identify love with consistency, hate with inconsistency. We have now delved deep enough into the foundations of mathematics and physics, sciences founded on consistency, to see what he was talking about from a new perspective.

Originally broadcast on 2BOB Radio, Taree, NSW on 23 July 1987

Books

Click on an "Amazon" link in the booklist at the foot of the page to buy the book, see more details or search for similar items

Related sites:

Copyright: You may copy this material freely provided only that you quote fairly and provide a link (or reference) to your source.