Lecture 7: Communication

Introduction

1 This is lecture 7. It is about communication. We have hit a magic number and a magic subject simultaneously, which I take to be a good omen.

2 My idea is that the universe is creative, and that in a creative universe there is always room for everybody to live in peace.

3 I have asserted that the universe is creative on mathematical, rather than observational grounds. Clearly, the universe looks creative. We see new things coming into being every moment. The problem is that many have attributed this creativity to a creator outside the universe.

4 The theory of transfinite numbers, as I have interpreted it, asserts that if the universe is consistent, it must be creative, and needs no outside creator. This means that we can study the creative power of the universe by looking at it, rather than by trying to communicate with an invisible creator outside.

5 Since we are all looking at the same universe, we must eventually come to peaceful scientific agreement about how the creative power of the universe works, and use that understanding to create peace.

Symbols

6 I have already pointed out that number is too narrow a word for the entities that mathematics deals with. Instead I prefer the term symbol, which I take to mean any definite and separate thing at all.

7 Remember the lecture about symbols? A symbol has duality. On the one hand, it is a definite and separate thing like a loaf of bread.

8 On the other hand it is a pointer, connecting to something else. The loaf of bread is connected with warm yeasty smells, jam and cream, the relief of hunger, a nice cosy kitchen full of good people, animals and wet washing; and many other things.

9 Duality has both a finite and an infinite quality. The loaf of bread is a finite thing. Our culture uses it to symbolise many things, but in theory it could symbolise anything at all.

10 We are free to establish our own links between symbols. That is what writing is, putting words together. Each word has its own dual, its range of meanings. When we string words together, their meanings, that is their duals, communicate with one another and intermesh, giving meaning to the sentence as a whole. This meshing is decoded symbolically in our minds, so that we understand the written or spoken word.

11 Mathematics strives to be completely general, and concentrates on the exploration of the infinities contained in the duality of its symbols. In general the mathematical specification of a point, that is a definite and separate object, involves an infinite series.

12 The mathematical dual of a symbol is everything in a universe that is not that thing, an infinity. I assume that the information in the definite part of the symbol is equal to the information in the infinite part.

Science

13 This equality of information is the lifeblood of science. Science works on both ends of the dual at once. Our experience, enriched by experiment, gives us a grip on the finite side of the universe.

14 We explore the infinite side of the duality with mathematics, discovering the most general structures that are possible. We know, because we hold both mathematics and the universe to be consistent, that they will fit together.

15 The history of science has been a continual dialogue between mathematics and experience. New experiences demand new mathematics. New mathematics makes sense of the relationships between experiences.

16 There has, nevertheless, been an imbalance in the picture. It is my feeling that most people who have written about the question hold that the world of mathematics is very much bigger and more perfect than the world of experience.

17 Mathematics proceeds without error, they say. The real world is full of error of all sorts. Mathematics is good. The world is bad. This we might see as the scientific implementation of the doctrine of original sin.

18 My assumption is that the universe is mathematics incarnate. More formally, I am saying that the world of things and the world of mathematics are two sides of one symbol, the universe.

19 Naturally I wish to exploit the duality of mathematics and experience to the hilt in my search for a theory of peace. A theory of peace, like any scientific theory, is nothing other than a set of connections between mathematically defined objects and objects defined by experience.

20 In a fully developed theory, we would expect to find a mathematical object corresponding to despair and a mathematical object corresponding to joy.

21 I mention these two feelings because they are close to creation. There are times when I sit here looking at the blank paper and feel that I will never write again. There are other times when things come so fast that I am too excited to write.

22 My personal ebb and flow of creative energy seems to be reflected around the world. Sometimes the economy drags and everybody goes into depression. Sometimes its boomtime, and we are all excited with the hope of wealth.

23 I am not talking about money alone. The pace of scientific discovery varies in a similar way. Will we ever see a time again like the beginning of this century, when physics and mathematics exploded beyond all classical bounds.

War and peace

24 I hope we do. We have got to break through the depression that the bomb symbolises. When we do break through I think the measure of our joy and relief will be the gap between the transfinite number that represents a world at war, and the number that represents a world at peace.

25 Imagine that we have found, beyond reasonable doubt, the mathematical object that corresponds to war and the mathematical object corresponding to peace. From the mathematical relationships between these two mathematical objects, we can deduce things about the relationships between the experiences of war and peace.

26 We may, by exploring the mathematical theory, find out how the mathematical object called peace can eliminate the mathematical object called war.

27 We can then try to find the actions in the world of experience corresponding to this mathematical conquest of war, and thus work toward peace with a sound mathematical foundation.

Communication

28 That is what I am trying to do. This activity, trying to establish correspondences between the duals of mathematics and experience is an example of communication.

29 Communication, for my purposes, is the action of establishing one to one correspondences between the elements of two sides of a duality. In arithmetic terms, communication is multiplication.

30 I wish to emphasise two features of communication. The first is that despite the uncertainty which we know to be part of the world, we can in theory make communication as perfect as we like. The second is that it is symbolic.

31 Let us start with the perfection of communication. We learn about it from the mathematical theory of communication, first published by Claude Shannon in 1948.

32 The theory deals with two elements, the source of the message, and the channel through which the message travels. Right now I am the source of the message and the radio system, from the microphone here to the speaker on your radio, is the channel.

33 The question that the theory of communication answers is this: how accurately can the information coming out of the channel represent the information entering the channel from the source? In radio, communication theory is concerned with the fidelity of sound reproduction.

35 If this radio system were perfect, the sound you are hearing now would be exactly the same as the sound you would hear if you came and listened to me in the studio.

36 If you are close to the transmitter and have a receiving system as good as our transmitter, the communication between me and you would be as close to perfection as we need to get. Our FM broadcast system is a high fidelity channel. Under ideal conditions (and with a few qualifications of course) it is better than the human ear.

37 Things are not always so good. If you are in a marginal reception area the sound you are getting will be full of noise. If you are a long way from the transmitter, you might be getting almost pure noise with a few human like gurgles in the background.

38 You could be getting nothing at all. Perhaps you are not even tuned to 2BOB. That means that I am not saying anything to you, and this communication channel between us does not exist.

39 From the point of view of communication theory, the fidelity of a communication channel depends upon its capacity to carry information. If the communication channel can carry information faster than the source puts it out, then communication can be perfect.

40 In theory, anyway. Shannon's theorems which prove this proposition formally, are somewhat non-constructive. They tell us the conditions for perfect communication, and they give us some idea how to realise these conditions, but they do not tell us exactly how to do it.

41 Channel capacity depends upon two things. The first is called the bandwidth. This is the frequency range available to the channel. In terms of numbers, it is the rate at which the channel can count.

42 The bandwidth of human hearing is about 20 kilohertz, that is twenty thousand cycles per second. We see immediately that AM channels cannot carry sound with perfect fidelity. Their bandwidth is less than ten kilohertz. On the other hand the bandwidth of FM channels is 150 kilohertz. There is bandwidth to spare.

43 The second influence on channel capacity is signal to noise ratio. Every communication channel exists in the midst of billions of other communication channels, and often they interfere with one another.

44 In communication the channel you want is called the signal and all the rest is called noise. The signal to noise ratio measures the strength of the signal compared to the strength of the noise.

45 A certain minimum signal to noise ratio is necessary for intelligible speech. In a quiet room, where there is little noise, you can achieve this minimum by talking quietly. If you are trying to talk to someone at a rock concert, however, the competing noise is very strong. You have to raise your signal level and yell.

46 If you are far from the FM transmitter, the signal is weak compared to the noise: the signal to noise ratio is low, and you get noisy reception. FM can overcome this to some extent. Remember than channel capacity depends upon both bandwidth and signal to noise ratio. Spare bandwidth can help to make up for poor signal to noise ratio.

47 This is why FM reception can be better than AM at low signal levels. The spare bandwidth in the FM signal is used to overcome the noise. Shannon's theorems tell us that this is possible, but it does not tell us how. The answer lies in coding, and finding good codes is often difficult.

48 The FM system uses rudimentary coding that yields some improvement but breaks down when things get too bad. The coding system used on the Voyager spacecraft is much closer to perfection. It enables a 20 watt transmitter on the spacecraft to send us almost perfect pictures of Jupiter, Saturn and Uranus over billions of kilometres.

49 The basic feature of coding is that the channel capacity is better used if you send messages in big blocks rather than small ones. Shannon's theorems tell us this, although they do not tell us how to construct the code.

50 Remember last week I pointed out that because everything has a finite lifetime, only those things that can reproduce themselves are permanent. Reproduction and communication are almost identical. At the moment, we are trying to reproduce structures from my mind in you mind.

51 The theory of coding suggests one reason why life forms become more and more complex as evolution progresses. A complex organism is in effect a big block of code. It is therefore more efficient in its use of the communication channel that joins its past to its future.

52 This enables it to bring down the error rate in its reproduction and so become more fit to survive.

Symbolic communication

53 The second feature of communication is that it it symbolic. In the course of these lectures we have symbolically taken the universe apart and put it together again. If we were to do this physically (with superhuman help, of course) it would take incredible amounts of energy.

54 In fact, during this whole lecture, your FM receiver will have picked up barely enough energy from the transmitter to shift a speck of dust.

55 Here we see physically why the pen is mightier than the sword, and why symbolic communication is so attractive. Energy is effectively the same as space. If you want more space to live, you have the choice of getting more energy, or using the energy you have more efficiently.

Love

56 This idea opens a window on the force that drives us to communicate. Why do we go to so much trouble to communicate with one another?

57 The drive to communicate, whatever it is, has obviously got something to do with love. This connection seems good enough to be a working definition of love: love is the force that impels things to communicate.

58 The best description of this force I know comes from probability theory. Imagine that you are a blind invulnerable fly, nuclear powered maybe, who for some manic reason flies all over the place at random, bouncing off walls, floors and ceilings, never stopping.

59 If you have ever done physics, you will see that I am describing the behaviour of a gas particle.

60 I, the mad scientist, trap you in a couple of rooms joined by an open door. One of the rooms is big, with a volume of 300 cubic metres. The other is smaller, with a volume of 100 cubic metres.

61 I spend a long time watching, timing how long you spend in each room. When I look at my results, I see that you have spent three times as long in the big room as in the small room. The big room must attract you, and yet you are blind and you only know the world by colliding with walls.

62 Probability explains the attraction naturally enough. Because your motion is random, you chances of being in any particular volume of space are the same. Since there is three times as much space in the big room, there is three times as much chance that you will be in there. Averaged over a long time, this chance becomes a fact, just as many throws of a coin will yield roughly equal head and tails.

63 In physical space, the big room attracts you the fly simply because it is bigger. The same idea carries over to more abstract spaces.

64 Communication creates space by uniting things. Remember it is like multiplication. So it is attractive.

65 The physical energy and spacetime available on the surface of the earth are limited. To escape this limitation, evolution has turned to symbolic communication where there is no effective limit to the space available to use.

66 How much energy did it take Georg Cantor to invent the transfinite numbers? If all the energy consumed in his brain went into the job day and night for twenty years, the transfinite numbers cost the world the equivalent of half a ton of firewood. In fact the computational functions of the brain consume only a fraction of the energy it needs to live.

Science and technology

67 Communication has two aspects, talking and listening; transmitting and receiving. Communication takes time. No message travels faster than the speed of light, and every act of communication requires encoding and decoding of the message. Encoding and decoding are a form of computation, and we know from Turing's theorem that this take time, in some cases infinite time.

68 It has taken me a long time to understand the ideas that I am trying to express here. It may take you a long time to understand what I am saying. Don't panic. Turing's theorem says that this is normal.

69 From a computing point of view, the hardest part of the process is decoding, that is listening. If you know the language, it is easier, although there is till a lot of work to be done to find out what each sentence and each paragraph mean.

70 If you don't know the language, the task is even more difficult and time consuming, but it can be done.

71 The biological component of our humanity is passed from generation to generation encoded on DNA. DNA is the text of the genetic language. As far as I know, all life on earth speaks the same genetic language or dialects of it.

72 When I was conceived the genetic statement that defined me was encased in a small decoding unit called an egg. This egg was encased in a large decoding unit called mum. Mum was encased in ever larger decoding units called the family, the society, the planet and eventually the universe. Mum and the egg enabled my DNA to talk to the rest of the universe and explain how it could shape itself into me.

73 The genetic system based on DNA transmits information with very little error. The rate of change in DNA, that is the rate of evolution, is very slow compared to the cycle of individual life. In the time since we departed from the ancestors we share with the chimpanzees, our DNA has become different from theirs in only about one percent of its characters.

74 Science listens to the universe, hoping to learn its language. Technology talks to it, showing it how to treat us more benevolently by yielding abundant crops and other good things. The language we use when talking technologically is the one we learnt when listening scientifically.

75 The language that science uses is very powerful. Look how much we have changed the planet in the age of science. The power comes from symbolism. Newton's physics can be summarised in half a page. Yet when interpreted physically it is the basis of most civil and mechanical engineering. Add quantum mechanics and statistical mechanics and you cover the whole of electronic and chemical engineering as well.

76 Science gives us words of power because it knows the language of the universe, or at least some of it. It seems to me that there is still part of the language missing. It is the most important part, analogous to grammar or syntax rather than to words or sentences. Although we live in a creative universe, we do not understand creation.

77 This question has been too hard, so we have put it aside, attributing creation to a creator, as yet unspecified and not understood.

78 Because we have not understood creation in a creative world, we have acted as destroyers rather than creators.

79 We need not be ashamed of ourselves for that. We were acting out of ignorance and exuberance. The technological achievements of the last few centuries have been spectacular. In the areas where they have had effect, they have got us fed, clothed, housed, healthy and reasonably well equipped with gadgets like radio stations and washing machines.

80 Before you judge the history of DDT or any technology, you should be aware what it was like in the old days. What would I say about nuclear energy if I thought my child's life depended upon it?

81 Environmental disasters, war and the painful segments of human relationships are communication problems.

82 I have defined communication as the action of establishing one to one correspondences between the elements of two sides of a duality.

83 I believe that the degree of peace we enjoy depends on the fidelity of our communication with the universe. Remember that in this broad statement, you are part of my universe, and I am too. Perfect communication is a matter of bandwidth, signal to noise ratio and coding.

84 On my interpretation, Cantor's theory of transfinite numbers guarantees that the bandwidth is available.

85 The signal to noise ratio measures the strength of the channel we want relative to all the channels we don't want. In a world of ignorance, poverty and violence, the channel of communication between ourselves and the universe remains very weak, despite all our science.

86 Because it is weak, we only hear the loudest things the universe says. Black holes, supernovas, thermonuclear weapons and other energy intensive processes have attracted our attention. Physical injuries and self inflicted diseases let us know we are not paying attention to nature.

87 This is good. We have to start somewhere. We are fortunate that energetic phenomena are relatively easy to understand because they evolved back when the universe was young and simple.

88 The weakness of our communications with the world should not lead us to despair. The theory of communication tells us that it is possible to communicate, no matter how loud the noise, provided we listen for long enough and scan all available frequencies.

89 The hardest part is decoding, that is learning the language of the universe, which I presume to be the language of creation. Here we are up against Turing's theory and the problem of computability.

90 In essence, Turing found that there are messages that can never be decoded because the gap between successive transfinite numbers in infinite and unbridgeable.

91 Maybe things are not that bad. If it is true that the universe is mathematics in the flesh, we can expect to find the transfinite numbers somewhere. When we do find them, they might not be as formidable as they sound.

92 Remember Goedel's theorem and the uncertainty principle from last week. It tells us that where there is definition there is fuzziness, where there is hardness there is softness, and where there is law there is love. In the physical universe you cannot have Cantor without Goedel and Turing at your side also.

93 The evolution of the universe to ourselves and beyond suggests that the decoding problem is not as intractable as it may sound. Perfect communication and perfect peace may still be in the cards. Next week I hope I can say something about the relativity of transfinity.

Originally broadcast on 2BOB Radio, Taree, NSW on 30 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.