INTERACTION, Volume I

Date of publication: 15th April, 1997

Proceedings of the First International Symposium, Oxford, December 1995

Topic: Structure: Principles and Applications in the Sciences and Music

Professor Mitchell Feigenbaum
(Theoretical Physics, Rockefeller University)

Unfolding Processes, Emergent Phenomena
and Numbers' Structural Legacy

"It is a pleasure to be here. As I was walking outside and smoking, I noticed that there is a motto on the building. The motto says "Get knowledge, get riches but with all thy gettings get understanding". Basically that is what I will try to talk about. The subject of physics concerns itself with things that are characterized by order, regularity and invariance. These are almost the same words, and I will try to address as I go along different aspects of that.

Since this discussion is related to notions of structure, I would like to make some comments on various facets of understanding that colour what we do when we offer a physical explanation. There is one ingredient of structure that is very clear throughout physics. That is the structure that physics inherits through its discussion through numbers. It would be fatuous to say that physics inherits structure through the existence of the world because, of course, that applies everywhere. The use of number is of fundamental importance. Let me just remind you that the beginning of science in the very serious sense of the word is owed to Pythagoras in his observation that a musical string makes a harmonious sound when it is cut in places that make whole number ratios. And from this deep observation Pythagoras realized that the role of number in describing, in this case harmonious sound, was a natural ingredient of the world. Now there are various ways one can take that. One can take that as number being a description of nature. One has to be a bit careful in just stopping at this point because even though it is true and perhaps only true that number is a description of aspects of nature, that nature is compatible with it, means that somehow number has something or other to do with nature. You probably know that Pythagoreans took that relation very seriously and regarded nature as coequal to number. Moreover, nature was regarded as a discretion, in the sense that it was discrete, built out of pieces, so to speak numbers. It is hard to know what these things mean, one has lost too many references over the millennia. You probably know that in general we don't believe that anymore, that is, we don't believe in general that the world is discrete. That doesn't mean we can't entertain the possibility that it is. The notion (of convention) of a continuum makes sense when one thinks of Zeno's paradoxes, which are paradoxes against the idea of the world being discrete, but rather needing a continuum for its description.

Whether that is the case or not is in fact unclear. Our present belief about the physical world is that it is a continuum, there is a vacuum, and that this vacuum is very rich indeed: each point in this continuum has an infinite number of potential excitations. If one believes that nature is doing something like computing, then there is a difficulty because for each point of the vacuum to know where it should be, what its destiny should be at the very next instant requires infinite computation. This thought occupied Feynman rather deeply for the last decade of his life. He was thinking more in terms of computation, so he came to believe that the world was finite, or again a discretion.

What is at stake in all of this is that as with music, this is a discussion about time. Physics, fundamentally, is a discussion of how things evolve in time. We can sometimes make comments about static things but we believe in the descriptions that follow when we understand also how they evolve in time. We don't have any experimental observation of an instant in time; we only know experimentally about duration in time. Should it turnout that the world has only a finite amount of process in it, then ipso facto time would have to become discrete.

That has some important consequences if it is true. It takes time for something over here to know about something over there. Time is a comparison of change done locally. If something changes over there it takes a while to know it over here, so time becomes a very complicated entity in such a world. Those are just speculative comments. It is to say that we have very many impressions about the world, we follow them so long as they are truthful, and sometimes we still pause and wonder if we are making the right comments.

I am sitting at this impasse. Is the world number or is number merely a description, and I will make some more comments about that later. Let me begin to track forward and say what physics has done. As you know there was a long discussion as to how the planets move, and there was a description of that put together finally by Ptolemy. It is a description that in general we don't believe anymore. One of the things that is important, in the first place, is what it means to talk about the motions of the planets. We can talk about the motions of the planets because the stars happen to be relatively fixed. The stars are fixed; there is an omnipresent background that you can learn to order; and our eyes are very very good at knowing how to see stars. In fact some of our conceptions about the world are inherited from our ability to see stars. Knowing that these stars sit in rather fixed positions allows us to use that background to discuss the motion of the planets.

Again, are the stars just things like numbers, are they just sitting at angles in the sky, or are they entities? The idea that stars are entities is of course a very good idea, because either they are sitting on a sphere very very far away with only angles to see, or they are bodies floating through space. Of course we have come to believe the latter, and we have very strong evidence that raises our belief that stars are, in fact, these big chunks floating in space with various distances between them. This is an important ingredient in physics, it is an ingredient of metrizing. To metrize is to say: how do we deem length. When I look at the stars, do I deem length to be angle, or do I deem it to be so many miles?

This is another important part of our human inheritance through evolution: when we look at very far away things like stars, we see angles. That is, the cortex metrizes the retina to measure angles. By measuring length along the retina, that is a measurement exactly of the angle involved. On the other hand, there is a consequence about this. This same eye that knows how to see stars very well, at finite distances, knows how to see planes. So we have inherited from our ability to see stars the ability to see things on a plane. However, the distances on a plane at a finite distance are not angles. The cortex has also learned how to metrize the retina to see distances on a plane. We have things in the world to which we can assign numerical meaning relative to various metrizations.

The story that unfolded starting around the 15th century, is that we have these planets that move against the background of the stars; "planet" means "wanderer". Perhaps the sun was not rotating around the Earth, but the Earth around the sun, as were the other planets. This is usually attributed to Copernicus, and it is worth knowing that this was not a particularly big advance in thought. Namely, if one looked at the Ptolemaic description and contrasted it to the Copernican one, the reduction in going to Copernicus was very slight. One still needed some 75 parameters to discuss what the planets were doing. This is a principle of parsimony. How important that is, I am not sure. In contrast, the present orbit of the moon is described with 250parameters. One has to be cautious in general of rather general principles.

While Copernicus was not much of an advance, the next step through Kepler and the understanding that Mars was describing an ellipse was a phenomenal advance. Now the number of parameters seriously drops, and one began to have a conception that one was talking about the truth. This is the problem in modern science. It was a problem already then, and it is a much more profound problem now. We have inherited a lot of mathematics. We have brewed more mathematics. We have gotten very clever at knowing how to play with it. Sometimes it is too easy to make a model and fit a number of parameters and embrace experimental information. That is a somewhat dangerous discussion which I won't pursue, but one can gobble up all of the data that one can elicit from a phenomena, and maybe not have learned very much.

The story from Kepler is that planets move along ellipses but there was not an understanding of why this was so. There is a critical intermediary which is Galileo. The thing I care most about Galileo is that he invented a new science, the science of dynamics, and he perceived the notion of inertia.

Inertia basically means that things continue to move unless you do something to them. Inertia is the same as mass, and as energy. The world we think we understand is a world with inertia at each point and energy at each point and that is part of the discussion of what physics is about. Galileo solved a particular problem: Given an object, how do you describe metrically, with numbers, distances, times, how this object falls when you let go? Galileo says many things. I should caution you if you read the original, many of the assertions are false. Sometimes the proofs do not stand up, and sometimes the results are false. Nevertheless, among the wonderful results, there are some truly exquisite.

Galileo claimed that we were going to do a new thing: we are simply going to guess that nature is easy, and so there should be easy laws. The first easy law he assumed was that when I let this thing drop, when it goes through one foot it picks up a certain velocity, and when it goes through two feet it picks up double that velocity. That is, the velocity might be proportional to the distance the object fell through. Presumably he checked that out experimentally. This turns out to be the wrong answer, and he offers some explanations as to why that couldn't possibly be the correct law. Since that one was wrong, he made the next obvious assumption, that the velocity is proportional to the time the object has been falling. That turns out to be the right answer. So Galileo obtained a correct description of how gravity affects things, at least near the Earth.

I should mention historically that Galileo had to do some rather remarkable things. Remember that he discovered the law of the pendulum, not quite correctly, but nevertheless he realized that for small excursions it beat regularly. That became the device to measure time accurately. The way that he measured time in his first experiments was with a very big water container holding water with water flowing out at a fixed rate, so that the amount of time was proportional to the amount of water that flowed out. So part of the development of physics was the beginning to carefully metrize time.

Galileo produced some solutions to this. People only seem to care about weapons, ballistics, trajectories of things; certainly Archimedes did and Galileo was no exception to that. Galileo spelled out the solution of what happens to a projectile. The answer is that the projectile follows a complete trajectory whose form is a parabola. (This neglects air resistance.) Now this is rather important: built into this is the understanding of inertia. It keeps going this way, so its horizontal speed is unchanged, and the speed vertically follows the law that the acquired velocity is proportional to the time of flight. That ends up producing a parabola.

Galileo attempted, for a number of problems, to produce entire solutions. This is not what we think anymore, and this leads to a big discussion. There is a serious next and final transition, which of course is Newton. One can ask the question: what if we have this projectile, and one constant value of gravity in this stratum, then another constant value in this stratum, and so on? Then, throughout each stratum, we know that the solution is a piece of a parabola. Now we have to go about and match up all these different parabolas and get one final complete solution. This is our general understanding of the idea of Newton that inertia is expressed locally: you can say accurately only what happens in a short interval of time. For a short time, the object is basically moving along a parabola, and now our job is to paste together all of those different pieces.

Well if it turns out the particle is free, it would just follow a straight line, then we can start according to Newton with a straight line and piece together all those little straight lines to get a big one. We can do it just as well with Galileo's problem: we start with little pieces of parabolas, paste them together, and get a big parabola. But the story, if you think about it, has totally changed. The conception of Galileo was to tell you that the description of nature was the entire trajectory. The description of Newton is that of a process. What is being said is that the thing that is regular is physics, i.e., the thing that is regular in nature, is the notion that over each small interval of time, processes act.

That process might do easy things, but there is no guarantee of that. It might turn out that as you try to connect the small pieces, the job gets very arduous, and you don't get a simple curve, you get something very complicated. If you follow it further it gets still more complicated, and that of course is completely within the set of possibilities as to what happens if all you know is this basic process.

So that is the main turning point to our modern knowledge. Instead of asking for these precise descriptions of the entire behaviour of something, we specify instead the process, but we do want to understand what the full outcome looks like, and so we put out great efforts to put these pieces together into some sort of form. If things are complicated, what might happen? It turns out that the way things almost always happen, amongst other things, is the problem that Galileo rejected. A general discussion says that we have a process happening (think of the solar system or of the parts of it), that depends only on where the things are. The laws of gravity are not changing from one moment to the next, just the disposition of all of these parts.

The rule of the process says how this part changes in time, and is related to where the parts are. In the most simplistic conception, which is a linearization of the problem, what we say is how something changes over an interval is proportional to where it is. This is exactly the story that Galileo rejected. That story said the velocity acquired is proportional to the distance it fell through. Very generally, that turns out to be a correct description, and it has some very important consequences. Anything that obeys such a law has the property that in this process acting again and again, the modifications that it makes do not grow arithmetically, they grow geometrically. That says that if there is some increase of speed over one second, over two seconds, however much it was multiplied by, it is again multiplied by that amount.

This is more or less what it means to be complicated. I can look at something like a planet that is going around, or all of the planets if I want to think about the solar system, and I can wait say a year, and see where the planets are, or perhaps just one planet. It did not come back to the same point. None of them has come back to the same point. I'll measure the deviation: how closely did it come back to the same point. Perhaps there is an error: there must be an error. I'll wait for another year. I'll discover that there is a new error from where it was two years ago. In general, what I would expect is if the second should be twice as big as the first then it would be twice as much again after three years. So we keep multiplying by two. For a constant increment in time, the errors not only grow in time, they grow exponentially. You keep multiplying and multiplying and multiplying. It turns out that the universe that we live in, the solar system that we live in, doesn't behave very regularly. This is one of the new advances made around 1986. The first indications are that the solar system is not behaving regularly: errors grow. Every five million years, the errors double. That means that over a period of say a billion years, the errors grow astronomically. That means that if we take the Earth, and shift it by a very small amount, say one Planck length, a very small number; then by the time you have waited a couple of billion of years, where this thing will be compared to where it would have been had you not shifted it by one Planck length, is the size of the entire universe. That is, there are things that grow complicated, and they grow complicated in the multiplicative fashion, in such a way that over time, information is eroded to ignorance. This is the discussion of what is called chaos. It is the sensitive dependence to initial conditions, and it means simply that very small deviations, because of that constant multiplicative behaviour, very small errors are very quickly made into very large deviations. We can only know things up to a certain resolution, and that means we don't know beyond a certain point what the destinies are going to be.

In fact, in the last year, a most extraordinary result has been unearthed, and that is not only that the solar system is chaotic, but it is unstable. That means the best we can say at the moment is that there is a good chance, perhaps in a few billion years, that Mercury will end up in a different place, or be expelled from the solar system. Going back over two millennia, we now have some very compelling evidence that things in fact behave very differently from the way we expected twenty years ago. The general belief then was that the solar system was a stable entity. Now it is rather clear that it is unstable. Not just unpredictable, but actually it could fall apart. So that is a rather strong transition coming from Ptolemy.

In trying to make this discussion that there is a multiplicative factor, this is part of a general discussion of "glasses"; what sort of glasses should we wear, in the sense of things that distort things, and distort the way we measure things. So we metrize a certain distance, but on wearing different glasses, I can see that distance to be different. When I am looking at something that isn't naturally reproducing itself, which is beginning to have disorder in it, if I am going to tell you the whole story, there are so many different details that I won't easily be able to tell you anything about what it is. On the other hand, if there are things that don't depend too much on precisely how I see them, these things might be very general. If we look at the paintings of Van Gogh for example, there are flowers, lots of flowers. No two of them are precisely the same, but a question immediately springs to mind: how many flowers, each different form the next, do we have to put in to give a simulation of reality. One of the last paintings that Van Gogh painted, a study called '"Ivy and Moss", is precisely a study in such an issue. We are often faced in the world with things that look very much the same, but only if we are a little bit sloppy. When we look sharper and sharper, we notice very subtle differences. The extent to which the world, that looks very regular to us, is chaotic, is hidden in these subtle things.

If we want to describe them, do I literally have to put down five million copies, or are there things that I can say in more generality? Part of the story of physics is not of telling everything about the world, because that might entail an infinite enumeration of details none of which have any particular meaning. What we strive for is a way to look through the right glasses so that we see things orderly. Physics in the end is a discussion of regular and orderly things. Number has an important role in the world, for some obvious reasons: one of the things that is true for ourselves and for our ancestors is that number count is a reliable thing in the world. If I look outside and there are seven trees, it is not true that at the next instant there are thirty six, and in the next instant one. Number is reliable, and so it is valuable to us.

In some sense, physics is metric for the reason that it is related to ramifications about count. Certainly those things about conglomerates, how many are in this group or that group, these are literally applications of simple arithmetic. If there be rich internal structure, one can notice more relations and so, in some sense, it is clear that number has something to do with description. Why time and things like that are naturally metrizable is a very subtle and complicated issue.

This general business of glasses - how should we see things - that does not make that much difference: we cannot look at all the details, because there are too many. One of the things that we can discuss this way are things that are called fractals. Again, when we look at a tree, there are a lot of leaves which all look the same, but are different. There is a texture that we see. When we look at a cloud, there is a texture that we see. As physicists, we don't want to arbitrarily describe the cloud, we would like that description to have something to do with the understanding of how the model is defined. Nevertheless, we have these complicated things that we can naturally assign texture to. Can we metrize this texture? The metrization of texture, in one sense, is the discussion of fractals.

The fundamental number that one assigns is called the fractal dimension, or Hausdorff dimension of such a thing, and it's just a way to say how much you have to puff it up to sort of get the same thing. This is not exactly right, but let that serve. This is not a brilliant result, but it is a very useful one because, of course, just in saying one number for trees and other things, they will end up having the same fractal dimension, but they are very different things. One wants to metrize over something richer than that.

Fractal dimension has a very simple property if you wear glasses like Coke bottles, glasses that distort everything but no part distorted by as much as say a factor of a hundred, and no part is squeezed by more than a factor of one hundred. If you put on these kind of glasses that make all things blurry, the fractal dimension is an invariant. Part of this discussion of what we do relates to when we talk about what the world looks like: how do we describe the world? This is an issue of finding these invariants, things that are reliable when the details are already becoming complicated. This is not so different from the idea of stationary stars: they are fixed, they are reliable, and we can discuss motions relative to them.

We need to find reliable things because then against those we can begin to find contrasts. In having made this discussion, this whole business of what nature is about, when we talk about things are we just describing them, or are they really that way? Are the laws of physics the way we describe the world, or are they actually what is the world? 1 don't know how to answer that question very well: it is obviously much more simple to say that it is merely a description.

However I would like to illuminate that by an example. This will take me to my last topic. You make a drawing: you draw a circle round a fixed point, having a radius the square root of one, which is just one. You draw another one at the radius square root of two, forty percent more, another at the square root of three, another four, etc.. The circles are getting closer together. I will keep doing this.

Now between the even and odd ones, I will fill them in. So I have not done very much. This object is called a zone plate, and it has some interesting optical properties. The rule of making this is truly simple: I draw a circle for square root of one, two, three, etc. and fill between even and odd ones. That is the rule.

I will now tell a computer to do this, and ask it to print it out on a piece of paper, or to display it on the screen. I will write a simple program. This is a little universe I am crafting, and I have told you its entire theory. This world is nothing more than these circles at these radii, filled in. There is one thing that happens on the computer, which I will try to explain. It can't draw a continuum: it simply does an approximation, so when it displays, it displays raster squares filled in or not. These pixels are painted black if more than half the square is filled in, and white if this is not the case. All we can do on this raster is illuminate a pixel, or not. The same thing is true if I have it printed on a high resolution laser printer. It is exactly the same discussion: within a cell, I can put a dot, or not.

So this is the world that I have made up. Now the question is: what do we see? We do not see this at all. What you see is, already by the second band, you see a ghost looking little thing. As you start moving at the side, you see a curiosity: you see a new copy of these alternating rings. It is a copy of the whole thing, but smaller! And as you move around, you see that there are more of them. They are all over the place, and just look like miniature versions of what the program would write.

(click for large image)

Each one of these things requires its own theory. You can go about and do that, but now the story is sort of amusing. There are an infinite number of these spots, each one to say what it is, requiring its own precise description. Is the theory of this object the theory of each of these spots? Well in some sense, the answer is no because I told you what the theory was: I just need to draw circles! This is a process that is unfolded, the process of drawing these circles and putting them on this raster. That is all there is in the world.

The question that, OK, you happen to see these things, was not the concern of the rule that made this. This nowhere had anything to do with the rule, the evolution, the scheme that crafted it. It is simply an artefact that, by seeing it, we can now ask: can I explain that? So there are things in the universe for which the rules are really simple; the laws of physics are relatively simple. Nevertheless, when these are allowed to act, they open for us all sorts of details. The laws themselves are just a process. Whatever happens to spin out that you notice, is now your job if you wish to discuss it.

This is one of the businesses that makes it clear that things are unclear: is it true that the laws of physics are what nature is, or is it just a description? Let me tell you what the nature of the theory of this is. These bands are not spaced regularly: they move closer and closer to one another; so the repetition rate increases as I move away from the centre. On the other hand, the raster has a definite frequency: how many dots per inch are there. what is happening is a phenomenon where the constant frequency of the raster is beating with the changing frequency of these rings. At those appropriate places where they are in ratios of whole numbers (going back to Pythagoras!) one sees these spots appearing. This is a so called quasi-periodic process, and on this one picture we get to see all the possible commensurations that we could make. This is not a particularly idealised discussion, because it is sort of the same as observing that if you put different clocks against the same wall, they synchronize. It is the same discussion to a good degree of what happens in the heart, which has some pace making cells that the other cells try to oscillate with, but sometimes are unable to: a second frequency arises and one starts seeing those quasi periodic oscillations, which in an advanced case are signs of rather bad fibrillation of the heart.

So this is again a piece of the inherited structure of science because it has inherited number. This is, it has inherited structure because it is describable by numbers. Completely different things show a relation to one another because their mathematical description entails the same things.

Now let me go on. I have started from very simple things, a straight line, a parabola, more complicated ones like the solar system and made-up things that have a large number of complications. What else can we say of the general mode of description of physics? Well, one of the things that we can think about is the gas in this room. We know that it is crafted out of the motions of an immense number of molecules. Now again, it would be pointless to tell you where each one was at an instant and with what speed. This would be an astronomical number of numbers. There is barely enough stuff in the world to even write down those numbers with any degree of precision. If you would see all those numbers, you would not conclude anything from them, and so we need another description. So the discussion we make is an average discussion: it's as though these things are bouncing around and behaving randomly, and rather than tell you what each one is doing, and so on. This is called a statistical description, and the branch of physics that describes it is called statistical mechanics.

This is an example of a historical description, and one talks here of reductionism. Statistical mechanics is the subject that tells us how we describe air, how we describe the materials around us, how we describe macroscopic things. However, those things inherited their behaviour from the fact that they were made up of those underlying microscopic things, which in turn were describable by versions of Newton's laws. So we don't expect by playing with materials, air, paper; etc., that we are going to learn things that will violate the microscopic principles. There is a standard discussion that physicists like to say that if we start over at one level, microscopic, we pass through some statistics, we pass through some chemistry, we pass through some atomic physics. Below that, we start passing through some nuclear physics, pass through some discussions of the constituents of particles, high-energy physics, until somehow in the end we reach some fundamental, indivisible thing. The idea of reductionism is that at some level, nothing should go wrong when passing above to the other. We should not be able to do something at the level of statistical mechanics and discover something that would contradict something in the underlying atomic physics. So far as we know, we have done our job well and no such thing is true. That is not saying very much. However, there is another standard discussion that then likes to say that once you have understood the bottom level, you have already understood the top level. That discussion l take exception with because it is to say that knowing the bottom level has armed you with all of the insights and all of the questions and behaviours that you can anticipate. This drawing of square root radii rings is again a story that violates that - simply crafting these things. It's at the higher level of our looking at these conglomerates, not just the rules that spun them out, that is beginning to reveal that there is a new structure. In some ways the story is not very good. It turns out that we have no way of deriving statistical mechanics from the underlying microscopic physics. We can sort of do it, but it's not the end of the story. It is regrettable, and there are conceivably deep reasons for that, but we don't know them.

But that is not perhaps the most interesting point. As I said, if you think that by just knowing what these point things are doing - the atoms - you will know what will happen at the next level, you are in for some big surprises. One of the surprises that you are in for is the notion of phases. A given material like water appears as vapour, liquid and ice. The transformations of passing from one of these to the other are called phase transitions. There is a kindred thing to that: if I take a piece of metal that is magnetizable, I can magnetise it, but if I heat it up too much it will lose its magnetisation. This ability to have ordered behaviour amongst all the parts is a curiosity that does not appear microscopically, it is a consequence of very many of these things being looked at when they are together.

So first of all, there are new phenomena. One can call these emergent phenomena. One could not have guessed by doing the quantum mechanics of atoms that you would get a transition from liquids to solids. It turns out that when you look better at how these phase transitions occur, the answers are universal. This is one of the grand results of the 60's, finally directly accounted for with an idea called renormalization group by Wilson in 1970. The upshot of this is I can look at water, nitrogen, magnets. It doesn't make a difference what I look at. When I see what happens in a phase transition, the numbers I measure are the same. Once I remove some sizes, of course: am I talking about volume, magnetisation, etc. Once I remove some overall size, when I look at what is happening, the results are metrically, numerically exactly the same, independent of what the starting material is. That says it's independent to a very high degree of the underlying microscopic behaviour. And so as an upshot of this, at a higher level not only are there new phenomena that exist, but their description would be pointless to understand from the microscopic level because all sorts of different microscopic theories can give rise to the exact same macroscopic behaviour of the object that you are looking at.

This idea is referred to as universality, and again when you look at how things move, it turns out that when things get more complicated in certain simplified circumstances it again turns out that it doesn't make a difference what forces you use. You write down Newton's equations for the dynamics, meaning that you specify the force: we now see the kind of motion that that force entails. Well, it turns out when things are complicated, although only in certain regimes so far as we understand at the moment, this identical kind of universality occurs. Just as things start getting chaotic in their behaviour it doesn't make any difference what the details of the forces are, and if you only look in that region, you would never be able so guess what the forces are, because the behaviour doesn't care about them.

One again runs into the game: the underlying process we know microscopically is right, however that doesn't necessarily assist us in understanding the so-to-speak emergent, more complicated phenomena which end up having their own interrelations, and those interrelations are substantially independent of that.

So I think that at that point I will simply make some conclusions. One of the things that I have tried to say throughout this discussion is that physics has one obvious structure, that is the structure of number that is inherited because we have noticed that there are metrical relations that maintain themselves, that are reliable, that we can see throughout the physical world, and so the relations between numbers themselves already gives some structure to physics.

On the other hand, if you think you know now what is happening, the most important part is, of course, that the world is full of surprises."

©1997 The Tureck Bach Research Foundation. All rights reserved.