Illusions of the Mathematical Imagination
A lecture given by Catherine Dalzell (Mrs Catherine Collins). November 19, 1994. Ontario Science Teacher's Conference' Making Connections.
Good Afternoon. I am very pleased to be able to address you this afternoon --- pleased, but also surprised. I am neither a scientist nor a teacher. I am a mathematician by training and profession. I understand that the math teachers have a conference of their own. So when your program chair, Elizabeth Dunning, invited me to speak, my first thought was to wonder what on Earth I could talk about that would be of interest to science teachers.
Well, after some consideration, I decided to talk about mathematics: what it is and what it does. In particular, I want to talk about the imaginative freight that it carries and that can distort the understanding of nature. This is what I call "the illusion". I know a lot about it from personal experience; thanks to the illusion my time in high school science class left me with no great appreciation for science, or understanding of nature. I was one of those good students who get all the right answers, while understanding none of it. I thought physics was quaint. This talk pertains to the conference theme of "making connections", and I hope to make a number of connections involving mathematics and nature that I hope may be of some use in the classroom.
The question about the nature of mathematics is not new, and the number of possible answers is small. I think that everyone would agree that mathematics has something to do with the physical world, since it has been so successfully applied in science. Regarding its precise relationship to nature, three answers are generally given, and two of these are wrong.
The first wrong answer identifies nature and mathematics. Mathematical space is the same thing as physical space. Physical quantity is the same as mathematical quantity. We could call this the Pythagorean answer.
The second wrong answer begins by assuming no relationship between mathematics and nature, mathematics being simply a human construct. Mathematics is then pasted onto nature, and provides models for understanding nature. It is a language for describing nature, and so on. We could call this the positivist error. Positivists can never explain why mathematics actually works.
The third and correct answer is found in Aristotle. It is the hardest to express precisely because it is correct, and it contains aspects of each of the wrong answers. According to this view, mathematical objects are "beings of reason" that is to say human concepts (as in answer 2) but they have been derived by abstraction from physical nature, and so bear an intrinsic relation to nature (as in answer 1).
So now we know what mathematics is; at least I have applied some labels to the issue. But the
errors themselves typically have a history in the development of the student. For instance, the Pythagorean error
is frequently committed by beginners with genuine mathematical aptitude. They get carried away by the equations,
and then see the mathematics instead of nature. The second error is a sort of undergraduate, pseudo-sophisticated
response to the discovery that nature and mathematics are not identical, typically occurring when one first meets
up with non euclidean geometry. In anyone older than that, it's an intellectual cop-out; regretably a cop-out that
you often find in those introductory chapters in textbooks all about the so-called scientific method: math is a
language pasted onto nature, and we paste a different equation every hundred years or so when we have a paradigm
shift. The third answer comes with
Today I want to talk about the Pythagorean error: seeing mathematics instead of nature. I am going to talk about it in connection with motion and the physics of moving particles. In the abstract I said I was going to talk about motion, probability, cause and effect, but I realized there would not be sufficient time to do justice to all of these topics. In any case, it is the same deal for all of them.
Mathematical objects, the concepts whereby we understand these phaenomena, have characteristics that the physical world does not: precision, necessity and immutability for instance. If you impute these to the natural world, you will be importing into nature a degree of solidity and immutability that nature simply does not possess. It would be like saying that because a photograph is fixed and two-dimensional, that nature itself must be fixed and two-dimensional. So what happens to someone who lives in a world of photographs? What happens when he suddenly realizes that he is been living with images and comes alive to the real thing?
I have a fine example here in this book by John Briggs on Fractals. I am sure you will have seen many publications of this kind, if not this particular example. The subtitle reads "discovering a new aesthetic of art, science, and nature." By "aesthetic" I assume that he means a way of seeing things, a way of seeing the beauty in things. The book contains pictures of natural phaenomena --- coastlines, trees, cracks, forked lightning, water and mountains --- together with amazingly convincing simulations of these shapes, using the mathematics of fractals and chaos. The extraordinary thing here is that Briggs is reacting to this as if he was seeing these things for the very first time. The text is awash with excitement. All of a sudden one is allowed to see waterfalls and mountains.
But this is crazy. How could he have failed to notice a tree or a crack on a wall? Well, I have to confess that I understand his excitement all too well. Naturally I have spent a lifetime seeing cracks and trees and waterfalls. But there was a sense in which I did not see these things either until I read about fractals. And then I was excited too. I had seen jagged objects in nature, but I saw them with my artistic eyes: the eyes that had been trained by Turner, Constable and the Group of Seven. They were there, but they were not scientifically there. A crack in a wall was not a scientific fact to me. In a sense, I never saw a jagged line until I studied fractals. I imposed on nature all the straight lines and smooth curves that science told me were there. This, in part, is what I mean by "seeing mathematics instead of nature". The physics I studied used smooth functions and ordinary differential equations to model motion and change, so all that I officially granted to the nature was smoothness and determinism. The jagged edges somehow didn't count.
It can be very insidious. Briggs, and the rest of us, would not have known that we were seeing mathematics instead of nature unless the mathematics had changed. Thanks to the development of computers, the past thirty years have witnessed a tremendous expansion in the repertoire of mathematical models that mathematicians are prepared to consider. Before there was simply no way. We knew these things were out there, but using them was out of the question, so scientists had to view the world through the lens of ordinary linear equations with constant coefficients and the whole bit. The models of classical physics were extremely simple, because they had to be.
But to return to Briggs, the amazing thing here is the mood of liberation that inspires his book. The new mathematics has become almost a principle of renewal, one that will solve all the world's problems of industrial greed and environmental destruction. Listen to what he says at the end when he is summing up:
He goes on to say that the philosophic lesson of fractals and chaos is that they show us "the inherent value of living in a world that springs beyond our control."
This is a pretty heavy load for a bundle of non-linear differential equations to carry, which is all we are dealing with here, together with a fast computer and some colour graphics. The load is also being carried by the wrong ship.
It is certainly true that we "live in a world that is alive, creative, and diversified because its parts are unified, inseparable, and born of an unpredictability ultimately beyond our control" but there is no mathematical result in existence that will prove to you that this is the case including fractals and chaos theory.
Basically Briggs is having a religious experience, and this is something that should be food for thought for those of you who teach in the separate school system. He realized that his old view of nature was a mathematical construct, and suddenly he saw nature as it is --- as something unified and diverse, moving and yet structured --- and it was a liberating experince. He saw that nature is a creation. Its source is outside our control. Something was wrong with his perception of nature, and it involved mathematics.
So I think that we need to look a little more closely at what mathematics is actually doing to the imagination. Not everyone, of course, falls into the Briggs trap. To return to my own experience, after high school I went to university, resolved to study no more science. I went to the University of Toronto and I studied pure mathematics. After the initial shakedown in first year, we were left with classes of around 25 students: half of us intended to pursue mathematical careers, and the other half were destined for physics and the more mathematical sciences. It soon emerged that there were two opposing cultures in the classroom. The mathematicians would look down on the physicists on the grounds that physics lacked rigour; and in reply, the physicists would speak darkly about something they called physical intuition. Apparently you needed it to do physics and if you were not born with it, well it was just no good. You might as well become a mathematician.
Physical intuition: what's that? Obviously something that I lack most of the time, but I think I had a glimpse of it once. I was driving along the St. John river West from Fredericton and I stopped at the Mactaquac dam. There is a park there and I stopped the car and got out to rest. As I stood there drinking a Coke and staring into space, I found that I was looking at a suspension bridge. I saw its graceful curve, and the two pilons --- which is what I normally see: a geometric shape --- and then suddenly I felt that I could actually see how it worked. It was as if I could see which way the forces were pulling, and the weight of the entire bridge was passed down through the pilons. It was a wonderful experience: for the first time, I could see the material world actually doing things: pulling, lifting, falling and withstanding. For a moment it ceased to be mere geometry and become something real. And yet, the geometry remained: the bridge still looked the same; it had the same pattern.
This, I think, is physical intuition. And if one could encourage such an experience in students one would, I think, be doing very well indeed. Encouragement, in my opinion, is probably necessary for most students. Most of us can identify life when we see it in a living animal: there is a power to move and to react there that is completely absence in inert things, and one has to be very well educated indeed before one loses the ability to see this and reduces life to mechanical principles. However inert objects have a sort of presence and a capacity for work, comparable in them to life in the living. Unfortunately, precisely because their activity is mathematical, it is easy to lose the intuition of physical being with the formalism that explains it.
Notice that physical intuition is something very positive and that it leaves mathematics intact. It is not achieved by the quick fix of error 2, of calling mathematics "a language used to describe nature" whose relationship to nature is not clear. I believe that knowledge of the material world is, at its most precise, mathematics. But it is true in much the same way that a photograph is true -- and just as the photograph freezes movement, so does mathematics. Physics is about things that move, to the extent that they move. But in mathematics, motion does not exist. Motion is not a mathematical idea. This is very important, because mathematics is constantly used to describe motion (rightly) and if one insists upon providing a rational and precise description of motion, mathematics is the answer --- but something has been lost in the translation. And this will be true of whatever mathematical models one chooses to use.
Briggs is euphoric because fractals have extended his scientific vision to phenomena not covered by classical physics or modern cosmology, but it is a temporary reprieve. The fractals are just as rigid and deterministic as the regular math that he is used to. In any case, the classical world view that he finds to be such a prison also began with an extension of mathematics to new areas -- namely to the study of accelarated motions, and in its day was seen, by some, as a liberation.
And this brings me to Galileo. When I wrote the abstract for this talk I did not know I was going to talk about Galileo, otherwise I would have advertized the fact. But it turned out that Galileo was precisely the man I needed to indicate the difference between physical intuition and its mathematical surrogates, because we see both in his work, and to a very high degree. Galileo was a great genius, and when a great genius makes mistakes he makes mistakes of genius. If you read his Dialogue on Two World Systems, or the later Dialogue on Two New Sciences, you will find extraordinary leaps of physical intuition --- and then on the next page, the most depressing substitutions of mathematics for nature that you can imagine. We have been living with these substitutions ever since.
Both dialogues involve three characters: Simplicio, Sagredo and Salviati. Simplicio represents at times the voice of Aristotelianism, and at times the voice of bone-headed main-in-the-streethood (or empiricism). Salviati and Sagredo represent alternatively the opinions of Galileo, and intelligent objections that one might raise against these. Simplicio represents the stupid objections.
The two new sciences are the science of strong materials (why things fracture) and the science of motion --- whose results Galileo intends to demonstrate rigourously, that is to say by geometric means. Geometry is used to prove things about nature. Geometry is crucial to Galileo, but the book begins with a great burst of physical intuition passing beyond geometry.
The dialogue opens in the arsenal of Venice, which is the site of numerous mechanical operations. They discuss the question of whether (and then why) if you scale up a machine its performance can deteriorate. This is very important: geometric figures are indifferent to scale. A big triangle has the same properties has a little one; but a big crane may fracture and a small one may work. The strength of machines is not scale invariant.
Now perhaps you are thinking, "of course machines are not scale invariant because you cannot scale up the force of gravity." Very good, neither you can. And it is thanks to Galileo that we know this. Aristotle actually thought that the motion of a falling body was scaled according to weight. I don't know what led Aristotle to this belief, but it seems a natural mistake in someone who wants very badly to explain everything by geometric patterns. Galileo, in his better moments, is highly aware that a moving object is not just a trajectory through space, but is a real body that does things.
There is a beautiful example of this further on (page 163) when he discusses the acceleration of falling bodies released from rest. Aristotle knew that falling bodies accelerate, by the way. But Galileo is making the point that in the initial stages, that motion is arbitrarily slow. He indicates the experience that a heavy weight dropped a short distance onto a plank may not even bend it, but the same weight dropped from a great height will break the plank. Here we have the concept of kinetic energy (a function of the weight and the velocity acquired) expressed not as a formula, but as an effective property of moving weights. This further clarifies the Aristotelian business. Everyone knows that there is a difference between dropping a heavy weight and dropping a light one, but the difference is in the damage they do, not the speed at which they fall.
When I was taught physics they told us this: v = gt. Based on the formula, if t = 0.0001 seconds from the start, then v = 0.0032 ft/s. We also had a horrible little experiment to do with wooden trucks and ticker tape. Mine didn't work the way it was supposed to, and it only served to remove the result further from ordinary experience. But Galileo relates the difference in speed to the effects produced by the falling bodies.
Galileo was a great scientist and it is because of moments like these. But there are problems here as well, where he seems to lose his physical intuition, and they involve mathematics. It turns on what happens when you try to use mathematics to describe motion. Physics is about moving bodies but motion is not a mathematical concept. Everybody knew this. Aristotle knew this; he defined the difference between physics and mathematics. But for ancient science this was not much of a limitation, since they did not use mathematics in order to understand motion but to describe trajectories. A trajectory is the idealization of the history of a motion. Galileo wanted to talk about changes in speed, and he wanted to use mathematics to do this. How do you use a camera to photograph movement?
Going back to the beginning of the Two New Sciences, as you recall, it begins with a discussion of breaking strength. They go on to talk about how you can construct a very strong rope from thousands of weak fibres twisted together. And then suddenly it gets very strange. They start talking about infinities and vacua. For example Sagredo has a construction, familiar to anyone who has studied calculus, showing a polygon approaching a circle as the number of sides increases. But the interesting thing here is the interpretation that he puts on this construction: this is to illustrate that a line can be broken into an infinite number of points. Each polygon is achieved conceptually by bending a line into a given number of sides: think of bending a coat hanger into a pentagon for instance. The circle is then an infinitely bent line. It has an infinite number of sides of infinitessimal length. Yes, truly. But it gets worse, because this mathematical construction is then applied to physical quantites.
"I think they would be content to admit that a continuous quantity is built up out of absolutely individisble atoms."
He is talking about real physical bodies here. Liquifaction, he claims, occurs as a material is broken into an infinite number of infinitesimal parts.
At this point we can begin to see where he is going with all these infinities. The infinitesimal is necessary to him as a way of describing motion. Liquids, after all, are characterized by their tendency to flow, a property he attributes to a continuum of indivisible atoms.
Later in the dialogue he takes up the issue of a mathematical description of the motion of falling bodies, and the infinitesimals recur. He repeats what everyone believes, namely that falling bodies increase in speed, and then argues that this increase is independent of the weight of the body. Thus with each increment of time, speed also undergoes an increment (or increase).
Note that he is not declaring any sort of physical law in terms of a formula. He has no grounds for suggesting the formula he does beyond the belief that nature will adopt the simplest solution. He observes an increase in speed and proposes a first order approximation. That is all he has shown, or that anybody has shown. Students might be less shocked by relativity if this distinction were made clear.
There then follows the discussion about the acceleration of an object from rest that I just mentioned, where he claims that it must pass through each "degree of slowness" as it speeds up. Simplicius has trouble with this, of course, and wonders how it can pass through an infinite number of speeds in a finite amount of time, but Salviati replies that it spends an inifitesimal amount of time in each. Each point of space is covered and each intermittent speed is achieved, as each point in time elapses.
So the physical reality of tossing a ball in the air (show example) has been converted to this: (show graph of linear speed and quadratic trajectory.) At least this is what it will look like a few years later when Descartes discovers analytic geometry. But it is already implied in Galileo.
I am greatly indebted to Ives Simon's fine book The dialogue of nature and space for an understanding of what is going on here. He argues that the notion of continuity is essential to any analysis of motion. But in understanding motion, which is a physical reality, the mind supplies something. Motion, as understood, has a mental component added. What is added is a conceptual unity between the "before" and the "after". When I observe a ball moving upwards and then back down, I do not think of it as resting in an infinitesimal number of parking places between my hand and its upper point. To move through a location is not the same as to be there --- not even the same as being there for an infinitesimal amount of time. But on the mathematical graph, we have lost the distinction between being somewhere and moving through. The trajectory has been resolved into a dense sequence of points: a point where you stay put is the same as a point moved through.
You know, we tend to get very visual about motion when mathematics is involved, but a plastic ball is blind and has no notion of what space looks like. Motion is not a visual experience to the one being moved. We have an alternate experience of motion in music: moving sound. In music you do not hear a discrete sequence of notes. In a sense you hear the current note together with the previous one, and while anticipating the one that will follow. You hear the intervals as much as the notes. Of course, rhythm and harmony help to create the impression of movement.
Motion, then, is not a mathematical concept; it is not a concept of any kind. Rather it is a physical fact. Galileo is trying to resolve this fact into a mathematical concept, and this is why he needs the infinitesimals: he models continuity by making the before and the after infinitely close to each other.
But is it necessary that the continuity implied in motion be caused by infinitesimal points in space? What relationship exists between these infinitesimals in their Cartesian space, and actually moving objects?
Galileo, I am afraid, seemed to think that the two were the same, and this is an error: an error of genius but an error nonetheless. It was shown to be an error from the philosophic standpoint in the 17th century. The German and English romantics decided that it was an artistic mistake in the middle of the XVIIIth century. At the same time it was shown to be a mathematical error by Euler and others. You cannot make mathematical sense out of the statement that a line is composed of infinitely many infinitesimal points. A mathematical solution was proposed by Cauchy in the 19th century, whereby all stages in the argument remain finite, but we can subdivide as often as we need. This is the basis of the infamous epsilon-delta proofs that have ended the mathematical careers of many undergraduates. And Finally, in the 20th century, it was shown to be a physical impossibility.
In classical physics, mathematical space is being used as a universal, indestructable, immobile, universal reference point for all motions. All motions are related to it, are measured against it, and understood through it. Einstein blew this out of the water and Quantum Mechanics sunk it without trace --- or should have done so, but usually it is the student who is sunk without trace. I am not sure that I know what Quantum Mechanics is supposed to mean, but if nothing else, it seems to imply that Galileo's space is not even locally true.
From a mathematical standpoint there is nothing wrong with Galileo's space. The constructions are logically consistent. We can handle dense sets, continuous functions and the Real numbers --- even complex numbers. It is all true and it works, but none of it is real.
Where students are blown away it is because they confuse the mathematical space with physical space. It is an easy confusion in part because we cannot do away with the mathematical space. If you look at what quantum mechanicists are doing you will see that they cannot even express what is going on without making appeal to entities whose existence they must later deny. For example, Richard Feynman, in his famous technique for calculating the probable destinations of particles, would sketch a host of possible paths for a particle to follow, from source to destination. The paths have a precise mathematical existence, and each particle has a neat, distinguishable label, as is the custum with mathematical points. Then he sums over these paths, each weighted by a complex valued weight function (also defined through a dense space) and emerges at the end with a probability distribution of what will be obvserved. But the experimental results that inspired quantum mechanics indicate that it makes no sense to talk about a particular particle following a particular, continuous path, nor is it even possible to distinguish one particle with another. They lose their mathematical identifiability along with their trajectories.
So although mathematical space is the mode through which these entities are understood by us --- and we will not do better by getting rid of it -- real space is not like that. I assume that what we experience as space is actually being constructed by these particles, and consequently space does not contain them. The furniture has a place in the room; the floor does not.
Quantum Mechanics produces no philosophic problems. What it does do, and we should be grateful, is to reveal the full strength of the illusion of the mathematical imagination: namely the confusion of the network of physical relations with mathematical space. According to the illusion a particle and its history must be in one-to-one correspondence with mathematical space in order to exist. Having co-ordinates becomes a condition for existence; Having a continous trajectory becomes a condition for motion. Mathematical space has become the universal backdrop against which all material objects move and have their being.
This is an illusion. Mathematical space may be the backdrop to our knowledge of motion. But a concept cannot be the backdrop to the real existence of anything, or to the real movement of anything. The illusion can be comforting or it can be stiffling. Briggs found it stiffling. On the other hand, we can derive a certain sense of comfort from the notion that there is a fixed and stable point out there, against which everything else can be measured and understood.
Still, it is an illusion. How bad is it? Since the illusion makes XXth century physics difficult to accept I should say that it is something that all of you should be interested in forestalling in your students. Separate school teachers have an added concern here in that the illusion will inevitably come into conflict with God.
To St. Thomas Aquinas, who had theological intuition, it was obvious that if something is real, eternal, immutable; if all motion relates to it and is measured by it; if it, in turn, is measured by nothing; then it must be God. St. Thomas has a proof for the existence of God that runs along these lines. When you take mathematical space to be a real, existing thing, then it is a thing with these properties and so it must be God; and a very dismal God it is. Space/matter has become your god.
My argument would rest simply on the observation that MacDonald's is in competition with Wendy's and not with General Motors. The fact that Quantum Mechanics raises difficulties for people about the causality and coherence of the universe (i.e. is someone in charge here) means that something associated with Quantum Mechanics is pretending to be God.
As a former victim of the illusion I know exactly how this works. I remember once worrying about how Divine Providence can apply to sub-atomic particles, if it is impossible to predict where a particle is, and how fast it is going. It seemed that the universe was leaking away from God's control. And no, you can't rescue Providence by saying that statistically, over the long haul, the results are predictable. Providence has to cover everything that exists, or it is not Providence.
Well, the problem here is in picturing two gods. I am picturing God over here, and space over there, with its Cartesian coordinates, and God has to relate the particles to their address in space in order to control their being. Space has become God, and God has become a postman. But this is nonsense. That space does not exist, and God only needs to relate the particles to Himself in order to make them exist, and know everything about them. The unmoved mover is God, not mathematics.
Now, I would like to be able to finish this talk by giving some snappy, no-fail techniques for math-proofing the next generation. I don't have any. The best science book I ever read was written by Anonymous of the U.S. Navy for teaching basic mechanics to seamen. The beauty of this book was that the seaman reading it was included as one of the physical entities. Everything was related to what he was going to do, so the concepts of work, energy, force and so on were explained primarily in terms of human work and heavy machinery. And since they didn't need to use the metric system, all the measurements were related naturally to the human body. We are back to Galileo watching the arsenal in Venice.
Section Contents Copyright © Mark Alder and Catherine Collins 2000
This Version: 4th March 2001