Time and Form in the Physical World

David Park

Whoever discusses conscientiously the development of ideas of scientific explanation must start by admitting that the story begins long ago, probably beyond the limits of written history, and that even in the age of Ionian philosophy we are making the acquaintance of an old intellectual tradition.

Time and Form in the Physical World

by David Park

Whoever discusses conscientiously the development of ideas of scientific explanation must start by admitting that the story begins long ago, probably beyond the limits of written history, and that even in the age of Ionian philosophy we are making the acquaintance of an old intellectual tradition. One might at first expect the earliest speculations that come down to us to be small and tentative; instead, we find that from the very beginning of the record philosophers of Ionia aimed at the heart of science with large and inclusive theories, expressed in more or less mythic language, on the ultimate structure of matter and on scientific method. I am interested in finding out whether anything they said was right, whether it can serve us as a guide and interpreter today. You cannot, of course, say that a myth is either right or wrong. It is pointless to discuss the correctness of the myth of Adam and Eve, but we can reasonable ask whether it helps us today to understand the human condition. In the effort to understand the scientific condition, I have found it both instructive and useful to think about modern physics in the light of my amateur Is knowledge of the ancients, especially Plato. Doubtless I shall appear historically naive, but perhaps naivety is not such a bad thing if we are to learn from the past and not be bound by it.

THERE ARE THESE REGULARITIES

Wittgenstein understood how philosophical inquiry begins. "A philosophical question," he wrote, "has the form: 'I do not know my way around.' " It does not even have to be a question; it is a sound one makes when one is uneasy. I want to begin an inquiry into the subject of time in physics with a fact that lay indigestibly in the minds of the first people who recorded for us their thoughts on intellectual matters: There are these regularities. What kind of regularities? There are rules (invented? imposed from outside?) that govern human nature and conduct. There are regularities in nature, and it must be remembered that the regularities of nature were a matter of concern to the ancients, and to the ancients' ancients, far more than to most of us today. Sailing, planting, and maintaining sacred calendars required difficult and exact scientific work. Regularities of the seasons are perfectly bound to regularities of the motions of the fixed stars. From the crafts, the making of metals and glass and dyes, from cooking, we derive a sense that nature not only acts, when left alone, in accordance with certain rules; she demands to be treated according to certain rules. What rules?

Not all rules work. Auguries deceive and plowshares crack, but year after year the rains begin at the time when the Hyades can first be seen rising in the East, and a triangle with sides in the ratio of 3 to 4 to 5 always contains a right angle. If the Fourth Dynasty Egyptians had not believed that the regularity of the stars has meaning they would not have bothered to orient the Great Pyramid, with its sides deviating by only by a few centimeters from what they believed, and we know, to be the cardinal directions, and the Mesolithic Britons would not have taken the trouble to align nine hundred rows and circles of standing stones to the risings and settings of sun, moon, and stars. Setting up an alignment requires surveying and a surveyor must be able to construct an accurate right angle. One way to do this is to use a triangle whose sides are the ratio 3:4:5. The Babylonians and Egyptians seem to have known the Pythagorean principle as if it were a fact of nature, but we do not know whether they used it to make right angles. By arguments that seem to me reasonable, Professor Alexander Thom(1) deduces that for surveying, the ancient Britons, earlier still, knew and used not only the 3:4:5 right triangle, but also those with sides in the ratios 5:12:13, 8:15:17, and 12:35:37. All these obey the Pythagorean rule for right angles: 5²+l2² = 169 = 13², etc. But to the Britons this was only a rule of thumb for constructing right angles, for they also used triangles for the same purpose whose sides almost but not quite satisfy the Pythagorean rule. At Penmaen-Mawr in Wales, for example, is an elliptical ring of stones that Thom finds was laid out using a triangle of sides 19:59:62. But 19²+59² =3842, while 62² = 3844. The error was undetectable through the methods of measurement then in use, and we conclude that these ancestors, so refined and careful in their geometrical methods, did not know the difference.

The idea of mathematical proof is supposed to have originated with Pythagoras. Before his time there may have been a sense that mathematical truth is of a different kind from that of astronomy or metallurgy, but it is probably significant that the beginning of Greek philosophy almost coincides with the beginning of mathematical proof. It was then that the human compass began to point towards truth, as such.

TRUTH AND MATHEMATICS

There are, then, these regularities, and with the advent of mathematical proof, a whole class of them could be explained: "I really do know it and if you will permit me to draw a few figures on this wax tablet I'll show you a proof." The meaning of the last word is clarified by Mark Kac's remark: "A demonstration is a way to convince a reasonable man, and a proof is a way to convince a stubborn one." Today the matter is not so certain. We know that the relation of geometrical proof to the actual experience of sizes and shapes belongs to the realm of physics, that is, to approximation. Though the ancients did not know this, they did understand that their notion of proof could not be extended to show that the Hyades bring rain, and still less to those phenomena that do not obey a perfectly strict law but only an approximate one. Apparently there are regularities of different kinds. This makes it more difficult to say how and why they are true, but it also suggests the best approach to follow: start from what you know, argue towards what you don't know. What the thinkers of Plato's time knew were some techniques of mathematical proof. Having installed a signboard at the entrance of the Academy saying "Let no one ignorant of mathematics enter here," they then applied themselves to more difficult questions of law and ethics.

Mathematical propositions are stated in the form of declarative sentences: "The sum of the interior angles of a plane triangle is equal to two right angles." The verb is in the present tense, but it does not really refer to the present, moment. It would be silly to replace it by "is now" or "was yesterday." It is in fact a tenseless verb, expressed in the form of the present tense only by linguisitic custom. "Was" or "will be" contain just as much truth. The propositions of mathematics refer to no particular time, and in this way, though they are not unique, they are certainly unusual. There are truths which exist outside of time.

Suppose we want to state the rule that governs some kind of natural process or event. The rule may be an invariable one, but it does not exist wholly outside of time, for events are situated in time; they are governed by past, present, and future. Now for the first time I pose a philosophical question in interrogative form: Is it possible to formulate, in the tenseless manner we have learned from the timeless universal truths of mathematics, the laws or rules governing events which occur in time? This question is perhaps easier if we answer another one first: Is there any timeless substratum of facts or things that pertain to the changing world? This question was discussed with astonishing power and originality, and with varying answers, by most of the pre-Socratic philosophers; I mention here only three answers, enclosed in the smallest of capsules: Parmenides taught that the world must be considered as a whole, that it is what it is and cannot turn into something else, and that change is, therefore, illusory. For Heraclitus physical reality consists of process, and the things of this world are only markers of its ceaseless flux. Finally, and more concretely, Leucippus and Democritus taught that the permanence of the world is its atoms. They never change and are too small for us to see, but the processes of the world around us reflect their continual rearrangements.

It is impossible to estimate the scope and fertility of the atomic hypothesis, since we may be only starting to explore its consequences. It commands, of course, "Test me," but nobody obeyed for two thousand years. It also warns "You can't always judge the truth from appearances." As Democritus said long ago, "In fact we know nothing at all; for the truth is hidden in the depths." At the Institute for Theoretical Physics in Copenhagen they still quote a couplet from Schiller(2) that Niels Bohr was especially fond of:

Nur die Fülle führt zur Klarheit Und im Abgrund wohnt die Wahrheit.

"Only fullness leads to clarity, and truth dwells in the depths." Physicists like this couplet because it adds to the discovery of the Greek atomists the one we have made since: that even though scientists no longer regard themselves as accumulators of knowledge, fullness is required - one must think widely.

Let me restate the central problem: How does one formulate, in the tenseless manner we have learned from mathematics, the laws or rules governing events which occur in time? It is my contention that Plato gave a correct answer to this question. I am not speaking of eternal truths, for I have no idea what is going to happen to our ways of thinking, but Plato's answer not only serves to describe the intellectual structure of modern physics, it also provides a useful standpoint from which to view some of its difficulties.

In trying to extract usable truth from Plato's theory there is no need to use Plato's mythic language. To understand and translate this language we must remember several things. First, the tradition of the past as Plato had received it was largely embodied in myth. Second, the entire intellectual situation was extremely confused. People were making fundamental conjectures about nature at a time when not one law of nature had yet been formulated in a general form. Thus there was no body of terminology that could be drawn on for clear and accurate expression. This is a perfectly familiar situation. The advent of quantum mechanics in the
1920's brought with it conceptual innovations so extensive that fifty years later we still lack a useful vocabulary of words and relevant facts with which to discuss them. But the theory is nevertheless a great success, for when we use it to predict exactly what will happen when a given experiment is performed, it always (if it is correctly applied) yields the correct answer. It seems to be perfect in the area of numbers, though still shadowy in the realm of concepts and words. As Werner Heisenberg wrote recently, "We may fully understand a connection even though we can only speak of it in images and parables."(3)

And finally, before we can dig into Plato to find what he has said in response to difficult questions, we must remember how very seriously he was concerned with literary form. just as Wittgenstein's philosophical question does not come in the form of a question, so Plato's answer does not come in the form of an answer. But he has much to tell us nonetheless.

PLATO'S THEORY OF KNOWLEDGE

For Plato, the world as we perceive it is a shadowy confusion in which truth and clarity are in some degree latent but from which they cannot be disengaged without long study and a certain amount of direct and unanalyzable inspiration. When the disengagement is made, it is found that truth consists of statements not about the world of sense but about a world of Forms, or Ideas, which, although abstract in themselves, nevertheless have rough counterparts in the world of the senses. It is often said that Plato was led to this formulation in the attempt to establish a rational politics, in which a few people would learn to understand the nature of Justice, Virtue, and the Good, and use this understanding for the good government of the State. This may be true, but I believe that the formulation itself owes much to Plato's mathematical studies. For we learn in the Timaeus that true knowledge of the natural world is knowledge of its forms, and has the necessary character of mathematical knowledge.

As an example of how this might work out in practice Plato gives us a "plausible account," as he says, not to be taken too seriously, of the ultimate structure of matter. Today we would say he proposes a model. Whereas for the earlier Greek atomists the atoms of a substance were very small lumps of that substance which for some reason could not be cut any smaller, for Plato they are not material objects at all, and in fact he avoids referring to them as atoms. Rather, they are mathematical forms composed of small numbers of lines. The basic units of earth, air, fire, and water are cubes, octahedra, tetrahedra, and icosahedra respectively. These units are not unbreakable; they are continually splitting each other apart into their component triangles and recombining into other elements. The whole process is governed by the laws of mathematics. Plato's concept of natural law is that it consists of mathematical statements, derived after long study through a direct intuition of the truth, concerning units of matter which are themselves abstractions, but which act on us so as to produce the phenomena of the world as our senses perceive them.

The whole is described in a language of symbols and metaphors, but there is a reason for this. The questions of what a conceptual structure has to do with fact, what fact is anyhow, what the criteria are by which we recognize a true statement, are very, very hard questions which we still cannot answer satisfactorily today. They had been posed by Plato's predecessors and he could not ignore them but neither could he answer them, and the last thing he wanted was to fill the gaps in the logical part of his argument with statements that were not solidly based. I do not wish to imply that myth for Plato was a last resort, but if he had imposed on himself the requirement of making definite statements in plain language he could not have proceeded. Today the literary form has changed, but we still solve this problem the same way. If one reverently tries to translate Plato's words into our own symbols and metaphors, Platonic natural law as I have just summarized it corresponds with the concept of natural law that emerges from modern theoretical physics, with the added bonus that we can now see more clearly the role played by the Forms. They correspond to certain abstract mathematical structures that mathematicians have created in the last century or so: for the cognoscenti I refer to the Lorentz group, the special unitary groups SU (2) and SU (3) and, let us say, the equations of Maxwell and Dirac together with their solutions.

For example, water is a Platonic Form, though Plato warns us that this Form is not at all the same as water as we perceive it. In modern atomic theory a water molecule is a relatively simple structure known to us not through observation but as the solution of a certain differential equation. Its properties are mathematical properties. Although the properties of ordinary water can be derived from those of the mathematical structure by theoretical arguments, the molecule itself as I have described it is not in any sense "wet."

At the end of his "plausible account" Plato invites anyone else to produce a better model, but in fact the thing could not be done. Plato's formal structure was bound to remain empty of scientific content until there were some facts to put into it. The lesson here is an important one. Plato's "eye of the soul" does not see much unless its blood supply is nourished with facts. "Nur die Fülle führt zur Klarheit. . ." But a physics based on reason rather than experiment has an immensely difficult question that it must deal with: today's theory is, in part, a self-consistent mathematical structure of great beauty and simplicity, and as such has a strong ingredient of a priori mathematical knowledge and taste, but it is based on experiment. In the logic of physics (as opposed to its practice) experimental fact has absolute priority over form. To Plato, of course, it was the other way around, and so he had to explain how it is that principles of reason and beauty act so as to control the motions of the celestial bodies and other things in the world. How does a lump of matter know about these laws? Why does it obey them?

Knowing that the laws are obeyed, that the connection must indeed be there, Plato expresses himself, as we do, in "images and parables," in myth-though his myths belong to his era just as ours belong to today. Adopting language handed down to him from antiquity, he said that the stars and planets are "living beings, divine and everlasting," and that they are governed by a universal soul.

THE MOVING IMAGE

Now we can go on to see how Plato introduced into his cosmology the slippery subject of time, since, if I am right, it was the contrast between the timeless truths of mathematics and the contingent facts of experience that led him to invent the Forms in the first place.

What he says about time in the Timaeus is so short that I can reprint almost all of it. He has come to the point where a Divine Craftsman has constructed the soul and body of the world.

When the Father who had begotten it saw it set in motion and alive, a shrine brought into being for the everlasting gods, he rejoiced and being well pleased he took thought to make it yet more like its pattern. So as that pattern is the Living Being that is forever existent, he sought to make this universe also like it, so far as might be, in that respect. Now the nature of that Living Being was eternal, and this character it was impossible to confer in full completeness on the generated things. But he took thought to make, as it were, a moving likeness of eternity; and, at the same time that he ordered the Heaven, he made, of eternity that abides in unity, an everlasting likeness moving according to number-that to which we have given the name Time.(4)

Time belongs to the world of created things. it furnishes the conceptual link between the timeless world of eternal truth and the changing world of our experience. It is the image of eternity, moving according to number. This, too, is mythical language, but it seeks to establish a connection that must somehow be made if one is to proceed. Later I shall say how most physicists would formulate it today.

Plato is very clear that not only the fact of time but the conditions for its measurement belong to the created world:

In virtue, then, of this plan and intent of the god for the birth of Time, in order that Time might be brought into being, Sun, Moon and five other stars-"wanderers," as they are called-were made to define and preserve the numbers of Time.

When we have seen how time enters modern physics, it will be evident how to translate this mythic statement into modern language.

The last thing I wish to do is to wander in the aisles of a museum of stuffed philosophers, each in a glass case with his doctrine spelled out beside him. Physics does its business in a laboratory, where vacuum pumps are running and the red High Voltage sign is on. where delay is costly and where people are using what means they can to solve difficult scientific problems. The problem is to find the right way to formulate laws of nature. Are space and time necessary parts of a physical theory, or do they belong only to the interpretive commentary that accompanies it? Is it possible to make a theory that contains only observable quantities, or does every theory contain elements that are in principle unobservable? Questions like these occur insistently in modern physics and are, implicitly and explicitly, under careful discussion by people who try to make theory agree with experiment.

There are two kinds of questions that one asks of a physical theory: one asks for precise quantitative predictions of what will happen if a given experiment is performed, and one asks for an explanation, a qualitative insight into what is going on. A rule, first expressed, I believe, by Niels Bohr, is imposed on the answers a theory may give: however abstract the hypotheses of a theory may be, and however mathematical its working out, it must always finally give its answers in plain, ordinary language. The theory, whatever else it may do, must therefore establish a link between timeless hypotheses and the time-bound perceptions of man. That is, the answer to quantitative questions has the form "If you measure this quantity you will get that result," while in reply to our search for insight it may have to speak, as Heisenberg said, in images and parables, but it is required to speak plainly. I believe it is correct to call considerations of this kind metaphysical, but I do not agree with those who equate metaphysics with meaningless noise. Physics swims in metaphysics as a fish swims in water, supported by it on all sides but unconscious of its existence until something goes wrong.

It seems a little strange, when one thinks how Plato and Aristotle and St. Augustine, among many others, compared knowledge of the eternal truth of man, nature, and God to knowledge of mathematical truth, that they made so few attempts at actual mathematical formulation. Plato was the only one who tried it at all. His atoms were simple geometrical forms. The mathematics related only to those forms themselves; none of the numbers were intended to be measured, and the whole theory (if you want to call it that) was put forward only as a "likely story." At the beginning of the Timaeus, Plato explains exactly why this is so: "An account of that which is abiding and stable and discoverable by the aid of reason will be abiding and unchangeable . . . while an account of what is made in the image of that other, but is only a likeness, will itself be but likely, standing to accounts of the former kind in a proportion: as reality is to learning, so truth is to belief."(5) The contrast between knowledge and belief, between ideal form and the embodiments of it that we experience, is fundamental to Plato's thought and, as we shall see in a moment, that of modern physics.

ATOMS IN THE SEVENTEENTH CENTURY

By the end of the Hellenistic period physics was stuffed with brilliant speculation and still starved for facts. The facts that nourish physics are quantitative. They are derived from experiment, and they did not begin to accumulate in significant numbers before the seventeenth century. Galileo was the first of the great experimenters. He has left us his description of natural law:

Philosophy is written in that great book which lies ever before our eyes-I mean the universe-but we cannot understand it if we do not first learn the language and understand the symbols in which it is written. The book is written in the mathematical language, and the symbols are triangles, circles, and other figures, without whose help it is impossible to comprehend a word of it; without which one wanders the dark labyrinth in vain.

Galileo was one of the best mathematicians of his time, and he left us quantitative laws that describe such phenomena as the motion of a falling object or a projectile, the sag of a loaded beam, and the dependence of the pitch of a vibrating string upon its length, density, and tension. In addition, of course, he speculated on matters that could not be decided by measurements-whether the sun or the earth stands still in the solar system, and whether there are atoms. It is worth noting that for Galileo, the ultimate atom is neither matter nor pure form: when its husk of substance is rubbed off, it bursts forth as pure light.(6) What he says is "Light is created," and I think the Scriptural overtone is so strong in this choice of words that, although one could hardly call Galileo a Platonist, his atom adheres to the plane of mental illumination at least as strongly as to the material one.

In the notebooks of Thomas Hariot, the English dilettante who was Galileo's exact contemporary, we find a similar idea expressed in mysterious language reminiscent of that of the Medieval doctrine of the Trinity: "Omnia fint ex nihilo & ex nihilo nihil fit-non contradicant."(7) ("All things are made out of nothing and nothing is made out of nothing-these statements are not contradictory.") The first proposition refers to the fact that Hariot's atoms are actual mathematical infinitesimals; the second states that nothing happens without a cause.

But atomism, when it came to England, came largely by way of Lucretius, who had it from the Greek atomists, and their atoms were little pieces of matter. Newton described them thus:

All these things being considered it seems probable to me, that God in the Beginning formed Matter in solid, massy, hard, impenetrable, movable Particles, of such Sizes and Figures, and with such other Properties, and in such Proportion to Space, as most conduced to the End for which he formed them; and that these primitive Particles being Solids, are incomparable harder than any porous Bodies compounded of them, even so very hard, as never to wear or break in pieces; no ordinary Power being able to divide what God himself made one in the first Creation.(8)

In the effort to explain the ceaseless flux of experience in terms of timeless concepts, we have seen two strategies: that of the Greek atomists, Lucretius, and Isaac Newton, in which permanence resides in the actual physical nature of the atomic substance, and that of Plato and Galileo, in which the concept of permanent substance does not appear, but instead the permanence is that of mathematical form and truth. Newton's theory did not, of course, stop at saying "there are atoms." He conjectured about the forces that act between them and gave the mathematical laws that he believed to govern their motion, but although his astonishing insight transcended the crudity of his techniques and the scantiness or ' his knowledge, he was unable to assemble any coherent theory of matter.

FIELDS

Newton's dynamical theory contained forces and objects, explaining the motions of objects in terms of the forces that act on them. It was not necessary for Newton to worry, like Plato and Kepler, about how the objects know what laws they are supposed to obey, for Newton believed his theory to be not legislation but merely a compendious description of what actually happens.

In 1864, James Clerk Maxwell produced the equations of a theory in which forces themselves are a dynamical system. For this purpose we make use of the idea of a field, which for people encountering it for the first time is an extraordinary hybrid of mathematics and physics. If an electrically charged object is situated near other bearers of electric charge it experiences a force, whose magnitude and direction depend on the position of the object. We abstract this fact into the notion that the nearby charges produce a condition in the space around them such that if a charged object is introduced it will experience a force, but that the condition exists even in the object's absence, when we see and feel nothing. This "condition" is called a field, and it is represented by a mathematical function, defined at each point in space and instant of time, from whose value the force on any charged particle that might be put there can be calculated. The field is known only by its physical effects, and all of these are implied in the mathematical expression. It is therefore only a step to saying that the field is the mathematical idea, and this is the general usage among physicists. It equates mathematics with reality, and it is very Platonic.

The field is more than an exerter of forces-it is a bearer of energy and momentum, just as solid bodies are, and like them its changes are governed by equations of motion. But to leave the description there is to mock the Muse of History. I said that Maxwell provided the equations of this theory. What he thought the equations meant is another matter. For him they were the equations of motion of a strange dynamical system, the luminiferous ether, a fluid everywhere present, whose flows and eddies and vortices, though not directly observable, were to be taken perfectly literally. It was only gradually that people felt safe in accepting the mathematics while rejecting the ether.

With regard to the structure of matter, Maxwell was careful, but he followed Newton's lead. In his elegant and learned article "Atom" in the Encyclopedia Britannica,(9) for example, he is very cautious:

We make no assumption with respect to the nature of the small parts-whether they are all of one magnitude. We do not even assume them to have extension or figure.

But Maxwell was a pioneer of the theory of gases, and in his theories the little lumps are there, just the same. They occupy particular positions in space, and they go where the forces push them.

In 1912 Ernest Rutherford and Niels Bohr proposed that an atom is not a solid piece of matter but a whole dynamic system, mostly empty space, consisting of electrons and a nucleus. Initially, these new particles were regarded as small, and permanent, pieces of matter. Bohr's theory ran into great difficulties, requiring ad hoc theoretical assumptions and persistently,. giving numerical results in 20 to 50 percent disagreement with experiment. Only after a decade of hard work was it discovered that Bohr s theory, which still visualized matter in the form of lumps, even though on a far finer scale, had pushed what we may call the Democritean picture or matter to the furthest limit of its development, at which point it failed for good.

In 1923 came Louis de Broglie's discovery: the elementary particles of matter (no longer what are now called atoms, but their smaller parts) can be represented by fields; and soon afterwards Erwin Schrödinger and Paul Dirac gave the equations of motion of these new matter fields. But they had a predecessor, for Michael Faraday, the English genius of electricity, had guessed twenty years before Maxwell's theory that matter is a field of force:

The view, now stated of the constitution of matter would seem to involve necessarily the conclusion that matter fills all space, or, at least, all space to which gravitation extends (including the sun and its system); for gravitation is a property of matter dependent on a certain forces and it is this force which constitutes the matter. In that view matter is not merely mutually penetrable, but each atom extends, so to say, throughout the whole of the solar system, yet always retaining its own centre of force.(10)

Few students today are aware of Faraday's speculations, and their surprise is painted on their faces when they first learn that, in the modern theory, an atom has no boundaries at all.

Faraday, who thought non-mathematically, gave no formulas. With the equations, the formulation of an abstract mathematical theory of matter was almost complete, and, in my opinion, it can be taken as an example of the Platonic doctrine in action.(11)

Modern atomic theory-its technical name is quantum mechanics-exists in several forms. There is for example a Parmenidean form, called the Heisenberg representation, in which the state of a system remains fixed- nothing ever happens,-and a Heraclitean form, the Schrödinger representation, in which "all things flow." But these representations do not stand for conflicting schools of thought, because they are fully equivalent mathematically, and any calculation that is possible in one representation is also possible and gives the same result in the other. Thus one thing we know now that the ancients did not is that Parmenides and Heraclitus can both be right. We can probably learn something from this, for if we can now see through to the end of the arguments and know how each is in its own way correct, it equips us psychologically to go back to the beginning and take them more seriously.

Further, there is a representation (more properly, a picture) which portrays an atom as an image in space and time and another which describes it only in terms of dynamical variables such as energy and momentum. Again, they are equivalent, but we should remember Bohr's dictum that the final results of a theory should be expressible in ordinary language. Our ordinary language is that of space and time, so I shall talk about that picture.

In each of its forms, the theory that gives the properties of an atom is expressed in terms of a partial differential equation. Each atom can exist in many different states, and these states correspond to the many different solutions that these equations possess. To every state corresponds a solution; to every solution corresponds a state. An atom, like, let us say, a clock, is an item of our mental furniture that is formed out of contributions from things people have told us, things we have read and thought, and the experience of our eyes, ears, and hands. The equation and its solutions form an abstract mathematical structure that has, in itself, nothing to do with atoms, for the variables and operators that occur in them are defined purely in mathematical terms and make no reference to anything in the world around us. These mathematical structures date from the eighteenth and nineteenth centuries, before quantum mechanics existed. They are transparent in form and it is no exaggeration to say that they can be perfectly understood. In contemplating them one feels very strongly the force of Galileo's remark that when we understand nature mathematically we are perceiving it as God Himself perceives it.

These self-contained mathematical structures are timeless in the sense that the symbols they contain are timeless, but one of the symbols, normally a letter t, represents time. Thus time, as one of the elements of our experience of nature, one of the dimensions of our consciousness, is captured in a formalism which is itself timeless. This modest intellectual device solves the ancient problem or relating the temporal to the timeless. Why it works is clear if we consider ourselves with honest candor. We are creatures of bone, flesh, and blood, of atoms, and all our senses, our perceptions, our consciousness itself, are the result, and only the result, of things that atoms do. If there is a formal harmony between the mathematics of our atomic theory and the experience of our lives, it is because in talking about nature we are talking also about ourselves, and it gives us confidence that we may hope someday to understand the actual relations that underlie the formal harmony of which we ourselves are a part.

FORMS

Each of the great theories of physics is embodied in mathematical structures that are self-consistent and, ultimately, of a certain kind of simplicity. The educated mind tends to perceive them as beautiful. They are not adjustable. If an experiment tomorrow proves that Einstein's General Theory of Relativity is wrong by 1 percent, then that majestic mathematical edifice crumbles to ruin; no tinkering can save it.

If these mathematical structures are not examples of Platonic Forms, then I simply do not know what Plato was talking about. They are abstract, they are unalterable; they are grasped by the prepared mind, after long study, in an act of intuitions They have no direct logical connection with the world of our experience; yet if we know them we can understand, predict, and control some aspects of the world around us. We can be philosopher-kings in the laboratory, even if not in the street.

Knowledge, for Plato, was knowledge of the Forms. He warned us about belief. Yet belief, in the system of ideas I have been describing, is a necessary element, for we do not experience mathematics; we experience our own reactions to lights, colors, sounds, and not only those but memories, hopes, prejudices, all at once. This is the world of the cave. It is idle to expect to deal with experience directly in terms of the Forms. Surrounding each of the mathematical structures I have described is a verbal commentary, an equally necessary part of the theory, which relates the mathematics to the rich and largely irrational contents of our minds. To this part of physical theory belong, for example, Einstein's Principle of Equivalence, Heisenberg's Principle of Indeterminancy, and Bohr's Principle of Complementarity, and it is here, of course, that the arguments take place. Nobody argues about the solution of an equation, but there is often a genuine confusion as to the best way to say what it means. Fifty years after the foundation of quantum mechanics, we are still uncertain how best to relate the mathematical symbols to experience, and hard work is being done to find out. The motive is partly a feeling that until there is general agreement the theory remains incomplete, and partly the hope that, if some logical flaw can be discovered in the verbal commentary, it may suggest ways of improving the theory so as to extend its range. There is still much to be explained. After fifty years we only dimly perceive some of the outlines of a theory of elementary particles, and we are essentially in the dark as to how our mathematical understanding of the external world can help in establishing a scientific theory of human cognition. We know enough, though, to suggest that if Plato had been able to imagine equations of motion, mathematical forms in which intervals of time are treated no differently than intervals of space, he might have defined the cut between the worlds of knowledge and belief quite differently than the way he did. Time is, for Plato as for Augustine and many who came afterwards, a dimension of human experience, indeed the sole dimension of our inner lives, which cannot be caught in the mathematical discipline of the eternal realm, and is therefore only its moving image. It is exactly the temporal dimension of belief that separates it from knowledge.

If one believes that part, at least, of the Platonic program can be and has been carried out, it is natural to begin to wonder about the rest of it. What are the possibilities for morals and politics?

I am aware it is a truism that these intractable disciplines cannot be brought under mathematical control. But I am also aware of the excessively narrow and historically conditioned ideas of mathematics that underlie this truism. Almost all the mathematics there is, except for some dreary and sterile statistics, was created by people trying either to solve problems in physics or to create formal structures of esthetic delight. Most of the people whom I have heard assert the intellectual limitations of mathematics have been people who have apparently never made any serious efforts to transcend them and so do not even know what the difficulties are. The point is not to create a quantitative theory of politics that predicts sizes of armies and numbers of votes-numbers provide no new kinds of understanding here-but to create a calculus of situational forms from which future events may be charted. In particular, it should warn us far in advance if a planned course of action will lead to a situation in which rapid and uncontrollable change-a catastrophe-is inevitable. Such a theory has been sketched out by the mathematician Rene Thom;(12) it is called catastrophe theory, though many of the sudden changes it describes have nothing catastrophic about them. Its Forms, if I may use the word so very prematurely, are curved surfaces in spaces of three or more dimensions by which the basic types of catastrophic situations are geometrically represented. They have names like fold, star, wigwam, butterfly, swallowtail. Some mathematicians are talking as if this notion were the first basically new idea in applied mathematics since calculus was invented in the seventeenth century. It is starting to make headway in biology and economics. Does it allow us to predict and control events that cannot now be predicted and controlled, or does it merely furnish a new formulation of things we already know? This is the critical question that now confronts the theory. To make a mathematical politics as effective as mathematical physics is, political scientists will have to be at least as mathematically creative as physicists have been. Plato's ghost is watching us and waiting, a little impatient with all this work in natural science. It was never his first interest. "Why don't they begin?" it squeaks. "Begin!"

NOTES:

(1) A. Thom, Megalithic Sites in Britain (Oxford: Clarendon Press, 1967).

(2) From the Sonnets to Confucius.

(3) In Physics and Beyond (New York: Harper and Row, 1971). The remark is from the essay "Positivism, Metaphysics, and Religion."

(4) F. M. Cornford, Plato's Cosmology (London: Routledge and Kegan, Paul, 1937), pp. 97, 105.

(5) Ibid., p. 23.

(6) Galileo, Opere (Firenze, 1844), v. 4, "Il Saggiatore," p. 338. The published English translation of this passage (Drake) is extremely faulty.

(7) Mathematical Papers (in Brit. Mus.), vol. 3, fol. 375.

(8) I. Newton, Opticks, ed. I. B. Cohen (New York: Dover, 1952), p. 400.

(9) 9th ed.: see also Scientific Papers, W. D. Niven, ed., Cambridge Univ. Press, 1890, vol. 2, p. 445.

(10) London and Edinb. Phil. Mag. 24, 136 (1844); Experimental Researches in Electricity, vol. 2 (London: Taylor, 1844), p. 293.

(11) The first statement of this view that I know of was made by W. Heisenberg, Physics and Philosophy (New York: Harper and Row, 1974), pp. 8, 104.

(12) R. Thom, Structural Stability and Morphogenesis (Reading: W. A. Benjamin, 1975).