nothingness
the science of empty space
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THE SCIENCE

OF EMPTY SPACE

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HENNING GENZ
Translated by Karin Heusch

BASIC
B
BOOKS
A Member of the Perseus Books Group New York •

Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book and Perseus Publishing was aware of a trademark claim, the designations have been printed in initial capital letters.
English translation copyright© 1999 by Perseus Books Publishing, L.L.C. Copyright 1998 © by Perseus Books Publishing
Originally published in German as Die Entdeckung des Nichts© 1994 Carl Hanser Verlag MOchen Wien
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First paperback printing, December 2001

CONTENTS
•
PREFACE Things, Just Things vii Acknowledgments xi PROLOGUE 1 Physics and Metaphysics 2 NOTHING, NOBODY, NOWHERE, NEVER 33 Philosophical, Linguistic, and Religious Ideas on Nothingness 3 PROBLEMS WITH NOTHINGNESS 97 How to Make It a Physical Reality 4 MATTER IN THE VOID 145 Ether, Space, Fields 5 CROWDED SPACE 179 Movement All Around-the Quantum Vacuum 6 SPONTANEOUS CREATION 209 Particles and Fields 7 LET NATURE BE AS SHE MA Y 227 Special Systems 8 NOTHING IS REAL 257 The Universe as a Whole 9 EPILOGUE 305 Physics and Metaphysics Notes 317 Figure Sources 325 Bibliography 327 Index 337

PREFACE
•
THINGS, JUST THINGS
•
THIS BOOK IS DEDICATED TO THE QUESTION "CAN THERE BE SPACE
independent of things?" Space that is immutable, like a stage that remains the same no matter what is being played on it? Space that may be empty and for all time remain empty? Such a space would be a "void" proper, or what we call "nothingness"--concepts that in antiquity were found by natural philosophers to be so controversial as to be unthinkable. Over the millennia, this void has evolved into what physicists call the vacuum-their term for empty space. Physics fills this vacuum with the progeny of quantum mechanics, of the general theory of relativity, of whatever the Big Bang left for us. This question then poses itself: How empty can space be and still remain in consonance with the laws of nature?
In the seventeenth century, Galileo's disciple Evangelista Torricelli was the first to succeed in the removal of everything material from an otherwise empty container. This experimental success made it harder for skeptics to insist that there couldn't be such a thing as empty space-a skepticism maintained by the followers of Aristotle, who dung to the tradition of the horror vacui, nature's supposed abhorrence of a vacuum. It was Isaac Newton who considered the meaning of space in the laws of nature; he also discussed the concept of the ether-not the substance used to anesthetize but some mysterious matter that was assumed to fill the universe. Newton found that these concepts helped him describe (if not understand) the motion of a planetary system through the longdistance action of gravity across empty space.
From the seventeenth century on, science has made tremendous progress based on the notion that there are just two fundamentals: matter and empty space. In the back of their minds, the natural philosophers-who slowly evolved into natural scientists in the modern sense-had long nurtured the concept of the mysterious ether, a substance that could not be chopped up into atoms but instead was strictly continuous.
It took Albert Einstein to do away with this idea. The details are complicated,
VII

viii

NOT H I N G N E S S

and we will discuss them. Today, we know that the laws of nature do not permit space that is absolutely empty. At high temperatures, space at its emptiest will be filled with thermal radiation. At low temperatures, structures will form in the void. According to quantum mechanics-more specifically, to Heisenberg's uncertainty relation-we can never precisely fix the amount of energy that fills a certain region of space in a certain amount of time. The amount of energy will fluctuate. Consequently, we will never be able to define a zero-scale for energy. One might say that the vacuum of physics emits energy-more of it the shorter the time span we define, less of it for longer times.
According to Albert Einstein's famous formula E = mc2, energy and mass
are the same thing. Mass therefore also fluctuates, and empty space will see a constant emergence and disappearance of particles that carry this mass. These particles don't last, and physicists call them virtual particles.
The physical vacuum is by no means empty and devoid of characteristics. Rather, anything that can exist at all will oscillate and spin in it in a random, disordered fashion. In this vacuum, quantities will emerge that, in an abstract space of particle properties, will define directions; these quantities, which in their abstract space act somewhat like magnets in real space, are called fields. Although these fields influence the way in which we perceive the physical world on all levels, the discipline of physics needs to examine minuscule regions in order to confront them directly. It might appear paradoxical-but the huge accelerating machines of elementary particle physics not only examine the particles they accelerate but also explore the emptiest of spaces we can imagine, and thereby some of the questions that the Greek natural philosophers bequeathed to us as problems still to be solved. But it is not only with huge accelerators that we investigate the void-we can perform less spectacular but enormously precise and complex experiments based on nothing but light.
Most of this book is devoted to what we know about the void. There is, however, another question intimately related to this void, a question that has fascinated humankind across the millennia, that has spawned myths and legends of creation: How did it happen that at some point in time, something appeared out of nothing? To this day, physics cannot give a definite answer to this question. The standard models advanced by cosmologists and elementary particle physicists permit them to reconstruct the history of the universe back to fractions of a second after the Big Bang. So much is for certain. But the closer we get to the Big Bang, the less certain our knowledge. We can only speculate about the Big Bang itself, and what happened immediately after it. This is the subject of the last two chapters of this book. Which ideas about creation do the laws of nature admit? These laws always and everywhere have been the same. This is a central tenet of physics: It is just the state of our world that has changed over time.
The main purpose of this book is the transmittal of scientific insight. Only such insight can foster the reali7.ation that nature is. in fact, understandable to

PRE FA C E

(X

humankind. I stand convinced that this is the most noble aim of basic scientific research. By this I do not mean to assert that the world in all its complexity will one day be completely understood. That, to be sure, will not happen. Rather, the implication is that natural phenomena are not the work ofspooks and demons but are due to rationally explorable causes. This is the attitude that launched Western cultural and scientific thinking six centuries before Christ. We owe it to the so-called pre-Socratic philosophers in Ionic Greece, who, as Erwin Schrodinger, recipient of the 1933 Nobel Prize in physics, has put it, "saw the world as a rather complicated mechanism, acting according to eternal innate laws, which they were curious to find out. This is, of course, the fundamental attitude of science up to this day."
My physics colleagues who browse through this book may be amazed at the long passages I devote to the ancient naturalists. I do this for two reasons: first, there is my curiosity, which goes hack to my own school days; second, there is my conviction that those ideas from antiquity do not differ much from what determines many of our contemporaries' notions of the natural sciences. It is therefore appropriate that we take them as a starting point for the communication of today's scientific insight. It should be possible to start from the views of the ancient Greek naturalists and move to those of modern science. This is what I have tried to accomplish here.
In the process, I have had to ask myself how far I can pursue this path without losing contact with the notions actually developed by those ancient naturalists. Take the ideas of Anaxagoras as an example: I started with his divisibility of ur-matter and interpreted it in terms of modern ideas on the formation of structures of matter. In doing so, I did not dare to go as far as taking what he calls the seeds of, say, hair, which are hairs themselves, to be an early form of self-replicating fractals.
In my original manuscript, I limited myself to the interpretation of those ideas from antiquity that were directly related to the topics I am covering. Other ancient notions I merely renamed in modern terms. This was not enough for Eginhard Hora, my meticulous editor at Hanser Publishing: He insisted on the insertion of minireviews on many physics subtopics, from A (as in antimatter) to Z (as in zero-point energy).
The philosophical and historical passages are based on an unsystematic perusal of the available literature. The term nothingness invariably evokes mythical psychological connotations. The reader will not find these connotations here, nor will terms such as nirvana, black-out, immersion, the nothing that acts out of nothingness be found here. I hope I have managed to write this book in such a way that these connotations cannot even be read into my text.
But I cannot be sure. The great English astrophysicist Sir Arthur S. Eddington wrote in the preface to his popular science book New Pathways in Science, published in 1935: co A hook of this nature has to evoke precise thinking hy means

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NOTHINGNESS

of imprecise turns of language." Therefore, the author may not always succeed in "evoking in the reader's mind the very ideas he is trying to convey. He certainly cannot do so unless the reader joins in an active effort." Eddington demands that the author manage to "relegate secondary complications to the background." Those complications will become obvious to anybody trying to describe the facts in a straightforward fashion. But what do we mean by "secondary"? That may well be a matter of opinion. Certainly, some scientists who have spent years chasing down one of those complications will be unhappy to see it classed as "secondary. "
Given such a book, how will it be read, and by whom? All authors naturally hope to see their readership riveted on their books, breathlessly engaged from start to finish, but such readers are rare. Most pick and choose, looking for what elicits their interest, their happy concurrence, or their violent protest. I tried to write the prologue and epilogue so that those readers who read nothing else will still gain an overview of the substance of the entire book. The prologue and epilogue should stand on their own as an intelligible and, I hope, captivating synopsis of our subject.

ACKNOWLEDGMENTS
•
Part of this book is the result of a six-month stay at the TRIUMF Laboratories in Vancouver, British Columbia, from 1991 to 1992. The author wishes to thank his Canadian colleagues for their hospitality. He acknowledges the support of the Volkswagen Foundation, which made that visit possible. He has greatly profited from discussions with, and from the reactions of, friends and colleagues who may have wondered about some of the topics that came up in the process. This book has benefited considerably from the helpful comments of Eginhard Hora, editor at Carl Hanser Editions, and Jeffrey Robbins of Helix Books. To all of them, and to many whom I cannot name but who are well aware of their contributions to this project, I feel deeply beholden.
XI

CHAPTER 1
PROLOGUE
•
PHYSICS AND METAPHYSICS
•
LET'S ASSUME WE CAN REMOVE ALL MATTER FROM SOME REGION OF
space. What will we be left witM A region of empty space? Not necessarily. In the universe, between galaxies, each atom is at a distance of about one
meter from its next neighbor. Still, the space between those atoms is not empty; it is bright with light and other radiation from very different sources. It is only in the absolutely empty space of our imagination that no light, no radiation penetrates-that space is as dark as the legendary rooms of Schilda, in the German fairy tale. A region of space is not really empty simply by virtue of not containing matter.
BLACKBODY RADIATION Ifwe wanted to produce a region of really empty space, we would have to remove from it not only all matter but also all radiation. To keep it from exchanging matter and radiation with the space around it, we would have to shield it effectively-say, by surrounding it with walls. We might then take an ideal pump to evacuate this enclosed space, hoping that the radiation it contained would gradually be absorbed by the walls and that the final result would in fact be a truly empty space.
Unfortunately, that is not how it works. First of all, walls not only absorb radiation but also emit it. Every enclosed space is filled with the radiation absorbed and emitted by its walls. That is why a space free of matter is not necessarily empty space.
The radiation we are discussing here might be thermal radiation; at higher temperature, it might be red light, like that emitted by an overheated electric stove; and at still much higher temperature, it might be the light of the Sun. This radiation weakens as the temperature of the emitting body decreases, but we would have to go to what the physicists call absolute zero-that is, - 273

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NOTHINGNESS

• •

•

? •

a

b

c

Figure 1 A container filled with air at temperature above absolute zero ( - 273 degrees Celsius) contains molecules and thermal radiation (1a). When we remove the molecules with an ideal pump, the thermal radiation remains (1b). The space that remains when, in a gedankenexperiment ("thought experiment"), we cool it to the (inaccessible) temperature of - 273 degrees Celsius is what we call the physical vacuum (Ic): Everything that the laws of nature permit to be removed has been taken out.

degrees Celsius (C)-to have it die off altogether. It follows that above the unreachable temperature of absolute zero, the radiation emitted by the walls will never permit an enclosed space that is truly empty (see fig. I).
THE TORRICELLI REVOLUTION
There is no way to produce a space emptier than the one that we approximate by lowering the temperature in, say, a box we have previously evacuated by pumping out its air content. It is not a matter of course that we use the words evacuation and pumping in this context. We owe it to what may well be the most significant event in the historical development of our subject. In the seventeenth century, the Italian naturalist Evangelista Torricelli, a student of Galileo's, was the first to formulate a correct answer to the question of why it is possible to suck the liquid content out of a vessel through a straw. He discovered that the weight of all the air above weighs down on the surface of a liquid. If we remove this weight inside the straw by sucking, that same weight, which continues to exist outside the straw, will push the liquid upward inside the straw.
The Torricelli experiment seems simple enough to today's reader: Fill an 80cm-Iong tube with mercury, invert it, and set its open end in a bowl of mercury. (See fig. 2.) Mercury from the tube will now flow into the bowl through the open end of the tube-but not all of it. The mercury level inside the tube will drop to 76 cm, and will stay there. The 4 cm above that level are empty of air and appear as a vacuum from which the mercury column is suspended.
In reality, of course, the mercury docs not dangle from the vacuum ahove;

PROLOGUE

3

Figure 2 Torricelli's experiment. It showed that air can be removed from a given volume. No air can be contained in the space denoted "vacuum," since air can penetrate neither glass nor mercury. The space is not, however, entirely empty, because it contains such prosaic substances as the vapor of mercury. (At the present stage of this book we are not taking into account the complications arising from present-day physics.) Right after Torricelli's experiment it was argued by Scholastic philosophers that the so-called vacuum was a plenum-a space full of a hypothetical, ever-present fluid that can penetrate even glass and mercury. This fluid was invented in order to save the hypothesis of horror vacui, that emptiness cannot be created.

rather, it is supported by the external air pressure, just like the liquid inside a straw. The Torricelli experiment showed, above all, that nature opposes the formation of a space devoid of air with a finite force that we can overcome. In other words, the experiment shows that space without air is possible after all.
When Torricelli's experiment became known to the scientific world, it moved to center stage immediately (see fig. 3); many investigations were triggered by it. Up to that time, there had been no way to link the characteristics of empty space with the consequences of our special position in the universe-our existence on the surface of Earth, but at the bottom of an ocean of air.
About one century before Torricelli, the Copernican revolution began to gain acceptance; it stipulated that we live on a planet that orbits the Sun. This made it possible to separate the influence of our specific position in the universe on our astronomical observations from the consequences of the natural laws that govern our solar system-and opened up the question of how physics explains the motion of the Moon, the planets, and comets around a Sun at rest. The progress in our thinking owed to this "revolution" is dearly demonstrated in today's planetaria, which often simulate a space voyage from Earth out into interstellar space: What we perceive as the irregular motion of the planets gradually settles into simple elliptical orbits about the Sun.
Where our topic, empty space, is concerned, the Torricelli revolution is analogous to the upheaval caused by Copernicus. The Torricelli revolution would not have been possible without the knowledge extant in Greek antiquity: that air is "something" rather than "nothing." A room full of air had long been known not to be an empty room. When we now discuss a space as empty as possible inside an evacuated box at ever-decreasing temperatures, we are in fact taking Torricelli's empty space one step further. It soon became clear that even after a giv('n volume had been eV,1(uated by pumping action, material from the walls

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NOTHINGNESS

Figure 3 This 1698 engraving illustrates the activities of the French Academy of Science founded by Louis XIV. The choice of subjects shows that the main interest of the academy was in the applications of science. The final influence of Torrice\li's experiment on the development of the sciences and their applications was more important than any of the practical devices shown here. Although in the seventeenth century there was no foreseeable application of that experiment with the exception of the barometer, it finds its rightful place near center stage in this scene. I was unable to discover any depiction concerning electricity or magnetism in this picture. If there is really none, this is a serious omission.
evaporated into it-in Torricelli's case, this was mainly mercury vapor. If we now insist on creating a space we can call empty, we have to pump out these vapors too. The realization that the universe beyond Earth's atmosphere should be a matter-free space in that sense began to take hold soon after Torricelli.
MAnER AND SHAPES
How do we characterize space in a state we approximate gradually by pumping out a container and lowering its temperature as best we can? Physics says that this container is as empty as it can get, but adds that it is not empty in the sense of a true, absolute vacuum. So what will necessarily remain in a space after we have taken out of it all that can be taken out?
The stories, myths, and legends of creation ask the opposite questionprobably the oldest question of natural philosophy: How did the world get started? From what did it originate? That would probably be the same as whatever remains after all has been removed that can be removed from a region of space. This

PROLOGUE

5

question has been answered in the most diverse fashion through the ages, starting with the earliest Greek philosophers before Socrates. Thales of Miletus, the earliest Greek natural philosopher we know of, thought that the universe originated from water. In his opinion, water filled the universe in its different forms-solid, liquid, gaseous. There is no such thing as empty space, since water pervades everything. If we interpret his ideas in terms of modern physics, Thales' space is as empty as it can get when it contains nothing but water in its primal form-presumably in its liquid form, as in the oceans. To remove all matter from the world would then be tantamount to reducing everything to this primal form, melting down the rocks, condensing the air. If all that exists is some form of water, the opposite must then also be true: Water contains every potential form of matter.

SOMETHING, NOTHING, AND THE VOID
In reality, Thales and his successors did not ask about empty space. Rather, they inquired about the concept of nothing, the opposite of something. The physics question about the existence and properties of empty space can, in principle, be answered by experiment. The philosophical concept of nothing, of course, is accessible only to logical analysis: Is it possible to imagine this nothing, the absence of anything, without violating the laws of logic? Will not this nothing become something simply by virtue of its being imagined? Is our problem in speaking of this nothing perhaps simply a problem of language, or is it one of logic?
No successor of Thales has ever given us an answer to what exactly defines nothing, other than characterizing it simply by negatives. The most important was the argument that nothing cannot exist-clearly a tautology. In the creation myths and in the ideas of the earliest philosophers, some "ur-matter" existed. Whence it came and what exactly it was-these questions were of less concern than the question of how it had acquired the form it possesses. This question remains today. By posing it, the originators of creation myths and the early philosophers developed modes of thought, and these are more important than the specific concepts they frame. Physical models of what is something and what is nothing can have only coincidental value if developed without the knowledge available to modern physics. Still, the modes of thought that hail back to the creation myths and to the early philosophers retain their significance.
WORLDS IN A VOID
Just as the metaphysical nothing presents us with a conceptual conundrum, so docs empty space. A wholc sClJucnce of naturalists and natural philosophers, from Aristotle through tht' medieval Scholastics to the great French mathematician

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NOTHINGNESS

and philosopher Rene Descartes and on to Albert Einstein, has tried to come to terms with it. In the process, we have come to see that truly empty space cannot exist-though the concept has to be further clarified before we can even enter into a more precise discussion of its meaning.
Let's start with a dilemma: Let a box be so big, and let its walls be so far removed from us, the observers, that our experiments on the properties of empty space cannot be influenced by the walls. Taking this approach to the extreme, we might remove the walls altogether; then the box in its infinite expanse has become a universe that contains nothing but us and our experimental gear. In that case, what can we find out about this space all around us, and about our motions inside it?
First, we want to find out whether the result of an experiment can be influenced by the location where it is performed. Suppose we have a toy planetary system at our disposal. In the first experiment, we situate the Sun in space somewhere in front of us, add the planets, give them a set of well-defined initial velocities, and record their subsequent motions with a movie camera. After a little while, we stop the experiment and redo it somewhere else. The great German philosopher and mathematician Gottfried Wilhelm Leibniz, a contemporary of Newton's, believed that the scenario I described does not permit the definition of "somewhere else." If everything that exists is embedded in an infinite empty space, any imaginable universe is identical with itself-so argues Leibniz in what he calls his principle of the "identity of the indiscernibles." It would be senseless to differentiate between a world "here" and another world "there."

BODIES AS PROBES
If we consider only logical differences, this is certainly correct. Physics, however, differentiates between objects that influence others and objects that simply mark a space without influencing others; let us call these latter objects "probes." In the language of physics, such probes are mostly idealizations that approach, but never quite reach, reality. Let two examples serve to explain this. Take Earth, with its magnetic field. If you measure that field with a compass carried from place to place, the influence of the needle itselfon Earth's interior, which generates the magnetic field, can be ignored. The influence is there, but it is so small that we cannot really measure it. The compass needle is a probe that is exposed to the influence of Earth's magnetic field, but its own magnetic field is so small that its action on Earth's field is negligible. The same would hold if we distributed a whole series of compass needles to map out the entire magnetic field of Earth.
If we now tried to map out the magnetic field of a compass needle instead of Earth's field, it is clear that we would need much smaller probes; otherwise, the field to be measured would collapse under the influence of the new probes. In high school physics, we tend to use small iron filings for this purpose.

PROLOGUE

7

Next, let us try to measure the gravitational field of Earth by approximate probes; for this measurement, we utilize the probes' mass. From the motion of the tides, we know that the Moon deforms the oceans and thus influences Earth's gravitational field. This means that the Moon, due to its own large mass, is not a very precise probe for measurement. On the other hand, the masses of communications satellites are so small that their presence exerts no noticeable influence on the shape, and therefore on the gravitational field, of Earth. The implication is that the closer we get to an ideal probe for the measurement of Earth's gravitational field, the smaller we must make that probe.
Leibniz does not differentiate between markings that do or do not influence physical behavior. He considers logical differences only, so that, for him, marked spaces are always different from unmarked spaces. But as physicists, we may imagine that a small probe has no influence on a faraway experiment. Thus we leave a mark at the position (and with the velocity) of our toy planetary system before moving it to another location-satisfied that the probe has no other effect than to indicate where the system has been. At the new location, we perform the same experiment on our toy planetary system and compare the results with a movie of the previous experiment. There should be no difference-this is what the reader expects. For Leibniz, however, once a mark has been left at the location of the first experiment, the second experiment will be entirely different. If no mark has been left, the first and second experiments, as Leibniz would see them, are entirely the same experiment; the ensuing courses of events must then be identical, on logical grounds. But if a mark has been deposited, Leibniz will not admit any relation between the two experiments. However small and far away the mark may be, the mark has altered the logical situation of the second experiment as compared with the first. Consequently, the two experiments have no relation whatsoever to each other; for Leibniz the first implies nothing about the second.

THE VELOCl1Y OF EMP1Y SPACE
Based on these experiments, we would imagine that there is absolute space, immutable everywhere. We can cover it with a network of probes that we call a coordinate system. Let these probes be at rest with respect to each other, and let their characteristics be indistinguishable from those of the space they cover, which they define. So far, we see no dilemma. That will arise only when we start talking about the velocity of space.
Let us now imagine we set in motion our toy planetary system for the second experiment relative to the markings of the first one. If we now ask again whether our observations in the second experiment concur with those of the first, we run into trouhle; If they do, it would not he possihle to tell a space in motion from a SpiKe ill rest. To be <thk· to talk aboul space, we would have to choose arbitrarily

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NOTHINGNESS

one space with a certain velocity relative to all others from infinitely many physically equivalent spaces, and define the one we chose as the space. Physics does not distinguish anyone of our choices of spaces from any other one. However, if the second experiment does not look like the movie we took of the first, then space must be able to tell one form of motion from the other; we would therefore have to say that space acts in this fashion, and this action makes it observable, if only indirectly. If we apply this train of thought to the space in an evacuated box-the one we want to define as "empty"-at ever-decreasing temperature, then we have to concede that this space behaves like a medium.
Newton's mechanics unites these two possibilities, in a curious way. As long as the second experiment moves with a velocity that is constant in both magnitude and direction when compared with the first one, there will be no difference between the actual event and the movie. Therefore Newton's mechanics can serve to define empty space in a given state of motion: All "spaces" that move at constant speed relative to each other are physically equivalent. In each one of them, the same laws of Newton's mechanics will apply. If, however, our motion in the second experiment is accelerated with respect to the first, then what is recorded on the film is not the same as what we observe in actuality.
This is so because we need forces to effect accelerations. All rotations belong to the class of accelerated motions, since the direction of the motion is constantly changing. The force active here is the centrifugal force. Let us assume we are located on a rotating platform when we position our toy planetary system. Initially, it will share with us not only location and velocity but also acceleration. The initial acceleration will, however, not influence its further behavior. Once released in empty space, each system moves at a constant-that is, unaccelerated-speed. Had we released another observer together with the system, that observer would see what our film shows. We, on the other hand, being further accelerated, will notice a different behavior. It is not the velocity but the acceleration ofan observer that figures in the laws of nature that describe what this observer will see. Therefore, neither the magnitude nor the direction of the constant velocity of some object has absolute significance; what does count is the simple statement that it moves in an unaccelerated state of motion.
Historically, the most important example of an object's moving at constant speed if not influenced by external forces was provided by Galileo. A stone released by a sailor from the top of a moving ship will fall parallel to the mast, just as though the ship were at rest (see fig. 4). The velocity in question is parallel to the surface of the ocean. The stone is being accelerated by Earth's gravity perpendicular to it.
Later, we will consider systems and their observers that are jointly subject to free fall. For instance, the toothbrush of an astronaut in a space capsule is suspended in front of him. A movie might show that he himself can turn freely,

Figure 4 The rock dropped by a sailor off the mast of a ship moving at constant velocity will drop vertically along the mast irrespective of the ship's motion.

PROLOGUE

9

- - - - - ----- --- ---- ------~ ----------------- ------------~ ~~
as though he were weightlessly suspended in space. Ifhe then moves in accelerated fashion, his toothbrush will not move with him if he leaves it to its own devices. This example is not unlike what we demonstrated with the toy planetary system earlier. The laws of nature that apply to an observer subject to rotation are more complicated than those that apply to an observer at an unchanging velocity.
The forces appearing in accelerated systems are well known. They cause cars to move out of lane on road curves, they lock seat belts, they distort the faces of astronauts at launch, and they lift the seats of rotating carnival rides. Newton's mechanics distinguishes not a particular state of motion but rather a particular state ofacceleration. This physical reality, which figures in Newton's law of motion, is described in a remarkable article of Einstein's as the ether of mechanics. Instead of ether-according to Einstein-one could just as well speak of the "physical properties of Space."
Leibniz, like Descartes, thought that motion could be defined only by observing the motion of material objects with respect to other such objects. This is correct for motions at constant velocity. Because of the forces that oppose acceleration, it is possible to differentiate accelerated from unaccelerated motion using the characteristics of the system at hand; there is no need to look elsewhere. Also, "empty space" acts like a medium. If there is a system accelerated with respect to this empty space, there will be centrifugal force. That it is otherwise impossible to physically define "space" creates a paradox, which Immanuel Kant addressed in the following way:
The circular motion of two bodies about a common center (including the axial rotation of the Earth) can be recognized even in empty space. In other words, there is no need for comparisons to external space based on experience; still, a movement comprising a change in the properties of external space can be expressed empirically, even though said space is itselfneither expressed empirically nor deserves to be an object of experience: here is a paradox that /leeds to be resolved.

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NOTHINGNESS

Let's admit that this tortuous statement of Kant's bears reading two or three times.

SPACE AND THE LAWS OF NATURE
Does it make sense to define empty space as a space from which "everything" has been removed? The answer to this question depends on the laws of nature. For example, we might never be able to remove substance A together with substance B; our choice might be to remove either A or B. Alternatively, those laws might prescribe that the remaining amounts of A and B form a product whose value cannot be less than some given quantity. They could also say that a remnant of A must remain in all enclosures. The first choice is, in fact, approximately the way it turns out in reality. Furthermore, the supply of a substance in anyone vessel might be inexhaustible; for that reason alone, it would be impossible to empty it altogether: It is, so to speak, bottomless from the start.
The definition of the vacuum given by the great nineteenth-century Scottish physicist James Clerk Maxwell is the following: "The vacuum is that which is left in a vessel after we have removed everything which we can remove from it." This definition encompasses our question but still leaves an opening: What is it we are unable to remove, and how do we know we have removed "everything we can possibly remove"?

QUANTIFYING "SOMETHING"
Physics quantifies "something" through its energy. A bowl is empty in the physics sense when it has given off all the energy it can. Since air molecules with their masses m, according to Einstein's formula E = me 2, stand for an amount of energy, we remove energy from a vessel when we pump air out of it. In this case, Torricelli's definition of empty space coincides with that of physics. But the latter surpasses Torricelli's. Energy is the "special stuff' of physics. Any system left to its own devices will give off as much energy to its surroundings as those surroundings can absorb. Let the pendulum serve as an example. In its state of lowest energy-which we might call the ground state or the vacuum state-the pendulum hangs motionless. Whatever its initial motion, it will eventually pass into this state. This applies to any pendulum that can give off its energy through any mechanism-for instance, through friction.
And so it goes with all physical systems: Left to their own devices in an environment to which they can pass their energy, they will eventually assume a state of lowest energy-hence the special position that energy holds in the characterization of "something" in physics. Wherever Torricelli's definition of "empty" is applicable, it coincides with its definition in physics. In some other cases, the physical definition of emptiness will lead to a surprising result. Take,

PROLOGUE

I!

for example, a glass filled with water at 0 degrees Celsius. Let this be our physical system; never mind Thales for now. Let's look into this system's state of lowest energy, its "vacuum." At 0 degrees Celsius, when water passes from the liquid state to the frozen state, it surrenders energy in the form of heat. When it melts, it absorbs energy-the so-called heat of melting-so that water in the state of lowest energy at 0 degrees Celsius is solid, not liquid.
Obviously, we can take the ice out of the vessel. That means we lower its energy again, according to Einstein's formula E= mc 2• Could it not be that there is something-some substance A (which surely could be neither ice nor water)-that we cannot take away from a given system without raising that system's energy? Or that we cannot remove from a given system at all? That could indeed be the case, and that is in fact what happens. This substance A is, to be precise, the subject of our book. At this point, we are ready to give it a name (or, rather, two names, since there are two variants to our substance). In the first variant, we speak of the Higgs field. Once this field appears in a vessel that has been pumped empty and whose temperature has been lowered as much as possible, its energy will be further lowered. This is similar to what we observe in water: When we lower the temperature of water from above 0 degrees Celsius to below 0, it will turn into ice at precisely 0 degrees. But in the process of turning from liquid to solid, it will give off "melting heat"; that is, we have to remove more energy so that the temperature can drop below 0 degrees Celsius.

PHASE TRANSITIONS
The freezing of water is called a phase transition. The temperature at which this happens is called a critical temperature. A different phase transition is the evaporation of water, at another critical temperature of 100 degrees C. Similarly, iron undergoes a phase transition at its critical temperature of763 degrees Celsius: At this temperature it loses its magnetization; when cooled below this temperature, it will regain its magnetization. The critical temperature of the phase transition at which the Higgs field appears is so high that only at a very early phase of the existence of the universe, fractions of a second after its creation, did higher temperatures exist. The Higgs field has pervaded the entire universe ever since, including our container--our so-called black box-the temperature of which we have been attempting to lower.

THE HIGGS FIELD AND VIRTUAL PARTICLES
There is no "something," no "water" whose "ice" could be tied to the Higgs field we mentioned above. This field pops up in our empty space simply because in its presence this space is in a state of lower energy than in its absence. How is it possihle for a "something" to conhlin less energy than i1 "nothing"? I will

12

NOT H I N G N E S S

explain shortly. But first I must return to the second variant of what we called substance A above, and give it a name: We will call it virtual particles. Such virtual particles exist always and everywhere. To remove them is impossible. The reader should think of them as droplets in a steam bath-droplets that materialize and dissolve in saturated steam. I will explain their physical meaning as we go on. I mention them here simply to complete the concepts that contribute to our picture of empty space-of space that is as empty as possible.

NEGATIVE ENERGIES
Now let's talk about a "something" that is characterized by an overall negative energy. Let's start with a rock so small that it fits into a single point in space, and let's place it into the gravitational field of an Earth that we also imagine to be pointlike. In our gedankenexperiment, we can pretend the rock is suspended by a thread from a cog on the axis of a generator; when we let it drop in the gravitational field, it will set the generator in motion. The closer it gets to our pointlike Earth, the larger the force with which it is attracted by the gravitational field; this means that the system consisting of rock and Earth is able to give off more energy the smaller the relative distance. We can extract an almost arbitrary amount of energy, and might take it away in electrical form by the use of an accumulator. Obviously, the more energy we take out of the system of rock and Earth, the more the energy of the system itself has to decrease-such that eventually it must become negative. On the other hand, when rock and Earth are also taken into account, Einstein's formula E = me2 sees to it that they add a very large positive amount of energy into the balance. And yet we can imagine things to be such that rock and Earth are so close together that the energy balance of positive and negative contributions adds up to zero or even to a negative value.
Negative energies are the focus of a unique fascination. Physicists disagree once the concept of negative energies is extended to the universe as a whole-a disagreement that emerged pointedly in a dialogue that two recently deceased Nobel laureates, Eugene P. Wigner and P. A. M. Dirac, engaged in: If energy can become arbitrarily negative, then negative energies may appear, regardless of which energy level has been defined as having value zero. Einstein's general theory of relativity, in fact, permits us to define that zero level as the energy that does not cause a gravitational force.
The thought process that led us to negative energies was touched on in previous paragraphs; it will reappear as we proceed in this book. Let us recapitulate: A space that contains a pointlike rock in the gravitational field of a pointlike Earth at close distances may well be at a lower energy level than if it were truly "empty." If only energy counted, rock and Earth could then spontaneously emerge in empty space. In reality, this won't work for rock and Earth-but it will work with the Higgs field: Below a certain very high temperature the Higgs field can,

Figure 5 Every water molecule consists of
two hydrogen atoms (small empty circles) and one oxygen atom (large gray circles). In the fluid state, the water molecules move at random. In ice crystals, they oscillate around their rest positions.

PRO LOG V E

13

and indeed must, appear out of the void, simply because the resulting contribution to the total energy is negative; there is no characteristic tied to the Higgs field that would impede its appearance.
MOTION AND ORDER
Why should that be true only below a given temperature? The reason is that this mysterious field connotes a highly ordered state-just as water molecules become ordered when they undergo a phase transition into ice. As long as they move in the liquid water phase, they wander about in random motion; in the crystalline ice phase, they oscillate- about a point of rest (see fig. 5): Their state of lowest energy is always the ordered state in the ice crystal. But they will reach this state only below the critical temperature, which is 0 degrees Celsius in this case. Their own thermal motion will not permit an ordered state at higher temperatures: The hotter a substance, the faster and the less ordered its molecular motion, and the larger the momentuin with which molecules collide. Above 0 degrees Celsius, the water molecules collide so frequently and so violently that no structure can appear or be sustained. Should there be somewhere in the water a small piece of ice, its molecules will be rent asunder by the momentum of the surrounding molecules. It is only at 0 degrees Celsius that the number of these hits and their impact no longer suffice to destroy incipient structures: The water can now freeze into ice.
The water molecules in the ice phase form a regular pattern; we might think of it as a pattern on a wallpaper, except that it is three-dimensional. Therefore, ice (as distinct from water) does not share the high degree of symmetry, with respect to arbitrary translations and rotations, that empty space possesses: Ice has only very limited translational and rotational symmetries, leaving its crystal

14

NOT H I N G N E S S

Figure 6 A snowflake.

structure unchanged (see fig. 5). This means that certain distances and certain directions in geometrical space are singled out as special. The same is true for the Higgs field; this field also distinguishes certain directions in its own abstract space-the space spanned by the properties of elementary particles.
SYMMETRIES OF OBJECTS
We will return to the Higgs field farther down the road. But we will keep talking about symmetries of individual objects, as well as of the laws of nature, as we go. Take the definition of the mathematician Hermann Weyl, as formulated by the physicist Richard P. Feynman: "[AJ thing is symmetrical if there is something we can do to it so that after we have done it, it looks the same as it did before." In this fashion, a snowflake is symmetrical with respect to the operation "rotation by 60 degrees," because a 60-degree rotation does not change its appearance (see fig. 6). Moreover, snowflakes are mirror symmetric: If we reflect a snowflake on anyone of six different straight lines across the center, its appearance will not change.
Figure 7 shows part of a pattern that extends to infinity toward both right and left. We call it translation symmetric, because the pattern can be shifted right or left by discrete amounts (which we can read off the figure) without changing it. Going a step further, we can apply the same thinking to figure 8: If we consider it part of a wallpaper pattern that continues to infinity in all directions in its plane, then the pattern we see enjoys multiple translational symmetry. Moreover,

Figure 7 This figure shows an excerpt of a pattern that extends infinitely to both sides symmetrically. It is translationally, rotationally, and mirror symmetric. The symmetry with respect to reflection at the horizontal median line of the pattern is obvious. It is also easy to note the two sets of turning points and vertical mirror lines. (The figure comes from a computer program written by the author.)
Figure 8 Excerpt of a large symmetrical pattern that is invariant under a number of different translations, rotations, and mirror imagings. (This image was also created by the author's computer program.)

PRO LOG U E

IS

the patterns of both figure 7 and, figure 8 are invariant when reflected on certain lines, or rotated at given angles about certain points.
ORDER, CHAOS, AND SYMMETRY
Although the symmetries we discern in figures 6, 7, and 8 are the most obvious features in these illustrations, they show evidence of the breaking of higher symmetries: those of the circle, the straight line, the plane. The circle remains unchanged if we turn it around its origin, or center, by any angle; the snowflake, by contrast, has to be turned by 60 degrees or a multiple thereof to remain the same. A horizontal straight line can be shifted any distance right or left without change. Figure 7 can be shifted only by certain given amounts. The same goes for the planar pattern in figure 8: While an empty plane can be rotated without change by any angle about any point, the pattern will remain the same only if we rotate it by certain angles about certain points. That should be enough: Patterns with symmetries that are based on the ordering of finite elements will, by that very definition, break higher symmetries based on continuous parameters.
Since the water molecules in liquid water move around without any order, water possesses, on the average, all the symmetries of empty three-dimensional

16

NOT H ] N G N E S S

space--among these, translation symmetry by arbitrary amounts. As soon as water freezes into ice, this translation symmetry is reduced to the lesser symmetry, by certain given amounts only. This obviously implies that the crystal can occupy an infinity of positions that can be distinguished by comparing the displaced crystal with the same crystal in its original position. The position of figure 5 is one of them. Others are generated by subjecting the crystal to any transformation
that is not one of its symmetry transformations.
As ice forms in the water, the emerging configuration has to decide which of these possible positions it will occupy. True enough, there may be prior asymmetries or external influences that influence this decision-the formation of the ice will make these influences visible by magnifying their effects. But for all practical purposes, the formation of ice breaks the continuous symmetries of water nondeterministically. This nondeterministic symmetry breaking is called
spontaneous symmetry breaking.

SYMMETRY AND EMPTY SPACE
Given that empty space, as we understand it, contains nothing that could be changed by translation, rotation, or mirror imaging, we may call it symmetric under translation, rotation, and mirror imaging. 'The void as such," according to Aristotle, "is unable to differentiate." In it, a stone cannot know what is above or below. Consequently, the stone cannot begin to fall. Newton called that the
stone's state of rest-and this is certainly what Aristotle meant. The symmetries
of physical empty space are not a matter of course. This space, the way we see it, remains the same when seen from a rotating merry-go-round. But here is the hitch: The merry-go-round is accompanied by centrifugal force; there is no such thing as a symmetry under a change of the velocity of rotation of our merrygo-round.

CLOSED SYSTEMS
The symmetries of a physical empty space depend on those of the laws of nature. We will be discussing their multiple relations as we go along. The symmetries ofthe laws ofnature can be read from the characteristics ofself-contained physical systems. The toy planetary system we experimented with was one of those, since we set it in an otherwise empty space. We can interpret it just as we do any closed system, as a probe for the properties of "empty" space. An arrow sent off by an observer on Earth's surface is not a closed system: It will not continue in a straight line, simply because Earth attracts it (and we haven't talked about the effects of air friction on its trajectory). The arrow's path provides enough information to observers, at least in principle, to tell them about their own elevation above sea level. At higher elevations, the force ofgravity is smaller, so that the path ofthe arrow will be less strongly

PRO LOG U E

17

Figure 9 The path taken by an arrow de-
pends on how and where the archer shoots it. Suppose he practices on the banks of Lake Erie (9a). In a subsequent competition at the top of Independence Pass, he will miss his target, because of diminished effects of gravity, which doesn't bend the • path as strongly as at the elevation of Lake Erie (9b). In empty space, in the absence of gravity, the arrow will fly in a straight line (9c). The archer can therefore infer certain facts about his location simply by the success of his marksmanship, without bothering to look at his surroundings.

---------- ---~

r---~_.l_..A.-----------------------

deflected. That applies equally to an arrow sent off inside a closed hall. To find out whether he or she is on the plains ofNebraska or in Independence Pass in Colorado, the archer need not look out at the surrounding landscape. The inner characteristics of the arrow's trajectory-that is, its curvature-will provide that information (see fig. 9). To hit a bull's-eye, the archer has to aim differently on the plains from the way he or she does in Independence Pass.
I do not claim that the effect is so large that the archer needs to consider it here on Earth, but it does exist. The same point can be made more strongly: It would make little sense to practice archery on Earth for a subsequent contest on the Moon-or, say, on the tiny planet of Saint-Exupery's Little Prince. Where there is no gravity to speak of, the armw will take off in a straight line.
That means the observer with bow and arrow in a gym on Earth does not form a closed system. Earth influences archery. The arrow behaves differently in Earth's vicinity from how it would behave in a hall somewhere in otherwise empty space. The laws of nature that are valid for the flight of the arrow close to Earth's surface are not fundamental: They depend on the presence of Earth, which we did not include in our system. Therefore, they are not translation symmetric. If we move the system from, say, the plains of Nebraska to Independence Pass in Colorado, we will have changed the locally valid laws of nature such that the trajectory valid on the plains is no longer possible in the Rockies, and vice versa. This fact makes the elevation ofthe imagined hall in which we practice archery an observable quantity. We don't have to look out the window to ascertain the elevation of the hall and the

18

NOT H I N G N E S S

relative change between the two locations: The laws that govern the trajectory of the arrow tell us about the altitude of the hall.
Let's return to the closed systems: A closed system behaves exactly as though there were nothing else in the universe. The universe as a whole clearly is such a system. Can systems that do not include the whole universe but do not show the influence of the remainder of existing matter be considered closed?
We cannot shield any system from the fundamental laws of nature. They act on every system, everywhere, at all times. Suppose those laws of nature that we consider fundamental based on our limited experience actually depend on the existence ofdistant galaxies somewhere out there in space: This would mean that these laws are in fact not fundamental and that only the universe in its entirety is a closed system. It does not preclude the potential existence ofother laws beyond our knowledge, more fundamental than those we observe; they might be independent of the state of our universe, and therefore truly fundamental.

SYMMETRIES OF THE LAWS OF NATURE
The hall in otherwise empty space we have been imagining for our archery experiments is, according to Newton's mechanics, a closed system. In it, the arrow, once sent off, moves with a constant speed in a straight line-unless, of course, it hits some obstacle. This is independent of the hall's position in otherwise empty space. When we say a law of nature has translational symmetry, we mean that translations transform every process allowed by the law into a process that is also allowed. Rotational symmetry, on the other hand, means symmetry with respect to rotations. It applies to the laws of nature, such as Newton's law for the motion of a planet around the Sun, but almost never to the individual processes the law allows, such as the motion of the planet on an ellipse, which is dearly not rotation symmetric. The type of motion that the arrow adopts in empty space is independent of the direction in which the archer shoots it. Its trajectory will always be a straight-line path, at constant velocity-conforming to Newton's first law. This law is therefore symmetric with respect to both translation and rotation.
Now let's place the hall on the surface of Earth. The laws of nature that now apply are no longer fundamental. They describe the trajectory of our arrow in a way that lacks both translational and rotational symmetry. Moving the scene of the shooting from the plains of Nebraska to Independence Pass would yield equivalent paths of the arrow only if the laws of nature allowed the arrows to travel the same path in different topographies-but we know they don't. Even more obvious is the laws' lack of rotational symmetry close to Earth's surface: The trajectory of an arrow shot off parallel to Earth's surface clearly differs from one that starts its motion vertically, either up or down.
The fundamental laws of nature-to which, as Newton was the first to realize,

PRO LOG U E

19

we can reduce the effective laws that are valid dose to Earth's surface-are both translationally and rotationally symmetric. We can visualize this by including Earth in our observed system. In the joint system of archery hall and Earth, the laws of nature no longer depend on where Earth happens to be in its orbit around the Sun, and how it is oriented in space on its own axis. It should be emphasized that we are not concerned with the rotation of Earth but rather its orientation in space at any moment due to that rotation. Strictly speaking, we should try to imagine that the motion of Earth about the Sun, as well as Earth's rotation, stop at certain points, at which we then inquire about the trajectory of the arrow. The laws of nature that are valid for the system containing Earth, arrow, and bow are subject to Newton's laws of gravity and of motion under the influence of forces. In contrast to the archery hall on its own, the combined system is a closed one, if we ignore a few corrections. One such would be the gravitational pull of the Sun's mass on the arrow-and to eliminate it, we would have to include the Sun in our system also. What we are really after is an idealized system consisting of arrow, bow, and all celestial bodies in an otherwise empty space. In this system, which is truly a dosed one, Newton's rotationally and translationally symmetric fundamental laws will be valid. They specify how different components of the system attract each other. They also imply how gravity influences the curvature of the path of the arrow. The precise shape of Earth with its mountains and valleys makes no difference as far as the laws of nature are concerned-it is simply part of the definition of the system.

ONCE AGAIN: SYMMETRY AND EMPTY SPACE
Although it may appear that we have moved away from our principal topicempty space-this is not really so. Between the symmetries of empty space and the symmetries of the natural laws, there are dose connections. Both empty space and natural laws are symmetrical with respect to translation, rotation, and certain changes of velocity. These changes in velocity exclude, in Newton's mechanics, acceleration or deceleration. When we turn, we experience centrifugal force; this means the applicable laws of mechanics are not symmetrical with respect to changes in the velocity of rotation (which is, technically, called angular velocity). The reason, according to Newton, is that we rotate with respect to absolute space-that space is therefore equally asymmetrical under the transformation of our example: When an observer who rotates in a space is subject to centrifugal force, the observer's relation to that very space implies the possibility of defining the fact that he or she is turning.
Position, direction, and constant velocity of space in Newton's physics can be freely chosen, because of the symmetries the laws of nature possess with respect to changes of position, direction, and constant velocity. Newton's laws do not depend on these three properties of his imagined space. To what extent can we

20

NOTHINGNESS

then say that a space with these very properties is, in fact, existent? No doubt, it can be defined by a set of coordinates, but those coordinates are not implied by the space. The only property of space that can be defined by itself objectively and free of arbitrary choices is the physical concept of its acceleration; this is true because Newton's laws are not symmetrical under changes in acceleration. Let us stress that the acceleration we are discussing is not a change of velocity with respect to some predefined space; rather, it describes the change of velocity of systems under the influence of external forces when compared with others that move in the absence of such forces. This comparison illustrates why we use the term acceleration; we could equally well describe the difference of the two systems in terms of the forces that pull on each of them.
We have no doubt that there is space in which we can embed things, objects, matter. But its status, ontologically and physically speaking, is far from clear. Our senses do not tell us whether the space is "curved," even less whether centrifugal force would act on a merry-go-round in a space devoid of stars or galaxies. These are physics questions that we will address-but we will not be able to answer in which sense space as such "exists."
In the physical void or vacuum-that is, in the lowest-energy state permitted by the laws of nature-the two properties with which we endow this very vacuum in our imagination do not necessarily coexist. These properties are that there is no matter in the vacuum's volume and that it shares the symmetries of the laws of nature prevailing in it. The postulate that our space be empty in a conventional sense, in fact, uniquely defines the state of the void. Symmetry transformations of the laws of nature cannot change the void-after all, there is nothing whatever in it that could be changed. Therefore any such transformation will leave our system unchanged: It shares the symmetries of the laws of nature. For the state of lowest energy, the empty space as defined by physics, this is not necessarily true. This state may, in fact, have some structure that is not invariant under symmetry transformations of the fundamental laws of nature. In other words, we can infer from its property of not sharing the symmetries of the fundamental laws of nature that there must be several possible states with the same lowest energy. Symmetry transformations of the laws of nature cannot change the value of the lowest possible energy.
We'll return to this subject in more detail later on. The basic idea is Aristotle's: In empty space, no stone can begin to fall, because it does not know which way is up, which way is down. In this sense, the vacuum of physics-the space that contains the minimal permissible amount of energy-cannot be empty in a general sense. If a space of lowest energy is in a state that is not invariant under symmetry transformations of the basic laws of nature, then it contains "something"; it is not a void. The transformations act on this "something" and change it. The very fact that this space has d!stinguishable states, equivalent to each other through symmetry transformations of the laws, permits the definition

Pit 0 LOG V E

21

of a direction, a distance, a concept that defines a particular rotational motion, and these very transformations can take the space from one such state to another. Take a rotation by 180 degrees of a space that includes our Earth, and let this rotation happen about a horizontal axis that goes through, say, a church spire. This rotation transforms our system into a different state-on in which our Earth stands, so to speak, on its head. We can now tell "up" from "down": It is because our space contains Earth that the symmetries of the fundamental laws of nature no longer apply-that we can tell that the space is not empty. Our rotation of space by 180 degrees permits us to introduce two states of that space, connected by the rotation of 180 degrees. They fix the up and down directions for a stone we may drop. Any space that distinguishes directions and accelerations cannot be "empty," in Aristotle's meaning.
Aristotle, in fact, is the outstanding figure in the history of the topic we are discussing. He was the first to deny flatly the existence of empty space on Earth's surface, elevating this denial to the concept of the horror vacui ("fear of the void") and determining the direction of occidental thought on this topic for more than two thousand years. In addition, he established the first image ofour present definitions ofa physically empty space with his concept ofur-matter, which his translators into Latin called materia prima. From this empty space of physics, as empty as the laws ofnaturewill permit, anything may arise as an excitation, just as from Aristotle's materia prima.
Aristotle thought that the universe must have a center toward which all massive bodies gravitate. Were that so, empty space could not exist in the sense that there is no preferred direction in it. Newton's physics knew no such concept as a center ofthe universe. A space that included Earth's gravitational pull could not be empty. Aristotle, in fact, had somewhat contradictory ideas: The center of Earth, for one thing, defined the center of the universe; for another, massive Earth itself must be moving in this very direction. But in no way did he think that Earth with its surroundings was a special system that hid the existence of the fundamental laws of nature-rather, Earth and its surroundings made up the universe. He thought there was nothing whatsoever, not even empty space, outside the shell of visible fixed stars. Bycontrast, our modern concept takes Earth and its surroundings-what we now call our solar system-as just one unit, which we integrate into "empty space." It serves as a kind of probe for the fundamental laws of nature, which must be valid always and everywhere.

FORCES IN A VOID
Aristotle would have agreed with Einstein, who did not want to call Newton's space that contained no "bodies," and therefore no gravitational force, an empty space, but rather the ether .of mechanics. Let's look at another example, in order to pin down the difference between real physical objects that act on their surround-

22

NOTHINGNESS

Figure 10 Oftwo identical spheres in an oth-

erwise empty space, one rotates about the

common axis, the other one does not. How

is it possible that the rotating one (the upper

one along line a) knows that it is turning and

•

that its neighbor is not, and that it should deform in response to centrifugal forces,

while the other should not? Newton's me-

chanics alleges that this is precisely what

happens. Mach's principle, as formulated by

Einstein, denies it. It says, instead, that all

the masses of the universe cooperate in de-

termining what rotational motion, or the

lack thereof, is. If, in an otherwise empty

space, a very massive sphere and a very small

one rotate with respect to each other, the

small one will deform, but the large one will not (lOb). Furthermore, according to this

view, a single sphere in an otherwise empty space will never deform. In 10c, we have

placed the small sphere inside a hollow space surrounding the center of the large one:

The larger one surrounds the smaller one just as the masses of the universe surround a

centrifuge. The general theory of relativity, incidentally, is not in complete agreement

with Mach's principle. Rather, it allows for many different realizations of empty space.

ings and idealized probes that merely register the forces acting on them. Take two identical spheres made of elastic material in an otherwise empty universe. Let them rotate about an axis that coincides with the straight line connecting their centers of gravity, but at different rates (see fig. lOa). Could it be that one of them will be deformed in some way but the other will not? According to Newton, that can be so. Let's leave the true answer open for a minute. Newton's rules of mechanics tell us which (if any) of the spheres remains free of centrifugal force-it is the sphere that does not rotate with respect to absolute space. If there is no centrifugal force, there will be no deformation of the sphere-no flattening at the poles, no bulging at the equator. Newton would interpret figure lOa as showing that the sphere on the upper right-hand side rotates with respect to absolute space but not the one on the lower left.
What exactly would happen to the spheres in our gedankenexperiment was beyond Newton's knowledge. He simply assumed that whatever his mechanics predicted would in fact happen. In reality, the planets, the moons, and the rocks whose behavior his mechanics describes are not situated in an otherwise empty space: The space surrounding them also contains distant stars and galaxies, and who can tell whether they alternately define the meaning of rotation? If that were the case, Newton's mechanics would constitute nothing beyond an effective description of our world; it would become inv<l;lid once we removed those distant stars. One fact speaks in favor of the latter interpretation (unless we want to

Figure 11 A long photographic exposure of the sky with a camera fixed in location.

PROLOCiUE

23

treat it as a mere coincidence): The system in which we observe no centrifugal forces is identical to that in which the firmament-that is, the observed set of fixed stars-does not rotate.
The reader can verify this easily. Figure 11 shows that our Earth rotates relative to the firmament. There is centrifugal force. We learn about its action in high school: Since Earth makes only one full rotation per day, we usually do not notice this force. If a reader in a swivel chair rotates five times per second, the stars in his view will perform the circular motion indicated in figure 11, five times per second. What does he notice? First, he himselfrotates relative to the firmament. Second, he notices that his arms are pulled up and out by centrifugal force. Could it be a mere coincidence that this force acts on him only while he rotates with respect to the stars?
Now back to our spheres: Up to this point, we have treated one of them merely as a marker for the other one-a system of reference that permits us to speak of relative rotation. Let's stick with the notion that space contains only these two spheres and that, contrary to Newton's mechanics, one of them does not realize that it is the one that rotates, that will deform. We now build up the universe, with its totality of physical masses, in a very special fashion: We add more and more mass to one of the spheres (say, to the one in the upper right of figure lOa); we will not change the other sphere-we leave it small and at constant low mass. If in fact it is the masses of the universe that determine the definition of rest versus rotation, it will be the upper-right sphere in figure lOa that does so in our gedankenexperiment. Anything that rotates exactly like this reference mass is defined as not rotating. Consequently, the small sphere will be deformed if there is relative rotation-the greater the deformation, the higher the rate of this rotation and the larger the mass of the upper-right sphere (see fig. lOb). We can treat the sphere with small mass as though it did not act on its surroundings, n~ though it were only a probe. The same is true in figure 10c:

24

NOTHINGNESS

Just as the firmament surrounds a centrifuge, the massive sphere surrounds, in this figure, the (nearly) massless one.
A related experiment to that in figure lob can be performed with a top to be spun inside a satellite in orbit around Earth. After all, Earth is part of the overall mass distribution in the universe, all of which together determines the meaning of what rotates and what doesn't. The top on its path around Earth will therefore have to find a compromise that takes into account the centrifugal force of the firmament and the forces that the rotating Earth exerts on its motion.

UR-MAnER AS A PHYSICAL CONCEPT
It is not clear what physics will come up with as a substitute for the concept the creation myths call ur-matter. That is because there has not been a final synthesis of the two great theoretical developments of our century: Quantum mechanics and the general theory of relativity each make statements about the vacuum, but each requires allowances to be made for the other. We think that the radical statements of quantum mechanics and of the special theory of relativity don't say enough about the vacuum of physics. We have already talked about them: First, if we remove from some space as much energy as possible, there may be a "spontaneous" materialization of structures that we have named Higgs fields without really understanding their nature. Second, virtual particles incessantly emerge and vanish in this space. Because of the indeterminism of quantum mechanics, we cannot say exactly how many of these particles exist in a given interval of time and space, nor what their energies and momenta are. But nothing prevents us from marking points in space and time to our liking. A theory that unifies quantum mechanics and the general theory of relativity will presumably differ in this respect: Very small distances in time and space will no longer be individually identifiable. Maybe the theory will even keep us from telling spatial from temporal distances. Maybe, as we go to very small distances, we will no longer be able to say how many dimensions there are to time and space, just as the number of particles becomes indeterminate in ordinary quantum mechanics.
If this sounds less than clear, don't attribute it to my inability to formulate. Rather, we will have to concede that, so far, there has been no successful way of expressing these ideas in a compelling theory. It is out of the question that the modifications to which the concept ofempty space has been subjected by quantum mechanics and the special theory ofrelativity might be reversed by future developments. The vacuum of physics contains in its faculties everything that the laws of nature will permit. It fluctuates-the virtual particles come and go. The only thing they may be missing is the energy it would take to make them appear as real particles. All that can appear in reality must be present as a possibility-as a state of virtual particles-in a vacuum. Add energy to the vacuum and those virtual states may appear as particles.

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Figure 12 The two spirals are the traces of an electron and a positron. A photon enters from the left; it is electrically neutral; therefore, it does not leave a track. As it collides with an atom, the latter absorbs it and emits an electron-positron pair. An external magnetic field makes the resulting tracks spiral in opposite directions, where we see a reduction of the radius of curvature as the particles slow down. The electron and positron lose motion energy and, thereby, velocity, so that the curvature increases. It is the charge of a particle that determines which way the spiral turns, in opposite directions for electron and positron. Another track, appearing at the same location as the electron-positron pair, is that of an atomic electron pushed out of its atomic orbit by the incident photon. During this collision, the particle-antiparticle pair appears as a by-product. The photon vanishes in the process; its energy is converted into the mass energy of the particle pair plus the motion energy of the particles in the final state of the entire process.
In coming chapters, the puzzling phenomenon offluctuation will be described further. Let's give an example here to illustrate what we mean by the.a priori mysterious concept of "vacuum fluctuations" (see fig. 12).
In the figure, we see two particles of opposite but equal electric charges emerge from the vacuum on the left-hand side. One of these is an electron, the other a positron. Since they are charged, these particles leave tracks. They were kicked out of the vacuum by a photon-a particle of light. This light quantum is electrically neutral and therefore leaves no track. What it did to make the particle pair materialize is this: It collided with an atomic nucleus. But the nucleus acts only as a catalyst in this process; it absorbs the recoil of the emerging particles. Bccalls(' of Ihal, Ihe photon is ahle to transfer ils energy 10 the eleclron-positron

26

NOTHINGNESS

pair in the vacuum. Thus a virtual excitation of the vacuum becomes real-one of the possibilities that are inherent in the vacuum is thus being realized.
Real systems are, in this sense, "excitations of the vacuum"-much as surface waves in a pond are excitations of the pond's water. Just as the properties of water determine what amplitude and speed of the waves are permitted, so the properties of the physical vacuum define the possible excitations-the possible systems that can emerge from the physical vacuum. Water, just like the vacuum, contains the final reality as an initial possibility. The vacuum in itself is shapeless, but it may assume specific shapes: In so doing, it becomes a physical reality, a "real world."

UR-MATTER AS A MODEl OF THOUGHT
We could use almost identical words to characterize both Plato's space and Aristotle's materia prima. Plato attributes actual existence not to matter as such, but to the structural plan of matter, to its form, its idea--today we would say to "the laws of nature." To recognize this plan, Plato thinks, we don't need to know the specifics of matter. The human spirit is part of the higher reality of forms and basic laws: That is why it is able to recognize these through a mere thought process. Matter is part of actual existence only as far as it can be understood. According to Plato, geometry finds the properties of space by means of human thought, human insight; that is why space is more "real" than the material world that we observe. He interprets the latter by means of a fairly convoluted construction as empty space within four of the five regularly shaped bodies that we associate with his name (see fig. 23a in chapter 2). He divides empty space into such bodies in order to make us see the world as we observe it in reality, in his terms. Plato's notion is that the world is part of our reality by virtue of the fact that his special construction subjects it to mathematical forms.
The fundamental aspects of the question of whether or not empty space exists were of no great interest to Plato. His philosophy knew empty space, and that was it. What we said about the vacuum of physics does, in fact, apply to his views: Space has no shape by itself, but it can assume form or shape-and that is how it becomes the observable universe. That also goes for the materia prima of Aristotle, the other great Greek philosopher, whose philosophies dominated Western thinking for two thousand years. The starting point of Aristotle's construct is always a specific concrete object with its own properties: This object he calls a substance. A substance is matter in a specific shape--for instance, in the shape of a statue of Socrates. The materia prima is defined as the remainder of a step-by-step process of analytic thought that starts from that substance, like a stage setting. All specific properties it possesses are disregarded one after the other-its specific shape, its material strength, whether it is made out of marble or out of some other material, and so on. In reality, matter and form always

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27

appear simultaneously. One form of matter may be replaced by another form. But there can be neither form without matter nor matter without form. The materia prima is therefore an abstract object: Real objects may approximate it, but they cannot turn into it.
By reversing the steps that we took to advance from the concept of substance to that of materia prima, we can imagine building up our universe: The homogeneous basic substance, materia prima, contains all the possibilities that the laws of nature will permit. There is no such thing as empty space into which our material world is introduced-that is what Aristotle preached, and modern physics agrees with him. The emptiest space known to it-the physical vacuum-has characteristics that Aristotle gave his materia prima.
Werner Heisenberg characterized Aristotle's matter in the following way: "Aristotle's matter is certainly not a given substance such as, say, water or air; also it is not just simply space. Rather, it is an indeterminate physical substrate which contains the faculty to assume some given form, and thus to enter into physical reality."

EXPERIMENTS ON THE VOID
After chance observations in antiquity and a few selective model attempts, the experimental investigation ofthe void started with Evangelista Torricelli's production of an airless space above a mercury column. Blaise Pascal continued Torricelli's investigations; he tried to prove by straightforward experimentation that it is the external air pressure that is responsible for the phenomena ascribed to nature's "abhorrence of the void," the horror vacui of the ancients. Otto von Guericke in Germany and Robert Boyle in England were the first to develop pumps that permitted the removal of air from larger containers and, thereby, experimentation in a vacuum. Th~se experiments contributed to important insights for any discussion that attempts to define empty space: first, that empty space is translucent; and second, that magnetic fields are not dampened by it. Beyond that, evacuated volumes were used to produce some spectacular effects. Guericke's so-called Magdeburg hemisphere was the first of those. Pascal redid Torricelli's experiment in front of an audience of five hundred, with water and wine instead of mercury. In place of Torricelli's meter-long tube, Pascal used hoses of 10 meters' length hoisted on a ship's mast for his experiment. He built a water barometer in Rouen (see fig. 13), and soon it was discovered that empty space could be used to do actual work (see fig. 14).
Torricelli's vacuum lost its philosophical interest soon after its discovery; it was seen as space devoid of air, nothing else. Gradually, the philosophical questions about the void became questions of physics. It was recognized that the space hetween the planets is empty like Torricelli's vacuum-but then how is it possible that the gravitational force of the Sun acts over such large distances?

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NOTIIINGNESS

Figure 13 Seventeenth-century street scene of the city of Rouen. It is here that Blaise Pascal measured the air pressure with a water barometer around the year 1650.

How does its light travel through empty space all the way to Earth? How does this empty space transmit magnetic and electrical effects? Newton's experiments on heat conduction demonstrated that space also appears to transmit thermal energy; how can that be?
To answer these questions, the theory of the ether emerged, and it took Einstein's 1905 special theory of relativity to finally rid us of it. Today, there is still a good deal of experimental and theoretical investigation dealing with space that is as empty as possible. One of these is an experiment on the so-called Casimir effect: It shows that supposedly empty space actually exerts pressure (see fig. 15). As an experiment, it may not be spectacular, but its importance for our understanding of empty space is obvious. The tremendous machines of experimental particle physics in today's research centers also investigate the vacuum, in their own way. They include the LEP accelerator at the CERN (European Council for Nuclear Research) laboratory for particle physics, in Geneva, which we will discuss in chapter 6. While they cause electrons and their antiparticles, which we call positrons, to collide at energies a hundred thousand times that which corresponds to their rest mass, it is still the vacuum that they are really helping us to understand.
When electrons and positrons collide, they annihilate in a flash of pure

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Figure 14 This contraption is the steam engine of the English inventor Thomas Newcomen, from the year 17l2. It pumps water from a well. The pumping rods on the left are actuated by external air pressure. Water vapor generated in the atmospherical volume B fills the cylinder C, where it is cooled down by the outside application of water. The external air pressure then forces down the piston. The machine was able to pump 1000 liters of water per minute, in twenty strokes.
energy. In the process, they concentrate their energy in a small region of space, the properties of which do not differ from the vacuum in any respect. This flash furnishes the energy that makes particles appear as "real," whereas in the normal, not energetically excited vacuum they are so-called virtual particles. As a consequence, these virtual particles emerge from the vacuum and interact, causing an annihilation that produces a flash, which can be seen by a detector. That is how they tell us ahout the structure of the vacuum.

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NOTHINGNESS

Figure 15 Two electrically conducting parallel plates, at a temperature of absolute zero ( - 273 degrees Celsius), attract each other. The space around them, said to be "empty as possible," therefore cannot be empty: Whatever remains in it exerts pressure on the plates. They attract each other, so the pressure outside them is greater than the pressure inside. Hence the space between the plates is "emptier" than the vacuum reaching to infinity on either side.

The excited state of the vacuum "knows" which particles it can generate with what probabilities. Among its many possibilities is the creation ofan electronpositron pair just like the one whose annihilation created it. But this is only one of many possibilities; the annihilation process makes the electron and positron lose their identity and generates a whole spectrum of possibilities from the vacuum.
THE VACUUM AND THE BIG BANG
On a very small scale, the accelerator experiment emulates the state ofthe universe as it was fractions of a second after its beginning. There is general consensus among physicists about how the world developed after this point in time, but not before it. One model for the evolution of the universe from the very start is the so-called hot Big Bang. According to this model, the universe began its existence as an infinitely hot and dense flash of pure energy. Because of this infinity, physicists speak of the singularity that marked the beginning of the universe. From then on, according to the model, the universe became colder and colder. I'm not saying that the model is uniquely compelling: There may never have been a hot Big Bang; maybe it took the universe a few small fractions of a second to reach the very high temperature-very high but not infinite-that we know it once possessed. But there is agreement both on that hot temperature's being much higher than any temperatures we will ever be able to duplicate in our laboratories and also on the steady cooling the universe has undergone ever since. The higher the energy density we can create in an experiment, the closer we approximate the state of the universe at that initial high temperature. Its state

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3 [

at that point, when there was nothing except that energy, is best described as "the physical vacuum plus energy."
One ofthe most important questions we expect future accelerator experiments to answer concerns the structure of the early universe. The hotter the universe is, the less structure it can have. This is just like what happens when ice melts into water, and water evaporates into steam. In a very early phase of the universe, it was so hot that no structure could evolve. But the cooling process began immediately. Theory tells us that only fractions of a fraction of a second after the Big Bang, the Higgs field emerged as an early structure-just as ice develops in water as it cools down. Inversely, it should be possible to melt the structures that are frozen into the vacuum. If the temperature is high enough, the physical vacuum must undergo a phase transition. It passes from a state with frozen structures to a state devoid of them. It is likely that the energy densities that can be produced at the LEP machine in Geneva will not suffice-although we don't know just how much energy we need to effect this transition. Neither theory nor experiment has given us a convincing indication for the structure of the vacuum. The inquiry about the state of the universe within fractions of a second after its very beginning is as exciting a question as I can imagine.

TIME AND/OR SPACE
If we knew the state of lowest energy ascribed to the world by a theory that unites both quantum mechanics and the general theory of relativity, we would be able to say much more about the origin of the universe than we presently can. This is so because our world has most probably had its origin in this state of lowest energy, or ground state, which may be identified with the absolute vacuum. We tend to associate with this origin a particular moment in time-the time at which our world started to exist. The origin of time is an almost ineffable idea; nevertheless, it might well be that time came into being together with our universe, instead of our universe starting at some point in a preexisting time. This is an important topic of this book. Via the quantum mechanical uncertainty relations, time and space are related to the fluctuating energy and momentum of matter so that all three-time, space, and matter-may share a common origin. However, as of yet, nothing definite is known on this topic. A theory of the physicists James Hartle and Stephen Hawking implies that time and space are not separable at the origin of the world. If we move backward in time, time and space blur into each other at some distance from today. Instead of speaking of "former times," we should speak of another place in space-time. We never reach some previous earliest point in time, because we can move in space-time just as in space alone. This means that we may have already started a forward motion in time while we still think we are moving back in this coordinate.
In spill.' of its finiteness, the surface of ollr Earth knows no limits: We can

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move on it without ever running into such a limit. The same goes for the combination of time and space according to Hartle and Hawking. If we substitute "north" for an advance in time and "south" for backward motion in time, moving across the South Pole on Earth's surface is equivalent to changing the direction of time from backward to forward. We can imagine that time and one dimension of space together form a surface like the surface of Earth. In this case, nothing distinguishes one point on Earth's surface from any other. According to Hartle and Hawking, the same goes for time and space. We can draw arbitrary curves into diagrams that depict space-time, and we can follow these curves in arbitrary directions-we will never hit a limit. It is a condition of our world that it has no confining margin, be it in space or in time. In Hartle and Hawking's model, the question of the earliest time is as pointless as it would have been to ask Roald Amundsen why, when he reached the South Pole on December 14, 1911, he did not go farther south.

NAIVE REALISM
The physical vacuum, which in the course of history has made its appearance in human thought under as varying a set of names as nothing, the void, space, materia prima (Aristotle), matter (Plotinus), and the ether, carries within itself the possibilities of everything that can exist in the physical world. Once we attain true knowledge of this vacuum, we will have a comprehensive knowledge of everything, including the laws of nature. It is as in the thinking of the ancient philosophers: The knowledge of the void, the "nothing," is intimately connected with the knowledge of the "something." Let the world image of the atomists serve as an example: There is nothing except atoms and the empty space between them. The void is part of our reality.
The worldview of the atomists has been a good starting point for a description of reality and continues to be so today. We take it as a matter of course that empty space is not really empty; also, that atoms are not indivisible, continuous, compact physical structures. Quantum mechanics rules the universe. If we st!1rt from quantum mechanics alone, we have no way of understanding the origin of the quasi-classical laws that govern our world as we know it. Why, and in what manner, did an early cosmos develop in such a way that our Earth orbits our Sun on a well-defined trajectory?

CHAPTER 2
NOTHING, NOBODY, NOWHERE, NEVER:
•
PHILOSOPHICAL, LINGUISTIC, AND
RELIGIOUS IDEAS ON NOTHINGNESS
•
GREEK PHILOSOPHY STARTS WITH A DOUBLE NEGATION: IT DENIES THE
concept of "nothing:' Thales of Miletus states flatly that a "something" cannot issue from a "nothing"; neither can it vanish into nothingness. He thereby denies the creatio ex nihilo-the world's creation out of a void. This concept played no part in the Greek mythology of creation. True, the ancient Greek Orphic poets sang songs of "chaos, from which all else issued"; but soon "black-winged night is delivered of a still birth": It then has "Eros couple with the winged nocturnal chaos." Thus, Greek mythology prefers images to philosophical concepts-images often violent and grotesque, even baudily fantastic. Take the example of Gaia, the Earth Mother, who has her son Kronos castrate her husband, the god Uranus. For fear they might depose him, Kronos himself devours four of the five offspring he has with his wife, Rheia. Only his last son, Zeus, is spared: In his stead, Kronos wolfs down a rock wrapped in diapers, falling for another ruse of Rheia's-this is as raucous a tale as you might invent and, fortunately, not the subject of our discussion. But it forms the backdrop for Thales' idea that the world arose from the waters. This ur-ocean, however, to him is not one more god among others, as is Hesiod's and Homer's "deeply vortexed Okeanos"; for Thales this ocean is the one and only source from which the world springs, by condensation and evaporation. And all the while the total amount of this ur-matter remains the same; it cannot be created or annihilated. It carries the force that moves both the cosmos and all organisms-so pervasive and so multi-faceted that it makes up the stuff of the world and fills it to the brim. In this worldview, there is no room for nothingness.
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MYTHS OF CREATION
Most stories of creation deal with the development of ur-matter into the world we see, leaving aside the question of where the ur-matter came from. Thales, atypically, makes a point of denying the creation of his ur-matter out of the void. His successors simply assume the presence of ur-matter ab initio. Heraclitus, who does discuss development and contradictions in this edifice, still leaves out the question of where his ur-matter, fire, comes from; he simply lets our world come from fire and turn back into fire cyclically. But fire itself is eternal-it is identical with the deity, with the basic cause of our world.
More highly developed creation myths-those that can make do with gods who do not share in the more animalistic functions of humankind-can be divided into two categories: those that have a god create the world out of nothing and/or out of some ur-matter; and those where creation happened spontaneously. "Nothing" stands for ur-matter in a completely amorphous state. A spontaneous creation-we might call it a self-creation-appears in the writing of the Taoist Chuang-tzu as follows:
If there is a beginning, there must be a time before the beginning; there furthermore must have been a time that preceded this very time before that
beginning. If there is being, nonbeing must precede it; and before this
nonbeing, there must have been a time where not even that nonbeing had started. Furthermore. another time before that which had not even seen the not-beginning of the nonbeing.
This is where, all of a sudden, nonbeing turns itself into being-"one cannot even say whether this being of the nonbeing is part of an overall being or nonbeing." Be that as it may, we may take it for granted that the perplexing paradox of being/nonbeing in the Taoist's tale will finally turn out to be the world.
The idea of something appearing out of nothing without the mediation of some creator is also present in the Kojiki, the oldest document in Japanese literature, dating from the year 712: "There was chaos-but who can say what shape it had? There was no shape; nothing moved, there was not even a name for it. But in all this emptiness, Earth and Heaven parted and something emerged between the two. What did emerge is a god-but a god who had no part in that creation, not even in his own."
Then there are the creation myths that contain a creator god. This god exists independent of the world, and the myths have nothing to say about his origin. Saint Augustine, in his interpretation of the biblical Genesis-the creation story of the Old Testament-puts it this way: "Heaven and Earth emerge. They call out in a loud voice that they have been created. . . . They also sing out that they did not create themselves. Thou, oh Lord, didst create them." I will not

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quote Genesis in this context, since it is one of the most diversely interpreted and commonly quoted books of the Bible. The creation story according to the Gospel of Saint John begins as follows: "In the beginning was the Word, and the Word was with God, and the Word was God.... And the Word became flesh, and dwelt among us." Whether or not we replace, as Goethe did, the term Word with Sense, Force, Deed, what the evangelist means to stress above all is the origin not so much of substance, of matter, but rather of form and structure.
The creation myth of the Maori aborigines of New Zealand starts with a god who, in the beginning, finds water in existence, and who then also creates structure-in time rather than in space: "In the beginning, there was darkness, and there was water everywhere. There was no light, and 10 lived alone in this immeasurable space. And from the deepest darkness came Io's voice that said: Darkness, light up! And there was light. Then the voice said: Light, turn into darkness! And it was dark again." The change of day and night was thus fixed. The creation myth of the Popul Yuh-the holy book of the Maya-starts with the universe at rest. "Not a breath-not a sound. The Earth was motionless and silent. And the skies were empty. There were only the gentle oceans and the vast spaces of the skies." From all of this, six gods create the world "in water, flooded with light." Among the six gods, the first was Tzakol, who created the world in the beginning; the second was Bitol, who subsequently formed it.
Considering the care with which they detail the concepts of the creation of shape, structure, and ultimately of life, creation philosophies and myths deal only lightly with the initiation of the substance that forms or takes shape. Saint Augustine, our authority for Genesis, spends many pages of his Confessions wrestling with the fact that the first book of the Bible does not even mention the creation of matter as such. Did God, when he "created Heaven and Earth in the beginning," just give shape to matter that was already present? And is there such a thing as matter without shape? Let's listen to Saint Augustine again: "I would rather judge that something that has no shape at all does not exist in the first place; I cannot· imagine a Something that provides a link between shape and nothing-a Something that is neither formed into some shape nor really nonexistent, Something I might call a shapeless almost-nothing." He adds to the question about substance and shape a theological one: Did any time elapse between the creation of substance and the creation of shape? His answer reads like what we would call today a carefully crafted press release. True enough, there is on one side the matter that makes up our Earth and fills our skies; there is also the very shape of Earth and the skies. "But Thou, 0 Lord, didst create both. Thou didst create matter out of the void, shape out of shapeless matter; both were created in one act: Matter took shape with no time loss in between."
Let's not haggle over the question of whether or not the various histories of creation have much, or anything, in common with today's scientific knowledge. That is not our point. Fundamentalists who claim today on biblical grounds that

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only a span of a few thousand years have elapsed since the creation of the world dearly expose themselves to ridicule. We might say that they put themselves in the tradition of James Ussher, archbishop of Armagh and primate of Ireland, who fixed the time and date of creation as the evening of October 23, 4004 B.C. Dating of this kind was done by counting the generations that, according to the Bible, creation followed. Actually, the universe is older by a factor of a million. But whoever forgoes science would have no trouble finding the world fashioned by the creator 6000 years ago so that today it looks precisely like a world that is some fifteen billion years old. The Grand Canyon, in this scenario, would have appeared simultaneously with the first totem pole-about 4000 B.C.-as would Crater Lake. In the same vein, the light that appears to originate from the Andromeda Galaxy, two million light-years from our solar system, would have been created in midflight. The same goes for the light from most stars in our galaxy, the Milky Way. We will also have to put aside our knowledge of fossils, which date as far back as three billion years (unless God went to the trouble of changing the laws of nature in the meantime).
Clearly, the various stories about creation don't elicit our interest because of the empirical truths they tell us about our world. Rather, they deserve our attention because they introduced the conceptual categories that still serve us today when we try to make sense ofthe world. Chiefamong them is the differentiation of "something" and "nothing"-about which the creation myths tell us very little. Another pair of concepts we owe them is that of the juxtaposition of chaos (absence of shape) and shape. Saint Augustine's musings, quoted earlier, are an instance of the usefulness of these categorizations.

WAS THERE TIME BEFORE THE BIG BANG?
Saint Augustine starts his discussion of the time when the world was created with a question: "What did God do before He created Heaven and Earth? Had He been idle. . . and not active, why did He not remain so for all times just as He had been idle and inactive before the creation?" Augustine's answer to his own question has theological as well as logical arguments, but it also opens up a new train of thought: Time started when the universe began its existence. If we track the development of the universe backward in time, we reach a point when both world and time made their appearance jointly. To seek a time before that makes no sense. In today's language, we would say: Events do not follow one another in a time that would also flow if they were absent; rather, they define the very concept of time. If there is nothing that can change, there is no way to make time an observable quantity. Take, as an example, the lOO-meter dash at a track meet: A runner's time over the distance is fixed by comparison with other processes, like the ticking of a stopwatch. In so arguing, we beg the question of whether, as Newton postulated, there is such a thing as an absolute, true mathe-

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matical time that, of its own accord and by its own nature, takes its course uniformly, unrelated to any external object. Time, in this reading, would not be defined by clocks; clocks would only read it, and make it observable through their reading. The highly speculative attempts at a modern quantum theory of gravitation don't know time in Newton's sense. Instead, that part is taken over by a parameter called "the content of the universe." Were this book dealing with time, I would now have to investigate whether time thus defined prefers something like an "arrow of time" over its inverse. But time is clearly also observable in the absence ofsuch arrows-say, if 50 percent of all clocks run with time moving forward and the other 50 percent with time moving backward.
The hypothesis of the Big Bang accords with a model in which time and the universe originate jointly. This is the standard model that physics has of the beginning of the universe. The galaxies are not distributed in the universe in a static fashion, like the cities on the surface of Earth; rather, they move apart as though they had been propelled from one and the same point in space by a catastrophic explosion some fifteen billion years ago.
The Big Bang hypothesis concludes from this distribution of matter that all of it was packed with infinite density at the instant of the explosion. The laws of physics say nothing about matter in that state. We can approximate its description, however, through a sequence of states in all of which the laws of physics are valid. To illustrate this sequence, think of all the universe's matter being concentrated, first, into a space the size of our galaxy, the Milky Way; second, into our solar system; then on to our Earth, to Mount Rushmore, to a nutshell, to a grain of salt. . . . In this fashion, we build up a sequence of states where the laws of physics apply; if we go on with such a sequence, we will approximate the state of matter at Big Bang time to arbitrary precision.
I am not claiming that we know the laws of physics for matter in that state; this isn't so. But we do think that there must be laws, albeit unknown to us, that do apply. Certainly, as the volume is reduced more and more, the matter must have moved faster and faster. We know that the galaxies, as they are moving apart, are slowed down by the force of gravity. This implies that the less space all this matter had at its disposal, the faster it must have moved. We also know that this motion must have been in the form of random relative motion. After all, the galaxies have different velocities; consequently, their building blocks must have moved at different rates in the beginning.
We conclude that immediately after the Big Bang, when all matter was infinitely closely packed, its components moved randomly with huge velocities. We do not know the makeup ofthese components, nor whether it makes sense even to speak of individual components. The important feature that characterized matter in the very early universe was its state of rapid random motion, whether we describe this motion in terms of discrete parts or as that of a liquid, with all the known features of fluid motion.

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For the sake of the present argument, let's stick with the image of discrete

particles of matter. The faster their random motion, the hotter this matter gets.

Temperature is, in fact, defined by the random motion of particles. Hot water

dissolves sugar better than cold water because the water molecules move faster

at a higher temperature and are thus better able to knock the sugar molecules

from their configurations. The closer we get to the Big Bang, going backward in

time, the hotter all matter must have been. At the instant of the Big Bang, the

universe was infinitely hot and infinitely dense. This is the state of maximal

disorder or, as we say, of complete chaos. If there was such a thing as a "before,"

that "before" cannot have influenced the "after." The infinitely high temperature

and density at Big Bang time erased all information about whatever state the

universe might have been in "before." Whether or not it makes sense to speak

of time "before" the Big Bang is immaterial for the development of the "after."

Anything that we speculate about the "before" cannot be checked and is therefore

not in the realm of science.

If the universe was, at one time, infinitely hot, that point in time can be

reached from our vantage point only by following our own timescale in a backward

direction. Obviously, it makes no sense to speak of an actual, physical motion

backward in time. What we can construct is a table of states that describes the

sequence through which the universe must have advanced in the course of time,

and then read it backward. This backward reading will have to stop at the state

marked by the description "infinite temperature." A table that does not stop at

that point cannot be based on observation. It would have to rely on revelation,

a concept well beyond the topic of our discussion.

If we take different substances with equal atomic composition and heat them

until their atoms move individually, we lose, in the process of thermal breakup,

all information on what fraction of these atoms originally belonged to any given

substance. Take graphite and diamonds: Both are made up of carbon. When we

heat a mixture of graphite and diamonds for a few hours to a temperature above

1500 degrees Celsius, there will be only graphite: All information about the

original amounts of the two components has been lost, as have the diamonds

we started with.

.

Back to the Big Bang: There is no harm, no contradiction to empirical fact,

if we assume that there was a time "before" or that there was no such thing.

The same goes for space. Space, rather than time, is closest to the topic of this

book; we will therefore return to the question of whether there can be space all

by itself-space without matter, without radiation, without energy. Is it legitimate

to imagine the universe before the Big Bang as empty space in the context of

considering the Big Bang as creat;o ex n;hilo---creation out of nothing? If this is

not legitimate, maybe empty space before the Big Bang was more than an absolute

nothing. Time and again, artists have tried to portray the "mighty Fiat of the

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39

Figure 16 An illustration of the creation of the universe by the "mighty Fiat of the Creator Spirit."
Creator Spirit" ofwhich Pope Pius XII spoke in a remarkable speech on November 23, 1951. See figure 16 as an example,
PROPERTIES OF SPACE We have to establish the rules of the game before we try to answer these questions. The scientific method demands that as we pose a question, we introduce a procedure that can be used, at least in principle, to answer it. If that proves impossible, the question is not a scientific one. For our topic, the questions that can be asked in this scientific way arc cloaked in intuition. When we think about

40

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--------------------------=-

North pole

a)

parallel Great ---t----f circle

meridian 1

polnto! intersection

equator

b)

c)

Figure 17 (17a) A flat bug believes the world to be a plane-the plane that it inhabits. This bug, coincidentally caDed Euclid, has just figured out that of aD the straight lines that can be drawn through one point it is contemplating, there is one, and only one, that will not cross the line drawn in the figure. (l7b) Another bug, called Non-Euclid, is two-dimensional, just like its cousin Euclid. But it is not flat: It has curvature, so that it hugs the spherical surface on which it lives. Having wandered across the markings we drew into the picture, it is pondering whether there is any straight line that does not cross the one connecting the points marked FRA and VAN. The solution it finds, illustrated in figure 17c, is described in the text.

space, we start from notions that originate in our everyday experiences. These notions contain a variety of ideas about the nature of space and were reduced to a few axioms as early as 300 B.C. by Euclid, in his treatise on the "elements." Among these axioms, a few deal in an obviously correct way with points, straight lines, and planes in the space that we inhabit. Here are two of them.
One axiom says that there are at least four points in space that do not lie' in the same plane. Another one, equally obvious, states that out of three different points on a straight line, one and only one will be located between two others. But there is also Euclid's (in)famous axiom of parallel lines: Consider any straight line and a point not on that line. Obviously (fig. 17a) there will always be one and only one plane that contains the line and the point-the plane of the page, in the case of figure 17a. The axiom of parallel lines then says that there is precisely one line within the plane that goes through the point and that does not intersect the first line. This new line is called the parallel of the first one.
The points, straight lines, and planes of our imagination make up the ingredients of these axioms. Our experiences, however, are restricted to the limited

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spaces of our observation, and we cannot examine the general truth of these axioms insofar as they deal with the infinite. After all, over what distance would we have to observe the two straight lines in figure 17a to make sure they would not intersect? To infinity, for sure-but that we cannot do.
This axiom of parallel lines is notorious, since it makes a statement that reaches out into infinity. Over the centuries, there have been attempts to reduce it to obviously correct statements involving only finite regions, but to no avail. Around 1800, it was postulated that there might be spatial structures where all of Euclid's axioms apply with the exception of the last one, the axiom of parallel lines.
Such a structure would have to involve a curved space of three dimensions. I cannot imagine this space, and I doubt that the reader can either. That leaves us in good company: Several physicists who have published papers on curved spaces with three, four, and more dimensions have openly stated that they cannot imagine any curved space with more than two dimensions. But in two dimensions, they can visualize such spaces, and so can we. A plane is clearly a two-dimensional space without curvature; the surface of a sphere is, equally obviously, a twodimensional space with curvature. Any reader who can imagine a two-dimensional bug that lives in a plane and has no concept of what lies outside that plane will also understand what we said about curved space. Our two-dimensional bugs cannot crawl in a direction at right angles to the plane in which they live; this direction does not exist for them. And in precisely the same fashion, there is no fourth spatial dimension for us inhabitants of three-dimensional space.
Now let's imagine bugs that live on the surface of a sphere (see fig. 17b); a point in the world of this bug is any point on the spherical surface; a straight line is a great circle on that surface. We call a great circle any circle on the surface of a sphere that has its origin in the center of that sphere. An example of such a great circle on the surface of our Earth is the equator; the meridians fall in the same category. Just as straight lines define the shortest distance between two points in a plane, so do great circles on a spherical surface. That's why air connections follow great circles rather than, say, lines ofequal latitude: Vancouver, British Columbia, and Frankfurt, Germany, share the same latitude; the shortest distance between them is the great circle through both of them. Following that circle, Air Canada and Lufthansa will fly over Greenland rather than following the parallel (see fig. 17b).
The great circles are the shortest lines connecting two points on any spherical surface; they are the "straight lines" for the bugs living in this curved space. It is easy to convince ourselves that the great circles can be substituted for straight lines in all of Euclid's axioms, with the exception of the axiom of parallel lines. For this axiom, great circles won't do. After all, the great circle that connects Vancouver and Frankfurt will intersect every other great circle on the surface of Earth; there is 110 "parallel line," according to our definition, in this space.

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NOT H I N (i N Ii S S

That is exactly what figure 17c is meant to make clear. The figure shows three great circles on the surface of Earth. Two of these are meridians, and one is the equator. Let's look at the first meridian and at the point of intersection of the second meridian with the equator. Obviously, every great circle that goes through this point of intersection will also intersect the first meridian. Our second meridian does so at the North and South Poles.
.The spherical surface that conforms to all of Euclid's axioms except the axiom of parallels is a curved two-dimensional space. The well-known eighteenthcentury mathematician Carl Friedrich Gauss was the first to conjecture that we are living in a curved three-dimensional space. In that space, we would exist a bit like our two-dimensional bugs in figures 17b and 17c. Gauss is said to have tried to prove this point experimentally. To do so, he obviously could not leave his-our!-space in order to take a look from the outside; that would be impossible. But we can determine the possible curvature of a space from its inner characteristics-characteristics that become apparent to inhabitants of a space without forcing them to leave that space for a look from the outside. Suppose they want to find out whether they can draw one or several straight lines through a given point such that they are parallel to a given strSlight line outside that point. Gauss, of course, was not able to investigate straight lines beyond the surface of Earth. Instead, he is reported to have triangulated three easily identified mountain peaks in the Harz Mountains of northern Germany. He joined these points by light rays instead of following the surface of Earth from one to the other; he correctly surmised that light rays define the shortest distances in our threedimensional space. Should these rays be curved, the sum of angles in triangular configuration will not amount to 180 degrees, as we learned in high school math, which, after all, deals with triangles only in flat Euclidean space. The sum of angles on a spherical surface is always larger than 180 degrees; in the specific triangle of figure 17c, it amounts to 270 degrees. Gauss found the sum of angles in the Harz Mountains to be 180 degrees. That means our space does not have enough curvature to make a deviation from flat space measurable with Gauss's instiuments for a triangle the size of the one he observed.
The larger the radius of our sphere, the less curvature its surface will have.. If we observe only small areas on the surface of a large sphere, we can safely ignore the curvature for all practical purposes. In the space we inhabit, this is exactly what happens. Space on a large scale may well be curved, but our everyday experiences take no note of that. Architects who build houses on Earth's surface need not worry about the spherical shape of Earth. They do quite well on the assumption that the surface is flat. On the other hand, it would be unthinkable to use a rectangular street pattern like Manhattan's for a very much larger city: It is impossible to divide up the surface of a sphere into rectangles bordered by straight lines that are the shortest connections between the corner points. Similarly, the spherical geometry of Earth influences the shape oflong bridges, tunnels,

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43

Baku
",~""a Aden

b)
Figure 18 If Earth were flat, as in figure 18a, the shortest flight connection between three cities would form a triangle whose angles add up to 180 degrees. On our Earth, the sum of the three angles in any such pattern connecting three cities is larger than ISO degrees (lSb).
and pipelines. And we know well that triangles made up of flight connections between three cities are a long way removed from triangular shape (see fig. 18).
The properties of space determine which path light rays will take. Light rays that form a triangle determine the sum of angles in that triangle; this fact alone means that space as such has properties. Pure logic tells us that a space containing light rays that are able to probe its properties is fundamentally different from a space without light rays. Quantitatively, true enough, space is incomparably less influenced by rays than it, in turn, influences them. Light rays should be seen as probes in an otherwise empty space--much like the magnetic needles and mass probes we used in the prologue of this book, in our gedankenexperiment for the measurement of the magnetic and gravitational fields of Earth.
Our space also determines the behavior of the probes we use to determine its properties. Let's again make them sufficiently small so that space acts on them but we can ignore their action on space. Let us release several such probes in different locations, with different velocities, and/or at different times; they will follow trajectories from which we can deduce the properties of that space. As long as Newton's mechanics are valid, our probes in empty space remain at rest or move at a constant speed in a straight line. However, if the probes are far apart, the expansion of space becomes an important factor. Given that there are these possibilities and many more, it is quite clear that space is anything but a "nothing" without properties.
One of the giants in the history of the topic of our book, Otto von Guericke,

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Figure 19 Just as Poincare's and Feynman's creatures (in figure 20), the angels and devils in Douglas Dunham's computer drawing diminish in size as they approach the confines of their world.

translated Aristotle's definition of the void as "a space that is not taken up by any extended object, but that is capable of being filled with such objects." We might speak of something like an empty box. Aristotle also thought that there could not be such a void-everYthing that can be filled has in fact been filled by nature. To the mathematician, space is nothing but an ensemble of points that are connected by some set of relations. In this interpretation, the laws of nature that apply to objects, including our probes, have nothing whatever to do with the properties of the space around them. The great French mathematician Henri Poincare is one of those who contributed significantly to this mathematical view. To physicists, on the other hand, the properties of space can be read off the trajectories of the test probes they release in it.
But we have more evidence than those trajectories. The shortest path from one point to another is defined not only by the trajectory of the light ray that connects them but also by its very length, measured in whatever units-say, feet or meters. Take two points in an ordinary Euclidean plane and an arbitrary amount of one-meter-Iong measuring sticks (see fig. 20a). Assume that the length of these sticks, defined by the so-called ur-meter, which is carefully stored in Paris, is very small in comparison with that of the path. The length of that path, measured in meters, equals the number of measuring sticks we have to lay down end to end in order to join the two points of reference. The shortest path from one point to another is defined by the fact that fewer measuring sticks are needed on this path than on any other. Obviously, the straight line defined in this fashion is the one we drew in figure 20a, for the motion of a bug in a Euclidean plane.
For a curved space, we have to define the meaning of a I-m measuring stick-the last paragraph showed that this is not easily done. Seen from the outside, lengths may well change when we move from one system to another.

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Figure 20 This figure illustrates the constructs realized by the bug in a plane (20a) and on a hot plate (20b), as described in the text, The circular Jines stand for constant temperatures. We assume that (20b, 20c)

if~f ,::::>E
sticks of

equal length

A

all distances shrinR to zero as soon as a 01 certain lowest temperature-that is, abso-

lute zero, or - 273 degrees Celsius-has

been reached, This means the world of the hot plate bugs has an impenetrable edge:

\

The circle with zero temperature cannot

BO·

be reached, because the bugs' steps become

infinitely short once they get arbitrarily

close to the edge.

hI

\
BO"

cl
Poincare gave a model for this, using spherical surfaces. Richard p, Feynman gave another detailed two-dimensional model in terms of a hot plate, In his famous Caltech lectures on introductory physics, he asked his listeners to imagine a hot plate inhabited by two-dimensional bugs, such as we discussed in the case of spherical surfaces. The temperature of the hot plate is not the same everywhere. It may be cold at the perimeter and hot toward the center (see fig. 20b). These bugs have measuring sticks that Feynman assumes change their lengths as a function of the temperature: The higher the temperature, the longer the sticks. This is borne out in reality by railroad tracks; spaces are left between adjoining track lengths so that their expansion in summer will not lead to a deformation, Feynman's bugs change their measurements as a function of temperature along with the sticks.
Feynman uses his hot plate model to illustrate many properties of curved spaces, Let's address only the topics of the shortest path between two points and of triangles, To join two points on a straight line on the hot plate, the bugs will need more sticks than for the curved line that deviates through the (hotter)

46

NOTHINGNESS

central part of the hot plate (see fig. 20b). Remember, the sticks lengthen with increasing temperature. But the bugs do not know that. They say, "The sticks are the same length; we know that from putting them down next to each other and comparing." The bugs are so obtuse that they now insist on calling their shortest connection a "straight line." Figure 20c shows that if they connect three points with what they call "straight lines" they will notice that the sum of angles in the triangle they just constructed is smaller than 180 degrees.
Our own space is in fact curved. This is certainly true for distances observable by us, in the presence of masses. Whether space as a whole is curved or not transcends our knowledge. Maybe the local curvatures due to distortions caused by individual masses are nothing but valleys and mountains in an overall flat universe-or, for that matter, in a spherical universe. We cannot exclude other forms of curvature beyond those mentioned. To make matters worse, we will have to include time as a fourth dimension in any complete description of the curvature of space as it is realized in our universe. But that goes beyond the ambition of this book.

SPACE AND BODIES
Whether we investigate the properties of space by means of light rays, mass probes, or measuring sticks, it is only by means of the objects that space contains that we learn about it. To probe it, we chose very small masses in order to avoid noticeable action of those masses on the space. Masses, after all, deform the space that surrounds them, just as additional heat sources on our hot plate would further deform the two-dimensional space in which the bugs of figure 20 live. Light rays that pass close by the Sun on their way from distant stars toward Earth will be deflected by the Sun. The experimental proof of this phenomenon made a huge splash in 1919 and served as pivotal evidence for the validity of Albert Einstein's general theory of relativity. We stick to our belief-as did Gauss and Einstein-that light rays follow the shortest path between two points. We have to interpret the fact that light, on its way from a star to Earth, deviates from a straight line as the effect of a deformation of space by the Sun.
As we mentioned above, galaxies move apart as though they had been propelled fifteen billion years ago from a common origin, and they move at velocities that increase with the distance between them. The interpretation of this finding, however, inverts the causal relation of "distance" and "velocity" implied by this correlation. The larger its initial momentum, the farther a galaxy has moved to this day, and the faster it is still moving.
It is improbable that our position in the universe is special in any way. That implies that all galaxies move apart with velocities that increase with the distances between them. We can convince ourselves of such a possibility by pulling evenly on each end of a rubber band. Were that rubber band infinitely long, it is quite

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47

evident that any two points would move apart with a velocity proportional to their distance on the band.
The space of that rubber band is one-dimensional. A frequently used twodimensional model of an expanding space is a balloon. If we regard small markers on its skin as galaxies, the balloon's inflation will make these galaxies move apart at rates that increase with their distance (see fig. 84 in chapter 8), In a threedimensional space, the galaxies move apart just like the raisins in a cake that expands in the oven. These images share one correct point: As the galaxies move apart, they will not expand themselves but rather maintain their sizes. We can treat them as probes that move along as space expands. The space between them increases-more rapidly in the early moments of the universe, more slowly today. The galaxies attract one another and thereby slow down the increase of the space between them. Gravity holds their own masses together and does not permit them to grow as the space around them does. We find analogous behavior in atoms, which are held together by electrical forces.
Bodies influence the space this surrounds them; they tell us whatever we can learn about that space. Does this mean that, possibly, there is no such thing as space by itself? Could it be that space is nothing but a theoretical construction invented for the purpose of giving the observed world an ordered framework? Could it be that what we perceive as the reality of space is nothing but the influence of abstract laws of nature on the behavior of massive objects or bodies? That space is wedded to these bodies and will vanish if they do? This is a respectable position to take. For more than two thousand years, it has coexisted with the view of space as the primary stage that permits material objects to make their appearance. Natural philosophers who theorized about space can easily be charted on a scale between these two extreme positions. On the left is Thales of Miletus, whose "space" is nothing but one shapeless fluid; on the other side, there is Democritus, with his empty space in which material objects whir around. Leibniz on the left, von Guericke and Newton on the right. Albert Einstein juxtaposed these two concepts of space as, on one hand, the positional qualities of the physical world (left) and, on the other, the container of all physical objects (right). In his left-hand case, there is no space without an object; on the right, such an object cannot be thought of except in conjunction with the space that surrounds it-thereby assigning to space a higher reality than that possessed by objects.

PRE·SOCRATIC IDEAS, MODERN IDEAS
The two concepts listed by Einstein may be seen to summarize the twenty-five hundred years' worth of discussions about space. When the pre-Socratic Greek philosophers were looking for unity behind diverse phenomena, they broke with the tradition of Clscribing eCleh phenomenon to an individual force of nature, an

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NOTHINGNESS

individual god, or the whims of a particular demon. Their novel idea was that the world can, in fact, be understood; that there are laws that run the mechanisms inherent in the workings of our world; that ur-matter is the basis for all substances observed.
The earliest pre-Socratic philosophers from Ionia-Thales, Anaximander, and Anaximenes-did not differentiate between substance and law. Their basic question was about being and its negation, nonbeing. Their train of thought had enormous influence: According to them, whatever truly exists, the "being" must be infinite and eternal and immutable. Were that not so, we would have to see the "being" before and after a transformation. That, however, would contradict the fact that there is only one basic substance. The transformation that we observe daily cannot concern the true being. The last consequence is that transformation doesn't exist; it is but a deceptive appearance.
This opinion, obviously, cannot possibly stand: After all, there is such a thing as change, transformation. The question must address the nature of that change: What does it leave untouched? What do we leave out when we deny change? It may sound paradoxical, but physics starts where that change is denied: Denying transformation is tantamount to the assertion that there is something that simply exists and is not subject to change. Take, for a random example, the dancer's two-step: The only permanence in its existence is its formula-one-two-change your step. Similarly, behind each phenomenon there are the laws of nature. Parmenides, a later pre-Socratic philosopher, put the laws of nature into a class of true existence aU by themselves and denied true existence to anything that can change. This idea, as we said before, has been superseded. There is, after aU, matter that is subject to change. Parmenides speaks of a single "being," thereby dividing our world into two categories-that of the laws of nature and that of random coincidences.
Champion for unity that he was, Parmenides did not intend that division. But modern natural science is based on the separation he defined. From our vantage point, the distinction between eternal laws beyond the range of our influence and coincidental phenomena that can change at random is the greatest achievement of Parmenides. It was, in fact, the seed of physics as a science. .
There is no fixed demarcation line that separates the laws of nature from the remaining phenomena. Parmenides' tiny seeds have spawned a vast field of action, a field that keeps expanding as fundamental research grows. The ultimate dream of physics is still that of Parmenides: to understand everything-that is, to elucidate the realm of the laws of nature to the point where nothing remains beyond. But that is still a long way from where we are today. Ultimately, we would like to have the many laws of nature replaced by a single fundamental law governing the behavior of the single fundamental substance of the world. We call that one law we are seeking a unified theory; its basic substance is the ground state, or the vacuum.

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Today, the world of physics can be divided into two areas. First, there are the laws of nature-timeless, immutable. We have no influence over them. Second, there are the initial (or boundary) conditions: Archers set the boundary conditions for the flight path of their arrows by choosing the location, the time, and the direction jn which they are sent off. Once these conditions are fixed, the laws of nature determine the flight path.
Those laws needn't be fundamental. What we see as a given by natural law and by the initial conditions depends, as our example will show, on our position in the universe as much as on our understanding of those conditions. The reason why the arrow does not take off to infinity along a straight line is, simply, the presence of Earth below it. That is not a law of nature; it is simply a consequence of our specific location in the universe-a boundary condition, not a law.
Where is the demarcation line between the two areas? Our very existence on Earth is due to our specific position in time. Ten billion years ago, there was no dump of matter fit as a domicile for intelligent creatures like humankind, and ten billion years hence, no such dump may remain. If we go back further and further in time, we will reach the early universe in its initial conditions. Nobody can say what difference there may be between the consequences strictly due to those initial conditions and the consequences due only to eternally valid laws of nature. If there is only one universe, both apply, both are unique.
The regime of eternal laws of nature within the boundary condition of a given point in time lends the world a measure of uniqueness. The laws of nature apply to numerous natural phenomena that differentiate themselves by their initial conditions. We don't know whether there are laws of nature beyond these-laws that set the initial conditions of the universe.
The laws of nature are defined in a causal relationship to time: First, the state of some physical system is fixed at a given time; then this causal relationship fixes the state of that system for all time according to the laws of nature. (Here is an aside for chaos buffs: There is a difference between determinism and predictability.) As we said before, the logical order then defines the two realms of the universe, that of the initial conditions and that of the laws of nature. The division between them is not arbitrary; it is such that the mutable initial conditions fix whatever follows according to the immutable laws of nature. It is the job of the natural sciences to find, to define, this division. The importance of this point was dearly stated by Eugene P. Wigner, a 1963 Nobel laureate in physics:
The world is very complicated and it is clearly impossible for the human mind to understand it completely. Man has therefore devised an artifice whicl, permits the complicated tlature ofthe world to [,e blamed Oil something

50

NOTHINGNESS

which is called accidental and thus permits him to abstract a domain in which simple laws can be found. The complications are called initial conditions; the domain ofregularities, laws ofnature. Unnatural as such adivision of the world's structure may appear from a very detached point of view, and probable though it is that the possibility of such a division has its own limits, the underlying abstraction is probably one of the most fruitful ones the human mind has made. It has made the natural sciences possible.
What we have discussed so far does not put us in a position to describe all of reality-only isolated systems that we have chosen for the purpose. In these systems, the methods of physics have enjoyed overwhelming success. Newton introduced them in his seventeenth-century treatise on mechanics. Before him, Ptolemy, Copernicus, Kepler, and Galileo looked at the skies and observed mainly the geometrical trajectories of planetary motion. Most important, prior to Newton, nobody conceived of the idea that the planetary system might function very much like a clock: If, like setting a clock, one sets the planetary system into motion at a certain time in a certain way, the laws of nature will determine its behavior for all times in the future. Initial conditions and the laws of nature are thus identified by Newton as the two very different types of causes, both of which are needed to predict the behavior of a planetary system in the course of time.
Newton's idea of separating the two regimes-initial conditions and the laws of nature-has survived to this day. It certainly is a brilliant concept, but, more important, it is successful. I am, however, not sure if Newton ever explicitly mentioned this separation at the basis of his work, or if he even was aware of it. Maybe he would not have been interested. What he did is this: He discovered laws on the basis of which he was able to understand the orbits of the planets. Finding these laws and applying them-that was the bulk of his epochal work. He realized that every planet moves, to a good approximation, as though there were nothing in the universe except itself and the Sun. The smallest physical unit that lets us understand the planetary system consists of one planet and the Sun. Kepler, on the other hand, thought the planetary system could be understood only as a whole-that everything was interdependent. To this day we have no proper understanding of all the observations on which Kepler based his interpretation of the planetary system: the number of the planets and the interrelations among the radii of their orbits. Some aspects of these observations are today understood in the framework of the theory of chaotic phenomena; on the whole, however, we have to invoke random coincidences that occurred during the formation of our solar system to explain the number of the planets and the radii of their orbits. Newton derived, from the laws of nature he thereby discovered, which orbits are possible for planetary motion-they are ellipses, and the Sun is

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located in one of their focal points. He had nothing to say about which ones out of all the possible orbits around the Sun the planets would occupy.

UR-MATIER
The historical development of thinking about empty space largely begins with the pre-Socratic philosophers' idea that all of the physical world issues from one and only one basic matter element-what we call ur-matter. This ur-matter cannot be broken.down or analyzed any further. Later pre-Socratic philosophers, in a logical leap, redefined flat negations of the "there is no such thing" variety into such (non- )concepts as the existence of nonexistent matter, only to lapse immediately into the denial that there could actually be any such thing as nonexisting existence.
The first to consider all aspects of nonexistence was Parmenides. He showed that Thales and his successors' thinking about "being" left plenty of room for what is not-for the nonexistent. Consider, for example, motion: Something that exists is by necessity fully defined-it cannot change, it cannot move. One of Thales' followers, Anaximander, is not willing to assign the distinction of being the ur-matter to anyone of the known elements. According to him, there is only one thing we know for certain. Ur-matter has no limit, it may even be-and this is not quite the same-infinite. Everything we can define originates in ur-matter; by inference, we cannot define that original form of matter. Everyday elements such as water, air, and earth come into being and then vanish again because of internal motions of ur-matter. Their emergence is due to change, which marks them as not really existing.
The existence of one thing means, to these ancient philosophers, the nonexistence of something else at the same location and at the same time. For its existence at the expense of something else's chance to exist, this one thing has to pay a price; Anaximander borrows the term "has to atone for" from judicial thinking. It does so by giving up a bit ofexistence, and that gives a chance to the nonexistent. In the process, the limitless ur-matter remains the same. It secretes matter according to its inherent rules and absorbs it again. True existence is limited to ur-maUer.
We are to understand that Anaximander's ur-matter enjoys real existence. Plato, in his dialogue Timaeus, introduced a related substance that is essentially like putty, out of which the world as we know it was formed. Anaximander's limitless ur-matter and Plato's putty are forerunners of Aristotle's materia prima. The difference is that Plato's substance cannot become apparent in its true form. It is a purely spiritual object. Natural scientists may not have to take cognizance of its specific properties. but they cannot rob matter of those properties.
Tht' idea that ur-matter has to be infinite is shared by the philosophy of

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Anaximander's follower Anaximenes. But the latter doesn't go as far as ignoring everything except the infinite: For his ur-matter he chooses air. It makes sense to him to consider air as extending to infinity-something that cannot be postulated for Thales' water. On the other hand, the conditions known on Earthtemperature, pressure--permit only the substance water to appear in three phases: It can be a rigid body (ice), it can be fluid, it can be gaseous (steam). Because of this quality, water is particularly successful as a substance that shows the unity of existence in those different phases. We are now pointing again toward Parmenides: For him, ur-matter has to be infinite; if it were not so, there would be room for an additional real existence.

TO BE OR TO BECOME
In pre-Socratic philosophy, things being are opposed not only by things nonexistent but also by things developing. From the vantage point of things existent, both things nonexistent and things developing (into existence) touch on the question of whether or not a being can grow out of a nonbeing. Xenophanes and Heraclitus, two immediate predecessors of Parmenides, introduced very different models of thinking about change into pre-Socratic philosophy. According to Xenophanes, "the universe remains immutable." His ideas tend to deny all change; he claims that "the universe is unique, of spherical shape; it is not infinite. It did not originate, but has always been there, always remaining at rest." Heraclitus, on the other hand, sees change everywhere, even though our senses may suggest evedasting constancy. "You cannot step into the same river twice," he says; "the river's flow makes it change rapidly, its currents moving apart and reuniting perpetually." He wants to unmask the seemingly immutable as never in fact remaining the same, and this view made him enormously popular. He did not theorize about the possibility that there is an existence of the nonexistent. Rather, his opinion makes all existence real in appearance only. Therefore, he wants to eliminate this question from our description of the world.
Heraclitus, like all of Thales' followers, wants to reduce the world to one guiding principle, and for this principle he chooses change. Depending from where we look, the principles of change and constancy have their merits and their problems. Take a river from afar: It looks constant-a ribbon that appears at rest all the way from the mountains to the ocean. But then take a look at dose range: Now it is a current of water in motion. If I am a follower of the principle of constancy, I will take the ribbon as describing the true nature of the river, as its natural law. If, on the other hand, I am partial to the principle of change, I have to explain this appearance of constancy in terms of the river's flow. I will elevate the river to a paradigm and use my eyes to detect all along its ribbon the flow it hides. Indeed, the Greek expression panta rhei-everything flows-is closer to reality. Ever since Leucippus and Democritus, physicists have seen the

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dance of the molecules in all matter, even in the most rigid bodies, such as rocks or metals. Or take a magnet and put it on a tabletop next to a plastic comb that we have rubbed so that there is an electrical charge on its surface: There will be energy flows on closed paths through the air above the tabletop. In both cases, we see the laws of nature running the show: One-two-change your step; the rules of the game determine which motion is possible.
Heraclitus did not imagine immutable laws of nature behind aU that change: "Change alone makes not a law." But the idea that we can create the appearance of constancy from substrates in perpetual motion is by itself a most useful one. A number ofdifferent aspects ofa river can be described in terms ofstatic qualities, but the flow of water is best interpreted in terms of a dynamic description. It alone can interrelate width, depth, angle of flow, and the overall shape of the river, and thereby facilitate our understanding of the whole system. For another example, and maybe a more impressive one, consider a cyclone: Seen from outer space, it will change its position and its form slowly; both position and form can be described without reference to the violent motion inside it. But to understand even the external perimeters ofthe cyclone, we have to study that internal violence; indeed, we have to start with that study.

PARMENIDES: TO BE, TO THINK, TO SPEAK
Although the modern concept of the laws of nature might be ultimately due to Parmenides' teaching, it constituted, scientifically speaking, a step backward. He does not make any attempt to discover unity in diverse phenomena. To do so, he would need detailed insight and he would have to propose concrete mechanisms. Instead, he gives up and announces that there is no such thing as mere phenomena:
What truly is, the truly existent, is immutable, in contrast to merely apparent
phenomena. Thus, he divides the world into two areas: First, there is the area of thought, which alone is related to the truly existent; then there are the mere appearances, or phenomena. The latter do not really exist, and are deceptive at that. The school of Elea assumed that there were two substances: "One of these has true existence and can be grasped only by the mind; the other is mutable, developing, and our senses can perceive it. This second one they do not recognize as being truly existent; its existence is only seemingly true." "That is why," Parmenides believes, "we can recognize truth in the truly existent, while we can form only opinions about the changeable."
While Heraclitus sees change everywhere, Parmenides recognizes none. To him, the worlds of thought and of true existence are identical. There is no path that connects the latter to mere appearances-not even a one-way road in either direction. This is exactly what distinguishes the philosophy of Parmenides from Wigner's division of the world into the realm of natural law and initial conditions. Wigner sees the laws of nature as immutable, just like the truly existent of

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Parmenides. In contrast to the latter, however, the laws of nature have their influence on the world of phenomena. It is the job of the natural sciences to gain insight into them. Wigner describes the natural sciences "such as they are." Greek philosophy didn't have the necessary knowledge and could therefore only develop a rough copy of a true science of nature. That was within its scope, but Parmenides' philosophy also falls short of this aim.
According to Parmenides, only the being-the truly existent-actually is. Its inverse, the nonbeing, does not exist, which bears heavily on the topic of this book. Let us repeat this statement: "Only the being has reality. It may well be tangible in front of us-whereas the nonbeing cannot possibly have any reality." This is tantamount to a negation of the world's having sprung from the void, the creatio ex nihilo.
How would you invent the origin of true existence? Whence did it come, how did it grow? I will permit you to think or say it came straight out of nonexistence. We must neither think nor say that it does not actually exist! What possible compulsion could lead to its startingfrom nothingand growing into something? In this fashion, we can extinguish all growth, get rid of all decay. The same goes for thought and for the object of thought. You will not find thought outside the realm ofexistence, where it finds its expression. After all, there is nothing outside that existence, and there never will be. Therefore, everything which has been put into language by mere mortals is nothing but empty names; they may well believe that there is some reality at the very basis: "growth" and "decay," "being" and "nonbeing, " "change of position" and "change of brilliant color."

NOTHINGNESS AND EMPTY SPACE
While Parmenides assigns the "being" to the realm of thought, he astonishes us by considering it, at the same time, as material, pervasive in space. This gives a completely new aspect to our topic, the "nothing," the opposite of the "being"-an aspect that makes it the same as empty space! By denying reality to the "nothing," Parmenides also negates that of empty space. Since space is an object of scientific study, he therefore elevates the nothing to an object of science, not just of philosophical speculation.
CHANGE BY WAY OF UR-MATTER
The fundamental ideas of the pre-Socratic philosophers were carried too far by Parmenides and Heraclitus. Both claimed that only their own view of the world could be true; in the process, they accepted astonishing contradictions to everyday

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experience. The philosophy of their successors Empedodes, Anaxagoras, and the atomists tried to join their points of view, and in the process dispose of latent contradictions to experience.
Change is a part of everyday experience. To account for it, Empedodes replaces one unique ur-matter with four ur-elements: fire, air, water, and earth. Since these four are existent, in Parmenides' definition they are immutable. The change they were introduced to explain is nothing but the external appearance of their internal intertwinings.
This means that something is happening in the world of Empedodes. The forces he sees at the basis of what is happening are "love and discord"-attraction and repulsion, in today's language. He interprets these forces as independent phenomena to the point ofaccording them, at least on occasion, material existence on a level with the elements. "Love" and "discord" are supposed to bring about, for instance, the formation of a sphere out of the elements, and its subsequent disintegration. This concept, obviously, is wrong; there are no such forces. But the idea of independent forces is important: It transcends the concept of immutable laws, since it awards concrete shapes to them. The laws of nature now define independent forces and thereby imply phenomena that can be observed. This is where progress sets in.
Independently existing forces determine the behavior of matter, but they are not matter themselves. Their characteristics can be detected by experience. It is not the work of the rock that attracts it to Earth; rather, gravity, the gravitational force, is at the basis of this observed phenomenon. Anybody who has ever fallen from the branch of a tree is witness to the independent existence of gravity.
Empedodes was the first to distinguish between matter and forces. There is no way to exaggerate the importance of this distinction for the development of natural science. It resurfaced in Newton's distinction between initial conditions and the laws of nature.
There are three details of Empedocles' thinking that bear on the topic of this book. First, he denies the existence of empty space, which had taken the place of the metaphysical "nothing" as a physical concept of natural philosophy. Empedodes, however, maintains that there is no such thing as empty space. In his own words, "The universe has no space that is empty nor space that is overcrowded." True, his denial of empty space belongs to the tradition he grew up in-but the idea as such does cause him trouble. He therefore resorts to his version of an experiment on the subject, which we will discuss below.
Second, he denies the formation and disintegration of elements. Again in his own words, "There is no empty space in the universe. How then should something be added to nothing?" And: "It is impossible that something grows out of nothing; and equally impossible that something present disintegrates into

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nothing at all." And: "There is a coming together and a growing apart of existing elements."
Third, Empedodes tries to define the hidden mechanisms by which nature accomplishes the physical happenings we observe. It is impossible to overstate the importance of this element of his thinking for the development of the natural sciences. Empedodes furthermore attributes the intermingling and separation of substances to what he calls excretions of solid matter; this concept implies that there must be pores penetrating all bodies. For our purposes, this theory of pores penetrating matter has one difficult aspect: How is it possible that there are these hollow channels of emptiness if there is no such thing as empty space? The originators of the idea of pores denied the existence of empty space; the pores must therefore differ from that concept-they must be filled with some invisible substance. This is where we find the first mention of a fluid medium that will dominate discussions of empty space for the next twenty-five centuries. This medium is the least massive of all matter, later on chosen by Aristotle as a fifth element. He called it ether. The ether, as we see, was invented to ensure that there could be no such thing as a true vacuum, a truly empty space. Out of Empedodes' four elements, only air might qualify for the role ofether. As Aristotle puts it: "The physicists who see pores penetrating solid matter don't suppose them to be empty, but rather filled with the very light substance, such as air."

THE HYDRA MONSTER AND AN EXPERIMENT
It is significant that Empedodes did not limit himself to idle speculation on the question of whether the air is a substance or nothing but empty space. He was the first to introduce a detailed experimental observation into that philosophical discussion. In a highly poetical fashion, he uses the example of the mythological nine-headed Hydra, a contraption that was known to raise the level of water in a controllable fashion, to explain that air is, in fact, a substance and not just empty space. (See fig. 21.) This is frequently recognized as his most important contribution. We can divide his observations into four steps. First, take the contraption shown in figure 21a as his version of a hydra dipping into water. If the top of the neck is dosed, no water will penetrate through the holes at its bottom. Second, open the top so air can escape (see fig. 2Ib). Water now enters from the bottom, and air escapes at the top in a tight flow. This shows that water and air cannot coexist in one and the same space-the water cannot enter the hydra before the air leaves it.
It is the third in his series of observations that is the most spectacular and has brought the hydra experiment fame for over two millennia: If we lift the hydra, now filled with water, out of the tank while keeping its neck tightly sealed, the water will not escape through the holes in the bottom (see fig. 21c). We have to uncover the top to let the water flow out of the bottom (see fig. 2Id), while,

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Figure 21 Torricelli's experiment (fig. 2) is basically nothing but a repetition of this experiment using mercury rather than water-and with one decisive difference: Unlike the column of water in this experiment, Torricelli's column ofmercury could not be supported by the pressure of ambient air.

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b

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according to Empedocles, "air wildly replaces it in a turbulent stream from the top."
The poetry here does not contribute to clarity. We can, however, convince ourselves from Empedocles' original description that he correctly understood the first two steps of the hydra experiment. The spectacular third and fourth ones he still explains correctly as showing that both air and water are forms of matter"substances." He avoids the ancient horror vacui. But the details of his description are a bit muddled; they cannot pass for a correct interpretation of the effect in physics terms--to wit, that it is the pressure of the surrounding air that provides the definitive interpretation. It remained for Torricelli, Pascal, and their contemporaries, who were able to observe the level of water (or mercury) inside their versions of the hydra, made out of glass, to supply that interpretation, a couple of thousand years later. Our figure 21 gives the erroneous impression that Empedodes might have similarly seen the level of water inside his hydra; but his was made out of "brilliantly shining iron ore," so he could not.
ANAXAGORAS
Anaxagoras was active at about the same time as Empedodes. In a number of ways, he was quite modern. He is, however, famous for introducing the "Mind" as the primal mover: "It seemed to me in a certain way right that Mind should cause all things, and I reflected that, if this were so, then Mind in ordering all things must order and arrange them in the best possible way." What is modern about his philosophy is his not taking recourse to such an ill-defined concept as long as he could find concrete causes for the phenomena he described. Plato later reprimanded him on this point: "My magnificent hopes were shattered, my friend, when, as my reading progressed, I found a man making no use whatever

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of Mind and ascribing to it no causal action in the ordering of things, but assigning such causes as air and aether and water and many other strange things."
Anaxagoras, just like Empedocles, dismisses the existence of empty space and the formation of "something" out of "nothing." He even formulates a conservation law: "We have to realize that when we take all things together, they will neither increase nor decrease; their totality will remain the same." He differs from Empedocles by assuming the existence of endlessly many elements, which he calls seeds: "Let us consider matter of a given kind as elemental-let's say, flesh, bones, and the like. . . . In anyone of these elements of living beings there are the seeds of hair, nails, veins and arteries, sinews and bones; they may be too small to be seen, tiny as they are. But let them grow, and you can tell them from each other. After all, how could hair grow out of matter that is not hair? Flesh out of nonflesh?"
This, of course, makes no sense at all. Still, it puts Anaxagoras on the trail of two important ideas: the concepts of what we call today the continuum and entropy. About the first of these, he says: "Among things small there is no smallest-there always will be something still smaller; it is unthinkable that something existent can be divided until it no longer exists." He adds: "And it is the same with things large: there is always something larger."
We do not know exactly how Anaxagoras meant to deal with the endless divisibility of all things being. Obviously, this makes no sense in terms of the seeds, where there is no infinite dividing without change. We have long known that we can break ice molecules into atoms, atoms into nuclei and electrons, and so on. But we don't know whether the physical space that surrounds the atoms and the constituents is infinitely divisible-remaining the same but becoming smaller and smaller in measure. Mathematically, space is, in fact, infinitely divisible, which means that the idea, at least, makes sense. To illustrate, let's take a ruler. Between any two points on a ruler you can always place another point. The distance between two neighboring points can be arbitrarily small-an inch, a thousandth of an inch, a billionth, and so on to infinity, to infinitely small distances. Mathematically, we call that a continuum of possible lengths: There can be arbitrarily short rulers. Physically, however, that doesn't work. There is no way of having a ruler shorter than the diameter of a single atom, about 10 - 8 em (one-hundred-millionth of one centimeter). This is not to say that distances smaller than this one cannot be measured; there are other ways to do the measurement, without a ruler. But we do not know whether there is a limit beyond which the divisibility of lengths ceases to make sense. We cannot exclude the possibility that, at some level, space becomes discrete. It is thinkable that some elementary length-say, for instance, 10- 33 cm-is the smallest possible separation of two points. We can imagine the smallest division possible along a straight line like the individual keys on the keyboard of a piano, in contrast to the positions of a violinist's finger along the fingerboard of the instrument. The length of the

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keyboard is divided into discrete smallest distances, the violin string into a continuum of possible finger positions.
Anaxagoras's second important idea concerns the relation between order and disorder, or chaos. He does not doubt the difference between "something" and "nothing," which he takes for granted. He believes that everything has been in existence from the beginning of the world, as some structureless medium. How then, he asks, has structured matter risen from this chaotic state? By gradual ordering, or unmixing, obviously. But how could that have happened? He was the first to recognize the problematic nature of this unmixing, or increasing order, and concluded that "spirit" played a role in it. This spirit, he opines, set the initial chaos into a turbulent motion that initiated the process of unmixing, of ordering.
Today, we know very well that the concept of unmixing presents a considerable problem in many ways. Increasing the order in one place implies destroying it somewhere else-but overall, disorder carries the day. That is what the law of entropy says: The degree of disorder either remains unchanged or increases. When we apply this law to the universe as a whole, we have the difficulty of not really knowing how to define the "overall" of entropy increase. What does "overall" mean in an expanding universe? We don't know all of that universe, and in particular we don't know the degree of order it had immediately after the Big Bang. But we maintain that the order in all kinds of structures here on Earth-be they living creatures, solar panels, cathedrals, accelerators for elementary particles, books about the void-originated only at the expense of undoing order someplace else.
Anaxagoras, of course, couldn't have known about any of this. He felt that unmixing presented a problem, and he was right. But he was far from any possible solution-and so are we to this day, especially when we include the universe as a whole. Even on a small scale, it is not obvious that order deteriorates in all processes. This is because energy in the form of heat contributes considerably to the balance of order and disorder. If a system gives off heat energy, it may be raising its own state of order. We have already mentioned water: When water freezes, it passes from a disordered to an ordered state, while at the same time giving off melting heat. This heat can be used to thaw ice someplace else. Should that happen, it will transform the water molecules in that faraway place from a state of higher order (ice crystals) to one of lower order (the ordinary liquid state). Ordering a substance in this case obviously causes disorder to grow in a substance someplace else. And overall, disorder has the upper hand.
Here are a few more examples, taken from everyday life. Let's say all the ancestors of the Salisbury family lived in and around Salisbury, all the Salazars around Salazar Castle in Portugal, and that all the men in the Cook family were cooks. In the course of time. this order was destroyed without necessarily being replaced by any new order. Or take the books in your library: If you rearrange

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all of them according to subject matter from a previous order based on book size, you destroy one order at the expense of another. Or consider bricks being unloaded from a truck in front of a building site: Masons pick them up and arrange them in precise patterns for the construction of a wall, a house. They appear to increase order, but in fact overall order decreases. Just think of all the gasoline or diesel oil that had to be burned to transport the bricks, and the resulting fumes that were released into the atmosphere. And think of where the bricks came from: maybe some old palace had to be taken apart to yield them, or beautifully undulating sand dunes were dug up to yield the sand out of which the bricks were then baked. And think in particular of the energy dissipated in the form of heat by that very baking.
We see that the creation of material objects cannot stand by itself, an ordering of matter, a building of structure, is also needed in the process. We can interpret Anaxagoras's arguments in this way: He considers the act of creation as equivalent to the formation of order out of chaos. And millennia had to pass before the juxtaposition of order and disorder came back into focus.
In Anaxagoras's world, filled to the brim with continuous matter where Empedodes spoke only of poorly defined pores, there is no room for empty space: "There is no such thing as empty space," he says. Still, he investigates the nature of air experimentally. Is air a substance? Is it just empty space? In his own version of Empedocles' experiment with the hydra, he uses air-filled wineskins. With these he demonstrates the pressure exerted by the air inside them; it is proof that air is not the same as empty space, and he infers that there is no experimental indication for the existence of empty space at all.
With the appearance of Empedocles' four elements and Anaxagoras's infinity of seeds, the unity of all things being that had governed the natural philosophy ofThales and the school ofElea was relegated to the dust bin. Parmenides showed that we can insist on the immutability of all things being only by declaring that all change and all motion are real only in appearance. But there is no denying change or motion in our world; in fact, Heraclitus saw change and nothing but change all over. It was all a bit hard to take: Empedocles saw his agents of change in terms of love and discord, which was still supposed to leave no trace on the sum of things existent. Just like Anaxagoras's "Mind" of the universe, this does not really make sense and should probably be seen as a consequence of one fundamental reluctance of early Greek philosophy: It was loath to give up its bastion built on the unshakable idea that there can be no such thing as empty space.

MOTION AND EMPlY SPACE
Motion brings about change. Couldn't it be that all change is based on motion? The principal tenet ofthe atomists is that matter consists of invisible tiny particles,

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and their motion brings about all change. This motion happens in empty space. If we accept the existence both of the particles and of empty space, it is easy to explain condensation and dilution of matter in terms of an assemblage and a dispersion of particles. The mixing of all substances is then nothing but the random motion of the particles they consist of, in the otherwise empty space they occupy.
The notion of the existence of empty space was first advanced by the atomist Leucippus. With this idea he undid a web of tangled thought as unwieldy as the Gordian knot, in the process providing for a mechanistic view of the world that is free of obvious contradictions. Atoms move in empty space; and that is all there is: The space the atoms occupy is filled, and the rest remains empty space. The changes we observe in the world do not imply changes in the atoms. They move, but they carry all the qualities of true existence as defined by the Eleatics-eternal, unchanged, immutable. They didn't spring from the void; that is impossible: Nothing can spring into existence from the void; nothing can vanish into the void from existence.
Atoms are compact and indivisible. It is the atomists' contention that matter cannot be divided below the atomic level into the infinitesimally small. (The Greek word atom, a-tomas, means "without parts.") Their indivisibility is fundamental; it is not just a practical property. The atoms have mass; they are not part of the void.

THE EARLY ATOMISTS
Leucippus and his student Democritus adopted the concepts of "what is" and "what is not" in terms of matter and void, of the "being" and the "nonbeing," from their predecessors. Their scientific intuition told them that they had to provide explanations for their observations of existence and change. If there were only one kind of atom, there would be no way to explain their observations: There must be various kinds of atoms. But how to tell them apart? They postulate that all atoms are made up of the same basic substance and differ only in their shapes, as pieces of gold might. Still, they do away with Parmenides' unity of being by situating the atoms, the "being," inside empty space, the "nonbeing." As a result, all being is made up out of the same substance; the atoms are merely the smallest units of Parmenides' "Sphere of the Being."
The atomists, of course, could not come up with a reason for atoms taking on only "certain shapes in preference to others, so they assumed that all shapes must be possible." Consequently, there is an infinity of possible shapes, with no restriction imaginable. But how big are atoms? That is where Leucippus and Democritus differ: The former takes them to be so small that our senses cannot discern them; the latter believes there might be at least some atoms of macroscopic size. It is not clear that Democritus followed this notion to its conclusion: He

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never addressed the question of why those visible units of matter would have to be indivisible.
Epicurus later interpreted the teachings of the atomists: Atoms, according to him, cannot take on arbitrary shapes and sizes, certainly not sizes large enough to be visible. To match the atomists' theory to our observations, Epicurus waters down their strict perceptions. He does not accept their rule, later called the "rule of sufficient cause" and which for the atomists implied that there cannot be any restriction on the possible shapes and sizes of the atoms.
The atomists also included the concept of empty space in the framework set by their predecessors. We know what they mean with their tenet: There is no more existence to "what is" than to "what is not," just as empty space is as real in its existence as matter (which equals nonempty space); but there is no need for the paradoxical expression "the existence ofthe nonexistent" for the expression of their thought. The main use to which they put empty space is to make room for the atoms and allow motion without resistance. Just as the atoms are too small to see, there is no way to perceive the empty space between them.

ATOMS AND EMPTY SPACE
The atomists recognize that change in the physical world and perception by our senses are real, but this reality is qualitatively different from that which they grant to the atoms and to empty space. Sensory perception is by no means deceptive, but it can be seen as just one aspect of ultimate reality: The atoms move, they form larger structures, they drift apart. As Democritus says: "Only seemingly does a thing have color, only seemingly is it sweet or bitter; in reality, we are dealing only with atoms and empty space."
Detailed criticism aside, the atomists' perception of reality marks an important step in the direction of scientific understanding in the modern sense. The fundamental questions they address have not been answered to this day. I'm not even thinking of the philosophical questions of "what is" and "what is not"; rather, here are important physical problems to be solved: Are there units of matter that are indivisible? At that level, are they all the same, or do they come in different varieties? Is there meaning to the idea of Anaxagoras that space forms one geometrical continuum?
Maybe some of the atomists' fundamental ideas were correct; if so, that is more or less coincidental. Considering what was known to pre-Socratic natural philosophers, there was no way to decide between the continuum theory of matter and the atomic model. Both are important constructs of their understanding of nature-no more, no less.
For the moment, let's leave the fundamental questions posed by Leucippus and Democritus aside, while admitting that they proved to be amazingly fertile ground for thought. After all, they hinted at elements of the theory of what

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Ludwig Boltzmann in the nineteenth century called the "ideal gas": The theory says that any gas of low density and of temperature that is neither very high nor very low can be considered as an assemblage of indestructible solid spheres that fly through an otherwise empty space and bounce off each other elastically. That's how the atomists thought of fire, where spherical atoms were engaged in something like a three-dimensional billiard game. Of course, this theory of the ideal gas is an approximate one; its validity is limited to a relatively narrow range of temperature and pressure. Once the temperature increases beyond that range, the collisions between the molecules or atoms of the gas become violent enough to destroy them: Their indestructibility no longer holds. At the other end of the temperature scale, the gas will condense to become a fluid and finally jell into a rigid body. The atomists thought up a detailed but incorrect image for this phenomenon: They supposed that atoms hook up to each other by means of mechanical hooks and eyes.
These hooks and eyes, in our modern understanding, are not mechanical: They are electromagnetic forces, governed by the laws of quantum mechanics. What is more important in the atomists' thinking for the topic of this book is their concept of motion in empty space. They correctly imagined that motion in empty space, once started, continues forever unless there is a force acting on it. To the best of our knowledge, nobody else had such clear notions on the subject prior to Newton or Galileo. Wisely, they abstained from speculation about the initial cause for the atoms' motion in empty space; never mind that Aristotle berated them for that reason.

MOTION IN THE ABSENCE AND PRESENCE OF MAnER
The tenet that motion, once started, keeps going forever holds true beyond what we usually call empty space and extends into the abstract vacuum defined by physics. This vacuum is better described in terms of a medium. The natural philosophers of antiquity developed their ideas of motion in it along the lines of the stirrings of a spoon in honey or of a feather in the wind. In fact, however, the notions of filling up a space and of displacing its contents cannot be applied to the ideas of motion in the empty space of physics.
As we understand physics today, these notions are not fundamental and don't lend themselves to detailed observation. The physical size of atoms, nuclei, and elementary particles is, to the physicist, an extremely complex and derived concept. Today's physics sees the interaction between the particles ofmatter in very complex terms-Qnly rarely, and in loose approximation, do they coincide with what the atomists saw as direct hits leading to displacement. To give an example: There is nothing wrong with having an elementary particle in a given location interact with an electric field while, at the same time, gravity pulls on it. It is therefore impossible to categorize gravitational forces and electrical forces as the ancient natural philoso-

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phers did. These forces act, which means they are "something," not "nothing." They cannot be bodies, because they are present in the same point in space. And since they are present there, that space cannot be empty.
The laws of physics will not admit the existence of a completely empty space. They say, at the same time, that bodies left to their own devices in the emptiest space possible will move as though this space were in fact empty. Their motion, once started, will keep going forever. Moreover, it took the instruments of modern physics to permit the observations that prove there can be no such thing as empty space. The motions that the ancient natural philosophers wanted to describe are still most accurately seen as happening either in empty space proper or in a medium that fills empty space, such as honey, water, or air.
In analogy, we can discuss a thickening or thinning of matter in space. By using concepts developed by the early atomists, we today not only understand condensation and dilution in empty space-which, as we know, is not really empty-but also why the bicycle pump heats up while pumping air into an inner tube. When we push on the piston, it will accelerate the air molecules inside the pump. They move about faster in random disorder; the temperature has risen. As with billiard balls, the air molecules kick against each other and are deflected by the walls of the pump. Otherwise they move as though there were nothing around them but empty space.
The atomists did not understand the concept ofheat. What theydid understand very well is the equivalence ofcondensation and dilution to atoms in a gas or liquid moving closer together or farther apart. Their insight challenged the accepted view that all motion must be described as that of a spoon in honey or a feather in the wind. Aristotle then developed the notion that there are several, sometimes mutually contradictory, reasons for declaring the impossibility of having something move in a true void: There cannot be a void, after all, where there is motion. The atomists thought that motion in a medium that fills all space cannot be possible; as a result, there must be empty space everywhere.
Within the framework that the atomists chose, this conclusion holds. We will cite Epicurus, and Lucretius, later followers of the atomists, on the impossibility of motion inside a filled space. They assume-in the language of physics-that no effect can propagate with arbitrarily large velocity. Take a fish swimming through water: Where does the water go when the head of the fish pushes it away? If there is no empty space-remember, the atomists want to prove that it does exist!-the water cannot be compressed. This means that the displaced water must in turn displace other water, and so on; finally, the displacement will reach the space vacated by the tail ofthe fish. This looks like an infinitely rapid propagation ofdisplacement in water. But the atomists don't believe it is possible.
The forms of motion in media such as honey, water, or air, disclaimed by the atomists, really do not exist. All substances are compressible, and no effect propagates with infinite velocity. For many actually existing forms of motion,

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this is almost irrelevant-bodies will move through media as though they were incompressible and permitted the effects to be transmitted with infinite velocity. This makes a certain amount of sense as long as the velocities of motion are small with respect to the velocity of propagation of the signals in water. When these conditions are met, the physics world speaks of dry water. In reality, there is no such thing; but real water can approximate its properties closely.
Upon patient observation, the observer will always notice that real media-air, "wet" water, honey, and so on-put up resistance against all motion of bodies inside their volume, and that includes resistance against stirring. Any motion not fed by outside energy will die off with time in a real medium. The moving body will lose energy to the surrounding medium through friction; it loses motion energy, and the medium absorbs the same energy in the form of heat. Real media are, in fact, compressible. In the process, they heat up, as does the air that we compressed in the bicycle pump.
The atomists needed their empty space so that the atoms would not stop moving. To date, we know only two real media that offer (almost) no resistance to moving bodies. One such medium was discovered in the 1930s: If we cool down helium, a noble gas, to temperatures close to absolute zero, it will flow through the thinnest of tubes with almost no friction. This phenomenon is called superfluidity. Another substance that shows superfluidity is a rare isotope of the same element, called helium-3. It takes experience to explore all the facets nature offers us. No form of logic can replace it.

DEMOCRITUS'S WORLDS, AND WHAT THEY MEAN
The atomists took a stab at cosmology, too. They started from the notion of an infinitely large empty space. Unending existence and infinite extent coincide for them. Hence the notion of limitless space: "Democritus took the universe to be infinite on the grounds that nobody could possibly have created it." This creation would not just mean that "something" replaces "nothing"; there must also be motion and, as Anaxagoras and the creation myths imply, ordering of shapes. "Democritus was one of the philosophers who took the cause of the heavens and all of the universe to be self-explanatory, a 'matter of course.' Turbulence, motion, they thought, started on their own; they brought order to matter, shape to the universe." Unlike Anaxagoras, Democritus goes beyond some of the creation myths in assuming that order might have sprung up spontaneously.
In his opinion, there are countless worlds in our universe. He has detailed views on their sizes and distances. As we saw previously, some of the opinions he passes on to us sound fairly modern. But he fails to invent mechanisms that could have contributed to his assumed structure of the cosmos; for that reason, his cosmological ideas are less illuminating than his musings about atoms in empty space. About those worlds in our universe, he has this to say: "Some

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worlds are still growing, others are in the prime of their existence; others again are on the wane. In one place, worlds originate, in other places, they disappear." Replace what he called "worlds" with the word stars, and the concept makes sense to the modern scientist.
The atomists developed detailed ideas about bodies in space, but not so about space proper. They never conceived ofthe idea that the properties ofthose bodies are based on the properties ofspace. Recall: To them, atoms were "something," and empty space was "nothing." This pragmatic differentiation was basically elevated to a fundamental tenet by their successors: As Grant wrote, "The atomist identificationof real, albeit empty, space with 'nothing' guaranteed that the history ofspatial concepts would, from its inception, be rooted in paradox and enigma."

PYTHAGORAS
It is Pythagoras who raised the notion of empty space to the level of an object of scientific inquiry. All statements about objects or bodies in space can be read as saying something about those objects just as much as about that space. Consider, as an example, the famous theorem on the sum of squares of the sides in a triangle, which takes its name from Pythagoras: Our space is made such that all objects it contains will be subject to this theorem. Pythagoras held that all can be expressed in terms of numbers--only to be shocked by the realization that he could not succeed in numbering all the points contained in a straight line. His proof for this observation I will describe; the formulation I choose is strictly modern. Since they cannot be numbered, those countless points between the numbers were declared by the Pythagoreans as their version of the void, the nothing. Thus, they had to believe in a nothing.
Pythagoras and his followers-unlike Thales-were interested in all possible forms of matter rather than in matter itself. They were fascinated that visible, tangible properties of matter can be formulated mathematically. In this way, properties can be transferred from one object to another; mathematical rules of geometry acquire their own abstract existence.
They were convinced that the essentials of the visible world can be registered only through abstraction. Later, Plato would emphasize that the essentials are not rooted in matter but in the shapes, the laws, the symmetries, the ideas. For the Pythagoreans, the numbers are what really count: Theirs is a real existence, unchangeable in space and time. All ideas that deal with the world of physical matter can and must be formulated mathematically.
Modern physicists would agree to the extent of finding my description astonishingly up to date. We do, after all, believe that the laws of nature determine which form or shape matter can adopt. Maybe there is only one ur-matter, and all that differentiates the matter we experience is the form those laws permit it to assume. That is today's scientific opinion-but we do not yet know that for

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sure. We think we know the basic stuff out of which all matter is made. But we know that there is positive and negative energy; this means that some amount of energy in one place may well cancel an equivalent amount of opposite sign in another place. If that is so, we do not need, in the final sum, any basic stuff at all. In that case the universe may be just one manifestation of empty space or, if we accept the definition of space in terms of a distribution of energies, of the void proper.
I will be more specific about such speculations as we go on. They are mentioned here simply because they appear as a new and different perspective on the thinking of the Pythagoreans and Plato. Their two schools of thought arrive at opposite conclusions on our topic-the void, nothing, or empty space. The Pythagoreans were the first occidental thinkers to concentrate on the relationship between numbers and space. They used numbers to enumerate points along straight lines or sides of triangles in order to reduce distances to integer numbers. However they approached that goal, there always remained a finite distance that separated one point from the next. But their philosophy, based as it was on numbers, had no use for a space between the points that were marked by those numbers. For them, the unavoidable space between points was identical with the void, which therefore exists: "The Pythagoreans said that void exists. . . . It is the void which keeps things distinct, being a kind of separation and division of things that are next to each other. This is true first and foremost of numbers; for the void keeps them distinct." This "void" or medium might be air or ether or any other dilute medium, as long as it leaves the numbers or the bodies that those numbers denote as discrete units.
The basic idea of the Pythagoreans, reducing ratios of specific distances to relationships between integer numbers, was ultimately unworkable for a very simple reason. We can demonstrate this easily by looking at the diagonal of a square: Every line can be subdivided into a number of short sections of equal length. Let's do this for two different straight lines, and those short sections will, in general, have different lengths. In figure 22a, a 20-cm-long ruler is divided into four sections of 5 cm each, and an 18-cm-long ruler is divided into three sections of 6 cm each. It is equally possible to subdivide both rulers into sections of equal length-say, into the centimeter scale also shown in the figure. The Pythagoreans believed that any straight lines of arbitrary length can be subdivided into shorter but equal units of length; if this doesn't work with inches or centimeters, we might have to resort to smaller scales such as millimeters or fractions. thereof.
This, however, is not true: There is no way to divide the sides and the diagonal of a square into sections of equal length. This is shown in figures 22b and 22c. Take the first of these: The three sides ofthe right-angled triangle shown there measure 3, 4, and 5 cm, respectively. The existence of triangles with these properties follows easily from the theorem whose discovery is at the basis of

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\::::;:: o

5

10

15

20

:;:;: ::: :~

Figure 22 The two rulers in figure 22a. of lengths 20 and 18 cm, respectively, can be 01 divided into length units of 1 cm. If we want to divide them into sections of 5 or 6 cm length, we can do so only for one of these rulers each. Similarly, the sides ofthe right-angled triangle with lengths of 3, 4, and 5 cm (22b) can equally be divided into I-cm sections. This is not true of the triangles that are formed by a diagonal divibl sion of a square (22c). As the Pythagoreans proved in antiquity, there is no way to subdivide the sides and the diagonal of a square into sections of equal length. This is illustrated in figure 22c.

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Pythagoras's fame. The lengths of the three sides correspond to the expectations of the Pythagoreans-their lengths can be subdivided into sections of identical extent.
But the same does not work for either of the two triangles in figure 22c. The Pythagoreans proved that there is no way to subdivide the diagonal and a side of a square into sections of equal lengths. No matter what length we choose for these smaller unit sections, our figure illustrates that at least one of the lines always has a length different from a multiple of these units.
The Pythagoreans were deeply shocked by this result. They had divided straight lines into sections; they had counted these sections in the expectation that any straight line would correspond to an integer multiple ofan appropriately chosen fraction of the length of any other line. That was the basis of their belief that numbers, either integers or fractions thereof, would suffice for the description ofspace. They took this abstraction seriously, only to find out that it was mistaken. The properties of integer numbers imply, instead, that it is not possible to express

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the lengths of arbitrary distances in space, and the relations among those lengths, in terms of integers.
The Pythagoreans held on to their fictitious ideas that integer numbers are all it takes to describe space. As a result, they had to correct the properties of space; their space, as a consequence, is not a continuum but a kind of lattice built up out of points at discrete distances from one another. Contrary to their initial intent, they demonstrated in this fashion that there is no such thing as a true continuum that can be completely described by integer numbers and fractions thereof: Every straight line whose length can be described in terms of these numbers contains infinitely many smaller sections for which this is not possible.

PLATO
The world is "artfully shaped such that it acts and suffers everything by its own devices." This is how-if we follow Plato's Timaeus--the Creator God, or Demiurge, arranged it. There is one central point, equally distant from every point at the world's edge; in short, the world is spherical and of finite extent.
Plato's Demiurge found the stuff from which he shaped his world in random, chaotic motion. That made him the agent of the ordering of the universe. There is no mention of his own origin, presumably predating all times. His world differs by its ordering from that of the atomists, who sawall atoms moving about in arbitrary patterns.
Timaeus gives a detailed image of the world as a whole: It realizes Plato's views of perfection. Starting with some hesitation, he develops a notion of having the Demiurge create the properties of the universe following his wish to shape perfection. Thus Plato arrives at four elements and their mixtures, at the spherical shape of the world, and at the orbits of the celestial bodies, from which he derives the notion of time. The universe is, in this fashion, the living realization of a preconceived idea: The Demiurge "intended the world to resemble the most beautiful of all ideas, as dose as possible to perfection in all respects, and thus he created this visible, living world."
The Demiurge also created time-as a "movable image of immortality." What Plato's predecessors had called the one and the empty, the being and the becoming, we find in his writings as the idea and the image. The manycorresponds to the world of physical phenomena, the one to that of ideas. Astonishingly, Plato dedicates only a short and rather obscure section of the Timaeus to the formidable problem of how time can be seen as an image of some eternal unchanging existence.
For the Pythagoreans, the numbers count as ideal building blocks. This idea is transferred by Plato from arithmetic to geometry, even physics. He considers ideas, abstract mathematical forms, to be the true objects of physical perception.

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Observed phenomena are of interest only as long as they partake in these ideas. This is well exemplified by the astronomer, whose most significant task is the reduction of the seemingly irregular motions of the planets to circles or ellipses, and thereby to ideal and "true" orbits of celestial bodies.
In this fashion, Plato formulates an important aspect of the inquiry into physical phenomena. In complete analogy to Plato's ideas, fundamental physics research is concerned with the laws at the basis of observed phenomena. The objects of research are chosen accordingly: It makes sense for the physicist to investigate the structure of crystals rather than the structure of, say, liverwurst. It is, in fact, one of the most important and difficult tasks of the experimentalist to create pure systems, those whose structure and behavior are determined solely by the natural laws to be investigated. But we differ from Plato in one significant way: To us, the laws of nature mean questions that we ask of nature. It was Plato's tenet that all statements concerning the physical world that conform to his criteria. of logical consistency and aesthetic beauty must count among the laws of nature. We cannot agree with him on this point. To us, natural law must be validated by experience--by observation or experimentation. In philosophy, this is called a contingency condition: The laws of nature are not true because of their logical deduction; they are contingent on verification. Things could be otherwise: The laws of nature, such as we see them, make statements about our world that could conceivably be found to be invalid by observation. We might even say that every so-called verification of a law of nature is tantamount to a failed attempt at falsifying it. There is no such thing as definite verification.
Of all conceivable reasons for some sequence of physical events, Plato recognizes only its ultimate aim as worthy of attention. The world is as it is simply because the Demiurge wanted to create something perfect. With this background, Plato's viewpoint on matter and empty space becomes understandable. Mathematical possibilities appear as ur-shapes of matter, giving structure to space. Matter and space are mathematical forms that distinguish themselves through symmetry and beauty. These forms, or shapes, belong to the realm of ideas; the ultimate aim of matter and space is the realization of these forms. Once they are perfect, they must be realized-that is the logic of the proof advanced in Plato's Timaeus.
To the four elements enumerated by Empedocles, Plato assigns four out of his total of five regular bodies, as shown in figure 23a: to Earth he assigns the cube, to the water he assigns the icosahedron, to the air he assigns the octahedron, and to fire he assigns the tetrahedron. Just like the atoms of the atomists, Plato's regular bodies, which stand for the elements, are surrounded by empty space. They differ from the atoms in the hollowness of their interior. They are ideal geometrical structures with infinitely thin surfaces; they have no reality that would give them mass, weight. They divide up space; and how they accomplish that reflects the implicate order of Plato's universe. They are not immutable and

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7l

fire

bJ
cosmos
cJ
oJ Figure 23 Of the five regular polyhedra that can be constructed, Plato assigns four to Parmenides' elements: water, air, fire, and earth; the fifth one he assigns to the cosmos as a whole. Aristotle stressed the connections among the four elements as indicated in figure 23a. Following a drawing by Kepler, a square can be subdivided into four identical triangles (b); similarly, an equilateral triangle consists of six smaller ones (c). The surfaces of the polyhedra assigned to water, air, and fire consist of identical equilateral triangles; this implies that the elements can be transformed irrespective of the subdivisions shown in 23c.

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indivisible, like atoms-their surfaces, and hence their shapes, can be built up out of two types of triangles.
A cube has six square surface areas. Each square is subdivided by its diagonal lines into four equal triangles (see fig. 23b). The surface of the cube thus can be divided into twenty-four of these triangles. The surface of the tetrahedron, consisting of four equilateral triangles, can be made up out of twenty-four right-angled triangles, as shown in figure 23c. Similarly, the octahedron's surface contains forty-eight such triangles; the icosahedron's, one hundred and twenty.
The triangles are the most basic of Plato's units. They are infinitely thin; therefore, they occupy only two out of three spatial dimensions. They do not appear individually, but only as the parts of overall surfaces of regular bodies-for example, twenty-four of them in the case of the cube representing "earth" in figures 23a and 23b. The modern reader might be reminded of the idea of quarks, since just like Plato's triangles the quarks do not appear individually. Rather, they are always contained in larger configurations, which we know as elementary particles. Three quarks make up a proton; the proton, in turn, is a building block of the atomic nucleus. Protons appear individually; quarks do not.
The quarks were originally introduced as mathematical symbols-the carriers of certain properties that would facilitate mathematical understanding of elementary particles. This very role was assigned by Plato to his infinitely thin triangles. But there is a difference: Plato's composite "elementary particles," the polyhedra that we discussed, are models of thought, not physical entities. To Plato, it made no difference whether their existence was ideal or real. Triangles and the structures built out of them, to him, served only to show that nature could be described in terms of such mathematical entities.

ARISTOTLE
With his notion that a body is equivalent to the space its surfaces surround, Plato meant to reduce matter to space, physics to geometry. He did not care whether or not empty space could exist on its own; he never addressed the question. As an object of geometry, Plato's space is closer to this idea than bodies are; to him, therefore, space is more "real." In an everyday sense, however, space "exists" only when surrounded by surfaces-that is, in terms of a body.
Plato's student Aristotle, on the other hand, did have very specific ideas about empty space: It cannot exist. He piles argument upon argument in order to prove this. These are best understood in terms of his preoccupation with the larger connotations that will help to establish his system rather than with any

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individual features. He wants to convince his readership by all means available to him that there cannot possibly be any such thing as empty space.

SPACE AND MOTION
Whatever does not fit into his system, his view of the world, Aristotle classifies as absurd. In the process, he comes up with concepts that reappear some two thousand years later as scientific truths, freed from the label of absurdity. Aristotle's formulation of Newton's first axiom may serve as an example: "Every body
perseveres in its state of being at rest or of ~oving uniformly straight forward,
except insofar as it is compelled to change its state by forces impressed upon it." Those were Newton's words. Aristotle's were: "Nobody can give a reason why a body that has been put into motion in empty space should stop on its own account. Why should it stop in one place rather than in another? Thus it will either remain at rest, or it will of necessity keep moving ad infinitum unless it is hindered from doing so." This theorem yields one of Aristotle's absurdities of empty space: His physics states that all motion will come to a standstill unless some external force keeps driving it-continual motion in empty space is therefore absurd; there cannot be empty space.
This vacuum or empty space ofAristotle's is isotropic; it contains no preferred direction. Today, we might say there is no gravity acting in or on it. But he also discusses gravity as the agent of free fall inside an otherwise empty space. His physics claims that heavier bodies fall more quickly than lighter ones, and that they do so in relation to their sizes, or masses. The same would have to be true for really empty space. But that cannot be. After all, for what reason should one body move more rapidly than another one? Inside a medium, that would be necessarily so: The "larger body" breaks through the medium more rapidly, because of its greater momentum. But in a void, all bodies will move with the same velocity. This is clearly not correct, however. Result: There cannot be a void, an empty space.
Here again, Aristotle speaks a physical truth, then proceeds to deny it. I fail to understand why he would insist on the dependency of the velocity of free fall in empty space on weights or masses-after having just argued that this observed dependency is naturally explained by the presence of a medium.
According to Aristotle, the world is finite; it is contained inside a large sphere defined by the firmament of fixed stars (see fig. 24). That sphere is also its limit: There is nothing outside it; there cannot be anything outside it, not even empty space. As he says: "At the same time, it is clear that beyond the heavens there is no place, no void, no time. A place is a location where a body might exist"; we might call it empty if there is no body there, but there could be one. We can put this idea in context with a different wording in Aristotle's famous definition

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Ga(oay clulter in comQ
Figure 24 Aristotle believed the universe to consist of ten concentric spherical shells surrounding Earth. The elements water, air, and fire made up the three "terrestrial spheres" delimited by concentric spherical 0 shells below the shell inhabited by the Moon. The spheres beyond the Moon sphere are made up of his fifth element, the ether. Those spheres are governed by celestial laws that permit no change: The only motion permitted there is rotation of spherical shells that do not change in the process. The motions below the sphere of the Moon, having a beginning and an end, however, do mean change. Contrary to this, ever since Newton, science has insisted that there is no subdivision of the world into regions where different laws of nature reign-the laws of nature must be the same everywhere. It was one of the foundations of Aristotle's view of the universe that it is finite. A physical argument in favor of this feature was this: The sphere of the fixed stars revolves once a day about Earth; an infinite universe would mean that there are fixed stars that move at infinite velocity. Figure 24a is from Kepler's Prodromus (1596). It shows Aristotle's finite world with three important modifications: The Sun has a fixed position at the center of the sphere of the fixed stars. Earth belongs to the planets and is orbited by the Moon. The skies do not revolve; Earth does. This means that the fixed stars, which are at an infinite distance, do not have to move with infinite velocity. Figure 24b is from a 1576 book by Thomas Digges. The universe may be infinite in extent-whether it actually is or not is beyond our knowledge; but it is certainly very large. That is stressed in a modern "scale-drawing" in figure 24c, where galactic superclusters are shown to be removed from our Milky Way Galaxy by up to three hundred million light-years. The Milky Way, on this scale, is nothing but a point of invisibly small size.

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of empty space, which he calls "a place that contains no body, but that could contain one." Clearly, then, there cannot be empty space beyond the outer confines of the world. If there were, his definition would require the possibility of bodies out there-and that he denies.
A world without limit cannot be reconciled with Aristotle's view. It is only through its finiteness that we can define a center or origin, that we can distinguish directions like "up" and "down." We can associate bodies with given locations in space only when they are permitted to reach those points. Aristotle defines the location for heavier bodies as "below," of higher ones as "above." This means bodies will sink like rocks or rise like smoke in order to reach the locations destined for them. We can speak of "free fall," and similarly of "free rise." These motions are the natural ones. In addition, there is unnatural motion-like that of a cart being pulled or an arrow being shot.
The forces that cause these "unnatural motions" issue from animate sources, such as the archer who shoots off an arrow; Aristotle feels they need not be explained beyond the animate prime mover who put the world as a whole into motion. In the absence of such a mover, earthly bodies consisting of the four elements-earth, water, air, and fire-assigned to them will move along straight lines. They will fall down (earth, water), they will rise straight up (air, fire), or they will move ahead along a straight line, only to fall or rise eventually. All this cannot exist in space that is really empty. If the space is not even the carrier of information of what is "up" and what is "down," no stone knows in which direction it should fall. Aristotle considers this consequence absurd; and again, he concludes that there cannot be such a thing as empty space. At the very least, the difference of the proportions that mark the various directions in space should be imprinted everywhere inside it. Once that is given, space will not be empty: "There is no getting around the alternative-either there is no natural motion or, if that doesn't hold, there can be no void, no empty space."
The consequence is that the occurrence of natural motions precludes the existence of empty space. Motions that do not distinguish particular directions do not belong in this category, as defined by Aristotle; they are therefore compatible with the existence of empty space. This is the kind of motion that the atomists postulated for the smallest particles: They interact only with one another when they come into contact, when they scatter. As a result, they move about in random motion-the kind of motion that Aristotle also considers compatible with empty space. If the existence of empty space is admitted and if there is only one body in that space, it cannot change its form of motion, which includes remaining at rest. Freed from Aristotle's absurdity, this statement will, some two thousand years later, reemerge as one of the foundations of Newton's mechanics.
Aristotle does not tire of arguing against the existence of empty space; his worldview would collapse otherwise. The directions "up" and "down," which play a vital rule in his natural philosophy, have no meaning in empty space.

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Were there such a thing as empty space, Aristotle would have to admit that the finite material world of his thinking could be embedded in it. But in that case, there would be the question of where exactly our world is situated-why in one place rather than in another? An infinite space cannot be built up around some center of forces toward which Earth, like all matter, would have to gravitate. But more than that, just as he denies the existence of empty space, Aristotle argues that there cannot be limitless straight-line motion, the only motion permitted in the void; it would, after all, necessarily lead to points beyond a finite world.
Aristotle argues that circular motion is the only motion that can persist unchanged forever without reaching unlimited space, which does not exist. Thus the fixed stars move in circles. More precisely, we can consider them to be attached to a rotating sphere. We recall that nothing exists beyond that spherical limit-not even empty space. Aristotle defines motion in terms of his concept of rest and location: Earth is at rest in the center of the universe; it is one of the massive bodies that are attracted toward this center. There is no law of nature that implies the identity of the center of Earth and the center of the universe. Rather, that is the way things must have arranged themselves in the course of time; otherwise, a massive Earth would have dropped to a different location. But Aristotle also argued in the opposite direction, stating that the center of Earth defines the center of the universe. He defines the location of a body as the "internal limit" of an extended physical entity. In this sense, he argues in only two dimensions: The location of wine, say, for him is the internal surface of the barrel; motion is equivalent to a change of this location; in this sense, a boat well anchored while a current of water moves by, according to Aristotle's definition, is in motion.
To Descartes two thousand years later, every motion is relative motion. He does not know of anything that is absolutely at rest; as a consequence, he can equally well speak of the boat at rest or the boat in motion. For Aristotle, the consequence of his definition of motion-that bodies move although they are at rest with respect to Earth-was an absurdity. He therefore specified his definition: "A fixed location is defined as the first immobile limit of an extended space or body"-that is, the location of the boat at anchor is defined by the banks of the river.

SPACE AND LOCATION
This definition is not quite clear; it has to be adapted to individual cases. In this spirit, Aristotle's thinking found many different interpretations. It is instructive to consider what the term location does not stand for, according to Aristotle: It is neither "shape nor form, nor the matter ofa body, nor the extension between the bounding surfaces ofa containing body." This space cannot exist independently; if it did, it would be the nonexistent empty space in Aristotle's definition. Take a

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barrel: Its inner surface can define the location of wine, of water, of air-the location of something that is material. Location is not defined by one kind of body rather than another but is a property specific to bodies only: The world is full of matter that fills all space, that touches all inner surfaces. Matter borders on matter, always and everywhere; there is nowhere and never a gap in between.
Space, in this sense, is equivalent to the full set of all locations; and as such, it exists. Empty space, however, does not. If it did, it would be another body. By equating a hypothetical empty space with just another body, Aristotle means to prove that it cannot exist. He argues that if it did, two bodies could occupy the same location: empty space and, let's say, a wooden cube. That is, of course, just as absurd as the idea that a wooden cube in water was not displacing some of that water but rather coexisting with it in the same location.
In addition, Aristotle feels no need for the existence of empty space. Even if it were there, there would not be any way of telling it apart from the volume taken up by bodies. Volume is one of the properties of bodies, regardless of where they are. But bodies are present all over. As a result, there is nothing that could be explained by empty space that is not already explained by volumes of bodies.
This reasoning is firmly grounded in Aristotle's teaching. Today, we might simply say that he differentiates between matter and shape. They cannot exist independently ofeach other-there is no matter without shape, no shape without matter. Matter necessarily has shape and thus becomes what Aristotle calls a substance. This substance is at the beginning of every natural philosophy. It is as concrete as can be; it simply is, say, a statue of Socrates that we might be looking at in a given museum at a given time. Proceeding from here, we may start to disregard details of the shaped matter we are looking at: the here, the now, the shaped image of Socrates, the marble it is made of.
The searching mind, according to Aristotle, can also disregard the rigid matter that makes up the statue without at the same time endowing it with another property-say, liquidity. He even believes in the possibility of abstractions that forgo all properties and forms, so that ultimately nothing but pure matter with no property or form remains.
This takes us back to ur-matter, materia prima. It is dearly nothing but an abstraction, not truly observable. It is impossible to remove all properties, all specificity, from a substance so that only pure, shapeless materia prima remains. This imaginary or idealized matter carries the potential of all forms and shapes imaginable; it possesses no individual property but is capable of assuming any one of them.
In other words, it may show up in arbitrary form. Aristotle denies the existence of matter without shape and, even more so, of shape without matter. Shape owes its existence to the fact that it may be assumed by matter; but as an idea, abstract and without ties to physical bodies, it is an impossibility. In this

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argument, Aristotle draws a line between himself and his teacher Plato. As the searching mind progresses from a given substance to materia prima, specific shapes recede into possible shapes. If there were an independent extended volume that existed without reference to any body, we would have empty space before us; but then again, empty space, according to Aristotle, cannot exist.
The materia prima will become a substance by realizing one of the forms or properties out of the totality that it contains "virtually." This is the only process admitted by Aristotle: Potential properties can be transformed into actual properties and vice versa; one shape may replace another; there is no such thing as a completely new property or shape. This includes motion as something that changes a potential new position into a real one: The potential property of the arrow to be "there" is realized at the expense of the previous property of being "here."
There is, according to Aristotle, no actual motion involved. The arrow, rather, replaces a label it carries that is marked "here" by another label marked "there." If real motion were nothing more than that, it would not need any space independent ofbodies in order to occur. Aristotle, in his completely occupied universe, can always tie a "here," a "there," to substances. Once this is done, the world looks like one immense three-dimensional pointillist image. Its points are made up by individual material objects that can be resolved individually, even without reference to their location. Note that, to make matters easier, we have been assuming a world composed of almost pointlike elements that have a finite size; it is easy to make a transition to the continuum by reducing that finite size further and further.
We can describe the world at a specific moment by identifying every pointlike element of matter with the location it holds at that time. Now let's turn to motion to change this picture: We observe the way all our points move in space to change the picture; distinguishable material points will change their positions. Aristotle describes it differently: He has the material points remain unmoved. It is their location that changes as one of their properties, just as a color might change.
Because Aristotle's ur-matter contains everything possible among its potential properties, it becomes closely related to the vacuum of quantum mechanics. We said so in the prologue and can now be more specific. But in contrast to the vacuum, the materia prima does not really exist; it is only a concept, a model of thought. It is an abstraction, fashioned after what Thales, Anaximander, and Parmenides imagined as some basic matter that filled the universe in its entirety.
The finite world of Aristotle is everything that exists: It is not an island inside some larger empty space. In this sense, our world has no location that could be defined as the "inner limit of a larger body surrounding it." In particular, if the world were infinite, this argument would hold; a surrounding body for an infinite world is not even thinkable. But if there is no surrounding body, then there is no definable location. "And thus we can speak of Earth as being surrounded by

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water, the ocean being surrounded by air, the atmosphere being engulfed in some ether, the ether filling our world. But that is where it stops: Our world is not further surrounded." And given that there is no location outside our world, even questions about the location of our world or its potential motion are an absurdity as far as Aristotle is concerned.

ARISTOTLE'S SYSTEM AS A STANDARD MODEL
Aristotle~s system was for two thousand years the standard model of philosophy and natural science. Just as we have our doubts about the ultimate validity of the modern standard models of elementary particle physics and cosmology, Aristotle's system has not been impervious to doubts. But those doubts notwithstanding, all scientific thinking up to the seventeenth century took as its reference this model-contradicting it, arguing against it, adding to it. It served as the basis for communication; today we would call it a paradigm. With two or three important exceptions, Aristotle discussed all the ingredients of later theories in the formulation of his own system-rejecting them (the concept of atoms) or accepting them (the horror vacui). We might think of the offerings a restaurant puts on its menu, where the variety of dishes is actually due to inventive combinations of very few basic foodstuffs: The philosophical and scientific theories developed in the two millennia after Aristotle are similarly based on a few original ideas formulated by him.
The main reason that his system proved so solid is not that it was correct. No, today we know that his system is fundamentally flawed. Soon after Aristotle's time, it became evident that his system could not be completely correct. But its success persisted simply for the reason that it is a self-contained system. You cannot replace an entire system by hauling in a variety ofdetailed new information; you have to bring in a completely new system. And that did not happen until Newton.

VARIATIONS OF THE STANDARD MODEL
Aristotle did not admit the possibility of empty space, or a vacuum, either inside our world or beyond it. An inner vacuum might manifest itself in two varieties. It might either be a "microvacuum"-in the version of the atomists separating minuscule units of matter-or a "macrovacuum," a macroscopic empty space. Today, we would have to call the latter simply a space without atoms and without radiation. The world of Aristotle is a continuum filled with matter, with no mention of atoms. It does not date back to some creation process; it will never cease to exist.
One early variant of the standard model was formulated by the Stoics. Their main preoccupation was a philosophy of life represented notably hy Seneca and

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Marcus Aurelius; they also developed a scientific view of the world. This was done notably by Zeno of Citium, Chrysippus, and Poseidonius. Their universe includes an external vacuum but not an inner one. Their world has a finite volume; it forms one and only one island of matter inside an infinitely large empty space, the external vacuum. The Stoics follow Aristotle in his belief that "empty space" would have to be capable of accepting the presence of bodies. To his decree that outside the world there can be no further bodies, they add the question of why that should be so. Of course, it is a trivial truth that there are no bodies beyond the confines of our world. But to what extent does Aristotle's decree extend beyond this simple statement? What precisely are those confines? Why can't we extend a hand, a projectile, beyond them?

TO THE EDGE OF THE WORLD
This question goes back to Archytas of Tarentum, a contemporary of Plato: "If I am at the extremity of the heaven of the fixed stars, can I stretch outwards my hand or staff? It is absurd to suppose that I could not; and if I can, what is outside must be either body or space. We may then in the same way get to the outside of that again, and so on; and if there is always a new place to which the staff may be held out, this clearly involves extension without limit."
The reader may compare Archytas and his staff to the bugs and sticks of Feynman's imagination on a hot plate (see figs. 19 and 20b): None of them reaches the outer edge of its world because they all diminish more and more in size the farther out they get. Similarly, a bug-size Archytas on a spherical surface (see figs. 17b and 17c) will never hit an edge; rather, he will find himself moving through the same space again and again.
These kinds of solutions that explain the riddle of the edge of our world in terms of non-Euclidian geometry have become accessible only in the middle of the last century. Discussions on the question of whether or not the space of our universe has all the properties that our immediate observations suggest were moot before that time; a mathematical model other than Euclid's had not been developed. The latter-day atomist Lucretius inquired about edges of the world just as Archytas had done. His answer was not dissimilar:
Let's assume, for now, space is limited, and somebody throws a javelin beyond its outer edge. What do you think the projectile that has been sent offat great speed toward the edge will do? Will it continue in that direction beyond the edge, or will it stop dead in its track when it hits that edge? However you might move or displace that outer edge, I will ask you again: "What about the javelin?" From this you should learn that our universe is limitless in all directions.

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The same answer about the limits of the universe is given by the Stoics: There are none, no matter what direction you choose. But the Stoics start from a finite world occupied by matter, or bodies. The space outside that finite world, however, is capable of accepting bodies, thus expanding the world; where previously there had not been bodies, and hence no material world, Archytas's staff created a new one. In Aristotle's definition, this means there must be empty space beyond the material world ofbodies-after all, the space beyond the material world does not contain such bodies in the first place, but is capable of accepting them. And this empty space extends without any limits to infinity. There cannot be a limit to how far the edge is extended by Archytas's staff to create an ever larger universe.
The idea that the world can expand simply because bodies it contains are moving farther and farther apart is astonishingly modern. The formulation I will use is, of course, different from that of the ancient natural philosophers. They could not possibly know what we have learned since about the makeup of the universe; their means of observation did not permit that. Their concepts of the qualities of the universe-continuous versus discrete, empty versus full, infinite versus limited-were developed into a consistent theory through many generations. But only experiment would be able to decide on the correct choices. Experiments--expensive ones at that, with telescopes, research satellites, particle accelerators-answer the questions that make up our cultural inheritance.
Let's return to Archytas's staff and to the expanding universe. A javelin that is thrown straight upward will drop back down a long time before we can sensibly ask the question about the edge of the universe. The javelin cannot escape the gravity of Earth. A spacecraft that runs out of propellant only after it reaches the velocity that permits its escape from the gravitational pull of our solar system will continue on its path, but it will remain inside our galaxy. The Milky Way's gravitational field holds on to it. Proceeding like this, we pass from a practical problem to a fundamental question. Can there be objects that may escape our group of galaxies, and that will ultimately leave our universe altogether? The answer must be no. The universe, after all, keeps expanding. Light that reaches us today was emitted shortly after the birth of the universe; its "headwaters" are moving away from us with a velocity close to that of light. There is no light older than what originated at the time of the Big Bang. Since no information travels at a speed greater than that of light, the observable universe is contained within the surface of a sphere with a radius defined by the distance traveled by light that was emitted at the time of the Big Bang.
Of all the material objects we know, quasars are closest to Lucretius's javelin. Quasars are galaxies in an early state of evolution. The light that tells us about them today has traveled for billions of years. They move away from us at a speed close to that of light; in that way, they mirror the expansion of the universe.

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Galaxies, quasars among them, are interpreted by astrophysicists in terms of probes that are attached to points in space and that move with these points. The increase of distances of the galaxies is seen as the growth of space. Space itself expands when galaxies move farther apart. The fact that quasars move away from us indicates that new space originates in between. Conversely, the masses of the universe help to determine both time and space. We might say that space expands because of the behavior of the masses; the masses of the universe, in their joint action, take the place of Archytas's staff. The expansion is caused by the initial momentum that was imparted to matter by the Big Bang.

THE STOICS' PNEUMA AND ETHER
The Stoics, like the atomist Lucretius, do not know about an edge of space, but they do know about an edge that confines the world of bodies. This edge opens up like a deep moat across which Lucretius might toss his javelin: The world of bodies with its given size could, at any time, increase when the javelin is tossed. It is embedded in empty space, but it does not contain empty space, neither microvacuum nor macrovacuum. The Stoics believe that the world is filled with what they call pneuma. This pneuma, in their imagination, is an elastic substance, a mixture of fire and air. It holds the world together; it keeps the world from diffusing out into infinite space. This pneuma is not a passive substance like Aristotle's ether, eternally at rest. It has "tension," which can change with location and time and transfer various effects from one place to a neighboring one. In modern terms, we would call the dynamic theory of the Stoics a theory of close interactions: If one body acts on another at a different location, this action is not instantaneous. It propagates with a given velocity from one location to a neighboring one. The pneuma "vibrates." This vibration starts at a given point; it expands into neighboring territory-just as a wave expands around a pebble we throw into a pond.
The Stoics noticed with remarkable scientific insight that sound is the same thing as vibrations of air. Their interpretation of the propagation of observable phenomena anticipates important aspects of field theory. To each point in space we assign a number; this number marks the strength of the field at that point. The field is then a map made up of all numbers assigned. One such example of a field is the distribution of temperatures on Earth: 98 degrees Fahrenheit in Los Angeles, 105 degrees F in Las Vegas, 61 degrees F in New York, and so on. This field, the temperature distribution, derives its physical reality from external sources: The molecules in air and ground are in a higher state of motion in Las Vegas than in Los Angeles.
Sound can equally be described in terms of a field. This field is another totality of numbers-air pressure as a function of location. The phenomenon of

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sound is based on a periodic change of pressure in a medium such as air. Pressure, of course, is another word for the density of molecules in a gas (in this case, air). Air molecules oscillate when a bell strikes. This is no different, in general terms, from the wake of a boat moving through the ocean: It causes a moving deformation of the water surface. We call this deformation "waves," and the waves are ultimately nothing but oscillations of water molecules. Whoever is familiar with the definition of fields will find examples in many places-in the distribution of velocities in the river flow from one bank to the other; in the inclination of rye stalks when wind passes over a field.
Once it became dear that the Stoics' interpretation of sound in terms of air vibrations was correct, it was only a small step to interpret light as an oscillation phenomenon. But of what? That is where the term ether came in. The theory that light is nothing but vibrations ofa hypothetical, omnipresent ether dominated physics in the late nineteenth century. Its triumphant advance started with Maxwell's equations, which describe electrical and magnetic phenomena jointly. In this context, light is easily seen as vibration-but Maxwell's equations have nothing to say about what precisely it is that vibrates. In fact, there is no such substance at all-something present in the dark that lights up once it starts oscillating. Maxwell's carrier substance of light, his "ether," was supposed to be capable of elastic oscillation. In that sense, it had more in common with the pneuma of the Stoics than with Aristotle's ether.

THE ATOMISTS EPICURUS AND LUCRETIUS VERSUS ARISTOTLE
The atomists needed the empty space between the atoms to permit atomic motion. They did not make a distinction between such concepts as inner, outer, microvacuum or macrovacuum. The world had to be infinitely large; otherwise there would not be enough room for the infinite number of atoms. Epicurus, successor and disciple of the early atomists Leucippus and Democritus, contrasts their worldview with Aristotle's in the following way:
Furthermore, the universe consists of bodies and the void. That bodies do ·exist, our senses attest to; they also permit ,us to reason about what is beyond our perception. . . . If there were nothing like the void, the empty space, a realm beyond what we can touch, then there would not be any room to locate bodies, to have them move the way we see them moving. . . . Also, the universe does not limit the amount ofatoms it contains on the expanse of the void. . . . And what is more, there is an infinite number of worlds, whether they resemble ours or not.
The Roman philosopher and poet Lucretius wrote an instructive poem, "On the Nature of Things," in which he summarized the teachings of the atomists. To

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show that motion cannot happen in space that is occupied, he refers to fish swimming in water:
First, let me make sure that you will not be taken in by misleading arguments advanced by many. They maintain that the water opens up for the swimming fish, closing up behind the fish once they have passed. In this manner, other objects may well move and exchange their positions though all space be filled a priori. This argument, ofcourse, is completely wrong. Where should the fish go if the water did not open up space to start with? But how can the water recede when the fish cannot move? We have to conclude that either there is no motion of bodies or that there is a void at the basis ofall motion of objeds through a medium.
This view of motion, with all its problems, is certainly superior to Aristotle's reducing motion to the exchange of labels. Prior to this, Lucretius had explained his interpretation of directional motion:
We cannot say that bodily objects fill space all over; there is empty space inside matter-empty space that we cannot touch. Were it not there, the objects could not move. After al~ resistance against motion is a natural property of all bodies, and it would have to show everywhere.
Nothing could move because nothing would be yielding. But we do see
a great deal of motion by land, water, or through the skies. If there were no void, all this motion would be impossible; and nothing could originate because of a lack of space-after all, matter would be densely packed, immobile everywhere.
The physics question concerning the void fuses for Lucretius and his contemporaries with the metaphysical question of the nothing. From this nothing, Lucretius says, nothing new can originate. His ideas are therefore incompatible with the Christian teaching about the creation of the world out of nothing. (It will take many centuries before Galileo's contemporary the French priest Pierre Gassendi will heal this rift.) "In the beginning of all rational observation of matter, there must be the recognition that nothing can be generated out of nothing; there cannot be a creation ex nihilo by, say, some divine creator." As Lucretius said, the world was created all right, but not out of the nothing. Conversely, matter cannot vanish into the nothing. For the topic of this book, it is interesting to see him restate Democritus's argument that only two components make up the world: the atoms and the void. He formulates it like this: "Let me add that we cannot find anything beyond bodies and the void; there is no third component of the world." Anything that acts is a body. Anything that does not act is empty space, "but space cannot be if there is no void. And thus the void and the bodies

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do not leave an opening for any third component that could manifest itself to our senses or to our minds."

TWO CONCEPTS OF SPACE
The two concepts of space considered by Aristotle also determined the philosophies of his disciples Theophrastus and Strato. The former defined space via the positioning qualities of the world of bodies; the latter assumed space to be the container of all bodily objects. We have previously seen these concepts contrasted in Einstein's thinking. Two millennia after Aristotle, Leibniz and the English philosopher Samuel Clarke, the latter inspired by Newton, squabbled bitterly, and not always on a scientific level, over these ideas.
Theophrastus replaces Aristotle's concrete space, which is defined by the bodies it contains, with an abstract space. By this he means the spatial relation among the bodies, their "positioning qualities": "Maybe space itself is not real; rather, it may be determined by the position and the ordering of the bodies with respect to their properties. This is just what we are accustomed to from animals and plants and other inhomogeneous bodies, bodies with structure. After all, these bodies have a well-determined ordering and positioning, by which their parts make up the total." The continuation of this citation of Theophrastus shows his Aristotelian schooling; he does not implement his own concept of space in the direction later taken by Leibniz, Mach, and Einstein. Instead, he says: "Every body that takes its rightful position has a specific ordering. Every part of a body seeks the location and the correlations for which it is destined." Space, in this reading, is not dependent on the actual positions of the bodies; it is to be abstracted from the locations that Aristotle's system assigns to them. According to our previous definition of fields, we can interpret space as a field that directs the motions of all bodies toward the locations assigned to them.
Theophrastus ascribes reality only to bodies; in the absence of bodies, there can be no space. Therefore he agrees with Aristotle that there can be no vacuumneither microvacuum nor macrovacuum, neither inside the world of bodies nor outside.
After Theophrastus, his follower Strato headed the school of philosophy founded by Aristotle. He adopted many of the atomists' ideas: Objects move in empty space without displacing a continuum; in dense matter, the atoms are closer to each other than in more dilute matter. He imagines that there is empty space between the building blocks of matter. His space is a container for bodies, and as such an absolute quantity in itself-its existence does not depend on whether it contains bodies or not. If not, it is empty space, a void or vacuum. He differentiates between what actually is and what could be. As his tradition passes on to Philo, then to Hero, these ideas develop so as not to permit the

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existence of a macrovacuum in nature; but they allow for the possibility that it could be produced by artificial means.
Strato, however, maintains that under normal circumstances "there cannot be a continuous vacuum; there may well be small vacua scattered throughout the air, the water, the fire, and all bodies. We want to adopt these ideas and we will seek experimental proof that this is indeed so."

STRATO'S EXPERIMENTS ON THE VACUUM
Strato then thinks it possible that, perforce, there may be an expanded empty space, a "macrovacuum"; but this would have to be under abnormal circumstances. And he does mean to prove it experimentally. As a method, this must be seen as revolutionary. He starts out with a sober lab protocol devoid of all poetic frills, describing the observations we already know from Empedocles' experiment with the hydra (see fig. 21). He summarizes his result: "We have thus proven that air is, in fact, a physical substance." He then addresses all those who categorically deny the existence of vacuum; he means to prove to them, again by experiment, that there is in fact a phenomenon that can be called a continuous vacuum, even if it does not normally occur in nature; and then again, that this vacuum may be normally present-but in small amounts and widely dispersed. These scattered microvacua will fill up with bodies under pressure. Our demonstration, he says, will make it impossible for all those idle wordsmiths to talk their way out of these observations.
With pedantic precision, he then describes an experiment in which air is blown through a pipe into a metal sphere-a space normally filled with air. His explanation for the phenomenon that more air can be added to the space already filled with air still makes sense to us: The air molecules move closer together. That goes against the grain of the logicians, those finaglers of words; they argue that since there is no empty space, there is no way to add more air without taking some of the already present air out of the vessel. We don't know how these critics interpreted Strato's experiment. They probably abstained from comment. Experiments designed to address a particular point did not, after all, belong in their arsenal.

PLOTINUS AND AN INTERPRETATION OF TODAY'S PHYSICS
We now have assembled all the concepts that were to dominate scientific discussion on empty space to the present day; physicists, of course, are interested in solutions and detailed answers rather than in mere questions and inquiries. But many of these questions, the answers to which they keep discussing, date back to Leucippus, Plato, Aristotle, Theophrastus, and Strato.
Plato holds a special place among them. His detailed interpretation of matter

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as a hollow space surrounded by regularly shaped bodies was too specific to hold up when scientific development advanced. On the other hand, his concept, taken over from the schools of the Eleatics and of Pythagoras-that only ideas possess reality-has persisted to this day. To him and his successors, matter really exists only by virtue of its form. It is only through its form, after all, that matter is tied to the ideas. It takes mathematics to describe form precisely. This means that the true laws of nature are expressed mathematically. We might say, the laws pin down the quantities that make up real existence.
Among Plato's successors, the most remarkable thinker to pick up this strain of argument is Plotinus. He offers little to those who would pin him down to rational argument. His intricately defined system combines ideas owed to both Plato and Aristotle. He categorizes our world in terms of four hierarchical steps, each of which is subtended between two quantities that have no existence of their own. At the upper end, there is the abstract quantity he calls the One, or the Good. It cannot be described by any sequence of positive terms. Every other existence can be so described, but the One, the Good, cannot; we'll leave that to Plotinus. Now to the lower end of our range of observation: That is where we find matter miserable, contemptible, poor to the degree that it, again, has no existence of its own. But it contains a potential for all physical phenomena. We might say that it is fighting for its existence: "Its essence can be described in some measure by such images as utter poverty, constant want, perennial longing for making its appearance in the realm of reality."
In and by itself, matter is shapeless; as Plotinus says: "The very idea of matter implies absence of form." As a result, it will not make its appearance in the real world. If, however, it does show up, that must be due to its having taken on a specific form, to its having changed. This feature of Plotinus's matter-its ability to take on shapes notwithstanding its own lack of all shape-reminds us of Aristotle's materia prima. Matter, according to Plotinus, is the carrier of all properties of bodies. This includes all physical extent: "Absolute matter must take its magnitude, as every other property, from outside itself."
Plotinus also mentions a different form of matter, a form tied to the spiritual world-but I will not discuss this here. "Matter," he says, "is understood to be a certain base, a recipient of Form-Ideas. . . . There is, therefore, a Matter accepting the shape, a permanent substratum. . . . The Matter must be ... ready to become anything. . . . Matter, not delimited, having in its own nature no stability, swept into any or every form by turns, ready to go here, there, and everywhere, becomes a thing of multiplicity: driven into all shapes, becoming all things. . . . The distinctive character of Matter is unshape, the lack of qualification and form. . . . Matter is therefore nonexistent." The concept of existence is rooted deep in Plotinus's mystical thinking. It can be rationally approached only to the extent that things immutable were distinguished by the ancient philosophers from things that are suhject to transformation, change, passing.

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Figure 25 Can a cat vanish, but not its grin? Can structure exist without matter? This may be possible in the abstract interpretation of Plato and Plotinus; in a concrete sense, it is impossible. Structured matter can be interpreted as an excitation of unstructured matter, just as the cat with the grin may be seen as an excitation of the nongrinning cat.
Plotinus may choose more poetically elevated terms about what he thinks of matter; but in essence, it is not much different from Aristotle's materia prima. The main difference is that form, for Aristotle, has no reality of its own but is inextricably tied to materia prima, while Plotinus accords an independent existence to form or shape. Maybe we are talking about a mostly semantic notion, somewhat akin to the grinning of Lewis Carroll's Cheshire Cat (see fig. 25) in Alice's Adventures in Wonderland: Does that grin have an independent existence of its own? The cat vanishes from the tree, disappearing piece by piece from our observation, "ending with the grin, which remained some time after the rest of it had gone. 'Well! I've often seen a cat without a grin,' thought Alice; 'but a grin without a cat! It's the most curious thing I ever saw in all my life!' "
I think Aristotle and Plotinus would have agreed that there cannot be any reality to a grin that is not tied to a creature that produces it. On the other hand, there is evidently the abstract concept of a grin; it is a form that can be assumed by all kinds of bodies-by candy animals, by flower arrangements, by all types of matter. In Plato's succession, Plotinus accords an existence of itself to form, a feature lacking in Aristotle.
Plotinus's ideas exceeded those of Plato's school in creating a psychological climate for disregarding--or, worse, looking down upon empirical investigation of-natural phenomena. I would not have dwelt on them in such detail were it not for the fact that Ugo Amaldi, the illustrious contemporary Italian experimental physicist, recently used them to illustrate a number of contemporary concepts:

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Figure 26 Chaotic or disordered distributions are symmetrical on the average. An observer cannot conclude anything about his location from looking around him. An example is the chaotic disposition of sand grains at the beach, or of flecks of light on a television screen when the station closes down. In the chaotic distribution, no location or direction is preferred. Only when there is structure appearing in a chaotic distribution does orientation become possible, as the overall symmetry is broken. Our figure shows a computer simulation of the formation of structures in the universe.

What Plotinus called the abstract One, or the Good, Amaldi interprets as Energy. Plotinus's "nonbeing" Amaldi compares "with the all-pervading fields ofquantum theory; indeed the fields are not matter, but their presence determines potentially the nature of the matter-particles which we directly observe. For Plotinus," according to Amaldi, "the overall movement from the One to matter gives rise to actual being, which matter by itself seeks in vain to attain." Energy and the fields of quantum theory are concepts to be exactly defined; there is no way of tying them one by one to Plotinus's ideas. Later on, after having considered fluctuations of fields and particles in the vacuum (chapter 6), where only the lack of energy keeps particles from making their appearance in reality, the reader may well reread the present section. At this point, I think we will agree that today's quantized vacuum and its chaotic fluctuations are but a modern version of Plotinus's image of matter in its "perennial longing to enter into reality" (see, for example, fig. 26).
is THERE ROOM FOR GOD IN THE UNiVERSE?
After this excursion into the poetry of physics, let us turn back toward "space as container for all material objects and as an expression for the positional quality of the physical world." Those, after all, were the concepts for which Strato and Theophrastus gave a dear formulation. True, through centuries they were modified and imbued with philosophical trappings of later interpreters. In particular, the rise of Christendom posed the question of where God might fit into the universe, into its spatial structure. I will stay away from this question as much as I can. To set the lone, we might listen to Philo of Alexandria, the Jewish

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religious thinker (born about 25 B.C.) so popular and influential with early Christian theologians: "Space has a threefold significance. First, it is the volume that contains all bodily objects. Second, it is the repository of divine order. Third, space is as God himself--comprising all, comprised by nothing." From this quote we can deduce that Philo of Alexandria understood space as a receptacle in the same way as Strato did. Beyond that, his contributions were in the realm of theology.

ARISTOTLE AND CHRISTIAN TEACHINGS
In the thirteenth century, Robert Grosseteste, Albertus Magnus, and Thomas Aquinas modified Aristotle's system to the extent that it became the dominant teaching of the Christian Church. There was one problem: For Aristotle, the world was not created-it had always been and would remain forever. Christian faith, however, held that the world had been created from nothing and would cease to be on Judgment Day. Moreover, Aristotle's determinism left no room for God's intervention in the affairs of this world. In these two cases, the church declared Aristotle to be in error; but it adopted his teaching that there could be no vacuum-as well as his view of Earth's standing still, with the Sun orbiting around it. Thus, the church opened itself up to the perils of being proved wrong by dint of declaring scientifically observable notions as matters of faith. This was bound to cause conflict in scientists' consciences and danger to those among them who arrived at conclusions that differed from church teachings.
All this is well known. The denial by this amalgam of Christian and Aristotelian teaching that there was no way a vacuum could exist was so complete that it had to be seen as the expression ofa consensus already existing. According to this teaching of the church, God-or, more specifically, God's immeasurability-must pervade all space, which therefore could not be empty. Not that this made God an object that could be spatially divided; the doctrine implied that God is fully present at all times and at every point in space.
An external sign that Aristotle's teaching on the impossibility of a vacuum had pervaded Christendom came in 1325: In that year, the church revoked a decree with which, in the year 1277, the bishop of Paris, Etienne Tempier, had intended to counteract the spreading of Aristotelian ideas in the church. He forbade the defense of 219 articles in theology and natural philosophy. For our discussion, article 49 is the relevant one. It was directed against an idea that might be due to a Christian Aristotle: God will not move the world along a straight line, because that would create a vacuum behind it. This did not sit well with Tempier, who clearly did not appreciate Aristotle's arguments against the vacuum: God in his omnipotence can obviously do what to Aristotle was absurdmove the finite cosmos inside an otherwise "empty" space. (The argument was revived centuries later in an exchange of letters between Clarke and Leibniz.)

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,The experts in the field are not agreed on the consequences of Bishop Tempier's edict. I tend to believe that it eased the tension on the point: He meant to do away with very specific statements that could hinder the unfolding of scientific activity in the wake of narrow adherence to Aristotle's views, such as the one denying the possible existence of the vacuum. His message was that if some phenomenon does not exist, this does not necessarily imply that it cannot exist; God's omnipotence could make it happen. Whether the powers of God Almighty could be replaced by experimental human endeavor is left an open question. Philo and Hero, two successors of Strato's of whom more will be said in the next chapter, assumed that while a vacuum does not exist, it can be built up.
As of 1325, Bishop Tempier's decree was declared null and void: Aristotle's teachings prevailed; moreover, the decree contradicted the positions of Thomas Aquinas, the main champion of Christianity's adoption of Aristotle's ideas.

SCHOLASTIC TEACHING ON EMPTY SPACE
The philosophy of the Scholastics was a wild mixture of preconceptions and logic. It was governed by conventions and compulsions in the wake of Aristotle's teachings and those of the church. It did not search for new, hitherto unheralded truths; rather, it was content with nailing down well-known revelations. Its evidence came in the form of indirect proof, which might appear either as contradiction or as an absurd consequence: The method was to make an assumption, then reject it if it led to a contradiction or to some nonsensical conclusion. To the latter category the Scholastics relegated everything that did not sit well with their conventions and with the narrow set of their beliefs.
One principle of Scholastic philosophy was the tenet that three-dimensional extension implies bodily existence. Similarly, it held that two bodies cannot coexist in the same location. As a consequence, three-dimensional space would have to be seen as a body; no other body could additionally occupy the same volume. Therefore, there cannot be three-dimensional space; alternatively, it would have to be seen as a physical extension of bodies, simply parading under a different name.
That is not very different from some of Aristotle's teachings. The Scholastics adopted from him the definition of location as the concave inner shell of surrounding space. The Scholastics did not share the audacity of the Neoplatonists, the followers ofPlotinus, who were willing to scuttle some ofAristotle's principles. This attitude might have led to trouble in the fourteenth century, given that the church had adopted many of Aristotle's ideas. Rather, the imminent task of the Scholastics at that time was a reformulation of Aristotle's teaching such that it would include the notion of the omnipotence of God.
All the Scholastics agreed on one point: There is no practical way to produce