The Design Argument and the Laws of Nature
TWO WAYS TO THINK ABOUT LAWS OF NATURE
There is something paradoxical in the long-running debate over design. On the one hand, we have the older view that the lawfulness of the universe implies the existence of a lawgiver. This is the perspective of the Bible and of ancient religious writers, as we saw from the passages from Jeremiah, Psalms, and Minucius Felix quoted in chapter 9. It was also the view of many pagan writers of antiquity. On the other hand, we have many modern atheists who claim that the laws of nature prove the very opposite: that there is no need of God. One thinks of the famous statement of Pierre-Simon Laplace, who when asked by Napoleon why God was nowhere mentioned in his great treatise on celestial mechanics, replied, “I have no need of that hypothesis.” (When told of this by Napoleon, another great physicist, Joseph-Louis Lagrange, is reported to have said, “Ah! But it is such a beautiful hypothesis. It explains many things.”1) Are the laws of nature evidence for God or an alternative to “that hypothesis”? Do they prove that God is needed or that he is not needed?
At work here are two very different ways of thinking about the order found in nature. The old Argument from Design is based on the commonsense idea that if something is arranged then somebody arranged it. The reasonableness of this idea can be seen from an everyday example. If one were to enter a hall and find hundreds of folding chairs neatly set up in evenly spaced ranks and files, one would feel quite justified in inferring that someone had arranged the chairs that way.
One can imagine, however, that a person might object to this obvious inference, and suggest instead that the chairs are merely obeying some Law of Chairs operating in that hall. Most people, I suspect, would regard that as an absurd suggestion. Of course, it is true that the chairs in the hall are obeying a “law,” in the sense that the positions of the chairs follow a precise mathematical rule or formula. But that is just another way of saying that the placement of chairs exhibits a pattern. The “law” in question does not explain the pattern, it merely states the pattern. It does not tell us why the pattern is there.
It is important to distinguish different kinds of “laws.” Some laws cannot help but be true, such as the law of non-contradiction in logic, which says that a proposition cannot be both true and false, or the Fundamental Theorem of Arithmetic, which says that every whole number can be factorized in a unique way into primes. But the “Law of Chairs” is very different, because it does not express some necessary relationship. It is true that in the particular hall in which we found the chairs so precisely arranged there is a mathematical formula that can be used to predict where the next chair is in a row or column. That is, the Law of Chairs works. But it did not have to work; it could have been otherwise. The chairs might just as well have been positioned in some other way. And this is one reason why in real life we would conclude from the orderly arrangement of the chairs that someone chose that arrangement. In such a case we would argue that the law was due to a lawgiver, whereas we would not say that of the fundamental theorem of arithmetic or the law of non-contradiction in logic.
The critical point is that the laws of nature are not like the laws of logic or the laws of arithmetic, which have to be true. Rather, the laws of nature are simply patterns which we discover empirically in the world around us, but which could have been otherwise. In that sense they are just like the pattern found in the chairs in the hall, and therefore, the religious believer says, they must be the product of a mind.
It would seem, then, that the atheist is making an absurd argument when he says that the laws of nature have some kind of ultimate status that would permit one to dispense with a lawgiver or designer. However, there is really more to the atheist’s point of view than would appear from this chair analogy. The atheist has a point to make which is worthy of serious consideration. To see what it is, it would help to consider a different analogy.
Let us imagine that we have a cardboard box with some marbles rolling around in the bottom of it. The marbles will tend to have a rather random distribution and roll around aimlessly. But, if we tilt the box slightly the marbles will all roll into one corner and we will see a pattern emerge—at least if we do a little jiggling of the box so that the marbles settle as much as they can. The pattern that will emerge is called the “hexagonal closest packing” pattern. It is the same pattern one sees in a honeycomb, or in the way oranges are sometimes stacked at a fruit stand (see figure 1).
Figure 1. The most compact way to pack spheres of equal size is called a “hexagonal closest packing” arrangement. Marbles at the bottom of a tilted box will tend to arrange themselves like this to minimize their gravitational energy.
In this example the marbles really do seem to be obeying a Law of Marbles in some more meaningful sense than the chairs in the hall were obeying a Law of Chairs. Whereas the chairs might have been found positioned in many different ways, the marbles in the tilted box have no choice: there are physical and geometrical considerations that force them to be arranged in a certain way. Gravity is pulling the marbles downward, and consequently the marbles tend to get squeezed into the tightest arrangement that is geometrically possible.
In this example, physical forces (gravity) and mathematical necessity (the closest way to pack spheres) combine to produce a simple pattern. This would appear, then, to justify the claim that order can spring forth “spontaneously” and “necessarily” from disorder by unconscious laws rather than by intelligent design. If this is so, then laws of nature can truly explain order rather than merely describe it.
This is how many atheists look upon natural law in relation to the universe. And, to be sure, there are many things in nature that seem to bear out this view. Indeed, the whole history of the cosmos seems to tell the same tale. Gravitation made the chaotically swirling gas and dust that filled the universe after the Big Bang condense into stars and planets. The same forces made those stars and planets settle into orderly arrangements like our solar system. On the surfaces of planets, chemicals clumped together under the influence of electromagnetic forces into smaller and then larger molecules, until at last molecules that could replicate themselves appeared, and biology began. This is the grand picture: order emerging spontaneously from chaos. And presiding over the whole drama, the atheist tells us, is not some intelligence, but blind physical forces and mathematical necessity.
While this history of the cosmos is undoubtedly correct, the lessons the atheist draws from it are based on a superficial view of science. It is a view that really leaves out a major part of what science has taught us about the world, perhaps the most important part.
The overlooked point is this: when examined carefully, scientific accounts of natural processes are never really about order emerging from mere chaos, or form emerging from mere formlessness. On the contrary, they are always about the unfolding of an order that was already implicit in the nature of things, although often in a secret or hidden way. When we see situations that appear haphazard, or things that appear amorphous, automatically or spontaneously “arranging themselves” into orderly patterns, what we find in every case is that what appeared to be amorphous or haphazard actually had a great deal of order already built into it. I shall illustrate this first in the simple example of the marbles in the box, and then later in more “natural” cases like the growth of crystals and the formation of the solar system. What we shall learn from these examples is the following important principle: Order has to be built in for order to come out.
In fact, we shall learn something more: in every case where science explains order, it does so, in the final analysis, by appealing to a greater, more impressive, and more comprehensive underlying orderliness. And that is why, ultimately, scientific explanations do not allow us to escape from the Design Argument: for when the scientist has done his job there is not less order to explain but more. The universe looks far more orderly to us now than it did to the ancients who appealed to that order as proof of God’s existence.
IN SCIENCE, ORDER COMES FROM ORDER
Before we look at examples taken from nature, let us revisit the example of marbles in a box. We saw that if we tip the box slightly the marbles tend to form an orderly arrangement. But why does that happen? If I were do a similar thing with my living room—if I were to hire a huge crane to come and tilt it so that everything slid into a corner—I would not end up with an orderly pattern. I would find, instead, that the lamps, the furniture, the toys, and so on, would pile up into a jumbled heap. Why, then, don’t the marbles form a jumbled heap? Part of the reason is that, unlike the objects in my living room, the marbles all have exactly the same size and shape. But this is not the whole story. After all, if I were to put a lot of identical spoons, say, in the bottom of a box and tilt it, the spoons would still form a jumbled heap. A crucial fact is that the marbles not only all have the same shape, but that that shape is a particularly simple and symmetrical one: the sphere. In fact, the sphere is the most symmetrical three-dimensional shape possible, because it looks exactly the same from any angle. So when the marbles fall into the corner it does not matter very much how they fall. Spoons or furniture pointing every which way will look like a jumble. But spheres cannot point every which way, because no matter which way a sphere is turned it looks just the same.
We see, then, that even before the box was tilted and the marbles lined up, there were two principles of order already present and at work: (1) every marble had the same size and shape as every other marble, and (2) each marble had the perfectly symmetrical shape of a sphere. And these principles of order are, in a basic way, very much like the principles of order that the chairs in the hall obeyed in our first example. When we found two neighboring chairs in the hall a certain distance apart, we knew that any other pair of neighboring chairs would be the same distance apart. And we also knew that if one pair of neighboring chairs were lined up in a certain direction, then any pair of neighboring chairs would be lined up in the same direction. In the case of the marbles, we know that if one marble has a certain size and shape, any other marble in the box will have the same size and shape. And we also know for each individual marble that if it looks a certain way from one direction it will look just the same from any other direction. All of these statements have to do with something being the same as something else, and as we shall see later they all have to do with various kinds of symmetry. In fact, the word symmetry is Greek for “same measure.” Both the arrangement of chairs and the properties of the marbles exhibit what I called “symmetric structure” in chapter 9.
So why are the marbles in the box symmetric? Why are they all identical and spherical? Is it because of some ultimate Law of Marbles? Not at all. We know exactly why the marbles are symmetric: they were designed that way. Someone chose that they should be and had them manufactured accordingly. So we are back to where we started: the commonsense idea that design comes from a designer. At first it looked like the marbles-in-the-box example pointed toward a different conclusion. It seemed that the orderly arrangement of marbles could be traced ultimately to physical laws (gravity) and mathematical necessities (the closest way to pack spheres). But such explanations, while perfectly true, were not ultimate explanations. They presupposed an orderliness even greater than that which they explained: they presupposed the orderly or symmetric properties of the marbles themselves. And that orderliness and symmetry, we know, came not from mathematical necessity or physical laws but from design.
Why were we misled at first in thinking about the marbles in the box? Why did we think we could do without design? We were misled by a very natural mistake, a mistake that all atheists make in thinking about nature: we took something for granted. We took for granted the characteristics of the marbles and focused solely on their arrangement in the box as if that were the most fundamental fact and the only fact that called for explanation. But further thought showed us that the thing we took for granted requires explanation every bit as much as the effect we set out to explain.
IN SCIENCE, ORDER COMES FROM GREATER ORDER
We have seen in this simple example of the marbles how scientific explanations of order work in practice. They take order that is observed at a more superficial level (the hexagonal pattern of marbles in the bottom of the box) and show it to be the consequence of an order that is presupposed at a more fundamental level (the sameness and spherical shape of the marbles themselves). That is, when order comes out it is only because order was put in; so we still have order to explain. And—in the marble example, at any rate—the order that was put in was put in by a designer: the designer of the marbles.
Maybe there is a way out for the atheist, however. Perhaps, a scientific explanation could explain a lot of order by presupposing only a little bit of order. And maybe that little bit of order could be explained by a more fundamental theory that presupposed even less order. And so it might continue until one got to some ultimate and deepest theory that had to presuppose almost no order at all. In other words, at the end of the day, perhaps science will show that starting with some very trivial or negligible amount of order, the immense amount of order we see in nature might be built up by successive layers of explanation. And perhaps the order that exists at that deepest and most fundamental level will turn out to be so trivial that we do not have to attribute it to a designer, but can dismiss it as an accident.
This is certainly a hypothetical possibility, but it does not seem to be borne out by any actual examples. In fact, both in our simple marble example and in examples taken from nature, one finds quite the reverse happening: the order that is presupposed by a scientific explanation is greater than the order it accounts for. As one goes deeper and deeper into the workings of the physical world, to more and more fundamental levels of the laws of nature, one encounters not ever less structure and symmetry but ever more. The deeper one goes the more orderly nature looks, the more subtle and intricate its designs.
I have just asserted that at the deeper levels of nature one encounters more, rather than less, order. In order to justify such a statement, there must be some way of quantifying the amount of orderliness that a pattern or structure has. In the case of the precise kind of orderliness that is called by mathematicians “symmetry,” it is indeed possible to do this, as we will now see.
How Much Symmetry?
Consider a small part of the hexagonal pattern of marbles that results from tipping the box. In particular, consider a little part consisting of seven marbles, as shown in figure 2.
Figure 2. Spheres arranged in a hexagonal array (“snowflake pattern”). This array has a six-fold symmetry because rotating it by 60 degrees or any multiple of 60 degrees leaves it “invariant,” i.e., looking the same.
Let us call this “the snowflake” for short. The degree of orderliness of the snowflake can be measured by its degree of symmetry. In what way is the snowflake “symmetric”? For one thing, we can see that the pattern would look just the same if the snowflake were rotated 60 degrees clockwise. To say that it would look “just the same” is to say it has a symmetry. To a mathematician or physicist “symmetry” is defined as follows: A symmetry of an object is some operation that can be done to that object that leaves it looking the same as before, i.e., that leaves it “invariant.” In the case of the snowflake, rotating it by 60 degrees leaves it invariant, and so the mathematician says that the act of rotating the snowflake by 60 degrees is a symmetry of the snowflake.
In fact, by this definition one sees that the snowflake possesses several symmetries. There are actually six different rotations that can be done to the snowflake that leave it looking the same: namely, rotations by 60 degrees, 120 degrees, 180 degrees, 240 degrees, 300 degrees, or 360 degrees. This means that the snowflake pattern has at least six symmetries. Actually, though, it has even more, because the act of flipping the snowflake upside down also leaves it invariant. Since one can combine the acts of rotating and flipping, it turns out that altogether the snowflake has a grand total of twelve symmetries. A mathematician would say that the snowflake has a “group of symmetries” consisting of twelve “elements.” This group of twelve symmetries has a name; mathematicians call it D6, one of the “dihedral symmetry groups.” The snowflake example illustrates how one can count or measure the amount of symmetry that an object or pattern has.
Now, I said that the underlying reason that the marbles arranged themselves in this symmetrical pattern was that they all have the same size and shape and are all spherical. Let us consider this more closely. If one took the snowflake and turned it by 60 degrees, what exactly would happen? Except for the marble in the middle, each marble would move to a place in the pattern where another marble had been before. Therefore, unless the individual marbles were indistinguishable from each other the snowflake would not look the same after it was turned: the process of turning the snowflake would replace each marble, except the one in the middle, by a marble of different shape (see figure 3). Having marbles of different shapes or sizes would consequently spoil the symmetry.
Figure 3. Unlike the snowflake pattern in figure 2, a hexagonal array of six dissimilar objects does not have a six-fold symmetry, because rotating the array by 60 degrees makes it look different.
Another thing happens when the snowflake is turned by 60 degrees: in the process, each individual marble (including the one in the middle) is turned by 60 degrees about its own center. Thus, unless each marble had a shape (such as a sphere) that made it look the same after rotating it about its center, the snowflake would not end up looking the same, and the symmetry would be spoiled. For example, in figure 4, all the marbles in the snowflake are cubes, so turning the snowflake by 60 degrees does not leave it invariant and is therefore not a symmetry.
Figure 4. Even a hexagonal array of six similar objects will not have a six-fold symmetry unless each of the objects has a six-fold symmetry. For instance, if the objects are cubes, as shown here, rotating by 60 degrees makes the array look different.
So the symmetry of the snowflake ultimately arises from the properties of the marbles, namely that they are all the same in shape and size, and that their shape is spherical. And these properties are really themselves also symmetries. This is most obvious when it comes to the spherical shape of the marbles. A sphere is symmetrical in the same way that the snowflake pattern is symmetrical: it looks the same when one rotates it. However, a sphere is much more symmetrical than a snowflake. The snowflake only looks the same if it is rotated by 60 degrees, 120 degrees, 180 degrees, 240 degrees, 300 degrees, or 360 degrees (combined possibly with a flip), whereas the sphere looks the same if it is rotated by any angle whatsoever. Thus, while the snowflake had twelve symmetries, the sphere has an infinite number of symmetries. In fact, one could say that the sphere has a doubly infinite number of symmetries, because a sphere looks just the same if rotated by any angle about any axis. The doubly infinite “group” of symmetries of a sphere is called by mathematicians SO(3), the “rotation group in three dimensions.”
Not only are there infinitely more symmetries of a sphere than of a snowflake, but, as we have seen, the symmetries of the sphere include all the symmetries of the snowflake. Therefore, it is true to say that the snowflake arrangement of marbles in the box is manifesting or exhibiting just a small part of the symmetries that were already built into the structure of the marbles themselves.
So far I have shown that the spherical shape of the marbles is an example of symmetry. But it is also the case, though not as obvious to a non-mathematician, that the indistinguishability of the marbles—the fact that they all have the same shape and size—is an example of symmetry. Suppose one has seven marbles of identical shape and size, not necessarily stuck together as they are in the snowflake, but even lying separately. One could take any two of those marbles and interchange them and it would make no difference; they would look just the same. So the act of interchanging two marbles is a symmetry. In fact, one could also interchange three or more of the marbles at a time. For instance, one could simultaneously replace marble A by marble B, marble B by marble C, and marble C by marble A. Such a “permutation” (as it is called by mathematicians) is also a symmetry. How many interchange and permutation symmetries do seven identical marbles have? It turns out that the total number is equal to 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040. Mathematicians call this group of 5,040 symmetries S7, the “seventh permutation group.” To say that the seven marbles have these 5,040 permutation symmetries is equivalent to saying that they are all indistinguishable or interchangeable.
Thus, underlying the twelve symmetries of the snowflake pattern of marbles that we saw emerge at the bottom of the box was a vastly greater number of symmetries intrinsic to the marbles themselves: an infinite number of rotational symmetries and 5,040 permutation symmetries. I have chosen to talk about seven marbles for concreteness, but the same points would apply no matter how many marbles were in the box.
Naively, when the marbles arranged themselves into the hexagonal array, it looked like symmetry was springing out of thin air. We were getting symmetry from chaos. But in reality what was happening was that a very small part of the symmetry that was already engineered into the marbles was manifesting itself in a certain way in their arrangement. Symmetry was not springing out of thin air, it was coming from greater symmetry. When the marbles chose to take on a particular hexagonal pattern, there was in a certain sense a reduction in the amount of symmetry. This is a phenomenon that in physics is called “spontaneous symmetry breaking.” Some, but not all, symmetries of the marbles are reflected in the pattern they form in space. The other symmetries are “broken” by the pattern. Symmetry is not being made, it is being broken. The idea of spontaneous symmetry breaking is one of the most important in modern physics. It is one reason for the fact that the deeper structure of nature has more symmetry than the more superficial layers.
Let us reiterate the main points. In the example of the chairs in the hall, it was obvious that the chairs could have been arranged differently, and so one inferred that the pattern was the result of choice, and that an intelligence made that choice. The pattern was there “by design.” In the example of the marbles in a box, it seemed at first that the marbles had to line up in a very particular way because of natural forces and mathematical necessity. Thus one might have concluded that no designer was involved. But, when this marble example was considered more carefully, it became clear that the pattern of marbles emerged only because of some very special prior facts about the marbles. And, when thought of even more carefully, these prior facts about the marbles were seen to be in themselves symmetric patterns that did not have to be as they were but were in fact the product of design. And, finally, it was found that the symmetry that was engineered into the marbles by their designer was of a higher degree than the symmetry of the pattern one at first thought arose spontaneously.
AN EXAMPLE TAKEN FROM NATURE: THE GROWTH OF CRYSTALS
As anyone knows who has ever seen frost on a pane of glass, very beautiful patterns can form automatically under the right conditions. No one imagines that water molecules are arranged in these wonderful feathery shapes by a miracle, or supernatural intervention. It is a completely natural process. The same is true of the lovely crystals that form deep within the earth. These assemble themselves spontaneously, with each molecule or atom attaching itself in the right place in the crystal without anyone telling it where to go.
The way this happens is quite similar to what happens in our marble example. The marbles in the bottom of the box line up in a hexagonal pattern because, under the influence of gravity, they are trying to find the lowest possible positions. In other words, the marbles are trying to lower their total “gravitational potential energy.” In a similar way, crystals form because the atoms or molecules are trying to lower as much as possible something physicists call “free energy.” It turns out that under certain conditions of temperature and pressure this free energy is lowest in certain materials if the atoms or molecules are arranged in crystals.
In a crystal, the pattern in which the atoms or molecules are arranged is called the “lattice.” There are many different kinds of crystal lattices found in nature. For example, the lattices of diamond, calcite, galena, and mica are all different. And just as one can use the mathematical language of symmetry to describe the orderly arrangement of the marbles, so every kind of crystal lattice has a specific “group of symmetries.” A specialized branch of mathematics is devoted to studying these “crystallographic symmetry groups.” Take, for example, crystals of diamond. A perfect diamond lattice has a group of forty-eight symmetries called by mathematicians Oh, or the “hexoctahedral group.”
In the marble example, we saw that the orderly arrangement of the marbles was a consequence, ultimately, of (1) the exact sameness of all marbles, and (2) the spherical shape of each marble. In a similar way, the possibility of diamonds forming presupposes (1) the sameness of all carbon atoms, and (2) spherical symmetry. [A technical point should be made for the sake of experts. It would be an oversimplification to say that the carbon atoms themselves were spheres. But there is a precise sense in which the laws of electromagnetism, which govern the behavior of the carbon atoms, and the physical space in which the carbon atoms are moving are spherically or rotationally symmetric.]
Just as in the marble example, the order or symmetry that is manifest in the diamond crystals is far less than the underlying order and symmetry that gives rise to it. For underlying the forty-eight hexoctahedral symmetries of a perfect diamond crystal lattice are a huge number of “permutation” symmetries of the carbon atoms (due to the fact that they are all exactly the same and thus interchangeable) and an infinite number of “rotational” symmetries of space and of the laws of electromagnetism.
Crystals are one of the more dramatic and impressive displays of orderliness that we see in the world around us. But when science attempts to understand this phenomenon it uncovers a much more impressive orderliness at the atomic level. Nor must we stop at the atomic level; we can trace this orderliness to its roots at even deeper levels of physics.
We can do this by asking why all carbon atoms are exactly the same. The marbles in our example are all the same because they were made in a factory to the same specifications: they are all the same by design. In the case of carbon atoms, however, we have a physics explanation. All carbon atoms are the same because they are all made up of identical electrons, identical protons, and identical neutrons, and because these particles act and are acted upon by forces that operate in exactly the same way everywhere in the universe.
That, of course, raises the question of why all electrons (or protons, or neutrons) are identical, and why the forces of nature act in the same way everywhere. The answers to these questions lie at a level deeper than atomic physics: the level of Quantum Field Theory. Quantum Field Theory is the mathematical framework used to understand fundamental particles, like electrons. It tells us that these particles come from “fields.” A field is something that fills all of space and can vibrate in certain characteristic ways; and those field vibrations can propagate along like waves in a pond. In quantum theory waves and particles are two different ways of looking at the same thing, so electrons can be thought of as particles or ripples in the “electron field” that fills all of space. This helps explain why all electrons are exactly alike. A good analogy is this: whether I stick my finger into a pond over here or over there, the same kind of ripples will spread out in either case, because the water of the pond is the same everywhere. Similarly, all electron waves (which is to say, all electrons) are the same because the electron field that fills space is the same everywhere.
Now it might seem that I have disproved the very point that I have been trying to establish. I said that the symmetries of diamond crystals were a consequence of even greater symmetries at the atomic level, among which were the fact that all carbon atoms are exactly identical. But now I appear to have explained away this symmetry of the carbon atoms by looking at the subatomic level and appealing to Quantum Field Theory. Whereas the marbles had to be engineered to be the same, all electrons have to be identical, because Quantum Field Theory says so. But this, again, is a superficial view. One must ask why the electron field is the same everywhere in space. That it is so is a very powerful fact which has to be assumed in constructing our theories of physics in order to agree with what we see in the real world. It is not an absolute logical or mathematical necessity by any means. One could imagine that fields had different properties in different places. True, that would strike most particle physicists as an ugly thing to imagine, but who said that the world had to be beautiful?
The fact that the electron field has uniform properties throughout all of space is itself a statement that the electron field possesses a very large degree of symmetry, in fact a much greater degree of symmetry than is enjoyed by any specific set of electron particles or carbon atoms.
Thus we see the same result repeated again and again as we trace phenomena down through layer after layer to the deeper levels of the world’s structure. The symmetries and patterns found at one level are manifestations of greater symmetries and more comprehensive patterns lying concealed at the more fundamental levels. In what sense more comprehensive? In this sense: the symmetry we see in a diamond has to do only with diamonds, whereas the symmetries of the carbon atoms apply to all carbon atoms, whether in a diamond, a lump of coal, a piece of charcoal, or a pencil point. Similarly, the exact indistinguishability of carbon atoms is just a fact about one particular kind of atom. But the underlying fact about electrons applies to all electrons, whether in carbon atoms, or oxygen atoms, or anywhere else.
Someday—perhaps quite soon—we may have a yet deeper theory that explains why the electron field is the same everywhere: maybe superstring theory, or “M-theory,” or some as yet undreamt-of theory. But we can be sure that whatever new and deeper theory comes along, it will reveal to us more profound principles of order and greater and more inclusive patterns.
What science has shown us is that most of the beauty and order in nature is hidden from our eyes. When ancient writers like Minucius Felix praised the “providence, order, and law in the heavens and on earth,” they spoke about something of which they could have only fragmentary glimpses. If we look at the world around us, we see only hints of order here and there peeking out from amidst a great deal of apparent irregularity and haphazardness. But science has given us new eyes that allow us to see down to the deeper roots of the world’s structure, and there all we see is order and symmetry of pristine mathematical purity.
I have been trying to show how the order which we see in nature at one level has its roots in a more mathematically perfect order that exists at a deeper level. This point is so important that I think it is worth illustrating with another example. A particularly noteworthy example, from the viewpoint of both religious and scientific history, is provided by the highly orderly movements of the heavenly bodies. As we saw, this “order in the heavens” was one of the pieces of evidence most frequently cited in antiquity that there is design in the structure of the universe, and therefore a designer. And it was the attempts to understand these astronomical patterns—by such men as Copernicus, Galileo, Kepler, and Newton (all religious men)—that ultimately gave rise to modern science.
When ancient astronomers looked at the heavens, they saw the stars, the Sun, and the Moon traveling around Earth with what appeared to them to be perfectly uniform circular motions. This, naturally, impressed them. The ancient Greeks were especially impressed, for they knew quite a bit of mathematics, and knew that the circle is a very special geometrical shape. One thing that is special about it, as I have already explained, is that it is highly symmetrical. In fact, a circle is more symmetrical than any other geometrical figure that can be drawn on a flat surface. Circles have other beautiful mathematical properties as well, many of which were known to the Greeks, who therefore regarded it as especially fitting that the heavenly bodies moved in circular paths.
The planets, on the other hand, seemed to move in a more irregular way than the Sun, the Moon, and the stars. (Indeed, the word planet means “wanderer.”) However, even the motions of the planets were understood as being made up of circles in Ptolemy’s theory of “epicycles.” All of this celestial harmony is in great contrast to what is observed to happen on Earth, where objects do not move in nice neat circles, but behave in exceedingly complicated and unpredictable ways. Moreover, the heavenly bodies never seem to undergo change or corruption, while on Earth everything eventually disintegrates or decays. These contrasts gave rise to the belief that celestial bodies were essentially different from terrestrial ones, and were even made up of an entirely different kind of matter.
Of course, most of these ideas have long since gone by the wayside. We now realize that, except for the Moon, the heavenly bodies do not go around Earth in circles, this being merely an illusion created by the rotation of Earth on its axis. What actually happens is that Earth and other planets travel around the Sun in orbits that are—as discovered in the seventeenth century by Johannes Kepler—elliptical in shape.
However, while some of the particular ideas of ancient astronomy have been shown to be oversimplified or simply wrong, the ancients were not at all wrong in seeing a great deal of orderliness in the heavens. In fact, this orderliness is greater than, and runs far deeper than, they ever imagined. Consider, for instance, Kepler’s discovery that the planets’ orbits are ellipses. This probably would have delighted the ancient Greek mathematicians, since ellipses were well known and much studied by them, and they were therefore aware that ellipses are mathematically beautiful shapes in their own right. An ellipse, like a circle, is an example of a “conic section.” A conic section is called that because it is the shape of the cross section of a cone when it is sliced through at some angle. There are five kinds of conic sections: circles, ellipses, parabolas, hyperbolas, and straight lines. (Interestingly, these are exactly the five possible orbits that an object can trace out if it moves under the influence of the gravity of another body, according to Newtonian physics.) Many of the mathematicians of ancient Greece, including Menaechmus, Aristaeus, Euclid, and Apollonius of Perga, wrote lengthy treatises about the properties of conic sections.
Kepler discovered, in addition to the elliptical shape of the planets’ orbits, two other “laws of planetary motion.” One of these has to do with how a planet speeds up as it approaches the Sun and slows down as it moves away. Kepler found that a planet moves in just such a way that a straight line drawn between it and the Sun “sweeps out equal areas in equal times.” The third of Kepler’s laws was a precise mathematical relationship between a planet’s distance from the Sun and the time it takes to complete one orbit. None of these three beautiful patterns discovered by Kepler in the motions of the planets were suspected to exist by ancient astronomers. Moreover, the Keplerian motions were far simpler and more mathematically elegant than the dizzying system of “cycles” and “epicycles” that had to be assumed in the older astronomy in order to accurately predict the motions of the planets. The order of the heavens as seen by the ancients fell far short of the reality found by Kepler.
In some of the details of their theories the ancient astronomers were not too far wrong. Although the planets do move in ellipses, these ellipses actually happen to be very close to being circles. Ellipses are ovals that come in a variety of shapes, from highly elongated to nearly circular. The amount of elongation is measured by a quantity called “eccentricity,” which ranges from 0 to 1. A highly elongated ellipse has an eccentricity that is close to 1, while an almost circular ellipse has an eccentricity that is close to 0. It turns out that the eccentricity of the Earth’s orbit is only 0.016. Neptune’s orbit is even more circular, with eccentricity 0.008, while Jupiter’s orbit has eccentricity 0.05, still quite close to circularity. Two other features of the ancient astronomical models turn out to be correct: the orbits of the planets all lie approximately in the same plane, and all of the planets go around in their orbits in the same direction.
There really is, then, as the ancients believed, a great deal of order and mathematical structure in the motions of the heavenly bodies. What we would like to know is, where does this structure come from? The answer, not surprisingly, is that it comes from the laws of physics. In fact, all of the features of planetary motion which I have mentioned can be explained quite straightforwardly from Newton’s theory of gravity and Newton’s laws of mechanics. But this does not mean that Newtonian science managed to explain away the “order in the heavens.” Quite the contrary. What Newton showed is that there is a more profound order, pervading all of nature, which reveals itself in a particularly transparent way in the celestial motions. Whereas Kepler’s laws applied only to the planets, Newton showed that they were a consequence of laws which governed the motions of all bodies, whether in the heavens or on Earth, whether great planets or little apples falling from trees. The grandeur of the orderliness of nature discovered by Newton moved Einstein to write the following verses:
Seht die Sterne, die da lehren
Wie man soll den Meister ehren.
Jeder folgt nach Newtons Plan
Ewig schweigend seiner Bahn.
Einstein’s biographer, Banesh Hoffmann, translates this:
Look to the Heavens, and learn from them
How one should really honor the Master.
The stars in their courses extol Newton’s laws
In silence eternal.2
Of course, Newton would have felt that another master, infinitely greater than himself, should be honored for what Einstein called “Newton’s Plan.”
In our earlier discussion we found that the symmetries that are evident upon the surface of nature reflect even greater symmetries that lie deeply buried in the underlying laws of physics. We can see this also in the present case. Consider the first of Kepler’s laws, that planetary orbits are ellipses. It turns out that this is a consequence of the fact that in Newton’s theory of gravity the force of gravity obeys what is called an “inverse square law.” This is a very simple mathematical relation, which also, as it happens, is obeyed by the electrical force. What it says is that the strength of the gravitational force varies inversely with the square of the distance between the two gravitating bodies. (For example, if one is three times as far away from the Sun the force of the Sun’s gravitational pull is only one-ninth as great.) If the strength of gravity depended in some other way upon distance, much more complicated orbits than ellipses would result. In most cases, in fact, the orbits would not be closed curves at all.
This inverse square law is a very special kind of law that results from the fact that the carrier of the gravitational force, the so-called “graviton” particle, is exactly massless. This masslessness of the graviton, in turn, is due to a very powerful set of symmetries called “general coordinate invariance” and “local Lorentz symmetry,” about which I shall have a little more to say in the next chapter. It is not important for the reader to understand what these symmetries are, just to know that the elegant elliptical shapes found by Kepler are only the tip of a huge iceberg of symmetric structure hidden in nature’s laws.
Let us turn now to Kepler’s second law, namely that planets “sweep out equal areas in equal times” as they go around the Sun. This fact can be shown to be a direct consequence of a very general principle of physics called the law of conservation of angular momentum. “Angular momentum” is a property of physical systems that tells how much they are rotating or swirling around. The principle says that under certain conditions the total amount of angular momentum a system has cannot change. The angular momentum of an object going around some center is given by the object’s mass times its distance from the center times its speed around the center. If the object gets closer to the center, it has to speed more quickly around to keep its angular momentum the same; if its gets farther away, it has to move more slowly around. In fact, it turns out that it has to do this in exactly the way described by Kepler’s equal-areas law.
The law of conservation of angular momentum is also a key ingredient in the explanation of why all the planets’ orbits lie almost in the same plane, and why they are approximately circular. It is believed that the Sun and planets formed from the condensation of a huge cloud of gas and dust that was swirling around billions of years ago. As this cloud condensed, its temperature went up and it radiated some of this heat into the colder surrounding space. The cloud thereby lost energy, but it was unable to lose whatever angular momentum it started with. It is a straightforward mathematical exercise to show that if one drains energy away from a cloud of particles while keeping its total angular momentum the same, the cloud will assume a more and more disc-like shape. That is, all the particles will tend more and more to orbit in the same plane. One can also show that the orbits of individual particles will tend more and more to approximate circles.
There is another fact about the celestial motions that is explained by the law of conservation of angular momentum. The fact that Earth rotates with uniform speed on its axis—which is the reason for the heavenly bodies appearing to have a uniform circular motion around Earth—is due to the conservation of Earth’s angular momentum.
We have just traced several of the regular patterns exhibited by planetary motion to a single law of physics, the law of conservation of angular momentum. But where does that law come from? It turns out that conservation of angular momentum arises, ultimately, from—what else?—symmetries. There is a very general theorem, proven in 1918 by Emmy Noether, which relates such “conservation laws” to symmetries of the laws of nature. Some examples of things that are conserved besides angular momentum are energy, ordinary momentum, and electrical charge. (There are many other examples, but they are somewhat esoteric.) All of these conservation laws are consequences of underlying symmetries.
The symmetries that are responsible for angular momentum conservation are of a kind I have already mentioned, namely “rotational symmetries” (the kind of symmetry a sphere has). What these rotational symmetries say is that the laws of nature do not distinguish one direction in space from another.
In sum, we have traced several patterns found in planetary motion to the fact that angular momentum is conserved, and that in turn to the fact that the laws of physics have rotational symmetries. We have traced other patterns of planetary motion to the inverse square law obeyed by gravitation; that in turn to the masslessness of the graviton; and that to underlying symmetries called general coordinate invariance and local Lorentz symmetry. But where do all those symmetries—the rotational symmetries, the local Lorentz symmetry, and so on—come from? Actually, we do not yet know, because we do not yet know what the deepest laws of nature are. However, there is no doubt that these symmetries of the presently known laws of physics have their roots in some still-greater symmetry or more profound principle of order that the as-yet-unknown fundamental laws of nature obey.