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Since the time of Newton, science has advanced by a strategy rightly called “reductionism.” This method, which explains things by analyzing them into smaller and simpler parts, has yielded a rich harvest of discoveries about the natural world. As a means of analysis, then, reductionism has certainly proven its value. But many wonder whether science is reductive in a more radical and disturbing way—by flattening, collapsing, and trivializing the world. For all its intellectual accomplishments, does science end up taking our sense of reality down several notches? One could well get that impression from perusing the writings of certain scientists. Francis Crick famously asserted that human life is “no more than the behavior of . . . nerve cells and their associated molecules.” Marvin Minsky, a pioneer in the field of artificial intelligence, once described people as “machines made of meat.” Neuroscientist Giulio Giorelli announced that “we have a soul, but it is made up of many tiny robots.” And biologist Charles Zuker has concluded that “in essence, we are nothing but a big fly.”

This tendency to downgrade and diminish reflects a metaphysical prejudice that equates explanatory reduction with a grim slide down the ladder of being. Powerful explanatory schemes reveal things to be simpler than they appear. What simpler means in science is much discussed among philosophers—it is not at all a simple question. But to many materialists it seems to mean lower, cruder, and more trivial. By this way of thinking, the further we push toward a more basic understanding of things, the more we are immersed in meaningless, brutish bits of matter.

The philosopher Georges Rey has written, for example, that “any ultimate explanation of mental phenomena will have to be in non-mental [i.e., sub-mental or material] terms or else it won’t be an explanation of it.” Of course, the logic of this could be turned around. One could just as well say that any ultimate explanation of the material world must be in nonmaterial terms. But for materialists the lower explains the higher; and lower does not just mean more fundamental but instead suggests a diminished ontological status. The presumption is that explanations move from evolved complexity to primitive stuff.

At first glance, the history of the cosmos seems to bear this out. Early on, the universe was filled with nearly featureless gas and dust, which eventually condensed to form galaxies, stars, and planets. In stars and supernovas, the simplest elements, hydrogen and helium, fused to make heavier ones, gradually building up the whole periodic table. In some primordial soup, or slime, or ooze on the early earth, atoms agglomerated into larger and more intricate molecules until self-replicating ones appeared and life began. From one-celled organisms, ever more complicated living things evolved, until sensation and thought appeared. In cosmic evolution the arrow apparently moves from chaos to order, formlessness to form, triviality to complexity, and matter to mind.

And that is why, according to philosopher Daniel Dennett, religion has it exactly upside down. Believers think that God reached down to bring order and create, whereas in reality the world was built—or rather built itself—from the ground up. In Dennett’s metaphor, the world was constructed not by “skyhooks” reaching down from the heavens but by “cranes” supported by, and reaching up from, the solid ground.

The history to which the atheist points—of matter self-organizing and physical structures growing in complexity—is correct as far as it goes, but it is only part of the story. The lessons the atheist draws are naive. Yes, the world we experience is the result of processes that move upward. But Dennett and others overlook the hidden forces and principles that govern those processes. In short, they are not true reductionists because they don’t go all the way down to the most basic explanations of reality.

As we turn to the fundamental principles of physics, we discover that order does not really emerge from chaos, as we might naively assume; it always emerges from greater and more impressive order already present at a deeper level. It turns out that things are not more coarse or crude or unformed as one goes down into the foundations of the physical world but more subtle, sophisticated, and intricate the deeper one goes.

Let’s start with a simple but instructive example of how order can appear to emerge spontaneously from mere chaos through the operation of natural forces. Imagine a large number of identical marbles rolling around randomly in a shoe box. If the box is tilted, all the marbles will roll down into a corner and arrange themselves into what is called the “hexagonal closest packing” pattern. (This is the same pattern one sees in oranges stacked on a fruit stand or in cells in a beehive.) This orderly structure emerges as the result of blind physical forces and mathematical laws. There is no hand arranging it. Physics requires the marbles to lower their gravitational potential energy as much as possible by squeezing down into the corner, which leads to the geometry of hexagonal packing.

At this point it seems as though order has indeed sprung from mere chaos. To see why this is wrong, however, consider a genuinely chaotic situation: a typical teenager’s bedroom. Imagine a huge jack tilting the bedroom so that everything in it slides into a corner. The result would not be an orderly pattern but instead a jumbled heap of lamps, furniture, books, clothing, and what have you.

Why the difference? Part of the answer is that, unlike the objects in the bedroom, the marbles in the box all have the same size and shape. But there’s more to it. Put a number of spoons of the same size and shape into a box and tilt it, and the result will be a jumbled heap. Marbles differ from spoons because their shape is spherical. When spoons tumble into a corner, they end up pointing every which way, but marbles don’t point every which way, because no matter which way a sphere is turned it looks exactly the same.

These two crucial features of the marbles—having the same shape and having a spherical shape—should be understood as principles of order that are already present in the supposedly chaotic situation before the box was tilted. In fact, the more we reduce to deeper explanations, the higher we go. This is because, in a sense that can be made mathematically precise, the preexisting order inherent in the marbles is greater than the order that emerges after the marbles arrange themselves. This requires some explanation.

Both the preexisting order and the order that emerges involve symmetry, a concept of central importance in modern physics, as we’ll see. Mathematicians and physicists have a peculiar way of thinking about symmetry: A symmetry is something that is done. For example, if one rotates a square by 90 degrees, it looks the same, so rotating by 90 degrees is said to be a symmetry of the square. So is rotating by 180 degrees, 270 degrees, or a full 360 degrees. A square thus has exactly four symmetries.

Not surprisingly, the hexagonal pattern the marbles form has six symmetries (rotating by any multiple of 60 degrees: 60, 120, 180, 240, 300, and 360 degrees). A sphere, on the other hand, has an infinite number of symmetries—doubly infinite, in fact, since rotating a sphere by any angle about any axis leaves it looking the same. And, what’s more, the symmetries of a sphere include all the symmetries of a hexagon.

If we think this way about symmetry, careful analysis shows that, when marbles arrange themselves into the hexagonal pattern, just six of the infinite number of symmetries in the shape of the marbles are expressed or manifested in their final arrangement. The rest of the symmetries are said, in the jargon of physics, to be spontaneously broken. So, in the simple example of marbles in a tilted box, we can see that symmetry isn’t popping out of nowhere. It is being distilled out of a greater symmetry already present within the spherical shape of the marbles.

The idea of spontaneous symmetry breaking is important in fundamental physics. The equations of electromagnetism have a mathematical structure that is dictated by a set of so-called gauge symmetries, discovered by the mathematician and physicist Hermann Weyl almost a century ago. For a long time it seemed that two other basic forces of nature, the weak force and the strong force, were not based on symmetries. But about forty years ago it was found that the weak force is actually based on an even larger set of gauge symmetries than those of electromagnetism. Because the symmetries of the weak force are spontaneously broken, however, they do not manifest or express themselves in an obvious way, which is why it took so long to discover them. (The strong force is based on a yet larger set of gauge symmetries, but this fact was obscured by a quite different effect and also was not discovered for a long time.)

This history illustrates a general trend in modern physics: The more deeply it has probed the structure of matter, the greater the mathematical order it has found. The order we see in nature does not come from chaos; it is distilled out of a more fundamental order.

Symmetry is just one kind of order. In the case of the marbles in the box, other principles of order were also at work, such as the principle that caused the marbles to seek out the configuration of lowest energy. This is an aspect of a beautiful mathematical principle, called the principle of least action, that underlies all of classical physics. When physicists investigated the subatomic realm, however, they discovered that the principle of least action is just a limiting case of the much more subtle and sophisticated path integral principle, which is the basis of quantum mechanics, as Richard Feynman showed in the 1940s. The lesson is the same: The deeper one looks, the more remarkable the mathematical structure one sees.

The mathematical order underlying physical phenomena is most easily observed in the motions of the heavenly bodies. Even primitive societies were aware of it, and it inspired not only feelings of religious awe (many expressions of which are found in the Bible itself) but also the earliest attempts at mathematical science. And when scientists began to study the solar system with more precision, they discovered unsuspected patterns even more beautiful than those known to the ancients.

Four hundred years ago, for example, Johannes Kepler discovered three marvelous geometrical laws that describe planetary motion. So impressed was he by the beauty of these laws that he wrote this prayer in his treatise Harmonices Mundi (The harmonies of the world): “I thank thee, Lord God our Creator, that thou hast allowed me to see the beauty in thy work of creation.” Decades later, Newton succeeded in explaining Kepler’s laws—but he did not explain them down, if by down we mean reducing what we observe and experience to something more trivial or brutish. On the contrary, he explained them by deriving them from an underlying order that is more general and impressive, which we now call Newton’s laws of mechanics and gravity. Newton’s law of gravity was later explained, in turn, by Einstein, who showed that it followed from a more profound theory of gravity called general relativity. And it is now generally believed that Einstein’s theory is but the manifestation of a yet more fundamental theory, which many suspect to be superstring theory. Superstring theory has a mathematical structure so sophisticated that, after a quarter of a century of study by hundreds of the world’s most brilliant physicists and mathematicians, it is still not fully understood.

It is true that science seeks to simplify our picture of the world. An explanation should in some sense be simpler than the thing it explains. And, indeed, there is a sense in which Einstein’s theory of gravity is simpler than Newton’s, and Newton’s theory of planetary motion simpler than Kepler’s.

As physics Nobel laureate Frank Wilczek notes, however, Einstein’s theory is “not ‘simple’ in the usual sense of the word.” Whereas Kepler’s laws can be explained in a few minutes to a junior-high-school student, Newton’s laws cannot be fully explained without using calculus. And to explain Einstein’s theory requires four-dimensional, curved, non-Euclidean space-time and much else besides. And yet, once we know enough, Einstein’s theory does have a compelling simplicity greater than Newton’s theory. The simplicity to which scientific reductionism leads us, then, is of a very paradoxical kind. It is a simplicity that is by no means simpleminded. It is not at all jejune, but deeply interesting and intellectually rich.

The same paradox can be found in many fields. The chess world champion Capablanca was admired for the purity and simplicity of his style. But to understand his moves one must have an understanding of the game that can be acquired only by years of experience and study. A later world champion, Mikhail Botvinnik, wrote of him, “In this simplicity there was a unique beauty of genuine depth.” Another world champion, Emanuel Lasker, observed that “[in Capablanca’s games] there is nothing hidden, artificial, or labored. Although they are transparent, they are never banal and are often deep.” Wilczek had just the right term for this kind of simplicity, which is also found in the fundamental laws of physics: profound simplicity.

Profound simplicity always impresses with its elegance, economy of means, harmony, and perfection. This perfection, as Wilczek notes, is such that one feels that the slightest alteration would be disastrous. He quotes Salieri’s envious description of Mozart’s music in the film Amadeus: “Displace one note and there would be diminishment. Displace one phrase and the structure would fall.” Applying this to physics, Wilczek says, “A theory begins to be perfect if any change makes it worse. . . . A theory becomes perfectly perfect if it’s impossible to change it without ruining it entirely.”

Symmetry is one of the factors that contribute to profound simplicity, both in the laws of physics and in works of art. Paint over one petal of the rose window of a cathedral, remove one column from a colonnade, and the symmetry is destroyed. Each part is necessary for the completion of the pattern.

The symmetries that characterize the deepest laws of physics are mathematically richer and stranger than the ones we encounter in everyday life. The gauge symmetries of the strong and weak forces, for example, involve rotations in abstract mathematical spaces with complex dimensions. In other words, the coordinates in those peculiar spaces are not ordinary numbers, as they are for the space in which we live, but complex numbers, which are numbers that contain the square root of minus one. Grand unified theories—which combine the strong, weak, and electromagnetic forces into a single mathematical structure—posit symmetries that involve rotations in abstract spaces of five or more complex dimensions.

Stranger and more profoundly simple are supersymmetries. There is much reason to think that supersymmetries are built into the laws of physics, and finding evidence of that is one of the main goals of the Large Hadron Collider outside Geneva, Switzerland, which has recently begun to take data. Supersymmetries involve so-called Grassmann numbers, which are utterly different from the ordinary numbers we use to count and measure things. Whereas ordinary numbers (and even complex numbers) have the common-sense property that a b = b a, Grassmann numbers have the bizarre property that a b = -b a. A simple enough formula, but hard indeed for the human mind to fathom.

Esoteric symmetries also lie at the heart of Einstein’s theory of relativity. These Lorentz symmetries involve rotations not just in three-dimensional space but in four-dimensional space-time. We can all visualize the symmetries of a sphere or a hexagonal pattern, but Lorentz symmetries, supersymmetries, and the gauge symmetries of the weak, strong, and grand unified forces lie far outside our experience and intuition. They can be grasped only with the tools of advanced mathematics.

Physicists have found beauty in the mathematical principles animating the physical world, from Kepler, who praised God for the elegant geometry of the planets’ orbits, to Hermann Weyl, for whom mathematical physics revealed a “flawless harmony that is in conformity with sublime Reason.”

Some might suspect that this beauty is in the eye of the beholder, or that scientists think their own theories beautiful simply out of vanity. But there is a remarkable fact that suggests otherwise. Again and again throughout history, what started as pure mathematics—ideas developed solely for the sake of their intrinsic interest and elegance—turned out later to be needed to express fundamental laws of physics.

For example, complex numbers were invented and the theory of them deeply investigated by the early nineteenth century, a mathematical development that seemed to have no relevance to physical reality. Only in the 1920s was it discovered that complex numbers were needed to write the equations of quantum mechanics. Or, in another instance, when the mathematician William Rowan Hamilton invented quaternions in the mid-nineteenth century, they were regarded as an ingenious but totally useless construct. Hamilton himself held this view. When asked by an aristocratic lady whether quaternions were useful for anything, Hamilton joked, “Aye, madam, quaternions are very useful—for solving problems involving quaternions.” And yet, many decades later, quaternions were put to use to describe properties of subatomic particles such as the spin of electrons as well as the relation between neutrons and protons. Or again, Riemannian geometry was developed long before it was found to be needed for Einstein’s theory of gravity. And a branch of mathematics called the Theory of Lie groups was developed before it was found to describe the gauge symmetries of the fundamental forces.

Indeed, mathematical beauty has become a guiding principle in the search for better theories in fundamental physics. Werner Heisenberg wrote, “In exact science, no less than in the arts, beauty is the most important source of illumination and clarity.” Paul Dirac, one of the giants of twentieth-century physics, went so far as to say that it was more important to have “beauty in one’s equations” than to have them fit the experimental data.

At the roots of the physical world, therefore, one does not find mere inchoate slime or dust but instead a richness and perfection of form based on profound, subtle, and beautiful mathematical ideas. This is what the famous astrophysicist Sir James Jeans meant when he said many decades ago that “the universe begins to look more like a great thought than a great machine.” Benedict XVI expressed the same basic insight when in his Regensburg lecture he referred to “the mathematical structure of matter, its intrinsic rationality . . . the Platonic element in the modern understanding of nature.”

Modern science does not directly imply or require any particular metaphysical theory of reality, but it does suggest to us that the picture presented by Daniel Dennett and Richard Dawkins is false because the picture is only partial. In the terms of Dennett’s metaphor of cranes constructing complexity, one sees what is built from the ground up; but delving beneath the surface, one finds an astonishing, hidden world—the underground mechanisms of the cranes, as it were.

It is true that the cosmos was at one point a swirling mass of gas and dust out of which has come the extraordinary complexity of life as we experience it. Yet, at every moment in this process of development, a greater and more impressive order operates within—an order that did not develop but was there from the beginning. In the upper world, mind, thought, and ideas make their appearance as fruit on the topmost branches of an evolutionary tree. Below the surface, we see the taproots of reality, the fundamental laws of physics that shimmer with ideas of profound simplicity.

To describe people as machines made of meat is as scientifically unsophisticated as to think of the sun as a heat-emitting machine made of swirling gas. It ignores the reasons why the machines function as they do—reasons that the explanations of modern physics reduce to simplicities as elegant as they are elusive. Peering into the hidden depths, we see that matter itself is the expression of “a great thought,” of ideas that are, as Weyl said, “in conformity with sublime Reason.” And we begin to discover that matter, although mindless itself, is the product of a Mind of infinite profundity and infinite simplicity.

Stephen M. Barr is professor of physics at the University of Delaware and author of Modern Physics and Ancient Faith and A Student’s Guide to Natural Science.