We should admire Ptolemy for many reasons, but chief among them is this: He did his job seriously and responsibly with the tools he had. Given what he knew, his system was brilliantly conceived, mathematically sound, and a huge advance over what had gone before. His observations were patient and careful and as complete as could be, his mathematical calculations correct. More, his mathematical system was as complicated as it needed to be and at the same time as simple as it could be, given what he had to work with. He was, in short, a real scientist. He set the standard.
It took a long time and there were some long arguments before astronomy could advance beyond what he offered—and that’s a sign of his achievement. But when advance was possible, Ptolemy had made it impossible for advance to come through wishful thinking, witch doctors, or fantasy. His successors in the great age of modern astronomy had to play by his rules. They needed to observe more carefully, do their math with exacting care, and propose systems at the poise point of complexity and simplicity. Ptolemy challenged the moderns to outdo him—and so they could and did. We owe him a lot.
QUASI-ELEGANCE
PAUL STEINHARDT
Albert Einstein Professor in Science, Departments of Physics and Astrophysical Sciences, Princeton University; coauthor (with Neil Turok), Endless Universe
My first exposure to true elegance in science was through a short semi-popular book entitled Symmetry, written by the renowned mathematician Hermann Weyl. I discovered the book in the fourth grade and have returned to reread passages every few years. The book begins with the intuitive aesthetic notion of symmetry for the general reader, drawing interesting examples from art, architecture, biological forms, and ornamental design. By the fourth and final chapter, though, Weyl turns from vagary to precise science as he introduces elements of group theory, the mathematics that transforms symmetry into a powerful tool.
To demonstrate its power, Weyl outlines how group theory can be used to explain the shapes of crystals. Crystals have fascinated humans throughout history because of the beautiful faceted shapes they form. Most rocks contain an amalgam of different minerals, each of which is crystalline but which have grown together or crunched together or weathered to the point that facets are unobservable. Occasionally, though, the same minerals form individual large faceted crystals; that’s when we find them most aesthetically appealing. “Aluminum oxide” may not sound like something of value, but add a little chromium and give nature sufficient time, and you have a ruby worthy of kings.
The crystal facets found in nature meet only at certain angles corresponding to one of a small set of symmetries. But why does matter take some shapes and not others? What scientific information do the shapes convey? Weyl explains how these questions can be answered by seemingly unrelated abstract mathematics aimed at answering a different question: What shapes can be used to tessellate a plane or fill space if the shapes are identical, meet edge-to-edge, and leave no spaces?
Squares, rectangles, triangles, parallelograms, and hexagons can do the job. Perhaps you imagine that many other polygons would work as well—but try and you will discover there are no more possibilities. Pentagons, heptagons, octagons, and all other regular polygons cannot fit together without leaving spaces. Weyl’s little book describes the mathematics allowing a full classification of possibilities; the final tally is only 17 in two dimensions (the so-called wallpaper patterns) and 230 in three dimensions.
The stunning fact about the list was that it precisely matched the list observed for crystals’ shapes found in nature. The inference is that crystalline matter is like a tessellation made of indivisible, identical building blocks that repeat to make the entire solid. Of course, we know today that these building blocks are clusters of atoms or molecules. However, bear in mind that the connection between the mathematics and real crystals was made in the 19th century, when the atomic theory was still in doubt. It is amazing that an abstract study of tiles and building blocks can lead to a keen insight about the fundamental constituents of matter and a classification of all possible arrangements of them. It is a classic example of what physicist Eugene Wigner referred to as the “unreasonable effectiveness of mathematics in the natural sciences.”
The story does not end there. With the development of quantum mechanics, group theory and symmetry principles have been used to predict the electronic, magnetic, elastic, and other physical properties of solids. Emulating this triumph, physicists have successfully used symmetry principles to explain the fundamental constituents of nuclei and elementary particles, as well as the forces through which they interact.
As a young student reading Weyl’s book, I thought crystallography seemed like the ideal of what one should be aiming for in science: elegant mathematics that provides a complete understanding of all physical possibilities. Ironically, many years later, I played a role in showing that my “ideal” was seriously flawed. In 1984, Dan Shechtman, Ilan Blech, Denis Gratias, and John Cahn reported the discovery of a puzzling man-made alloy of aluminum and manganese with icosahedral symmetry.* Icosahedral symmetry, with its six fivefold symmetry axes, is the most famous forbidden crystal symmetry. As luck would have it, Dov Levine (Technion) and I had been developing a hypothetical idea of a new form of solid we dubbed quasicrystals, short for “quasiperiodic crystals.” (A quasiperiodic atomic arrangement means the atomic positions can be described by a sum of oscillatory functions whose frequencies have an irrational ratio.) We were inspired by a two-dimensional tiling invented by Sir Roger Penrose known as the Penrose tiling, comprised of two tiles arranged in a fivefold symmetric pattern. We showed that quasicrystals could exist in three dimensions and were not subject to the rules of crystallography. In fact, they could have any of the symmetries forbidden to crystals. Furthermore, we showed that the diffraction patterns predicted for icosahedral quasicrystals matched the Shechtman et al. observations.
Since 1984, quasicrystals with other forbidden symmetries have been synthesized in the laboratory. The 2011 Nobel Prize in chemistry was awarded to Dan Shechtman for his experimental breakthrough that changed our thinking about possible forms of matter. More recently, colleagues and I have found evidence that quasicrystals may have been among the first minerals to have formed in the solar system.
The crystallography I first encountered in Weyl’s book, thought to be complete and immutable, turned out to be woefully incomplete, missing literally an uncountable number of possible symmetries for matter. Perhaps there is a lesson to be learned: While elegance and simplicity are often useful criteria for judging theories, they can sometimes mislead us into thinking we are right when we are actually infinitely wrong.
MATHEMATICAL OBJECT OR NATURAL OBJECT?
SHING-TUNG YAU
Mathematician, Harvard University; coauthor (with Steve Nadis), The Shape of Inner Space
Most scientific facts are based on things we cannot see with the naked eye or hear with our ears or feel with our hands. Many of them are described and guided by mathematical theory. In the end, it becomes difficult to distinguish a mathematical object from objects in nature.
One example is the concept of a sphere. Is the sphere part of nature or is it a mathematical artifact? That is difficult for a mathematician to say. Perhaps the abstract mathematical concept is actually a part of nature. And it is not surprising that this abstract concept actually describes nature quite accurately.
SIMPLICITY
FRANK WILCZEK
Theoretical physicist, MIT; corecipient, 2004 Nobel Prize in Physics; author, The Lightness of Being
We all have an intuitive sense of what “simplicity” means. In science, the word is often used as a term of praise. We expect that simple explanations are more natural, sounder, and more reliable than complicated ones. We abhor epicycles, or long lists of exceptions and special cases. But can we take a crucial step further, to refine our intuitions about simplicity into precise, scientific concepts? Is there a simple core to “simplicity”? Is simplicity something we can quantify and measu
re?
When I think about big philosophical questions, which I probably do more than is good for me, one of my favorite techniques is to try to frame the question in terms that could make sense to a computer. Usually it’s a method of destruction: It forces you to be clear, and once you dissipate the fog, you discover that very little of your big philosophical question remains. Here, however, in coming to grips with the nature of simplicity, the technique proved creative, for it led me straight toward a (simple) profound idea in the mathematical theory of information—the idea of description length. The idea goes by several different names in the scientific literature, including algorithmic entropy and Kolmogorov-Smirnov-Chaitin complexity. Naturally I chose the simplest one.
Description length is actually a measure of complexity, but for our purposes that’s just as good, since we can define simplicity as the opposite—or, numerically, the negative—of complexity. To ask a computer how complex something is, we have to present that “something” in a form the computer can deal with—that is, as a data file, a string of 0s and 1s. That’s hardly a crippling constraint: We know that data files can represent movies, for example, so we can ask about the simplicity of anything we can present in a movie. Since our movie might be a movie recording scientific observations or experiments, we can ask about the simplicity of a scientific explanation.
Interesting data files might be very big, of course. But big files need not be genuinely complex; for example, a file containing trillions of 0s and nothing else isn’t genuinely complex. The idea of description length is, simply, that a file is only as complicated as its simplest description. Or, to put it in terms a computer could relate to, a file is as complicated as the shortest program that can produce it from scratch. This defines a precise, widely applicable, numerical measure of simplicity.
An impressive virtue of this notion of simplicity is that it illumines and connects other attractive, successful ideas. Consider, for example, the method of theoretical physics. In theoretical physics, we try to summarize the results of a vast number of observations and experiments in terms of a few powerful laws. We strive, in other words, to produce the shortest possible program that outputs the world. In that precise sense, theoretical physics is a quest for simplicity.
It’s appropriate to add that symmetry, a central feature of the physicist’s laws, is a powerful simplicity enabler. For example, if we work with laws that have symmetry under space-and time-translation—in other words, laws that apply uniformly, everywhere and everywhen—then we don’t need to spell out new laws for distant parts of the universe or for different historical epochs, and we can keep our world-program short.
Simplicity leads to depth: For a short program to unfold into rich consequences, it must support long chains of logic and calculation, which are the essence of depth.
Simplicity leads to elegance: The shortest programs will contain nothing gratuitous. Every bit will play a role, for otherwise we could expunge it and make the program shorter. And the different parts will have to function together smoothly, in order to make a lot from a little. Few processes are more elegant, I think, than the construction, following the program of DNA, of a baby from a fertilized egg.
Simplicity leads to beauty: For it leads, as we’ve seen, to symmetry, which is an aspect of beauty. As, for that matter, are depth and elegance.
Thus simplicity, properly understood, explains what it is that makes a good explanation deep, elegant, and beautiful.
SIMPLICITY ITSELF
THOMAS METZINGER
Philosophisches Seminar, Johannes Gutenberg-Universität Mainz; author, The Ego Tunnel
Elegance is more than an aesthetic quality or some sort of ephemeral uplifting we experience in deeper forms of intuitive understanding. Elegance is formal beauty. And formal beauty as a philosophical principle is one of the most dangerous, subversive ideas humanity has discovered: It is the virtue of theoretical simplicity. Its destructive force is greater than Darwin’s algorithm or that of any other single scientific explanation, because it shows us what the depth of an explanation is.
Elegance as theoretical simplicity comes in many different forms. Everybody knows Occam’s razor, the ontological principle of parsimony: Entities are not to be multiplied beyond necessity. William of Occam gave us a metaphysical principle for choosing between competing theories. All other things being equal, it is rational to prefer the theory that makes fewer ontological assumptions.
We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances—Isaac Newton formulated this as the First Rule of Reasoning in Philosophy, in his Principia Mathematica. Throw out everything that is explanatorily idle, and then shift the burden of proof to the proponent of a less simple theory. In Albert Einstein’s words: The grand aim of all science . . . is to cover the greatest possible number of empirical facts by logical deductions from the smallest possible number of hypotheses or axioms.
Of course, in today’s technical debates new questions have emerged: Why do metaphysics at all? Isn’t what we should measure simply the number of free, adjustable parameters in competing hypotheses? Isn’t it syntactic simplicity that captures elegance best in, say, the number-fundamental abstractions and guiding principles a theory makes use of? Or will the true criterion of elegance ultimately be found in statistics—in selecting the best model for a set of data points while optimally balancing parsimony with the “goodness of fit” of a suitable curve? And, of course, for Occam-style ontological simplicity, the big question remains: Why should a parsimonious theory more likely be true? Ultimately, isn’t all of this rooted in a deeply hidden belief that God must have created a beautiful universe?
I find it fascinating to see how the idea of simplicity has kept its force over the centuries. As a metatheoretical principle, it has demonstrated great power—the subversive power of reason and reductive explanation. The formal beauty of theoretical simplicity is deadly and creative at the same time. It destroys superfluous assumptions whose falsity we just cannot bring ourselves to believe, whereas truly elegant explanations always give birth to an entirely new way of looking at the world. What I would really like to know is this: Can the fundamental insight—the destructive, creative virtue of simplicity—be transposed from the realm of scientific explanation into culture or onto the level of conscious experience? What kind of formal simplicity would make our culture a deeper, more beautiful culture? And what is an elegant mind?
EINSTEIN EXPLAINS WHY GRAVITY IS UNIVERSAL
SEAN CARROLL
Theoretical physicist, Caltech; author, From Eternity to Here: The Quest for the Ultimate Theory of Time
The ancient Greeks believed that heavier objects fall faster than lighter ones. They had good reason to do so; a heavy stone falls quickly, while a piece of paper flutters gently to the ground. But a thought experiment by Galileo pointed out a flaw. Imagine taking the piece of paper and tying it to the stone. Together, the new system is heavier than either of its components and should fall faster. But in reality, the piece of paper slows down the descent of the stone.
Galileo argued that the rate at which objects fall would actually be a universal quantity, independent of their mass or their composition, if it weren’t for the interference of air resistance. Apollo 15 astronaut Dave Scott illustrated this point by dropping a feather and a hammer while standing in near-vacuum on the surface of the moon; as Galileo predicted, they fell at the same rate.
Many scientists wondered why this should be the case. In contrast to gravity, particles in an electric field respond in various ways; positive charges are pushed one way, negative charges the other, and neutral particles not at all. But gravity is universal; everything responds to it in the same way.
Thinking about this problem led Albert Einstein to what he called “the happiest thought of my life.” Imagine an astronaut in a spaceship with no windows or other way to see the outside world. If the ship is far away from any stars or planets, everything i
nside will be in free fall; there will be no gravitational field to push them around. Now put the ship in orbit around a massive object, where gravity is considerable. Everything inside will still be in free fall, because all objects are affected by gravity in the same way; no one object is pushed toward or away from any other one. Given just what is observed inside the spaceship, there’s no way we could detect the existence of gravity.
Einstein, in his genius, realized the profound implication of this situation: If gravity affects everything equally, it’s not right to think of gravity as a “force” at all. Rather, gravity is a feature of spacetime itself, through which all objects move. In particular, gravity is the curvature of spacetime. The space and time through which we move are not fixed and absolute, as Newton had it; they bend and stretch because of the influence of matter and energy. In response, objects are pushed in different directions by spacetime’s curvature, a phenomenon we call “gravity.” Using a combination of intimidating mathematics and unparalleled physical intuition, Einstein was able to explain a puzzle unsolved since Galileo’s time.
EVOLUTIONARY GENETICS AND THE CONFLICTS OF HUMAN SOCIAL LIFE
STEVEN PINKER
Johnstone Family Professor, Department of Psychology, Harvard University; author, The Better Angels of Our Nature
Complex life is a product of natural selection, which is driven by competition among replicators. The outcome depends on which replicators best mobilize the energy and materials necessary to copy themselves and on how rapidly they can make copies which in turn can replicate. The first aspect of the competition may be called survival, metabolism, or somatic effort; the second, replication or reproductive effort. Life at every scale, from RNA and DNA to whole organisms, implements features that execute—and constantly trade off—these two functions.
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