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Last week, my wife, a painter-friend of ours, who wishes to be anonymous, and I did the Friday night walk down Canyon Road, the site of numerous galleries in Santa Fe, New Mexico, a small town that is the third-largest art market in the United States. Halfway down Canyon Road, we stopped in at a contemporary gallery that had a new show featuring conceptual art that our painter friend was dying to see. Inside, I gazed at one wall of the gallery painted red with white block letters pronouncing “Belief + Doubt = Sanity”; on another wall I encountered a photograph of a human brain with the caption “A Thinking Machine”; next to the photograph was a large white canvas with the black letters “Pure Beauty.” Puzzled by what I saw, I mumbled, “What pointlessness. What happened to beauty?”

Our painter friend wheeled around to face me, and I was surprised to see black anger in his eyes. He shouted at me and apparently at everyone else in the gallery, “Beauty is an old-fashioned, idiotic concept. Representational art is dead, killed by the camera, by technologists, and by scientists. We contemporary artists are painting ideas.” Then, he pointed at me and yelled, “What do you theoretical physicists know about beauty! Nothing!”

I shrugged my shoulders, and if I hadn’t placed friendship above the truth, I would have said, “More than you artists, apparently.” I surmised that theoretical physicists talk more about beauty than present-day visual artists. I recalled that even as an undergraduate hardly a class in physics or mathematics went by without the word “beautiful” spoken.

During the summer before my senior year at the University of Michigan, I took Introduction to Theoretical Physics, taught by Professor Otto Laporte. For some odd reason, the class met at 7:00 PM. One night, Laporte walked in, with two martinis under his belt, and announced, “Tonight gentlemen, I’m going to show you something beautiful.” He proceeded to elegantly lay out several fundamental theorems about vector spaces. When finished, he stepped back from the blackboard and said, “Isn’t that beautiful.” One student in the class, the very worst one, asked, “What is so beautiful about that?” Laporte was taken back, and after a moment of silence, asked, “Do the rest of you see that this is beautiful?” We all nodded, and several students replied, “Of course.” The professor then turned to the student who was blind to the beauty of vector spaces and told him, “You be quiet. The rest of us see it.” Laporte told us in his blunt way that intellectual insight and the apprehension of beauty are not democratic, that poor cultural formation, political and economic ideology, and a habituation to lies and ugliness can cut a person off from truth and beauty.

Strangely, neither Laporte nor any student in his class could give the characteristic marks of beauty. Except for the worst student, we all recognized mathematical beauty without having ever been taught to. In the terminology of ancient philosophy, our souls by nature were attuned to beauty.

When young, I relished beauty wherever I found it—relativity and quantum physics, the chamber music of Mozart, the great drawings of Picasso. I wanted to be close to heaven in this world, and never thought once about what beauty is or why it was so important. Years later, after I saw beauty put to evil ends at Los Alamos National Laboratory and everything in my life was called into question, I set myself the task to discover the marks of beauty, not that I wanted to establish a rigorous definition. I was already persuaded that any formula for beauty is a shadow of the experience of a rose, a Mozart sonata, or the Pythagorean Theorem; however, I believed that if I knew the marks of beauty both my experience of art and my understanding of physics would deepen.

The Beautiful

The beauty physicists talk about is not the product of private or idiosyncratic emotion. Ein­stein gives the three elements of scientific beauty in a single sentence: “A theory is the more impressive the greater the simplicity of its premises is, the more different kinds of things it relates, and the more extended is its area of applicability.”[1] Simplicity, then, is the first element of beauty. The “different kinds of things it relates” means how the theory harmonizes disparate things. Thus, we may label the second element harmony. And the extended applicability is a theory’s brilliance; that is, how much clarity it has in itself and how much light it sheds on other things. Simplicity, harmony, and brilliance. Each of these calls for a brief explanation.[2]

Simplicity. There exist today other theories of gravity besides Einstein’s general relativity, but because they lack simplicity, none are taken seriously. “Most rival theories are convincingly disproved,” observes mathematician Roger Penrose. “The few that remain having been, for the most part, contrived directly so as to fit with those tests that have been actually performed. No rival theory comes close to general relativity in elegance or simplicity of assumption.”[3] All rival theories of gravity lack what Werner Heisenberg called “frightening simplicity and wholeness.”[4] Without wholeness, or unity, a theory is a collection of disconnected ideas and observations that often borders on the frighteningly ugly.

The principle of simplicity implies completeness and economy. “It is because simplicity and vastness are both beautiful that we seek by preference simple facts and vast facts,” mathematician-physicist Poincaré explains.[5] A theory beautiful by this standard must account for all the facts with a few simple principles. A demanding standard, indeed! Heisenberg recalls that quantum theory was “immediately found convincing by virtue of its completeness and abstract beauty.”[6]

Harmony. “Without the belief in the inner harmony of the world there could be no science,” Einstein declares.[7] Heisenberg defines harmony as the “proper conformity of the parts to one another and to the whole.”[8] In any science a good theory will harmonize many previously unrelated facts, for without the harmony of its parts, a theory lacks unity. Harmony also implies symmetry. There is a pleasing symmetry to all the laws of physics. “Every law of physics goes back to some symmetry of nature,” observes physicist John Wheeler.[9] Heisenberg adds, “The symmetry properties always constitute the most essential features of a theory.”[10] Newton’s third law is a well-known example of symmetry in physics: To every action there is always opposed an equal reaction. A different mirror symmetry is found on the subatomic level, where to every kind of particle there corresponds an anti-particle with the same mass but with opposite characteristics. In fact, this symmetry successfully predicted the existence of many subatomic particles.

Brilliance. A theory with brilliance has great clarity in itself and sheds light on many things, suggesting new experiments. Newton, for example, astounded the world by explaining falling bodies, the tides, and the motions of the planets and the comets with three simple laws. Physicist George Thomson states, “In physics, as in mathematics, it is a great beauty if a theory can bring together apparently very different phenomena and show that they are closely connected; or even different aspects of the same thing.”[11] General relativity does precisely that in an elegant and surprising way, as astrophysicist S. Chandrasekhar explains: “It consists primarily in relating, in juxtaposition, two fundamental concepts which had, till then, been considered as entirely independent: the concepts of space and time, on the one hand, and the concepts of matter and motion, on the other.”[12] Moreover, general relativity has proven extraordinarily brilliant, shedding light on cosmology and the fate of the universe.

Both general and special relativity, like all theories that possess brilliance, have what Francis Bacon calls “strangeness in the proportion,” or fitting surprises.[13] From the principle that the velocity of light is the same for all inertial (non-accelerating) observers, the surprises of relativity—time dilation, length contraction, and the equivalence of inertial mass and energy—follow as inevitable consequences. Einstein’s 1905 paper on special relativity is a series of fitting surprises, not unlike what is found in the great music of Mozart and Beethoven.

Music conductor Leonard Bernstein’s comments on Beethoven’s Eroica Symphony apply equally well to Einstein’s 1905 paper: “The element of unexpected is so often associated with Beethoven. But surprise is not enough; what makes it so great is that no matter how shocking and unexpected the surprise is, it always somehow gives the impression—as soon as it has happened—that it is the only thing that could have happened at that moment. Inevitability is the keynote. It is as though Beethoven had an inside track to truth and rightness, so that he could say the most amazing and sudden things with complete authority and cogency.”[14] The truths of relativity, or quantum physics for that matter, are stranger than we imagined, but their inevitability convinces us that Nature is that way. Strip away fitting surprises and beauty is reduced to monotonous unity, mechanical symmetry, and uninteresting predictability. Nuanced broken symmetries in art and science command our attention and force us to contemplate or to seek deeper, underlying truths.

Beauty is not confined to theory; an experiment can possess a simplicity of apparatus, a brilliance that results from the transparency of data analysis, and an outcome that reveals something fundamentally new. The most beautiful experiments transform the way scientists think about Nature.[15]

Until Galileo experimented with rolling balls down inclined planes in the early 1600s, scientists believed terrestrial phenomena lacked mathematical intelligibility. Galileo showed experimentally that the acceleration of a ball rolling down an inclined plane obeys simple mathematical rules. This crucial discovery changed how scientists thought about matter and led Newton to discover the three general laws that govern all mechanical motion. Until 1803, when Thomas Young experimented with light beams, physicists believed all natural phenomena obeyed Newton’s laws and thus held that a beam of light was made of material particles. Through a simple experiment, Young observed that two light beams interact like waves, not particles. He divided a light beam in two by passing it over a card, about one-thirtieth of an inch in breadth. A series of light and dark bands resulted, not unlike what happens when a water wave on a lake encounters a square dock post. Young’s key experiment altered forever how physicists think about light and eventually led James Clerk Maxwell to discover the equations that describe all electromagnetic phenomena.

The ten paragraphs I wrote on the marks of beauty I circulated to former colleagues at Los Alamos National Laboratory and at the University of Michigan. In the past, over beers and nachos, I heard them voice cultural platitudes: “Beauty is in the eye of the beholder;” “Beauty is very personal;” “Beauty is not objective and thus varies from culture to culture.” But I discounted the voice of culture, for I knew that such remarks were contrary to their experience of doing physics and mathematics. My ten paragraphs drew mainly positive responses; the most negative comment was an emotional rant about how I was thinking like a philosopher, not a physicist. Except for this one person, no one objected to the four marks of beauty that I proposed—simplicity, harmony, brilliance, and fitting surprise.

I suspected that all my former colleagues in their youth were shaken and transformed by an experience of some profound beauty. I know I was. In the tenth grade, I was changed forever by Euclid’s proof that the prime numbers are infinite, an exquisite proof that surprisingly showed in six lines of text an eternal truth. Until that point in my life, I thought truth did not exist; everything about me changed, the seasons, my body, and people. My experience of the human world was that everything was in flux, sometimes bordering on the absurd. Suddenly, mathematics presented me with one thing that was unchanging, a timeless truth, demonstrated in an exceedingly beautiful way; the 2,500 years between Euclid and me were of no consequence. My encounter with Euclid was not unique. Bertrand Russell described his first encounter with serious mathematics: “At the age of eleven, I began Euclid, with my brother as my tutor. This was one of the great events of my life, as dazzling as first love. I had not imagined that there was anything so delicious in the world.”[16]

Years later, after I left Los Alamos, I was stunned by a colossal failure: Modern science cannot explain why Nature overflows with beauty, both to the eye and to the mind. Leptons, quarks, the chemical elements, the DNA molecule, crystals, snowflakes, sand dollars, hummingbirds, hawks, mice, chipmunks, jaguars, lions, horses, zebras… nothing in Nature is not beautiful in some way.

The only proposed scientific explanation of this fundamental aspect of Nature is that the origin of beauty is sex. Sigmund Freud reduced beauty to an instinct: “Psychoanalysis, unfortunately, has scarcely anything to say about beauty… All that seems certain is its derivation from the field of sexual feeling.”[17] In the same vein, Charles Darwin declared that the beauty of present-day animals resulted from sexual selection: “the more beautiful males having been continually preferred by the females.”[18] Sexual selection, of course, does not explain why we humans find nightingale songs, pheasant tails, and rainbow trout beautiful. Or, why we delight in the sparkle of a diamond, in the shape of the filigreed wing of a housefly, or in the elegance of Euclid’s demonstration of the Pythagorean Theorem.

Darwin acknowledged, “How the sense of beauty in its simplest form—that is, the reception of a peculiar form of pleasure from certain colors, forms, and sounds—was first developed in the mind of man and of the lower animals is a very obscure subject…but there must be some fundamental cause in the constitution of the nervous system in each species.”[19] Consequently, Darwin held that the “sense of beauty obviously depends on the nature of the mind irrespective of any real quality in the admired object.”[20] But the beautiful equations of general relativity, which can be written in the palm of a hand, “contain” such “objects” as black holes, the Big Bang, and the fate of the universe.

Darwin was too great a scientist not to be enthralled by the abundant beauty of Nature. “Delight itself,” he said, “is a weak term” to describe the deep pleasure a naturalist experiences in a Brazilian rain forest.[21] In the concluding words of The Origin of Species, Darwin invoked beauty as a proof that his theory is true: “from so simple a beginning endless forms most beautiful and most wonderful have been, and are being evolved.”[22] But to speak of the “most beautiful and most wonderful” organic forms is nonsense, if beauty “depends on the nature of the mind irrespective of any real quality in the admired object.” To get out of this contradiction, beauty must be a real property of Nature.

The True

The answer to the question “Why does Nature abound with beauty?” begins where Darwin ends, that is, with a working principle of science: “You can recognize truth by its beauty.” Physicist Fred Hoyle relates that when he was young and not many years into research, his thesis advisor Paul Dirac once told him, “Hoyle, you are much too empirical. Look more closely at the mathematical structure of what you are doing.” When Hoyle asked what he should look for, Dirac answered, “You have to learn to recognize what is beautiful.”[23]

The more something strays from simplicity, harmony, and brilliance, the less beautiful it becomes and the less intelligible. A circle is manifestly intelligible; we recognize it immediately and can define it. But a circle with an ever so slight asymmetric irregularity lacks intelligibility—it does not even have a name. Without simple, symmetric patterns, Nature would not be intelligible; without the brilliance of underlying, universal causes, Nature would not be intelligible. If Nature were ugly, then it would not be intelligible. But Nature abounds with beauty, and that is why it is supremely intelligible.

The Good

If existence were not good, we would be indifferent about death; we would greet a diagnosis of a fatal disease with a yawn. But we have developed medical technologies to fight disease and to extend longevity. Our bodies, like all organisms on Earth, maintain life as long as possible; physically, emotionally, and intellectually, we desire life to last forever. On our best days, we awake and tell ourselves a profound truth: “It is great to be alive.”

We know that this good existence is not forever. Even considering the end of the universe saddens some physicists. Theorist Edward Witten, contemplating how with the Big Freeze the universe ends like a wisp of smoke, said such a fate “is not very appealing.” Astrophysicist Glenn Starkman concurs that such a universe “would be the worst possible universe, both for the quality and quantity of life,” and his colleague Lawrence Krauss adds, “All our knowledge, civilization, and culture are destined to be forgotten. There’s no long-term future.”[24]

Divine Mind

Physicist Eugene Wigner succinctly describes the three miracles that science relies upon: “The miracle that the human mind can string a thousand arguments together without getting itself into contradiction…[and] the two miracles of the existence of laws of nature and of the human mind’s capacity to divine them.”[25]

Wigner calls these “miracles” because no scientist has even a bad idea to offer about how mind could evolve out of matter and then come to comprehend the deep mathematical structure of matter. Not one neuroscientist has proposed even a goofy mechanism for how a complex arrangement of neurons in the human brain can give rise to the Ricci tensor of general relativity or to the Hilbert spaces of quantum mechanics, nor has any evolutionary psychologist shown how the Schrödinger equation resulted from the reproductive success of our ancestors on the African savannah.

Once the truth that no arrangement of matter can produce mind is acknowledged[26], then the most straightforward explanation of these three miracles is that the universe results from Divine Mind and that in some way the human mind is akin to Divine Mind. Reflecting on the mathematical structure of the universe, Dirac suggests, “It seems to be one of the fundamental features of nature that fundamental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it… One could perhaps describe the situation by saying that God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe.”[27]

The Greek word for beautiful is kalos, which bears an uncanny resemblance to the verb kaleô, “to call.” A beautiful object calls us to it, and beyond to its maker. In this way, the beauty of a red rose, of a soaring hawk, or of a natural law calls us to Divine Mind. Pointing to the deepest aspects of the human soul, Socrates said a lover of wisdom questing for universal beauty will find himself or herself “ever mounting the heavenly ladder, stepping from rung to rung—that is from one to two, and from two to every lovely body, from bodily beauty to the beauty of institutions, from institutions to learning, and from learning in general to the special lore that pertains to nothing but the beautiful itself—until at last the lover of wisdom comes to know what beauty is.”[28] The quest for beauty ends when the lover’s eyes are opened to gaze upon Divine Beauty in true contemplation.

The universe has precisely the properties that we would expect if it were brought into existence by Divine Mind—the universe is supremely intelligible, exceedingly good, and abundantly beautiful. If we take the universe to be an expression of the transcendent, then Divine Mind, or God, is intelligible, good, and beautiful.[29]

Books mentioned in this essay may be found in The Imaginative Conservative Bookstore

Notes:

[1] Albert Einstein, “Autobiographical Notes,” in Albert Einstein: Philosopher-Scientist, ed. Paul Schilpp (New York: Harper & Row, 1959), p. 33.

[2] Stephen Daedalus, the protagonist of James Joyce’s A Portrait of the Artist as a Young Man, argues that the three universal qualities of beauty in the arts are wholeness, harmony, and radiance, his translation of Aquinas’ integritas, consonantia, and claritas.

[3] Roger Penrose, “Black Holes,” in The State of the Universe, ed. Geoffrey Bath (Oxford: Clarendon Press, 1980), p. 128.

[4] Werner Heisenberg, Physics and Beyond: Encounters and Conversations, trans. Arnold J. Pomerans (Harper & Row, New York, 1971), pp. 68-69.

[5] Henri Poincaré, Science et Méthode (Paris: Flammarion, 1949), p. 16. Our translation.

[6] Werner Heisenberg, “The Meaning of Beauty in the Exact Sciences,” in Across the Frontier (New York: Harper & Row, 1974), p. 167.

[7] Albert Einstein and Leopold Infeld, The Evolu­tion of Physics (New York: Simon & Schuster, 1938), p. 313.

[8] Heisenberg, “The Meaning of Beauty in the Exact Sciences,” p. 167.

[9] John A. Wheeler, “The Universe as a Home for Man,” American Scientist 62 (Nov.-Dec. 1974): 688.

[10] Werner Heisenberg, Physics and Philosophy (New York: Harper & Row, 1958), p. 133.

[11] George Thomson, The Inspiration of Science (Oxford: Oxford University Press, 1961), p. 18.

[12] S. Chandrasekhar, “Beauty and the Quest for Beauty in Science,” Physics Today 32 (July, 1979): 29.

[13] Francis Bacon, “Of Beauty” in The Essayes or Counsels, Civill and Morall, ed. Michael Kiernan (Cambridge, Mass.: Harvard University Press, 1985), p. 132. Available http://www.bartleby.com/3/1/.

[14] Leonard Bernstein, The Infinite Variety of Music (New York: Simon & Schuster, 1966), p. 198. Italics in the original.

[15] The ten most beautiful experiments in physics, according to a poll of Physics World readers, are: (1) Young’s double-slit experiment applied to the interference of single electrons (1961); (2) Galileo’s experiment on falling bodies (1589); (3) Millikan’s oil-drop experiment (1909); (4) Newton’s decomposition of sunlight with a prism (1665-1666); (5) Young’s light-interference experiment (1801); (6) Cavendish’s torsion-bar experiment (1798); (7) Eratosthenes’ measurement of the Earth’s circumference (3rd century BC); (8) Galileo’s experiments with rolling balls down inclined planes (1604, published 1638); (9) Rutherford’s discovery of the nucleus (1911); (10) Foucault’s pendulum (1851). Also see Robert P. Crease, The Prism and the Pendulum: The Ten Most Beautiful Experiments in Science (New York: Random House, 2004).

[16] Bertrand Russell, The Autobiography of Bertrand Russell: 18721914 (Boston: Atlantic Monthly Press, 1967), pp. 37–38.

[17] Sigmund Freud, Civilization and Its Discontents, trans. and ed. James Strachey (New York: Norton, 1961), p. 33.

[18] Charles Darwin, On the Origin of Species (London: Murray, 1872), p. 161.

[19] Ibid., p. 162.

[20] Ibid., p. 160.

[21] Charles Darwin, The Voyage of the Beagle (New York: Dutton, 1967), p. 8.

[22] Darwin, On the Origin of Species, p. 429.

[23] Fred Hoyle, Highlights in Astronomy, (San Francisco: Freeman, 1975), pp. 35–36.

[24] Edward Witten, Glenn Starkman, and Lawrence Krauss quoted by Dennis Overbye, “The End of Everything,” The New York Times, Science Section, January 1, 2002.

[25] Eugene P. Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” Communications in Pure and Applied Mathematics 13 (February 1960).

[26] For a detailed discussion of why the human mind is not the result of a complex arrangement of neurons, see George Stanciu, “Materialism: The False God of Modern Science” (July, 2016).

[27] P.A.M. Dirac, “The Evolution of the Physicist’s Picture of Nature,” Scientific American 208 (May 1963), p. 53.

[28] Plato, Symposium in The Collected Dialogues of Plato, ed. Edith Hamilton and Huntington Cairns (Princeton, NJ: Princeton University Press, 1989), 211c. Italics in the original.

[29] The conclusion God is intelligible, good, and beautiful, if read literally, is false. No formula, no set of words, no elaborate theology can contain God, the Unnamable. See Pseudo-Dionysius, The Divine Names in Pseudo-Dionysius: The Complete Works, trans. Colm Luibheid (New York: Paulist Press, 1987), 596A and 872; and Pseudo-Dionysius, The Mystical Theology in Pseudo-Dionysius: The Complete Works, trans. Colm Luibheid (New York: Paulist Press, 1987), 1033B.

 

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5 replies to this post
  1. With God, nothing is impossible. The end of the universe? Liken it to the ripples on a body of water that once dissipated appears as a mirror, everything in harmony. The source of all simply harmonizes all, forever.
    God, from privileged position, simply expands His domain.

  2. There is apparently some painter who has made a career painting natural numbers in the correct order. He started from the beginning and continued until he ran out of canvass. His next work picked up where the first left off and on it goes. Idiots pay top dollar for this and say “I have his 8000 to 8345” painting. Now I see – thanks to your article- why this seems to be fashionable in art. If the painter of numbers was in the least bit clever he might have painted half the canvas with the infinity of Real numbers and the other half with the infinity of Natural numbers to illustrate Cantor’s theory of diagonalization and the notion of one infinity smaller than another, bijection, injection etc…but that would require painters to know elementary math.

    Heidegger rightly claimed philosophy was over and only the poet could truly continue it.

    Likewise art is dead and only the Mathematician can continue it.

  3. If I may ask: When no scientist could explain lightning, was that justification for believing Zeus was responsible…?

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