Beauty for Truth's Sake

by Stratford Caldecott

  If, as many of the ancients believed, the material and changeable world is merely an imperfect shadow of the unchanging eternal, we can hardly hope to find matter conforming perfectly to mathematical laws. So it was that by Kepler’s time it was accepted that mathematical devices (such as epicycles) could be used by astronomers to predict the movements of the heavens, without anyone really believing that they existed in reality. Astronomy was not concerned with finding physical explanations for things. After all, what cause is needed, other than God? Astronomers were concerned exclusively with “saving the appearances” by finding accurate ways of analyzing heavenly movements into their component circles—that is, devising mathematical descriptions of these appearances using perfect circles without assuming any physical explanation of them whatever.37

  Kepler’s breakthrough came because he introduced a “why?” question where the astronomers of his day didn’t see the need for one. He sought physical causes for heavenly motions. And that was not because he believed less in God as the cause of everything, but because he had more respect for the physical world as God’s creation and as the image of God’s mind. It was the first step toward Newton’s cosmos, in which the same universal laws (such as gravity) governed both the earth and the heavens.

  Take the orbits of the planets, for example. At the age of twenty-five Kepler thought he had discovered that the orbits of the six visible planets orbiting the sun (Mercury, Venus, Earth, Mars, Jupiter, Saturn) fit beautifully within the five Platonic solids, arranged one inside another. This would explain why there are precisely six, and not more, and why they orbit where they do. Of course, we now know there are more planets than those visible to the naked eye. Koestler calls this a classic example of “false inspiration” that nevertheless triggered a series of breakthroughs that proved to be of lasting importance.

  Unfortunately for Kepler’s peace of mind, the fit between the orbits and the Platonic solids proved to be inexact, according to the data he had taken from Copernicus. His continuing unease with these discrepancies drove him to seek out the much more accurate observations of Tycho Brahe, whom he met early in the year 1600. Out of this meeting of two great astronomers eventually came Kepler’s New Astronomy, his physics of the sky, in 1609. In that book he reports his discovery that the planets move around the sun not in the perfect circles that seemed most appropriate to celestial bodies, but in ellipses. At one point he compared this to finding a “load of dung” in the heavens. Pure deduction from an aesthetic ideal had gone astray, but careful observation and measurement had led to a correct conclusion seemingly at odds with traditional cosmology.

  The irony is that if we contemplate the result of this observation, we find that an unexpected beauty reveals itself. For the medieval astronomers were wrong: there is actually nothing imperfect about an ellipse. It differs from a circle by having two centers or foci rather than one (the sun occupying one of them), so that the sum of the distances from any point on the circumference to the two centers remains constant. Thus the planetary orbit is determined by two centers, the visible and the invisible, just as the life of any creature must revolve around the incarnate Logos and the invisible Father. What could be more elegant? Or as Kepler came to see, if the circle represents “transcendental” perfection, and the straight line represents the created world, an ellipse (as the combination of the two) represents perfectly the incarnation of the ideal in the created order. Kepler’s original mistake did not lie in his Christian Pythagoreanism, but in his attempt to prejudge the mathematical forms he would find in nature. He should have been happy to be led by observation, confident that what he discovered would (eventually) turn out to have appropriate symbolic properties.

  In order to make Pythagorean sense out of these strange elliptical movements in the sky, Kepler tried to reconcile the orbits of the planets with the classical harmonic proportions. He eventually found the correspondence he was looking for in the variations of the angular velocities of the planets as seen from the sun, by comparing the speed at which they were traveling at different parts of their orbits. This is an example of the right way of doing things: to look at what really happens, and discover the beauty in it.

  Kepler had now discovered the first two of his immortal “three laws” of planetary motion. The third came to him as he tried to find the relationship between a planet’s period around the sun and its distance. Kepler thought there had to be a connection, if the sun was indeed master of the solar system. It turned out that the square of the period is proportional to the cube of the mean distance. Not intuitively obvious, but beautiful nonetheless.

  As Koestler tells Kepler’s story, the harmonies he searched for and eventually found were psychologically but not otherwise particularly significant. They lured him onward, but the more important Pythagorean insight that he had revived after a millennium and a half of neglect was simply that mathematical relations hold the secret of the universe—the whole universe, above and below the moon—and need to be uncovered by precise empirical observation.

  The Ptolemaic astronomers, assuming circular motion with the earth stationary at the center, had tried to account for the retrogression of the planets against the stars by means of a complex pattern of epicycles and deferents—wheels within wheels. We do not have to revert to their geocentric description to appreciate simple patterns that reveal themselves in the relative motion of the planets, as they pursue their separate eccentric paths around the sun, each at a different speed. Not only are the periods of the planets related to each other in fairly precise harmonic proportions (2:5 in the case of Jupiter and Saturn, for example, and 1:Φ in the case of Earth and Venus), but each traces a lovely sequence of loops around the other that reveal aspects of their geometrical relationship. Venus and Earth produce a particularly beautiful five-petaled “flower” containing a pentagram of close conjunctions over 8 years (13 Venusian years).

  Thus the solar system as understood and measured by modern astronomers abounds with beauty that would warm the heart of any Christian Pythagorean. One final example: ancient and medieval geometers were tantalized by the problem of “squaring the circle,” which meant finding the square whose perimeter (or alternatively area) measured exactly the same as that of a given circle. It is one of the strange coincidences in which the solar system abounds that the problem is “solved” by the respective sizes of the earth and the moon, which are in the ratio of 11:3. Thus if the moon were rolled around the earth’s surface, its center would describe a circle equal to the perimeter of a square inscribed around the earth (31,680 miles).38

  The harmonics of the planetary orbits are well known. Something analogous must surely apply within the subatomic world (quantum harmonics?), since all energy is a kind of vibration. These different levels of creation—the macro- and the micro-world, with humanity between—are bound up in a single whole. It has often been remarked that if any of the main physical constants (such as one of the four fundamental forces of nature, Planck’s constant, or the speed of light) had been slightly different than they are, the universe could not have developed into a suitable habitation for life. In this sense too the cosmos is a beautifully ordered whole, precisely tuned to permit human existence.39

  Everywhere we look in nature, we tend to find structure or form. The planets occupy distinct orbits, rotating in close numerical relation to one another. Materials vibrate at certain frequencies that harmonize together. Electrons fall into distinct “shells” around the nucleus of the atom, jumping from one level to another depending on the units of energy they acquire. In evolutionary biology, creatures do not appear randomly across the whole range of physical possibility, but fall into families and species each of which expresses a particular type of creaturehood. All these distinct forms we observe in the universe indicate what Pope Benedict calls the “inner design of its fabric.”40 They can be read as approximations to the fundamental “ideas” that lie behind the creation. Thus even today the concept of harmony developed by the Pythagoreans ca
n help us understand the way the unity and diversity of the world unfolds.

  The End of the Road

  Music, architecture, astronomy, and physics—the physical arts and their applications—demonstrate the fundamental intuition behind the Liberal Arts tradition of education, which is that the world is an ordered whole, a “cosmos,” whose beauty becomes more apparent the more carefully and deeply we study it. By preparing ourselves in this way to contemplate the higher mysteries of philosophy and theology, we become more alive, more fully human. This beautiful order can be studied at every level and in every context, from the patterns made by cloud formation or river erosion to that of the leaves around the stem of the most obnoxious weed, from the shape of the human face as it catches the light, or the way keys are ordered in a concerto by Bach, to the collision of stellar nebulae and particles in an atomic furnace.

  Yet at the same time, while studying and appreciating the intuitions that lay behind the cosmological sciences of the quadrivium, we cannot today simply revert to the worldview of the Middle Ages. The ancient mathematical theories of music and astronomy contain elements we need to retrieve, but they were not themselves entirely adequate. In his theological study of Western tonal music, Jeremy Begbie asks, in the Great Tradition stemming from Pythagoras,

  Is the created world being treated as able to glorify God in its own way, by virtue of its own distinctive patterns, rhythms, and movements? Many have argued that the streams of thought that guided much medieval thinking about music did not pay enough attention to the distinctive order and harmony of the universe as it is and as it could be. Out of a keenness to assume direct and necessary correspondences between the created world and God, to preserve (in some cases) a “hierarchy of being,” it is debateable whether the structures of creation were always being respected in their full integrity and potential.41

  We have seen that question arise in connection with astronomy too. Yet in noting the shift in our modern thinking “from the cosmological to the anthropological, from justifying music in terms of the cosmos at large to justifying it solely in terms of human needs and aspirations”42 (and for all his slight suspicion of the influence of “Platonic” otherworldliness on the arts), Begbie wonders if something immensely valuable has been lost along the way.

  For all that we might smile benignly at in the mathematical clumsiness and rhetorical hyperbole of the classical philosopher of music or the intellectual abstractions and tetchy fussiness of the medieval theorist, is there not something in the notion of being “cradled” in God’s created harmonia that is worth recovering?43

  In late 2007, the themes I have been discussing in this book hit the headlines all over the world when a maverick physicist, Garrett Lisi, published online a paper entitled “An Exceptionally Simple Theory of Everything.”44 In it, he suggested all known subatomic particles and forces (and a few unknown ones, which he predicted would be found by the Large Hadron Collider in Switzerland) could be located on a matrix provided by E8, an eight-dimensional shape discovered in 1887 that is regarded as the most elegant and intricate example of mathematical symmetry. As Lisi put it in the online paper, he had devised “a comprehensive unification program, describing all fields of the standard model and gravity as parts of a uniquely beautiful mathematical structure. The principal bundle connection and its curvature describe how the E8 manifold twists and turns over spacetime, reproducing all known fields and dynamics through pure geometry.”

  The theory was incomplete, and the predictions still to be tested, but what makes this interesting is the philosophy of science that lies behind it. “We exist in a universe described by mathematics,” wrote Lisi. “But which math? Although it is interesting to consider that the universe may be the physical instantiation of all mathematics, there is a classic principle for restricting the possibilities: The mathematics of the universe should be beautiful. A successful description of nature should be a concise, elegant, unified mathematical structure consistent with experience.”

  Lisi’s particular theory failed, but the drive of science is in this direction. Others will try, guided by the same intuition that the truth is beautiful, the same compulsion to discover the truth in beauty. But as Stephen M. Barr has pointed out, if science can explain the design of the world by discovering a deeper and simpler design among the laws of nature, it still “has no way to explain the ultimate design of nature.”45 Armed with a convincing Theory of Everything, it will have reached the end of the road of science. But the end of the road is the beginning of another and wider landscape. Science can discover the laws of nature, but not why they are that way, nor why there is anything to obey them. That is why cosmology leads only to the threshold of theology.

  1. Philo 1981, 115.

  2. Weil 1956, 2:512. For Weil prayer consists of attention to God and the concentration required to solve (or even attempt) mathematical puzzles is never wasted, since it develops the soul’s capacity for the higher attentiveness—not to mention (in the case of those of us who find mathematics difficult) humility!

  3. Pieper 1998, 22.

  4. It is called “octave” because the classical scale was divided into seven notes. By raising the pitch from one note to another you could progress through the scale contained between the first and second harmonics. The eighth step up would commence another octave, just as the eighth day of one week is the first of the next. (Similar principles seem to apply to color, for just as our ear naturally distinguishes seven notes in a scale, our eye tends to distinguish seven colors in a rainbow.)

  5. Platonists following the Timaeus expressed this model in the form of the Greek letter lambda (Λ). At the top they placed unity, source of all other numbers, with immediately below that the first even and first odd, followed by squares and cubes corresponding to the first, second, and third dimension of space: line, plane, and solid.

  From the ratios between these seven fundamental integers they could deduce (by taking the arithmetic and harmonic means of the numbers) musical consonances, the music of the heavens, and the harmonies of the soul.

  6. The “comma” seems to correspond to the discrepancy between the months of the solar and lunar calendars, and so may have been taken to represent a distinction between heavenly and earthly harmony. See Barker 2003, 275–76.

  7. Hugh of St. Victor 1991, 69.

  8. Ibid.

  9. Recent research suggests human beings may be unique among animals in having a sense of rhythm, learning to synchronize their movements with an auditory beat. Neurologically this might be explained by a connection between the part of the brain concerned with movement and that concerned with hearing. No doubt an evolutionary reason can be invented to account for this. But the same facts can be “read” from the other side as expressing a metaphysical truth about man, whose attunement to the cosmos is part of his nature and necessary to his purpose.

  10. Cited in Begbie 2008, 169. Jeremy Begbie’s book is recommended for the reader who wants to investigate the religious dimensions of music.

  11. Tavener 1999, 73–74, 98.

  12. Ibid., 135–36.

  13. Ibid., 48.

  14. Jerome Taylor, in a note to Hugh of St. Victor 1991, 196.

  15. Things are, of course, more complicated than this, and Manhattan’s Chrysler and Empire State owe their iconic status partly to the fact that they do possess a form and decorative features that speak of integration and thus of beauty. Le Corbusier’s design for the United Nations building, like much of his work, was based on the golden rectangle. Even the Trade Center did not lack a certain beauty (the beauty of a machine or a crystal), though less related by its proportions to the human body than the large buildings of earlier civilizations.

  16. Again, a qualification is necessary. In many parts of the world, especially in hot climates, we see that flat roofs can be beautiful too, but there I would claim the instinctive or traditional design of these buildings tends to make use of the vertical dimension in other ways, rather than simply ignore or supp
ress it.

  17. Pickstock 1998, chap. 2. Cf. Guénon 2001b and Guardini 1998.

  18. Louis Dupré’s Passage to Modernity gives a brilliant analysis of all these developments.

  19. Charles 1989, 76–97.

  20. Rose 2001, 15–29. He compares these to the traditional principles of Utility, Strength, and Beauty.

  21. Ratzinger 1996, 88.

  22. See Ratzinger 2000, 62–84; Lang 2004.

  23. On all this, with the relevant quotations from Abbot Suger and Bishop Durandus, see Schloeder 1998, 187–208. Much more detail on the Gothic specifically is provided in Burckhardt 1995, and Mâle 1958, as well as the classic thirteenth-century work by Durandus himself, The Rationale Divinorum Officiorum. Hani 2007 provides an impressive synthesis of this tradition.

  24. Schloeder 1998, 199.

  25. Wittkower 1998.

  26. Wölfflin 1984, 64.

  27. Cited in Ward 2008, 24. Michael Ward shows that each of Lewis’s seven Narnian chronicles was organized around characteristics associated with one of the seven traditional “planets,” which he regarded as spiritual symbols of permanent value.

  28. Dan. 3:57–88, 56.

  29. Modern thought tends to regard man as nothing more than an animal, but implicitly admits his centrality by making him solely responsible for the destruction of the biosphere.

  30. The Catholic Mass or Orthodox Divine Liturgy is the highest form of such invocation, anamnesis, and mediation.