Beauty for Truth's Sake Page 9

  This fact Weil regarded as perhaps the most beautiful expression of the interplay between circular and linear motion. And into it she reads nothing less than the meaning of the Incarnation, the “God-Man” of Christianity.

  As the circle encloses the moving point upon the diameter, God assigns a term to all the becomings of this world. As the Bible says, He rules the raging of the sea. The segment on the right angle which joins the point of the circle to its projection upon the diameter is, in the figure, an intermediary between the circle and the diameter. At the same time, from the point of view of quantities, it is, like the mean proportional, the mediation between the two parts of the diameter which are on either side of the point. This is the image of the Word.24

  Here we see the power of the poetic imagination to unite truths that our minds usually keep separate. For Weil, if the circle expresses the infinite motion of God (or the heavens), and the straight line down the middle expresses the world of creation (the earth), then the line linking the circle’s perimeter with the diameter at a right angle, the geometric mean, is the world’s Mediator, the incarnate Logos.

  The Golden Circle

  There is more to say about the figure of a circle divided into two halves—into which we have now introduced a right-angled triangle. This next step will require us to draw upon the symbolic properties of the irrational numbers. Let us extend the idea suggested by Simone Weil using Thales’s triangle, in which the diameter of the circle projects to the perimeter and a perpendicular is dropped from there back to the diameter, by adding another such triangle in the other half of the circle.

  These two right-angled triangles, stuck together along any diameter of the circle, make up a rectangle. If we choose the lengths of the sides correctly, we can make it a golden rectangle. That is why I call the figure a “golden circle”: it is a circle made golden by the rectangle inscribed within it.

  The golden circle is a beautiful synthesis of straight and circular motion, of phi and pi. The latter, of course, represented by the Greek letter π, is the irrational number (approximately 22/7 or 3.14159) by which the diameter of any circle must be multiplied to find its exact circumference. As the shape of the letter suggests, π is like a “gateway”—in this case between the domain of straight lines and the domain of circularity. We remember that π goes on to infinity when expressed in integers, whether we write it as a fraction or a decimal. In the golden circle, Φ and π are connected together by the fact that the golden rectangle’s diagonal forms the diameter of the circle.25

  If we meditate on this in relation to Simone Weil’s circular diagram, in which the diameter represents creation and the circle God, we can see in the infinity of π an expression of the limitlessness of God’s act of creation, whenever the divine circular motion gives rise to the straight line of creation.

  If we wish, instead of identifying the circle with the divine and the diameter with creation, as Weil does, we could apply the model at a higher level to God as he is in himself—that is, to the Trinity—as we did earlier. In that case the point from which the circle begins or from which it projects is the Father, the line that extends from the point to make the radius or diameter represents the Son, and the circle made by swinging the radius around represents the Holy Spirit. Then π could be read as describing the relationship between the Persons, a relationship that is infinitely fruitful and never ending. Thus the endlessly flowing numbers of π suggest the super-abundance of God’s mercy, the infinite quality of his love, and the unlimited space opened up within the Trinity for the act of creation.26 We see reflected in π the ocean of potentiality, the “waters” over which the Spirit moves at the beginning of Genesis, and into which the Son is projected as a light in the darkness.

  But there is another fact about the relationship that is particularly striking. We recall that for Weil the perpendicular is the Logos, linking divinity (the circle) with the created world (the straight line). It turns out by a remarkable coincidence that this “Logos line” is linked to the circle by a number associated above all with the creation of the world by God in the book of Genesis: the number 7. For if we multiply our Logos line by 7 we obtain a close approximation of the circumference of the circle (the exact multiplier is more like 7.025).

  Why only an approximation? If it were an exact number that would imply that the “gap” between lines and rectangles on the one hand, and circles on the other, could be bridged without going through the transcendental π. In that case the problem of “squaring the circle” (see next chapter) could easily be solved. But the reason it cannot ever be solved in that way is that, as Michael S. Schneider writes,

  true pi reaches for the infinite and never fully engages with things mundane and rational. The transcendental and rational, Heaven and Earth, can approach each other but cannot, by definition, fully meet, as they shouldn’t. No ruler and compass construction can truly square a circle. When the divine Architect separated Heaven and Earth it was permanent.27

  But it seems to me there is another reason why this number is only approximate, and this reason touches on the deepest mystery revealed by Christianity. Mediation between heaven and earth is accomplished by the seven sacraments issuing from the actions of the Logos during his life on earth, echoing the seven mystical days of creation.28 But we have just found that the “Logos line” needs to be multiplied by a number very slightly more than seven to arrive at the Divine Circle. Reading this fact theologically, it seems justifiable to speculate that it is because the mediation of the God-Man will always leave room for the vital but infinitesimal human contribution, our free cooperation with grace.

  As St. Paul says: “Now I rejoice in my sufferings for your sake, and in my flesh I complete what is lacking in Christ’s afflictions for the sake of his body, that is, the Church” (Col. 1:24, my emphasis). This sentence has often been taken by theologians to refer to our cooperation with God in our own salvation and deification (although admittedly I have never before seen a mathematical representation of it). As the Catholic Catechism puts it: “God grants his creatures not only their existence, but also the dignity of acting on their own, of being causes and principles for each other, and thus of cooperating in the accomplishment of his plan.”29 It is represented in the Catholic Mass by the drop of water with which the priest slightly dilutes the cup of wine that is about to become the blood of Christ. This is the tiny and indispensable human contribution needed if heaven is truly to descend to earth, and earth finally to be integrated with the everlasting Trinity.

  Speculations like those I have mentioned in this chapter will appear forced to many. Yet we must return to the central idea that God’s archetypal forms or Ideas are inevitably found within nature at every level, reflected with greater or lesser degrees of accuracy. That is not pantheism but Christian Platonism, perfectly compatible with the insights of theology and the revelations of scripture.

  1. Christian trinitarian doctrine and its history are usefully summarized in O’Collins 1999.

  2. Damascene 1999, 63–64.

  3. Of course, there are triads and triads. Not all are equally analogous to the Trinity, or analogous in the same respect. For an exploration of symbolic triads across many religious traditions, see Guénon 2001a. For a Christian perspective on this, see Bolton 2005, esp. chap. 3.

  4. Stein 2002, 463.

  5. ST 1, Q 45, art. 7. Augustine famously finds an analogy to the Trinity in the psychological triad of memory, understanding, and will, to which one might add the corresponding dimensions of culture—mythos, logos, and ethos. For an exposition of the point that Trinity is the most general characteristic of being, see Florensky 1997, esp. 420–24.

  6. Al’Arabi 1980, 141, 272. As Toshihiko Izutsu points out, however, for Ibn Arabi the triplicity of the Creator is not a triplicity of the One, which transcends number. The creation is possible because of its receptive triplicity that corresponds to an active triplicity implicit in the “singleness” (not Unity) of the Absolute (Izutsu 1984, 198–99
).

  7. So “x” symbolizes the generation of the Son, and “÷” the spiration of the Spirit. But this is only a suggestion. Tom McCormick, in private correspondence, has raised the question of what science and mathematics would look like if based upon a trinitarian conception of the number 1, rather than a monolithic unity derived from Greek and Hindu sources.

  8. One of the 99 Beautiful Names of God in Islam is Al-Wadud, The Loving. Allah loves his creatures, and is loved by them. But there is all the difference in the world between a God who is loving but needs the creation in order to have an object for his love, and a God who is in himself love (1 John 4:8). This is not to say that Allah is a different God from the one worshipped by Christians (the word Allah is simply the Arabic for “God”), but he is understood differently in the two traditions. Something has been revealed by Christ that is not revealed explicitly in Islam, namely the interior life of God.

  9. This is the divine name that is represented in Hebrew tradition by the four letters JHVH.

  10. John Paul II 1985.

  11. The Orthodox claim that the insertion of the word filioque into official versions of the Creed by the Latin Church was illegitimate—and the more extreme among them even claim that it makes the Catholic notion of God radically different from that of the Orthodox believer.

  12. Bouyer 1999, 232.

  13. Meerson 1998, 182–83.

  14. Durrwell 1990, 44.

  15. A circle bisected by a vertical line is also, coincidentally, the Greek letter phi (Φ or φ), which is used in mathematics to describe the first creative division based on Unity (see previous chapter).

  16. The direction of the linear motion is irrelevant (it could be up, down, or sideways), so let me put this another way. Imagine a point. The point contains implicitly two forms of motion: the line and the circle. Project each from the point and you arrive at our bisected figure. But there is nothing to prevent the circular motion from extending into every possible dimension. So, with the original point remaining as a fixed anchor, imagine the “disk” formed by the circle swinging around in the same plane to form a larger disk centered on that point. Now imagine this disk swinging around its diameter in the third dimension, to form a sphere. And so on. Whether we represent the Father as the starting point of the smaller circle or as the center of a larger one incorporating it, or even of the sphere that incorporates all possible circles, the fundamental relationships remain the same: the Father as the starting point for both straight and circular motion, representing the Son and the Holy Spirit.

  17. Cited in Ruhr 2006, 46. For more on proportion and harmony, see Critchlow 1994.

  18. Weil 1957, 171.

  19. Ruhr 2006, 48.

  20. Weil 1973, 208.

  21. In God, the Father and Son, though one in nature, are also other as Persons. The Holy Spirit eternally overcomes that otherness in a unity that never dissolves the relationship. A married couple become “one flesh” without losing their individual identities, and the presence of the “third” may later reveal itself in a child of the union.

  22. Weil 1957, 176.

  23. From “The Love of God and Affliction,” cited in Morgan 2005, 148. Vance Morgan explains that, for Weil, creation’s very existence is a suffering and dying motivated by love, and requires an eternal crucifixion of God (147), thus “all geometry proceeds from the Cross.”

  24. Weil 1957, 192. See also Morgan 2005, 113, 144–45.

  25. By the way, pi is related to phi by the following formula: π ≈1.2 Φ². The presence of the number 12 divided by 10 is perhaps worthy of note given the ubiquity of 12 as a symbol for the circular cosmos in ancient traditions and 10 as the Tetraktys.

  26. The mathematician Clifford A. Pickover argues somewhat fancifully that “somewhere inside the endless digits of π is a very close representation for all of us—the atomic coordinates of all our atoms, our genetic code, all our thoughts, all our memories” (2005, 300).

  27. Schneider 2006, 40; cf. Lawlor 1982, 74–79.

  28. See Caldecott 2006, chap. 6.

  29. Catechism of the Catholic Church, par. 306. It is hard to resist mentioning other examples of a slight difference from 7 that become significant if we read them with an eye to this theological mystery of the incommensurability of heaven and earth. Michael Schneider has pointed out in private correspondence that if you add √2 + √3 + √5 + Φ, the total also comes close to 7—in fact to something like 7.0003. The first three of these irrational numbers, which are also the first three primes, symbolically represent (respectively) Generation, Formation, and Regeneration (see Lawlor 1982, 31), while Φ represents growth, as in the Fibonacci sequence. Together they give a very good description of what the sacraments are supposed to be and accomplish supernaturally in man. It seems fitting then (at least to someone who believes there are necessarily seven sacraments!) for their total to approximate 7. Another example occurs in music, and is known as the “Pythagorean comma” (see next chapter).

  5

  “Quiring to the Young-Eyed Cherubims”

  For man is in receipt of a singular prerogative beyond all other animals, to worship the Existent, but heaven is ever making music, producing in accordance with its celestial motions the perfect harmony.

  Philo of Alexandria1

  If mathematics is inherently theological, it is also mystical. Writing of the contemplative function of symbolic mathematics, Simone Weil says in her Notebooks: “Only such a mystical conception of mathematics as this was able to supply the degree of attention necessary in the early stages of geometry.”2

  As we have seen, the “Liberal” Arts are precisely not “Servile” Arts that can be justified in terms of their immediate practical purpose. “The ‘liberality’ or ‘freedom’ of the Liberal Arts consists in their not being disposable for purposes, that they do not need to be legitimated by a social function, by being ‘work.’”3 As Josef Pieper argues, the reduction of the liberal to the servile arts would mean the proletarianization of the world. At the heart of any culture worthy of the name is not work but leisure, schole in Greek, a word that lies at the root of the English word “school.” At its highest, leisure is contemplation. It is an activity that is its own justification, the pure expression of what it is to be human. It is what we do. The “purpose” of the quadrivium was to prepare us to contemplate God in an ordered fashion, to take delight in the source of all truth, beauty, and goodness, while the purpose of the trivium was to prepare us for the quadrivium. The “purpose” of the Liberal Arts is therefore to purify the soul, to discipline the attention so that it becomes capable of devotion to God; that is, prayer.

  Having said all of that, we have also seen that there are indeed a myriad practical applications and implications of symbolic mathematics. I suggested in the second chapter that to appreciate the aesthetic and symbolic dimension of numbers and shapes would be the first step in transforming science itself, that most practical of human pursuits. The recovery of a contemplative appreciation of numbers and shapes would also herald a renewal of the arts (painting, sculpture, music, architecture, even film). For it is the contemplative dimension that connects us with the source of inspiration and beauty in the cosmos and our own souls.

  Let us now look more closely at the concept of harmony that is central in the Pythagorean tradition—illustrating this in terms of music, architecture, ecology, and astronomy.

  Good Vibrations

  Every material object is capable of vibrating, and furthermore has a “natural frequency” at which it does so, determined by its physical constitution—in fact most have several such frequencies, called “natural harmonics.” Thanks to the phenomenon of resonance, whereby the vibration of one thing sets going a vibration in another, the vibration of (say) a guitar string communicates itself to the air and creates a sound wave that we can hear, because the eardrum starts to resonate in sympathy with it.

  Harmony—the perceived agreement or “concord” between different frequencies—was first analyzed
mathematically by Pythagoras. He noticed that the sounds made by different hammers hitting an anvil depended on the relative weight of the hammer, and that the sounds seemed to fit together in a pleasing way when the respective weights were in certain ratios to each other (so, for example, when one hammer was exactly twice as heavy as another). The lowest note (or pitch) produced by an instrument such as a string or hammer, or a column of air in the case of a wind instrument, is called its fundamental frequency or “first harmonic.” The wavelength of this note will be exactly twice the length of the string. By shortening the string, or holding it partway along its length, another harmonic is produced. In fact any string will naturally vibrate first as a unit, and then in halves, thirds, quarters, and fifths—producing a series of “overtones.”

  The difference in frequency between one pitch and another is called an “interval,” the “octave” being the name given to the interval separating the first and second harmonics (2:1).4 Within the octave the Western classical tradition recognizes seven intervals in a major scale: “unison,” “second,” “third,” “fourth,” etc. up to “seventh,” each slightly bigger than the one before: