Beauty for Truth's Sake

by Stratford Caldecott

  17. The use of Arabic/Hindu numerals was encouraged by the election of the brilliant mathematician Gerbert as Pope Sylvester II in the year 999, and a century later by the translations of Arabic scientific texts by Adelard of Bath. It was Adelard who brought to Europe the text of Euclid’s Elements of Geometry in the early twelfth century.

  18. Note the concept of “zero point energy” or “vacuum energy” in quantum mechanics (the energy in supposedly “empty” space), which is related to the cosmological constant. The quantum vacuum is not an absolute nothingness.

  19. Lawlor 1982, 19–20.

  20. Stewart 1996.

  21. Although this sequence of numbers contains no discernible pattern, it is not random, for if any of its integers were deleted or moved the number would no longer be π.

  22. Transcendental numbers form a subset of the irrationals, in that they are not even “algebraic”: that is, they cannot be expressed by any finite equation using rational numbers (integers) as coefficients. Pi and e (the exponential constant, the base of natural logarithms) are both transcendental numbers in this sense, whereas √2 and Φ are irrational but not transcendental.

  23. An endless decimal, because Φ is an irrational number.

  24. Lawlor 1982, 46.

  25. Ibid., 47. In the paradigm envisaged here, A to B is like B to 1.

  26. Schneider 1994, 115. Please read the pages that follow in Schneider’s book for a full introduction to the golden ratio and its applications.

  27. Barr 2003, 97–98. Mathematicians now claim to have identified the complete range of possible symmetries right up to a form termed the “Monster,” which contains more elements than there are quarks in the sun.

  4

  The Golden Circle

  Mathematics connects directly with theology. If this seems a bizarre notion, it is only because we are so fragmented in our thinking that God and mathematics appear to belong to completely different worlds. If there is to be a revival of the Christian Pythagorean-Socratic tradition along the lines suggested by Pope Benedict, we need to bridge that gap.

  In order to make the jump from number to God as understood in the Christian tradition, we have to attune our minds to the idea that God is not just the infinite, or the One, but the infinite Three-in-One. One God. Three Persons.1 What on earth could such a statement mean? Hieromonk Damascene ponders this in his extraordinary work Christ the Eternal Tao:

  The Triad contains itself in perfection,

  For it is the first that surpasses the dyad.

  It lies beyond the duality of matter,

  Of subject and object,

  Of self and other.

  The Triad is beyond the distinction of the one and the many;

  Its perfection goes beyond the multiplicity of which duality is the root.

  Two is the number that separates,

  Three the number that transcends all separation.

  The one and the many find themselves gathered together in the Three.

  For the Triad, being many, is also a Unity:

  Not a unity of self-absorption, but of love.

  For the Three have one nature, one will, one power, one operation.

  As One, They do not blend or become confused,

  But They cleave to each other, having their being in each other.

  This is the perfect love, the original unity, the original harmony,

  the final mystery

  To which no human thought has ever succeeded in rising.2

  A Journey into God

  Tradition tells us that the threefoldness of God is not like that of a triangle or a shamrock or any other created thing. It is not the threeness of three objects that can be placed side by side and counted. Threefoldness in this world—in everything we can see—points toward divine triunity, but never reaches that far. Yet once we know that God is three Persons, we can see the mark or image or shadow of triunity in things that are made.

  We see that mark or echo, for example, in the three dimensions of corporeal space and the three dimensions of time (past, present, and future).3 Arguably, we see it also in the three notes of the chord, the three grammatical “persons” (I, you, and he), and the three elements of the human being (body, soul, and spirit), corresponding to the three “worlds” (material, psychical, and archetypal). Edith Stein writes:

  The threefold formative power of the soul must be regarded as a tri-unity, and the same is true of the end product of its forming activity: body—soul—spirit. If we attempt to relate this tri-unity to the divine trinity, we shall discover in the soul—the wellspring that draws from its own sources and molds itself in body and spirit—the image of the Father; in the body—the firmly designed and circumscribed expression of the essence or nature—the image of the eternal Word; and in the spiritual life the image of the divine Spirit.4

  Following Augustine, St. Thomas argues that every creature bears a trace of the Trinity in being created as an individual, having a form, and being related to other things.5 Even in Islam, a religious tradition that we tend to think of as anti-trinitarian, the greatest Sufi interpreter of the Qur’an, Muhyiddin Ibn Arabi, notes that three aspects of every creature (and especially the Prophet) correspond to a triplicity in the Creator, which he terms Essence, Will, and Word.6

  We cannot “prove” (let alone understand) the Trinity of God from such evidence, but these phenomena are illuminated for us by the knowledge of faith: that the maker of all is himself triune. Even mathematics is illuminated by it. It is not simply that numbers can be used as symbols. Numbers have meaning—they are symbols. The symbolism is not always merely projected onto them by us; much of it is inherent in their nature.

  So is there any way to find the “mark of the Trinity” in the domain of numbers itself? You might do it like this. Every natural number is a multiple of 1. But 1 multiplied or divided by 1 is 1. This makes 1 × 1 ÷ 1 a kind of arithmetical “icon” of the Trinity. It begins with the Father, who generates the Son as his own image without adding anything to the divine nature. The relationship of each to the other is then expressed as a ratio (1 ÷ 1), which symbolizes the Holy Spirit who is the “unity” of Father and Son.7

  There are more sophisticated mathematical images of the Trinity—one of them is in the golden ratio itself—but this will do for a start. Once you start looking, you can see pointers to the Trinity everywhere—and that is because Three-in-One is actually more fundamental to existence even than One. For Oneness has no real place for multiplicity, while Trinity does. This doesn’t mean that there is multiplicity in the One, but that multiplicity has its root in the One’s relationship with itself.

  All this may seem a bit abstract, but it is concrete enough to mark important divisions between one religion, one civilization, and another—Christianity and Islam being the most obvious example.8

  Theology of the Trinity

  Moses, when he asked the identity of God, received this reply: “Say this to the people of Israel, ‘I AM has sent me to you’” (Exod. 3:14).9 When theologians talk about the Trinity, what they are trying to describe is a dynamic but trans-temporal Act “I AM” that is the highest essence and source of both Being and Love.

  Pope John Paul II expresses the trinitarian mutuality of the eternal Act of love as follows:

  The Father who begets loves the Son who is begotten. The Son loves the Father with a love that is identical with that of the Father. In the unity of the divinity, love is on the one side paternal and on the other, filial. At the same time the Father and the Son are not only united by that mutual love as two Persons infinitely perfect. But their mutual gratification, their reciprocal love, proceeds in them and from them as a person: the Father and the Son “spirate” the Spirit of love consubstantial with them . . . The Spirit is also called Gift.10

  The French theologian Louis Bouyer tries to define the trinitarian relations in a way that overcomes the longstanding dispute between Catholic and Orthodox traditions concerning the filioque clause of the Nicene Creed (that
is to say, the question of whether the Holy Spirit proceeds just from the Father, or from the Father and the Son as the Latins affirm):11

  Everything comes eternally, within God as well as outside him, from the Father alone, the one invisible in himself as St. Irenaeus would say. Everything that can possibly be comes from him in the Son, comes as eternally enfolded within the Son albeit infinitely surpassed by him. But—or because—everything the Father has gives itself, realizes itself by giving itself through the Son so completely . . . everything also returns to him, reascends to him, recapitulates itself in him in the Spirit. The whole divine life is nothing other than Love given eternally, or rather giving itself, but this love lives only in the interchange by which everything flows from the Father through the Son and flows back to him in the Spirit.12

  Michael Aksionov Meerson summarizes the answer of Orthodox theologian Sergei Bulgakov to the filioque controversy as follows:

  The Spirit proceeds from the Father neither in general, nor because of some metaphysical necessity, but as the hypostatic movement of love. Thus “the first movement of the Spirit who proceeds from the Father is upon the Son, as the hypostatic love of the Father.” But the second movement of the Spirit as the hypostatic love of the Son for the Father is “from the Son to the Father.” Thus the eternal “circular movement of the Spirit from the Father by the Son, or in other words, from the Father and the Son,” is completed. Since the procession of the Spirit, as with all relations in the Holy Trinity, is timeless, which is to say, without beginning or end, both formulas are justified if taken together.13

  Finally, Roman Catholic theologian Francois-Xavier Durrwell gives this succinct formulation: “The Tri-unity has two poles, the Father and the Son, and the eternal movement goes from one to the other: the Spirit is this movement which encompasses and unites them.”14

  In Search of the Logos

  These attempts by theologians to describe the Christian Trinity almost require us to move from the realm of number to that of shape—from arithmetic to geometry. Number alone is inadequate to define relationship. We have to recall that mathematics begins not just with counting, but also with measuring. To perform an act of measuring one has to do two things. One has to count, but one also has to compare one thing with another (for example a ruler or measuring tape). These two functions are distinct and fundamental, for counting has reference to pure quantity, whereas measurement is concerned with the realm of extension, determined by form or “quality.” Geometry, then, is more than counting. It has to do with a world of more than one dimension, a world that has shape and form.

  In that world, we can try to draw a diagram to capture some aspects of the doctrine of the Trinity. The simplest involves a triangle to show the relationship of the three Persons in God, thus:

  Here we can see the Son and the Holy Spirit coming from the Father. The dotted line represents the Spirit coming also from the Son—this is the filioque relationship disputed by the Orthodox. One weakness of the diagram is that there is no indication of any essential difference between the Son and the Spirit. Each is represented by a straight line originating in the Father.

  More sophistication can be achieved by combining the straight line with a circle to capture the difference between the Persons: the fact that the Son is “generated” but the Spirit simply “proceeds” or is “spirated” (breathed). Here the Father is represented by a point, the Son by a point extended to form a line, and the Holy Spirit by the circle that joins them together.

  The simple figure of a circle bisected by a straight line is often used to represent the primordial Dyad, the archetypal number Two produced from the One by the act of dividing into two halves. In nature it is seen most clearly in the first division of a fertilized cell.15 We use it here to represent each divine Person by a different kind of movement. The Father is stillness, represented by the point. The Son is linear motion from the Father. The Holy Spirit is circular motion.16

  So the Father breathes forth the Holy Spirit as a movement that by definition returns to him. The Spirit traverses the same “distance” as the Son (the distance from Self to Other, thus constituting a third, distinct Person), but in a different way, represented by the circle. In a sense it is the Spirit who brings the Son back to the Father, in a communion of love that overcomes without destroying the distance of personal distinction. (It might equally be said that the Spirit leads the Son away from the Father before leading him back.) At any rate, the Spirit is one with the Son at his own most extreme departure from the Father, before the curve starts to bend again toward its Origin. The Father, of course, remains “prior” both to Son and to Spirit—not in a temporal sense, but in the sense that the processions are rooted in him alone, as source and fount of the Holy Trinity.

  Thus the diagram helps us “encode” the statements that various theologians of both East and West have wanted to make about the Blessed Trinity:

  the Father is the unbegotten Source of Son and Spirit,

  the Spirit originates from the Father as sole principle,

  the Spirit proceeds from the Father,

  the Spirit proceeds from the Father and the Son,

  the Spirit proceeds from the Father through the Son,

  the Son returns the love of the Father in the Spirit,

  the Son is begotten by the Father in the Spirit.

  Geometry as Prophecy

  One twentieth-century writer who was adept at translating theology into geometry is Simone Weil, a skilled mathematician (and sister of one of the greatest mathematicians of the twentieth century) as well as a profound religious thinker. Among all those who have studied Pythagorean geometry, she more than any recognized that it is marked deeply by the Trinity, for its central idea is that of mediation (metaxu), which she identifies with the Logos (Son). Her approach—introduced, explored, and ably defended by Vance G. Morgan in Weaving the World: Simone Weil on Science, Mathematics, and Love—will take us a stage further in the unfolding of symbolic geometry.

  For the Pythagoreans, Weil noted, harmony is based on a geometrical mean that establishes the proportion between two different elements. If the two elements are ratios (like musical intervals) that share a common term, they can be fitted together. So, for example, in A/B = B/C, “B” is the mediator or “proportional mean” between the ratios. To use a concrete example, a grandfather is to a father what a father is to his son (where the mediator in this case is the father). Medieval philosophers accounted for all knowledge on the basis of a similar analogy: they said that “knower is to knowing what knowing is to known.” A theologian might also say that “God is to Christ what Christ is to man,” where the mediator is Christ, who is both divine and human. Weil writes: “‘As my Father has sent me, even so I send you, etc.’ A single relationship unites the Father to Christ, Christ to his disciples. Christ is the proportional mean between God and the saints.”17

  Thus to Weil, Greek geometry seemed like “the most dazzling of all the prophecies which foretold the Christ.”18 One recent biographer remarks,

  That most of her contemporary readers will find it difficult to follow her in this, also shows why Weil’s reflections on the Pythagoreans are so pertinent to our own times. Apart from telling us something about the genealogy of Christian thought, they invite us to assess the depth of our own conception of the cosmos, show how the loss of a certain perspective may lead to religious belief that is spiritually impoverished, and challenge us to think about ways in which at least a part of that perspective may be regained.19

  Weil notes that just as mediation from a domain other than number per se, namely geometry, is needed to reconcile incommensurable magnitudes in mathematics, so mediation by a divine Other is needed to reconcile contrary human beings in the world of human relationships: “It is impossible for two human beings to be one while scrupulously respecting the distance that separates them, unless God is present in each of them.”20 For every human being is a self, face to face with others who are also selves. Each of us is the
unique center of our own world, an “I.” This gulf of absolute otherness can only be overcome with the help of a Third Person, who is the Spirit of love.21

  In each of the three relationships indicated by the word friendship, God is always mediator. He is mediator between himself and himself. He is mediator between himself and man. He is mediator between one man and another. God is essentially mediation. God is the unique principle of harmony. That is why song is appropriate for his praise.22

  The figure of a circular movement bisected by a straight line is probably the most perfect representation we could find of harmony as conceived by the Pythagoreans and raised to its highest power in the Christian Trinity: maximum distance combined with maximum unity between contraries. For Simone Weil, the circle represents God, and the diameter creation, or the “distance” between the divine Persons, within which all things exist: “This universe where we are living, and of which we form a minute particle, is the distance put by the divine Love between God and God. We are a point in this distance. Space, time, and the mechanism that governs matter are the distance.”23

  She then introduces a right-angled triangle into the circle, using the diameter as the hypotenuse (longest side) of the triangle, whose corner lies on the perimeter of the circle. In fact if you project any triangle from the diameter of a circle to its circumference, the largest angle will always be 90 degrees. (Another way of putting this is that a circle is made up of the apexes of an infinite series of right-angled triangles whose hypotenuse is the diameter of the circle.)

  Of course, a right-angled triangle has a special significance for the Pythagoreans, as we know from the famous theorem proving that the square on the hypotenuse is equal to the sum of the squares on the other two sides. This implies that the mean of two quantities can always be derived by representing them as the two sides of a right angle. But it is also the case (as Thales showed even before Pythagoras) that the perpendicular line drawn from a right angle touching the circumference back to the hypotenuse will always equal the mean proportional between the segments into which it divides the diameter.