5. Leclercq 1978, 170–71.
6. Huerre 1994, 88. There is a classic expression of the tradition in St. Bonaventure: “When the soul by faith believes in Christ as in the uncreated Word, who is the Word and the brightness of the Father, she recovers her spiritual hearing and sight, hearing to receive the words of Christ, and sight to view the splendours of that Light. When the soul longs with hope to receive the inspired Word, she recovers, because of her desire and affection, the spiritual sense of smell. When she embraces with love the Incarnate Word, inasmuch as she receives delight from him and passes over to him in ecstatic love, she recovers her sense of taste and touch. Having recovered the spiritual senses, the soul now sees, hears, smells, tastes and embraces her beloved, and can sing as a bride the Canticle of Canticles, which was composed for the exercise of contemplation” (Bonaventure 1956, 75).
7. Coleridge 1983, 304.
8. Ward 2008, 237.
9. To Coleridge’s primary and secondary Imagination Tolkien added the category “Art,” the ability to give to a collection of images the inner consistency of reality, and its highest expression in “Fantasy,” where we come closest to imitating God as the creator of worlds. For more on the imaginative grasp of truth particularly in Tolkien, see Caldecott 2005.
10. Maritain 1954, 126, 131–41. See also 123n17.
11. Lewis 1947, 89–90. Cf. Goethe’s philosophy of science as described in Bortoft 1996.
12. Taylor 1998, 49.
13. A “cosmology” in modern science is an account of the universe and its structure. The word may equally be used of ancient and traditional societies, provided we remember that the structure in question was not exclusively a physical one.
14. Taylor 1989, 379.
15. Schindler 2004, chap. 3. Schindler explains how gestalt is close in meaning to, but not exactly the same as, the Platonic eidos, Aristotelian morphe, or Scholastic forma.
16. See, e.g., the discussion of analogy in Cunningham 2002, 181–89.
17. The phrase is from Schindler 2006, 528, an article that eloquently defends Plato from the accusation that he was a dualist who despised the body.
18. Guénon 2001c, 9–10.
19. Pope John Paul II offers a profound modern reading of this “language of the body” and specifically the language of gender as expressing our calling to interpersonal communion and self-gift. In this we are marked by the image of God as Trinity. He shows further how theology becomes pedagogy: the gendered body is assigned to man as a task and therefore a spirituality. The development of our biological knowledge at the expense of our consciousness of the body as a “sign of the person, as a manifestation of the spirit,” plays into the hands of a dualism that alienates us from our own bodies and reduces them to property (John Paul II 2006, 360–61).
20. They were not the only or exclusive ways that grace could reach men, even if many Christians naturally assumed that they were. God could not be bound by his sacraments—hence the possibility that good pagans could be saved, provided they had not knowingly closed their hearts to whatever grace was offered them in their own lives and traditions.
3
The Lost Wisdom of the World
But thou hast arranged all things by measure and number and weight.
Wisdom of Solomon 11:20
That you, being rooted and grounded in love, may have power to comprehend with all the saints what is the breadth and length and height and depth.
Ephesians 3:17–18
The four disciplines of the quadrivium—arithmetic, music, geometry, and astronomy—had one thing in common: they were based in mathematics. Music was the expression of numerical harmonies in time, geometry the exploration of relationships in space, and so on. The assumption of this system of education was that by learning to understand the harmonies of the cosmos, our minds would be raised toward God, in whom we could find the unity from which all these harmonies derive: Dante’s “love that moves the sun and the other stars.” Thus the quadrivium would prepare the ground for the study of the highest contemplative sciences: philosophy and theology.
The idea that the cosmos is built on mathematical harmonies, and that numbers themselves can be a path to God, flowed from Pythagoras and Plato down to the Middle Ages, where it influenced the cathedral builders and later the artists of the Italian Renaissance.1 It was also one of the essential factors in the birth of science, as we shall see in more detail later, and it continues to influence and intrigue physicists today. Werner Heisenberg writes that “modern physics has definitely decided in favour of Plato. In fact the smallest units of matter are not physical objects in the ordinary sense; they are forms and ideas which can be expressed unambiguously only in mathematical language.”2
In 1937, another great modern scientist, Paul Dirac, speculated that “the ancient dream of philosophers to connect all Nature with the properties of whole numbers will some day be realized.”3 Heinrich Hertz, the German physicist who demonstrated the existence of electromagnetic radiation in 1888, provides further evidence of the Pythagorean spirit in modern science when he comments: “One cannot escape the feeling that these mathematical formulae have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than we originally put into them.”4
Michael S. Schneider writes:
Numbers are a map of the beautiful order of the universe, the plan by which the divine Architect transformed undifferentiated Chaos into orderly Cosmos. Cultures didn’t necessarily learn this from each other but only had to look at numbers and their relationships to see how they revealed harmonious models which are the same everywhere and at all times.5
Yet our present education tends to eliminate the contemplative or qualitative dimension of mathematics altogether, reducing everything to sheer quantity. Mathematics is regarded as a form of logical notation, a mental tool with no relation to truth except the fact that it assists us in manipulating the world. This elimination of the symbolic dimension of mathematics is largely responsible for the divorce of science from religion, and art from science. But rather than continue arguing that case in the abstract, I want to immerse us in an alternative vision of mathematics. Let us learn for ourselves the beauty to be found in this world of patterns and relationships.
Sacred Number
There is something mysterious about numbers. One of the greatest mathematical geniuses of all time, Srinivasa Ramanujan (d. 1920), was able to see the solutions to complex mathematical problems without needing to work them out. He was a devout Hindu and attributed his insights to revelation. He is famous for saying “An equation for me has no meaning, unless it represents a thought of God.” Even a materialist must regard numbers as part of “reality,” while puzzling over the fact that they are made neither of matter nor energy. He can’t observe them through any scientific instrument, yet they somehow underpin everything we see around us and help to define its form.
The Pythagoreans regarded each number as an expression or facet of Unity (the Father of all things) projected through Duality (the Mother) to create multiplicity.6 The multifaceted nature of number “appears most clearly when one transposes each number into its corresponding geometrical form: three into an equilateral triangle, four into a square, five into a regular pentagon, etc. In each of these figures, innumerable relationships appear, which multifariously throw light on the inner law proper to the figure in question.”7
The Pythagorean-Platonic tradition at the core of the Liberal Arts developed the following symbolic associations with the natural numbers:8
One
The Unity of being, transcending all that exists. It is often represented by a circle, or else by a point. One is the number that when “squared,” i.e., multiplied by itself, produces itself. Symbolically, One is not the first in a series of numbers, but the number-beyond-number that includes all others, equivalent in that sense to the modern conception of infinity. There are circular windows in Got
hic cathedrals that symbolize this unity, for example at Chartres, Notre Dame, and Rheims.
Two
If one is the source and archetype of Unity, two is the beginning of Diversity. It represents polarity and division, and also feminine receptivity and fruitfulness. In a Christian context it often signifies the separation of matter and spirit. Duality can also symbolize the beginning of the process of creation, which in the book of Genesis is described as taking place through a series of separations or polarizations (heavens and earth, light and dark, etc.). The division of Adam’s unity into duality gave us male and female. Geometrically duality appears as a line between two points, or else as the central point and circumference of a circle, joined by a radius.
Three
Unity and Diversity are reconciled in Harmony. The triangle that is the geometrical translation of the fundamental musical chord with its three notes is called a Triad. It presents a graphic image of how the number three returns polarity to unity. The number Three was often regarded as the first real number, after the two principles that were the source of number. Plato thought the particles of all natural elements were constructed from triangles (see below). An equilateral triangle is the simplest shape that can be repeated on a two-dimensional plane without leaving any space (just as the tetrahedron can be stacked in three dimensions without leaving space between). As such it is necessarily one of the fundamental building blocks of both art and cosmos. The world and man—macrocosm and microcosm—were both held to be of a threefold nature (matter, soul, spirit). Naturally in the Christian tradition the mark of the Trinity is everywhere. An equilateral triangle inside a circle is sometimes used to suggest the one God in three persons.
Four
Quaternity, expressed as a square, the first “solid” number, represents the earth, or the entire material plane. For the ancients, the world below the stars was composed of four elements: earth, water, air, and fire (perhaps equivalent in modern physics to the four basic states of matter: solid, liquid, gas, and plasma). Earth and fire were opposed to each other (contraction vs. expansion, solidity vs. radiance) while water and air mediated between them. According to Aristotle, there were four types of causation (end, agent, matter, and form). In Christian tradition, there are four rivers of paradise, four senses of scripture,9 and four evangelists whose foundational writings are spread throughout the four corners of the world.
Five
Five is the marriage of two and three, even and odd, female and male, the mid-point of the Decad, associated with the human body as center of the world. The body has five senses, and five fingers on each hand. Whereas inanimate nature conforms to the order of four- and six-fold symmetry, as in crystal structures, living nature appears to be ordered more often by fives. Five is manifested geometrically as a regular pentagon or a five-pointed star, perhaps more often used in Islamic art than Christian. The pentatonic scale in music is represented by the black keys on a piano. Five has a particularly close relationship to the golden ratio, as we shall see later.
Excursus: The Five Platonic Solids
If you start with a point representing one, extend it into a line representing two, then swing one of the two ends of the line around the other to create a circle representing three, and finally rotate the circle to make a sphere representing four, you have the simplest possible representation of the dimensions of space. It so happens that only five geometrical figures with identical plane faces and equal edges can be made to fit exactly within a sphere. These figures therefore symbolize the complete potentialities of three-dimensional space. They are known as the Platonic Solids, though judging from artifacts in Oxford’s Ashmolean museum dating back 4000 years they were discovered long before the Greek philosophers donned their sandals.
Plato hypothesized in the Timaeus that the world is made not just of four but of five elements in various combinations. Particles of each element are formed in the shape of one of the five solids—Cube (Earth), Icosahedron (Water), Octahedron (Air), Tetrahedron (Fire), and finally Dodecahedron. The fifth element (to which Aristotle gave the name Aether, though he was uninterested in Plato’s geometrical hypothesis) is what God uses to “embroider” the sky with stars.10 (The Aether was in a sense disproved by the Michelson-Morley experiment in 1887, but later tacitly reinstated by Einstein under the new guise of the space-time continuum itself.)
Six
Six is called a “perfect” number because it is the sum as well as the product of its divisors one, two, and three.11 It is represented geometrically as a regular hexagon, or by the Star of David also known as the Seal of Solomon (two superimposed triangles, one pointing up, the other down). Comprised of six equilateral triangles, the hexagon is the third shape that can be repeated in a plane without leaving any space. If equal sized spheres are allowed to stack along any plane they will form patterns of either six-fold or four-fold symmetry. It is the number of creation (six days), and of the four plane directions plus up and down.
Seven
The days of creation (six days of work and one of rest) are reflected in the seven days of the week. This number appears many times in scripture, being a particular favorite of St. John, for whom it is the number of the Holy Spirit. There are seven sacraments, seven gifts of the Holy Spirit, seven deadly sins, and seven virtues in the tradition of the Catholic Church. Seven represents totality, since it is the sum of four (the number of the material order) and three (the number of the Trinity). Philo calls it the “birthday of the world.”12 It is the largest prime within the Decad—known as the “Virgin” number because it alone does not give rise to any other number between one and ten by multiplication or division. As the combination of a square and a triangle it often appears in the composition of both iconographic and Western naturalistic art. In Jewish tradition it is represented by the seven-branched candlestick or menorah. According to the book of Tobit (12:15), Raphael is one of the “seven holy angels who present the prayers of the saints and enter into the presence of the glory of the Holy One.” An eleventh-century hymn which relates the ratios of the plants to musical tones tells us that “The nucleus (nodus) of the universe is the number seven.”13
Eight
Since the Sabbath or seventh day of creation brought the natural world to completion, that day on which God became man in the womb of Mary to remake the world from the inside is sometimes called the “eighth day.” Mary may be identified with the perfection of nature and therefore with the Sabbath, and her son Jesus with the eighth. Similarly the number eight may be identified with the day of Christ’s resurrection from the dead, which took place according to the Gospels on the day after the Jewish Sabbath (in this way Sunday, not Saturday, becomes the first day of the Christian week). Eight is represented geometrically as a doubled square, or as a regular octagon. So, for example, the baptistry of the Duomo in Florence is an octagonal building, as are many such buildings throughout Christendom. In a major scale the eighth note, the octave, marks a completion of the scale, a repetition of the keynote, and signals movement to a new level.
Nine
As the product of three multiplied by itself, and at the same time the sum of three threes, nine echoes three (for example, the musical triad), indicating the impress of the Trinity on creation. It features most famously in the nine choirs of angels in the celestial hierarchy of the fifth-century monk who purports to be a convert of St. Paul, Dionysius the Areopagite. There are nine eleisons in the traditional Mass. The seven heavens of Babylonian religion, each associated with one of the moving heavenly bodies or “planets” (Sun, Moon, Mercury, Venus, Mars, Jupiter, Saturn), become nine with the addition of the sphere of the fixed stars and, above that, God’s eternity.
Ten
There were ten Commandments given to Moses and ten Sephiroth or archetypes from which the world is created, according to Jewish mysticism. The fact that we have ten fingers made it the natural basis for counting systems around the world. The sacred Tetractys of the Pythagoreans is the sum of the first four numb
ers, represented by ten points in an equilateral triangle: one over two over three over four—the “four lettered name” of God equivalent to the Jewish “JHVH.” It can be used to construct many different geometrical forms in two or three dimensions, including the five Platonic Solids, and the basic harmonies of music.14
Here it is: the ancient “theory of everything,” a mathematical representation of God:
Twelve
Outside the Decad, twelve is probably the most richly symbolic of numbers. The product of three and four, it is often associated with their sum, seven. There are twelve lunar months in the year, and the heavens have been divided into twelve signs of the zodiac, or major constellations along the path of the sun. The heavenly Jerusalem glimpsed at the very end of the Christian Bible has twelve gates corresponding to the twelve tribes of Israel and the twelve apostles of the Lamb.
The intrinsic relationships of 3, 4, 6, 7, 10, and 12 are all beautifully manifested in one of the simplest geometrical forms, composed of nineteen equally sized circles or circular pebbles. Place one circle at the center, and six will fit around it, each touching the others—recalling the six days of creation and one of rest in Genesis. Around that circle of circles, only another circle of twelve circles will fit, the whole six-sided figure containing numerous figures including the Tetraktys itself.
Cardinal Ratzinger explores the biblical account of creation and discovers there the concept of sacred number. For, he says,
the biblical creation account is marked by numbers that reproduce not the mathematical structure of the universe but the inner design of its fabric, so to say, or rather the idea according to which it was constructed. There the numbers three, four, seven, and ten dominate. The words “God said” appear ten times in the creation account. In this way the creation narrative anticipates the Ten Commandments. This makes us realize that these Ten Commandments are, as it were, an echo of the creation; they are not arbitrary inventions for the purpose of erecting barriers to human freedom but signs pointing to the spirit, the language, and the meaning of creation; they are a translation of the language of the universe, a translation of God’s logic, which constructed the universe.15
Beauty for Truth's Sake Page 6