The Quadrivium and the Character of God
By Greg Wilbur
From the Winter 2017 Issue
Sometimes when reading or studying or listening, I catch a glimpse of ideas and connections beyond my current understanding. At times these ideas are nascent and ephemeral—mists that have not yet solidified (if they ever will). It is as if figures are coming together on the periphery of my vision, and I fear that if I look at them directly, the forms will dissolve into vaporous ribbons and float away. Some passages of scripture are like that—for a moment I can grasp a spiritual truth or significance that swells beyond my comprehension. I am encouraged by the fact that even the Apostle Peter said that Paul was hard to understand.
But sometimes a convergence of thought, words, Spirit, conversation, circumstances, and time solidifies those periphery shapes into concrete understanding—into wisdom that causes a change of heart, habit, or practice.
I had this experience when I first read Flannery O’Connor. I knew she was doing something, but I could not quite understand. I could tell that her stories were packed with symbols and ideas, but I failed to grasp the sacramental significance. By reading and re-reading the stories, reading O’Connor about her stories, and reading and listening to others talk about her stories, I began to see and to recognize what she was doing, what she had created, what she had to say, and how she said it.
Flannery O’Connor insisted that stories are more than just narrative. In Mystery and Manners she wrote, “People have a habit of saying, ‘What is the theme of your story?’ and they expect you to give them a statement: ‘The theme of my story is the economic pressure of the machines on the middle class’ or some other such absurdity. And when they’ve got a statement like that, they go off happy and feel like it is no longer necessary to read the story.” (i) Her point is that how a story is told is integral to the entirety of the art. Story cannot be reduced simply to theme or narrative.
In our mechanistic, reductionist age, too often the grand story of the universe—of all creation—gets reduced to bits and parts, atoms and quarks, facts and factoids. We miss the beauty and wonder of the story and fail to see and recognize what the Creator is doing, what He created, what He has to say, and how He has said it. We are too easily satisfied with the cosmic themes, and we “go off happy and feel like it is no longer necessary” to read, to comprehend, to study, to truly know the handiwork of the Creator or to see His glory declared in that which He has made.
But this is a problem the Quadrivium can help us solve.
Historically, students studied the Quadrivium in order to explore the fabric of the cosmos. The language of this discovery is number—the meaning of number, numbers in relation, numerical relations in time, temporal numerical relations in space. Or, Arithmetic, Geometry, Harmonia, and Cosmology.
Michael S. Schneider writes that “Numbers are a map of the beautiful order of the universe, the plan by which the divine Architect transformed undifferentiated Chaos into orderly Cosmos.” (ii) As C.S. Lewis writes in The Discarded Image, the medieval understanding of the world, as inherited from the ancients and translated through the lens of Christianity, assumes a divine hand at work in an ordered world that is imbued with meaning. He describes it as “tingling with anthropomorphic life, dancing, ceremonial, a festival not a machine.”
However, Christians are not immune to falling into the kind of mechanistic approach of which Lewis warns. In Redeeming Mathematics, Vern Poythress writes, “Christians have sometimes adopted an unbiblical concept of God that moves him one step out of the way of our ordinary affairs. We ourselves may think of ‘scientific law’ or ‘natural law’ or mathematics as a kind of cosmic mechanism or impersonal clockwork that runs the world most of the time, while God is on vacation. God comes and acts only rarely through miracle. But this is not biblical. ‘You cause the grass to grow for the livestock’ (Ps. 104: 14). ‘He gives snow like wool’ (Ps. 147: 16). Let us not forget it.” (iii)
If we think of the cosmos from a mechanistic starting point, this changes the very nature, purpose, and process of discovery by focusing on disconnected pieces. However, by looking at the universe as interconnected and structured, we find its root in the creative order of an Almighty God who made the heavens and the earth. For thousands of years, this was the dominant idea and foundation of intellectual and theological thought. Since God created an orderly world, by studying this world through the means of number, we would not only learn more about the creation but we would learn about the Creator.
The Liberal Arts Tradition by Kevin Clark and Ravi Scott Jain is a helpful and comprehensive approach to classical education that places the Quadrivium in the broader context of a fully realized vision for Christian Classical education. They write that the original role of the Quadrivium was “to lead the mind to the realm of eternal and unchanging truths,” but that was “eventually displaced by the amazing power of mathematics to describe the physical world.” (iv) Both are necessary: “the useful and the formative.” (v)
However, as Clark and Jain remind us, Plato speaks in The Republic about the importance of the formative aspect of mathematics above its usefulness. “Plato believed that the study of mathematics leads the mind toward pure reason and cultivates the true love of wisdom (the origin of the term philosophy). By training one’s thoughts on the perfections of mathematics, the mind learns to transcend the level of changing opinions to identify objective truth.” This wisdom and truth is found in none other but God Himself.
“Thou hast ordered all things in measure and number and weight.” —Wisdom of Solomon
Most, if not all, of our study of arithmetic involves the use of Arabic numerals: 1, 2, 3, 4, and so on. As such, our minds more naturally gravitate towards the use of numbers, their function, and utility. In Redeeming Mathematics, Poythress explains multiple doctrines of the character of God based on simple arithmetic by analyzing 2+2=4. The truth/existence of that basic addition problem is immaterial and invisible, transcendent and immanent, rational, personal, beautiful, and displays righteousness (“arithmetical laws lead to consequences”). (vii)
Calvin has 4 apples and throws 2 of them at Susie Derkins. How many apples does he have left? 4-2=2
Simple, straightforward, and helpful, these laws of arithmetic ignite the philosophical and theological discussion that Poythress expresses. However, this word problem fails to address the concept of “two” (or “two-ness” or the dyad), the meaning of “four” (or the tetrad), or the relationship between them. The ancient Greeks’ use of pips and symbols allows for contemplation of a different idea of number.
The monad (one), expressed this way , signifies stability, essence, foundation, and unity. All numbers are related to the monad because it forms a part of all other numbers. The triad suddenly has a different connotation than “3” does. Unlike the Arabic numeral, the connection within the number itself becomes more apparent—each pip is related to the other two pips individually and collectively; the composition of the triad is the combination of the monad and the dyad; the presence of the monad itself; the aspect of tri-unity or trinity becomes more clear.
In addition (pun intended), arithmetic and the meaning of numbers present the philosophical question of the one and the many. For Aristotle, this line of thought caused him to realize that something could be one in essence and quality. By recognizing the unity and diversity inherent in the world, theologians directed their thinking to the unity and diversity of God as Trinity.
Clark and Jain explain this connection in that:
“the early Church fathers elaborated the doctrine of God’s Trinitarian nature as the true foundational mystery grounding the problem of the one and the many. How God can be one nature and three persons is certainly inscrutable, though demonstrated as true in Christ. Augustine then recognized that if the Creator were a Trinity, creation might bear that imprint. Exploring the vestige trinitatis, the marks of the Trinity upon all created reality, then, suggests a path connecting this ancient Quadrivial tradition, not only to wisdom, but to worship as well . . . Arithmetic then offers a timeless introduction to the Quadrivium.”
“Art is an experience of balance, of the relationship of its parts to the whole. Perceiving it as anything else is missing its most fundamental component. A fine painting, a piece of sculpture, a work of architecture, music, prose, or poetry is organized and gracefully balanced around a hidden sense of proportion.” —Priva Hemenway
Geometry consists of numbers in relation or proportion to one another. One of the beauties of Euclid’s Elements consists in immersing a student into ratios, lines, shapes, and angles instead of just solving numerical equations. The logical progression and principles of discovery in Euclid can apply to any particular number, but the ideas behind them have deeper significance. He begins with the definition of a point—“that which has no part.” A point occupies no space and no dimension. It has no length, height, or breadth.
Two points connected create a line, or, in Euclid’s words, “breadthless length.” With two points there is no height or breadth but there is a dimension of length.
Three points connected create a triangle in two dimensions with length and height. A suspended, connected point over the middle of the triangle creates breadth and, thus, the third dimension (a tetrahedron).
Paul gives us a bit of a geometry lesson in Ephesians 3:17-19: “So that Christ may dwell in your hearts through faith—that you, being rooted and grounded in love, may have strength to comprehend with all the saints what is the breadth and length and height and depth, and to know the love of Christ that surpasses knowledge, that you may be filled with all the fullness of God.” Thus Paul uses a geometrical image to express the love of Christ.
With regards to biblical typology, understanding ratios, proportions, and relationships significantly increases our ability to comprehend the symbols inherent in Scripture. There is correspondence between the Garden of Eden, the tabernacle, Solomon’s temple, the structure of the cosmos, and heaven itself. As Poythress puts it, “The tabernacle of Moses and the temple of Solomon accordingly have symbolism that has affinities with the creation as a whole, and in particular with heaven as the dwelling place of God.” (viii) He goes on to say that, “The simple proportionalities belong to the small house, which is an image of heaven and in fact of the whole universe as the large house filled by God’s presence (Jer. 23: 24; compare 1 Kings 8: 27). The fact that the small house is a copy or image of the big house suggests that the big house may also display harmonious proportionalities. And indeed this turns out to be true, as the mathematical character of basic physical laws attests.” (ix)
God designed the cosmos and gave instructions for the building of His earthly habitation to imitate in proportion His heavenly dwelling. There is a reason for the created order that reflects divine design. In her essay, “The Pythagorean Doctrine,” Simone Weil illustrates this point by saying, “In a general way, and in the widest sense, mathematics, including under this name all rigorous and pure theoretical study of necessary relationships, constitutes at once the unique knowledge of the material universe wherein we exist and the clearest reflection of divine truths . . . It is this same mathematics which is first, before all, a sort of mystical poem composed by God himself.” Through the study of the Greeks and geometry, Weil places great significance in the idea of Divine mediation as demonstrated through geometrical proofs of circles and right triangles.
Which leads into the study of music and sound.
“We shall therefore borrow all our Rules for the Finishing of Proportions, from the Musicians, who are the greatest Masters of this Sort of Numbers, and from those Things wherein Nature shows herself most excellent and compleat.” —Leon Battista Alberti, 15th Italian architect, poet, and Renaissance man
As opposed to treating this discipline in the Quadrivium as simply music theory or music instruction, using the term Harmonia opens up this study to musica humana and musica mundana—the music of man and the music of the spheres. Harmonia is the study of harmonious relationships that reflect the biblical idea of shalom—peace, rightness, concord, or the way things are supposed to be. This idea of shalom operates on multiple levels—within a person (in their physical, emotional, and spiritual well-being) as well as within relationships, organizations, society, or the movement of the heavenly spheres. We use this language all the time when we say that our car needs a tune-up, that an eco-system is balanced, or that “The man that hath no music in himself,/Nor is not moved with concord of sweet sounds,/Is fit for treasons, stratagems and spoils (Merchant of Venice, Act V, Scene 1).”
Augustine put it this way:
The peace of the body, therefore, lies in the balanced ordering of its parts; the peace of the irrational soul lies in the rightly ordered disposition of the appetites; the peace of the rational soul lies in the rightly ordered relationship of cognition and action; the peace of the body and soul lies in the rightly ordered life and health of a living creature; peace between mortal man and God is an ordered obedience, in faith, under an eternal law; and peace between men is an ordered agreement of mind with mind. The peace of a household is an ordered concord, with respect to command and obedience, of those who dwell together; the peace of a city is an ordered concord, with respect to command and obedience, of the citizens; and the peace of the Heavenly City is a perfectly ordered and perfectly harmonious fellowship in the enjoyment of God, and of one another in God. The peace of all things lies in the tranquility of order, and order is the disposition of equal and unequal things in such a way as to give to each its proper place. (xi)
Harmonia can best be understood, illustrated, heard, and experienced through the means of music. Even non-musicians have an innate sense of the movement of a melody and when it comes to a place of rest. They can recognize something that is discordant or harmonious. They can anticipate musical order and how music fulfills or plays against that ideal expectation.
Reality is divided between the way things were meant to be and the way things are. We catch a vision of that in Genesis 1-2 with a world created good—where Adam and Eve have harmonious relationship with God, each other, and the surrounding world. This is shalom: the way things were meant to be. Sin wrecked that order and added a discordant tuning to that which God had made and thus reflects the way things are in our broken world.
Steven Guthrie discusses that idea by including the concept of ratio and the disruption that sin causes:
The fundamental human dilemma is that by aspiring to equality with God (Genesis 3:5) humanity has abandoned its well-ordered place within God’s Song of the Universe [carmen universitatis]. There has been a universal loss of ratio. We no longer stand in right relation to God. We no longer stand in right relation to the non-human creation (“Cursed is the ground because of you, through painful toil you will eat of it.” Genesis 3: 17), and we no longer stand in right relation to one another. More than this, we no longer stand in right relation to ourselves. (And so Paul for instance can speak of the members of his body waging war against the law of his mind, making him a prisoner of his own members. (Romans 7:23).) As Augustine recognizes, though body and soul are part of a differentiated unity, our experience in this life is often that of a hostile and mutually antagonistic plurality: body and mind, affections, appetites, and imaginations, resisting, refusing, and tormenting one another. In each of these arenas, we see that Satan is the Father of separation; that the dynamic of sin is division; that the hallmark of corruption is the distortion of relation. (xii)
The study of music and harmonia instructs us to identify the ideal, the discordant, the reality of sin, and the need for redemption. As Christ brings reconciliation and makes peace by the blood of His cross, He does so as the image of God by whom all things were created and in whom all things hold together (Col 1:15-20). Christ brings order and harmony to the cosmos.
“Just as when one hears from afar a lyre, made up of many different strings, and wonders at their harmonious symphony, that not only the low one produces a sound, not only the high one, and not only the middle one, but all sound together in balanced tension, and one concludes from all this that the lyre neither operates by itself nor is played by many, but rather that there is one musician who by his art blends the sound of each string into a harmonious symphony—even though one fails to see him—so too, since there is an entirely harmonious order in the world as a whole, without things being at odds with those below, and those below with those above, but one completed order of all; it follows that we know there is one leader and king of all creation, not many, who illuminates and moves everything with his own light.” —Athanasius, Oratio contra gentes 38
Cosmology is not only the study of astronomy but also of the great cosmic ordering sustained by Christ through the word of His power. Clark and Jain explain that, “While grounded in wonder, respecting work, and leading to wisdom, natural philosophy also situated its endeavor in the context of worship. The entire medieval vision of reality was conceived from the perspective of an ordered unity—a cosmos comprised of both whole and parts. While the ancient Pre-Socratics and Pythagoreans developed the concept of an ordered cosmos, the medievals understood it as creation ordered by Christ, the logos, under the creative impulse of God.” (xiii)
Steven Guthrie sums up this idea:
“In fact (as the Pythagorean tradition acknowledges), the very acoustical and harmonic foundations of music reflect principles of mathematical proportion and balance—principles expressed in every aspect of the universe, animate and inanimate. While one of the purposes of the cosmos is the creation of music, the ultimate purpose of all things, including music, is the glory of God. It is this common telos which unifies creation and invests every act and object with purpose and meaning. All things are good and significant, because they all may contribute to the Highest Good. Additionally, there is unity to the cosmos, because all things have been created for one purpose and end.” (xiv)
Thus, the movement of the spheres expresses the beauty of The Great Dance—the interplay, motion, direction, and order of a Divinely orchestrated dance in which every step and every dancer, no matter how minor or periphery, contribute to the overall beauty and purpose of the whole. The most beautiful things are those most reflecting of God (xv). Along with the saints and angels, we are called into a gallery to behold the glory of God on display. The heavens declare the glory and the beauty and the splendor and the infinite variety of God, and we are called to dance in that creation.
Christ is glorified as the Word of God—everything in existence was called forth by His word, is held together by His word, and is itself the communication of God: in number, in space and relationship, in harmonia and the music of the spheres, in time and space. God is all in all: sovereign over all, creating all, employing all for His glory and beauty. Arithmetic points us to the Trinity; Geometry shows the need for mediation; Harmonia displays redemption and reconciling peace; Cosmology calls us to move in sanctifying submission to the steps of the Lord of the Dance.
Dante traveled through the seven spheres of heaven from sun and moon and planets and stars to encounter the living light of Divine Love “which moves the sun and the other stars.” The Unified Theory of Everything is Christ and the Word of His power. As we begin to apprehend this with knowledge, we are led to wonder and awe and worship.
Returning to Ephesians 3:17-19, Paul calls us to be rooted and grounded in love, to have the strength to comprehend the breadth and length and height and depth, to know the love of Christ, and to be filled with the fullness of God. A line, a triangle, a tetrahedron—from one dimension to three dimensions. But Paul doesn’t stop there: by adding depth he moves us into the fourth dimension—to new discovery and to greater fullness of Divine Love.
The purpose of education is love—to see God in such a way as to love Him—to see the King in His beauty, the craftsman in His work, and the poet in His theme. The forms that flutter at the edge of our sight are the workings of Divine Love. This is Aristotle’s unmoved mover—no desire, no lack—yet He sets all else in motion, and our hearts are restless until they find their rest (their peace, their shalom) in Him.