In Defense of the Quadrivium
Roots and Reason
This summer, all across the country, homeschool educators are attending Classical Conversations’ conferences called “Returning to Roots and Reason,” which focus on how to effectively teach math to the next generation.
The following is an adaptation of the devotion I gave on the final day of the San Francisco Practicum.
‘Come, let us return to the Lord; for he has torn us, that he may heal us; he has struck us down, and he will bind us up. After two days he will revive us; on the third day he will raise us up, that we may live before him. Let us know; let us press on to know the Lord; his going out is sure as the dawn; he will come to us as the showers, as the spring rains that water the earth.’
The theme of our Practicum is “Roots and Reason,” and it’s appropriate to recognize that all things have roots: numbers; your family; your homeschooling community; our country. Our country’s roots are numerous—Jefferson ensured they would be when he penned in the constitution that ALL men have been endowed by their creator with inalienable rights—not just American citizens; not just taxpayers; all men. And so, since then, our roots have grown strong and plentiful.
Our Nation’s Roots
I want to argue today that there are two roots of particular depth, two that have grown and nourished our nation into what it is today. America is in fact rooted equally in two ancient cities: Jerusalem and Athens.
Jerusalem gave us morality. The Israelites, inspired by the Holy Spirit, gave us scripture, through which God revealed himself to humanity as the Author of Truth. America’s Judeo-Christian roots run deep, and as tempting as it to concede that we live in a post-Christian society, the reason we fight against slavery and oppression, and the reason we defend the poor and marginalized is because we’ve grown up in the soil of Jerusalem.
Meanwhile, and just as importantly, Athens gave us reason. The Greeks coined the Seven Liberal Arts, which Thomas Jefferson would later dub the arts required for free people to govern themselves.
So where Scripture has given us tools for morality, and in fact the Way to righteousness—Christ Himself—the Liberal Arts have given us tools for freedom. For this reason it might be better to think of them as the Seven “Liberating” Arts: the arts necessary for freedom.
Trick question: Would you agree that math is important because it’s one of the Seven Liberal Arts?
Nope!—Math isn’t one of the Seven Liberal Arts; Math is FOUR of the Seven Liberal Arts! If you’ve been around the Classical Education Resurgence for any length of time, you’re probably already familiar with the first three Arts, which we call the Trivium: Grammar, Dialectic, and Rhetoric. These are the Arts for being human—what tech companies are calling “soft skills” and which they are now saying are more important than STEM. Dorothy Sayers was right to place emphasis on these three as they are indispensable and correlate nicely with the stages of childhood development.
But we must remember that these are Arts, not stages. And in addition to the first three Arts, we have four more—the Math Arts, or the “Quadrivium:” Arithmetic, Geometry, Harmony, and Astronomy. Arithmetic is the study of value. Geometry is the study of value in space. Harmony is the study of value in time. And Astronomy is the study of value in time and space.
Aristotle taught that education is about knowing “who you are and where you are.” If the Trivium is about knowing who you are, then the Quadrivium is about knowing where you are. That is, the Quadrivium familiarizes its students with the world around them that they might find themselves comfortable in any situation or topic of conversation. As classical educators, we teach our students to recognize and love the transendentals: truth, goodness, and beauty. But I would argue that you can’t even begin to explore the transendentals without first mastering the Quadrivium.
Let me explain.
Arithmetic: the study of value
We have a lot of value debates happening today. In many branches of academia, value assertions change from decade to decade or even month to month. But in arithmetic, the values themselves are constant. It is for us to discover them, not design them. God designed them, and all that is left for us is to be impacted by their constancy: ultimate truth, made accessible to a four-year-old.
Arithmetic is not about me. It exists absolutely and eternally whether or not I study it or discover it. The value of a thing, then, must be assigned by virtue of some correlating absolute: its goodness, beauty, truth, or, more crudely perhaps, its utility. The pro-choice/pro-life debate asks, “An embryo + x is equal to the life of the mother. Solve for x.” That was what Roe centered upon. The debate today of course has shifted to “An embryo + x = the happiness of the mother. Solve for x.” These are value assessments.
The immigration debate asks, “An undocumented immigrant + x = an American citizen” as well as “A dreamer + x = an American citizen.”
Solve for x.
If you have not yet studied Arithmetic, you cannot yet have these conversations.
Geometry: the study of value in space
As you may know, a sign hung over the entrance to Plato’s Academy which read, “None but Geometers Enter Here.” Geometers? I thought they were talking about philosophy!? Yes. And Plato wouldn’t have made a distinction. To study geometry is to draw connections between theoretical, abstract knowledge and plant it firmly into real, three-dimensional reality. Likewise, to discuss ethics and human flourishing is to draw connections between theoretical, abstract knowledge and plant it firmly into real, three-dimensional reality.
Additionally, the implementation of Euclid’s geometrical method, the practice of showing your work beginning with a series of unknowns, all the way down to a collection of realized values, has been the standard in philosophy for the last several hundred years. What good are your assertions about truth, goodness, and beauty unless you can prove them? For this we look to geometry. We rely on definitions, axioms, postulates, propositions, and demonstrations to quantify value in the space in which we live.
If you have not yet studied Geometry, you cannot yet have these conversations.
Harmony: the study of value in time
What is harmony? Harmony is a right relationship between notes (some which sound concurrently and others which follow each other) or people (some who share the same moments and others whose lives follow one another). There are rules of structure and timing between them (which can always be broken). Harmony is also temporary; it exists within the confines of a moment or a series of moments—followed, perhaps, by a new harmony.
Nearly everything that can be learned about the nature of human relationships can be demonstrated within the study of music.
If you have not yet studied harmony, you cannot yet bring harmony to your community.
Astronomy: the study of value in time and space
If you want to be convinced that the Liberal Arts are concerned with freedom, observe how unanimously we have all conceded to being slaves to our calendar and our GPS! If you had studied astronomy, you wouldn't have to be. You could look up at the night sky and know where you are and when you are—and no one could take that away from you.
The Greeks believed that humanity could get our heads into the heavens by contemplating the universe around us. What was attributed to superstitious “acts of the gods” by all nations up until the advent of astronomy, suddenly became knowable and explainable to humanity. (For example, in 585 B.C. Thales of Miletus became the first person to predict an eclipse. What should have elevated him to the status of the gods he chose to attribute to astronomy; ‘I didn’t make it, I only beheld it.’) Humanity has always been prone to superstition, whether writing off events to “acts of the gods” or to luck or chance. But Astronomy teaches that through patient contemplation, humanity can perceive the very hands of God working for the good of his people. Astronomy is an art—not a science—because we are studying an intelligent, intentional design.
If you have not yet studied astronomy, you cannot yet attribute the universe to an Unmoved Mover.
Let us know; let us press on to know the Lord; his going out is sure as the dawn; he will come to us as the showers, as the spring rains that water the earth.’
I opened us with this in which Hosea offers a reason for his hope in the Lord’s healing. What is his hope? It’s grounded in the tangible world around him (geometry). Hosea had studied the dawn (astronomy) and contemplated its constancy, so he’s able to say, “That’s the Lord’s handiwork; that’s how he’ll return.” And by watching the patterns of the spring rains (arithmetic and harmony and other math) he was able to say, “That’s who my God is; he will come to us like that.”
In Aristotle's opinion of education, a student had to demonstrate mastery in all seven of the Arts before moving on to an apprenticeship in philosophy or theology. My hope is that the Classical Resurgence would embrace the Quadrivium as it has embraced the Trivium: the Trivium for Virtue, work done for the thriving of our own souls; and the Quadrivium for Wisdom, work done for the thriving of the community.