In our most recent First Things, we explored the nature of the liberal arts and their role within classical Christian education. By way of review, the liberal arts include the seven subjects of grammar, logic, rhetoric, arithmetic, geometry, astronomy, and music. The first three—grammar, logic, and rhetoric—make up what has been called the Trivium, or the “three ways.” In this First Things, we turn to the remaining four ways, also called the Quadrivium.
Let’s once again begin with our definition of classical education:
“Classical education is the pursuit of wisdom through a cultivation of intellectual virtue and an encouragement of moral virtue by means of habits rooted in the historic scholastic practices of the Church and the West and by a rich and ordered course of study, grounded in the liberal arts; ascending through humane letters, mathematics, natural science, philosophy, and theology; and yielding informed self-rule and a well-ordered understanding of human nature, the cosmos, and God. It is aimed at the thoughts and desires of heart, soul, and mind (the whole person) for Christ and his Kingdom.”
“All things that can be known have number; for it is not possible that without number anything can either be conceived or known.”
—Philolaus the Pythagorean, fifth century BC
In our very own Founders Hall sits a bust of Pythagoras, the Greek philosopher who gave us the Pythagorean Theorem, by which geometry students are able to determine the lengths of the sides of a right triangle. However, Pythagoras had a far greater influence than most people realize. He had a significant academic and religious following in ancient Greece, and his disciples established what has come to be known as the Quadrivium, which is our topic in this week’s First Things. One of his disciples, Archytas, is reported to have designated arithmetic (numbers at rest), geometry (magnitudes at rest), music (numbers in motion), and astronomy (magnitudes in motion) as the Quadrivium, or the mathematical seeds of learning. Just as the Trivium of grammar, logic, and rhetoric are the linguistic seeds of learning, arithmetic, geometry, music, and astronomy are the mathematical seeds of learning.
In the 21st century, the Quadrivium is probably held in much higher esteem than the Trivium, hence the rise of STEM (science, technology, engineering, and mathematics) as the most en vogue topic of study in progressive schools. The classical school certainly embraces the application of the liberal arts in the applied sciences typically taught in STEM courses. However, before we consider the usefulness of math, we first consider the primary role of mathematical education in human formation and in developing the human mind toward clear thinking and wise decision-making.
STEM without the liberal arts approach to mathematics is like grammar without logic and rhetoric. Plato taught that the study of mathematics leads the mind toward pure reason and cultivates a true love of wisdom. Like him, classical schools believe that by training students’ minds on the perfection of mathematics, they learn to transcend changing opinions in order to identify objective truth. The well-known apologetic of the fine-tuned universe pointing to the God who tuned it is derived from the precision and order found within the universe when it is studied through the lens of mathematics, or the Quadrivium as these subjects are grouped in the liberal arts.
In fact, it was the positive Christian attitude toward the goodness of the created order that allowed thinkers such as Copernicus, Kepler, and Galileo to discover this mathematical language woven throughout nature. Prior to their work, many Greeks saw mathematics as a realm of perfection that could not be found in the material world. But the Incarnation of Christ and the Christian understanding of creation paved the way for uniting the realm of mathematics with the study of the physical world. From this harmonized view of math and matter, the STEM fields were born. Sir Isaac Newton’s investigations into astronomy most profoundly demonstrated the power of mathematics to explain the natural world. Because math was considered philosophical, Newton—and many in classical education today—refer to the natural sciences as “natural philosophy.”
For us, math and science are certainly theological. The Christian classical school reflects on the reality that Jesus Christ, the Word or Logos made flesh, is in all and through all and holds all things together. The universe was made through him and is sustained by him. For this reason, the study of the Quadrivium, or mathematical arts, in the classical Christian school is just as spiritual as the study of language in the Trivium. In our schools, we pursue mathematics as wonder, wisdom, work, and worship.
Even in our earliest grades, our use of Singapore Math is rooted in what the ancients believed about numbers: that they are not merely practical algorithms but realities that should be understood deeply in their relationships and connections. This is what your child’s teacher may call “number sense.” Arithmetic once began in wonder and culminated in wonder, which is why we routinely ask students to wonder with us and imagine what might happen if we tried this or that with numbers.
In the Quadrivium, wonder leads to wisdom as the right questions are asked about topics such as the mystery of the one and the many. How is an organism one when it can be divided into cells, organs, and systems? How is a family one yet composed of individuals? Entire political and philosophical systems have emerged from this single mathematical question. Things can be understood as distinct entities or as multitudes when considered together. A line is one continuous magnitude, yet it is composed of distinct segments and singular points. The four subjects of the Quadrivium were developed by distinguishing these aspects of the one and the many: arithmetic deals with relationships among distinct quantities, geometry with extended magnitudes, music with discrete quantity in time, and astronomy with continuous magnitude in space.
Once wonder about numbers leads to questioning, exploration, and playfulness that culminate in wisdom—such as the theorems, laws, and postulates of mathematics—this wisdom can be applied in work through science, technology, and engineering. We begin with wonder and wisdom because the mathematical arts shape the mind and even the soul, forming intellectual virtue that yields work rightly understood as worship. This is the fourth goal of a classical Christian study of the Quadrivium. Seen rightly as part of a glorious whole, arithmetic, geometry, music, and astronomy point to the Triune God, who is One in three persons and by whom, for whom, and through whom all things are created and sustained. The only proper response to a reality this majestic is worship.
In the 21st century, we still highly treasure arithmetic and geometry, but we might wonder why astronomy retains a prominent place in the liberal arts. Today it is often an elective in high schools and colleges. In classical schools, we often return to the Latin phrase ad fontes, or “to the sources.” To develop a deep, holistic, and critical understanding of any subject, we must understand its origins and its logical relationship to other disciplines. Otherwise, subjects are learned in isolation and life appears fragmented.
Astronomy is a foundational science and therefore an excellent ad fontes study for STEM students. It was through astronomy that medieval scholars catalyzed the development of mathematical empiricism—the idea that mathematics can be used to discover physical realities. Early and modern astronomers alike collect mathematical observations and organize them into systems that explain the natural world. Scholars of the Scientific Revolution of the 1600s point to this practice, particularly in astronomy, as a catalyst for modern science. Astronomy, which is studied by Capstone seniors, teaches that mathematics is the language in which the universe is written. This understanding transfers to all scientific disciplines and forms students’ minds toward greater precision of thought.
As Gottfried Wilhelm Leibniz, the cofounder of calculus alongside Isaac Newton, once said, “Music is the joy that a soul takes in counting without realizing it.” In Leibniz’s day, music was not limited to singing or instruments, though those are part of it. Music theory belongs to this category, but music is also the study of proportionality. Whenever two quantities exist in a proportional relationship, they are considered harmonious or musical in the classical sense. Such proportionality is found throughout the natural world, in human beings, and in society itself. Thus, the liberal arts tradition understood audible music as only one expression of a broader musical reality. Indeed, the universe itself exhibits musical qualities beautifully embodied in the laws of physics and mathematics and reflected in instrumental and written music.
It is my hope that after spending a few minutes with this First Things and contemplating the four liberal arts of arithmetic, geometry, astronomy, and music, we might better appreciate the importance—and even the centrality—of the Quadrivium to the classical Christian school. Too often, parents recognize the uniqueness of the Trivium but overlook the equally humane, rigorous, and holistic approach to math and science found in a true liberal arts education like ours.
At Capstone, we eschew the statement, “My child just isn’t a math person.” All minds are mathematically inclined and dwell in a universe ordered by mathematics. The subjects of the Quadrivium are not reserved for STEM specialists but are time-honored tools that train all students—regardless of aptitude or interest—to think more precisely, wonder more deeply, worship more fully, and work more effectively.
Until next time, keep pursuing the Good, the True, and the Beautiful to the glory of God and the good of your neighbor.