Math and science: unpleasant hoops through which we must jump or a ticket to the upper middle class? Contemporary American education presupposes mathematics and science have as their purpose utility. They can get you through school and maybe even make you comfortably wealthy. However, math and science once occupied a different place in educational theory. The greatest proponents of math education emphasized its benefit for the soul. I myself somewhat owe the heritage of my faith to the intellectual rigor and beauty of math. Circa 1970, in his late teens, my father thought that Christianity was passé and that he could obtain truth through reading mystical texts of eastern traditions such as the Vedas of Hinduism. Simultaneously and providentially, a classically-trained high school teacher of his recommended Euclid’s Elements, which he was working through in the old school style with a straight-edge, compass, pencil, and piece of paper. As my father read the Vedas, he experienced dissatisfaction with the lack of proof and clarity that he encountered therein, proof and clarity which were so evident in Euclid. That led to his rejection of eastern religion and his subsequent re-embrace of traditional Christianity. He attributed his re-conversion to the Holy Spirit’s providential work through an ancient Greek mathematician. The intellectual precision of mathematics partially caused his return to the church. Our students benefit similarly when they experience the self-evident truth and beauty, and even the rigor, of mathematics. Learning the quadrivium puts us in touch with bracing reality. That is a point which Dorothy Sayers, a medievalist and lover of truth, would likely affirm. However, Sayers’ presentation of mathematics in her essay “The Lost Tools of Learning” might harm those newly coming to classical education by reinforcing our culture’s utilitarian approach to mathematics and obstructing the transcendental characteristics of the quadrivium. Specifically, Sayers potentially misleads readers in three areas: first, she miscategorizes the quadrivium as subjects rather than arts; second, she subordinates the quadrivium to the trivium, and third, she emphasizes the pragmatic purpose of the mathematical arts over their ability to form students’ minds and lead them to beauty.

Quadrivium as Arts Rather Than Subjects

Sayers mentions the quadrivium first when she presents the reader with the medieval scheme of education. She characterizes the quadrivium as “subjects” put in quotation marks.¹ Unfortunately, she does not unpack that idea any more than to state that the quadrivium ought to be studied by students at the end of their pre-university-level education² and that subjects proliferated beginning in the modern period.³ From these remarks, it is not clear that Sayers understood the quadrivium to be associated with mathematics and science, but as a medieval scholar, she surely knew they were.

In fact, the term quadrivium entered into western parlance in Boethius’ On Arithmetic. Boethius, a Christian philosopher, served as a statesman in the shambles of the Roman empire in the early sixth century. At the beginning of his seminal text on the preliminary mathematical art, he writes:

Among all the men of ancient authority who, following the lead of Pythagoras, have flourished in the purer reasoning of the mind, it is clearly obvious that hardly anyone has been able to reach the highest perfection of the disciplines of philosophy unless the nobility of such wisdom was investigated by him in a certain four-part study, the quadrivium, which will hardly be hidden from those properly respectful of expertness. For this is the wisdom of [the] things which [truly exist]… .⁴

Boethius refers to the quadrivium as a four-part study that leads to unchanging truth. These four parts were well known as arithmetic, geometry, music, and astronomy. Boethius clarifies the mathematical character of these four disciplines when he explains that all physical reality can be divided into multitudes—which are groups of individual things such as flocks or a crowd of people or pebbles—and magnitudes—which are continuous entities not separated into various parts such as a mountain or a tree.⁵ Think of multitudes as points that are indivisible and magnitudes as lines that can be cut into smaller parts.⁶ Arithmetic deals with multitudes considered in themselves. Music is defined as the study of multitudes in relation to one another, especially proportionality. Geometry consists of magnitudes at rest, while Astronomy concerns magnitudes in motion. By investigating each of these modes of reality, the mind is prepared to find the truth.⁷

Definitions of each of these liberal arts, synthesized from a number of medieval and modern sources can be found in [Table 1.1].

It is significant that Boethius coined the term quadrivium to name these four studies. Significantly, they are not ends in themselves. They are studied in order to get to something higher. The Latin etymology reveals as much: quad is related to four, and vium is related to the

Table 1.1: The Seven Liberal Arts

word for “road” or “path,” via. The quadrivium are thus studies that lead the student on a path. In speaking of the seven liberal arts, Hugh of St. Victor, a great 12^th century expositor of education, declared that by these arts, “as by certain ways [viae], a quick mind enters into the secret places of wisdom.”⁸ The academic word that pairs with “ways” is “arts” and not “subjects.” Subjects imply a body of knowledge. It is true that the disciplines of the quadrivium concern certain content areas having to do with number in its various forms. So each of the quadrivium could be considered a body of knowledge in that sense. But the same is the case for the trivium. Grammar, logic, and rhetoric each had their own famous books that teach the skills to be gained from those arts. We can conclude, then, that the seven liberal arts are bodies of knowledge put into action systematically in a way that prepares one to learn other bodies of knowledge and experience the true and the beautiful.

Therefore, Sayers may mislead when she characterizes the quadrivium as subjects, if we take her to mean that the specific areas of study in mathematics are bodies of knowledge that are ends in themselves and not preparatory for other kinds of learning. As we have seen with both Boethius and Hugh of St. Victor, the four parts of the quadrivium are arts that aim towards something higher: wisdom. Shifting our understanding of the math that we teach our students from inert subject to dynamic and purposeful art helps us remember that the goal toward which the student is striving is a lofty one, not just mastery over certain formulas and calculations. As Hugh writes, only those who could “claim knowledge of these seven” liberal arts “were thought worthy of the name of master” because “by their own inquiry and effort, rather than by need of a teacher” they could go on to learn other bodies of knowledge.⁹ Our students gain humility and our teachers are oriented to how their teaching fits into the big picture when we understand how the quadrivium leads us onward and upward when we venture out onto its ways.

Subordination of Quadrivium to Trivium

Now that we have identified the disparity between how the liberal arts tradition understood the quadrivium and how Sayers characterizes it, we should consider Sayers’s portrayal of the relationship between the trivium and quadrivium. Sayers diverges from the liberal arts tradition, when she subordinates the quadrivium to the trivium in younger education. She asserts that studies in the quadrivium were secondary to the trivium in medieval education and in her ideal education, quadrivium studies are only for “scholars,” and therefore not an essential part of a classical education. Sayers’s “tools of learning” are identical to the trivium, not the quadrivium, as she writes: “The interesting thing for us is the composition of the Trivium, which preceded the Quadrivium and was the preliminary discipline for it.”¹⁰

On the one hand, Sayers makes a fair point. Students must have the ability to read to advance to word problems in mathematics or to read any mathematical text with ease and profit. Facility in logic enhances students’ skill in solving the problems that are the substance of mathematics curriculum. But on the other hand, it is hard to imagine not teaching students arithmetic until they are ready for the university! What parents would send their children to a school in which the rudiments of arithmetic were purposely avoided until junior year of high school? Even the great 5^th century B.C. philosopher Plato advocated for the study of calculation “at the same time as [students] learn how to read and write.”¹¹ The Protestant reformer Philip Melanchthon attests to similar timing in the learning of math as Plato almost two millennia later in his introduction to arithmetic of 1536.¹² In fact, to my knowledge, no thinker in the liberal arts tradition banished the quadrivium to the time after the trivium was mastered.¹³

In fact, Sayers seems to recognize this problem and course correct as the essay proceeds. When she explains the trivium not in terms of tools but in terms of stages, she mentions mathematical work both in the Grammar Stage and in the Logic Stage.¹⁴ She even implies that in her revival of medieval curriculum, the older students in the Rhetoric Stage who lean towards the humanities should be required to take some math and science courses as well.¹⁵

This leaves us wishing that Sayers had simply allowed the quadrivium to sit alongside the trivium in K-12 education. I would wager all the schools represented in this room adhere to this view. We probably agree that students should be taught as much arithmetic as early as possible, provided that the teachers are equipped to do so and good learning in the language arts is preserved. Despite her suggestions of mathematical work in the three stages, Sayers nevertheless exhibits a bias for the trivium over and against the quadrivium. This bias becomes more apparent when we consider what she thought the quadrivium was for, which moves us to the third point: the purpose of the quadrivium.

The Purpose of the Quadrivium

Sayers gives us scarce testimony about the quadrivium’s purpose in the “Lost Tools of Learning” essay. In the Poll-Parrot period, Sayers says that learning science will “give a pleasant sense of superiority,” and multiplication tables will “be learnt with pleasure.” Her remarks on math in the Pert stage emphasize their connection with logic, which she hopes will allow students to see how mathematics have intrinsic connections to other disciplines. Lastly, Sayers states that the quadrivium is properly studied by those on the “scholar” track in the university.

From this evidence, we can conclude that Sayers believed math and science would give the student illumination and pleasure and that it would be useful. Indeed, Sayers’s essay strongly emphasizes the pragmatic benefit or expedience of mathematics. Sayers emphasizes the utility of mathematics in two ways. The first is by the overall pragmatic tone of the essay. Sayers establishes this tone in the introduction, in which she appeals to the reader’s despair over the illogical, incompetent state of social discourse. While Sayers identifies the inculcation of the trivium as the main practice that will improve argumentation and dialogue, her idea that the quadrivium-ish activities follow from and bolster the trivium suggest that math and science, too, contribute to a better society. The overall metaphor of education as recovering tools of learning also gestures toward expedience. Tools are useful. They help accomplish practical ends for Sayers such as not falling prey to advertising and propaganda, causing committees to run efficiently, writing with logical clarity, etc. Scientific and mathematical skills belong in the toolbox.

The second way Sayers stresses the usefulness of mathematics is in her presupposition that university students can learn mathematical bodies of knowledge, presumably in ways related to their careers. Perhaps Sayers’ notion of studies in higher education was more freed from concern about specific vocations in the late 1940s England than we contemporary Americans have. But even in her wonderful Lord Peter Wimsey mystery novels, such as Murder Must Advertise or Gaudy Night, Sayers portrays her non-aristocratic characters advancing in their careers on the strength of their university degrees.¹⁶

When Sayers lays emphasis on the usefulness of mathematical studies, she is in fact consistent with many voices in the liberal arts tradition. For instance, when writing to his young students about to study arithmetic, Melanchthon appeals to “economy.” He urges the boys in his school in Wittenberg to study mathematics diligently so that they do not become like the ignorant Thracians whom Aristotle claimed could not count beyond four. Melanchthon points out how important it is to be able to do calculations for running one’s own private estate or conducting the business of the public.¹⁷ Plato himself, that philosopher famous for his obsession with the heavenly realm of the forms, enumerates the mathematical arts pragmatic uses for both the individual and the city in Republic and Laws.¹⁸

However, Sayers puts us on dangerous ground when she stresses the expedient features of the quadrivium. The modern secular world is characterized by its wild hope that in getting systems and techniques right we can set our problems to rights. Or at least by mastering aspects of math and science we can earn a lot of money and secure a comfortable existence for ourselves. The great thinkers in the liberal arts tradition offer an antidote to the malaise of materialism surrounding us. Nearly all these thinkers orient the study of mathematics beyond mere pragmatism to include the two transcendent aims of the formation of the mind and beauty.

First, thinkers in the classical education tradition revere math for its ability to create mental discipline. The etymology of mathematics suggests as much. The Greek word μαθηματική (mathēmatikē), from which the English word “mathematics” is derived, simply means “studies.”¹⁹ Perhaps the first century rhetoric teacher Quintilian spoke most clearly for the tradition when he wrote of math that “it exercises [the] minds [of children], sharpens their wits, and generates quickness of perception.”²⁰ Quintilian recognizes that mathematics has a special ability to provide mental gymnastics that seem to prepare the mind for all sorts of other mental work. Melanchthon sums it up: “And those who are even moderately trained in arithmetic will easily understand many things.”²¹

Second, the quadrivium was thought by many to point to beauty and the source of all beauty: God. Boethius observed, “From the beginning, all things whatever which have been created may be seen by the nature of things to be formed by reason of numbers. Number was the principal exemplar in the mind of the creator.”²² Boethius here claims that God somehow combined number with matter to give the world the shape and order that it enjoys. While Christian philosophers still debate over the ontological status of numbers, the idea that number has its source in God and that, therefore, contemplating number can draw one’s mind upward to God permeates the thinking of Christians in the classical Christian tradition. That the creation reflects God’s own thoughts finds expression in various Scriptures, but perhaps none so vividly as Psalm 19:1, “The heavens declare the glory of God, / and the sky above proclaims his handiwork.”

This is why Christians like Boethius and Hugh of St. Victor were so drawn to some of Plato’s ideas about number. Plato’s characters in the Republic, Glaucon and Socrates, agree that studying mathematics purifies the eye of the soul and thereby draws us upward out of the cave of images and unrealities and toward reality itself.²³ The way that mathematics causes us to move from the tangible, invisible, changeable realities around us to intangible, invisible, unchanging ones prepares the mind for the deep, sometimes abstract contemplation of Beauty and even of God Himself. To echo Hugh of St. Victor, the quadrivium indeed prepares the mind for the apprehension of wisdom. Our youngest students work with manipulatives to learn addition, then the more abstract concept of place value. Eventually, they build the skills necessary to do geometry, advanced algebra, and even calculus. They start with what is visible and tangible. Then they move to a grasp of Truth and Beauty.

Conclusion

Dorothy Sayers in her famous “Lost Tools of Learning” essay says only a little about the quadrivium. What she does say might misdirect those new to classical Christian education. Instead of the quadrivium being subjects, the broad and deep classical education tradition thinks of them as ways or arts by which knowledge and wisdom are attained. Instead of the quadrivium being subordinated to the trivium, the liberal arts tradition generally endorses teaching the mathematical arts at the same time as the verbal arts and rates education in the quadrivium as highly as that of the trivium. Instead of thinking of the quadrivium as primarily useful, the liberal arts tradition thinks of study of the mathematical arts as excellent mental training and even as a road to experience Beauty and God Himself. Sayers wrote a short essay and did not have time to elaborate on these ideas, many of which I suspect that she would agree with. Whatever the depth and accuracy of Sayers’ thought, those of us who are responsible for our students and who want their good, must be cautious about making curricular and pedagogical choices based on half-worked-out ideas on the quadrivium. If we harness the insights of the liberal arts tradition, our students might have their minds formed in the rigor and excellence and beauty of number, and perhaps experience God more richly and deeply than they had before. Perhaps it will even save someone’s faith as it did for my father. May the God who is three-in-one bless us in that noble endeavor.

Notes