Mathematics and Liberal Education

I am not good at maths, but I keep a compass and straightedge on my desk, and I have three copies of Euclid's Elements.

From The Imaginative Conservative

By Arthur Hippler

Without doubt, one of the most important courses I had at Thomas Aquinas College was freshman mathematics—a year-long study of all thirteen books of Euclid’s Elements. While very often we were discussing Euclid’s principles and reasoning, much of the course was occupied with demonstrating the propositions from memory on the chalkboard—no notes and only minor nudging. I was not a gifted student in math, so I spend over an hour every night, writing out the propositions to make sure I had grasped all the steps. (My sister, by contrast, would look first at the propositions on the way to class, and then, if called upon, would figure out the proposition from the diagram. I am envious to this day.) The course was not just a training in geometry, but in logic. You could not just “cram” a proposition. The whole thing fit together, one logical link after the next.

The college’s emphasis on mathematics (which included four years of primary texts in mathematics, including Euclid, Archimedes, Apollonius, Descartes, and Lobachevski) is unusual among liberal arts colleges. Only the secular St. John’s College (Annapolis, Santa Fe) offers the same program. The college’s rationale for this emphasis came from the fact that liberal education traditionally included not only the trivium (logic, rhetoric, grammar) but the quadrivium (arithmetic, geometry, music, astronomy). For most liberal arts colleges, the mathematics courses are simply modern math stuck onto a “humanities” program of literature, history, and philosophy.[*]

When I graduated from college and later found myself teaching high school, I persuaded the headmaster to let me teach Euclid to upperclassmen. (This tends to be a trademark of TAC graduates—they will always smuggle Euclid into the program given half a chance). I covered far fewer propositions and gave a lot more nudging, and on the whole it went well. Later on, we studied conic sections out of Apollonius and the calculus out of Newton. I saw the same thing happening with them that I had undergone myself. They started to think of math in terms of principles and arguments, and the “logical links” that allowed a proposition to be proved.

Years later, I met a teacher who had been teaching Euclid to “Honors Geometry” students in a private school, who shared with me the common complaint of even high performing math students—”I don’t get proofs!” This is well worth reflecting on. What do they think is happening in mathematical reasoning if they “don’t get proofs”? “I will tell you what’s happening” said the math teacher, “they think of math as an arbitrary system full of meaningless formulas that we plug numbers into and get the approved result.” This fit with an observation that came from my sister who, as I observed earlier, was always good in math. She told me that when she was in high school, algebra was her favorite subject. Why? “Because it made no sense at all—you just have to memorize the formulas and go with it.”

By contrast, one of my favorite moments in college was reading Descartes’s Geometry, and seeing the geometrical diagram for the quadratic formula. (Yes, there actually is a diagram). This formula that I had accepted on faith all through my second year of algebra was visible and provable. More often than not, algebra teachers do not condescend to show students diagrams. Believe! Believe!

Liberal education is not just familiarizing oneself with classics of the humanities, or the polish of culture. Its ultimate goal is to free the learner to find the truth for himself. Algebra is certainly a useful tool for so many tasks in modern life, which indeed are impossible without it. But geometry, studying as it does objects that are evident to the mind and imagination, serves the reason directly, apart from technical application. It is not logic, but one of the easiest ways to experience how logic works. Without this, an important experience is missing. Instead of math helping the student see the need to define, to judge the evidence of principles, and to find logical links in reasoning, students are given what feels like an arbitrary system of symbols. This is the very opposite of liberation.

Notes:

[*] Marcus Berquist’s essay “Liberal Education and Humanities” is particularly clear and illuminating in defense of the traditional approach.