τὰ µαθήματα – A colloquy on mathematics’ role in the Program

By Robert Castle

Preparing for reaccreditation, SJC-SF is hosting a series of colloquia throughout the school year. Each meeting addresses a particular aspect of the college’s unique curriculum: Laboratory, Mathematics, Language, Seminar, and Music.

This Wednesday the colloquy addressed the math program. Tutors Michael Grenke and Jack Wales and senior Grace Grubb opened the meeting with brief remarks. The remainder of the time was spent in dialogue between the audience and the three speakers.

Ms. Grubb was the first to speak. She addressed questions and issues specific to the experience of the student body in the math program. Tensions in the math program (later qualified by Ms. Grubb as “lines” that have to be “walked”) were raised. The central tension brought up (which remained a theme throughout the conversation that followed) was between the value of struggle in the classroom and the necessity of “grasping” the material. A tutor in the audience addressed this tension with the question (here paraphrased), “Are we trying to experience the thought process of the thinker whose work we study, or are we trying to comprehend the content more simply in order to situate this thought in a historical context?” Both tasks are evidently part of the project of the college, and so the force of the question was actually which of these tasks should be understood as primary in the classroom. As might be expected, no easy answers were reached on this question.

This difficulty gave rise to another question, which is the use of non-primary material in the classroom. The mathematics program, along with the Lab program, is a place where students often struggle to comprehend the material directly from the primary source. For example, Owen Gemmer (SF ‘19) pointed out that a rudimentary understanding of trigonometry is necessary for understanding the astronomical work currently being studied in sophomore math, Kepler’s Astronomia Nova. But Kepler does not teach trigonometry. In fact, his work presumes a knowledge of it. So students are forced either to use supplementary material provided by the college, or to use the internet or other means to supplement their studies. This was recognized by several tutors in the room as something that ought to be addressed. Mr. Gemmer was primarily concerned with whether or not the sophomore algebra test should be more rigorous and include trigonometry.

Mr. Wales was the second to speak. He began with an anecdote that demonstrated the special place of math classes in the St. John’s Program. He said there was once a time when every undergraduate was given a Greek-English Lexicon upon acceptance into the college. A group of alumni were interested in reviving this tradition, but they were told–because so many students use electronic/online tools for their Greek studies–lexicons (much like good old English dictionaries) are becoming less ubiquitous in undergraduate study. They then decided to offer copies of Euclid’s Elements instead. His point was this: You can’t Google your way through a geometrical proposition. You just have to figure it out. You have to reason your way through it. This also remained a central part of the conversations that followed: The Elements (a book that develops logically from page to page, section to section) teaches its students more than just geometry. It teaches its students the practice of logical reasoning. There is something irrefutably useful and important about being able to prove something definitively.

Tutor Frank Pagano chimed in from the back and commented on the unique construction of Euclid’s Elements. The book itself is useful and fundamental to the program not only because it teaches logical thought, but because it showcases a quality that serious readers of the Great Books presume about every book they read with a mind to learning: severe intentionality of construction. A platonic dialogue is full of false-starts, complicated social and political interplay, obfuscation and misunderstanding of opinion, etc., and for this reason, the reader of Plato is often at a loss for what to “take away.” In his dialogue the Phaedrus, Plato hints that all of these little details are intentional, and that the serious student can learn not in spite of, but because of these confusions. Mr. Pagano pointed out that the Elements is a work that showcases this same sort of intentionality but more clearly. The development of that work as a whole relies on this intentionality: You can only finally learn (in Book XIII of the Elements) how to construct an icosahedron inside of a sphere once you’ve first learned (in earlier Books) about simpler elements like ratios, circles, angles, perpendicularity, etc. So we have to consider each proposition and each lesson of the Elements as functional in relation to the whole. In this way, the Elements serves to teach us not only how to reason but also how to read. We might encounter a detail that is perplexing, even frustrating, in a platonic dialogue, but the assumption that it was placed intentionally and functionally necessitates that we as readers reason through why the author might put this particular detail in this particular place, just as if we were trying to figure out why a middle step follows from the beginning and leads to the end of a mathematical proof.

Mr. Grenke was the last to present before the meeting opened up into a conversation. He began by saying that he tries to read every book on the Program as if it were correct in all respects, as if it really had something to teach him. He mentioned the immediate perplexity of imagining the world as spinning, saying that rough calculations show that “it must be moving rather quickly.” He then called to mind this very perplexity raised by the Greek astronomer Claudius Ptolemy in the Introduction to his work the Almagest. In response to this perplexity, Ptolemy says something like, “What would happen to the birds if the earth were to be spinning?” But when one slows down to doubt the surety of their modern opinion, they might realize that they don’t exactly know how it is that the earth could move so quickly without throwing around everything sitting on its surface–or without ‘outrunning’ all the birds flying just above its surface. Ptolemy raises this question for us and gives the answer that the earth must be motionless. Copernicus, some several centuries later, realizes that if he considers rotation of the earth to be the cause of the rising and setting of the sun, and the orbit of the earth to be the cause of the annual cycle of the sun, that the geometrical descriptions become much simpler and more intuitively correct. And it turns out that we moderns agree with Copernicus, at least with respect to those big details. But in spite of this development of geometrical nuance, we are still left with the original perplexity.

Mr. Grenke went on to talk about the importance of proof in a time when so much is easily dismissed as mere opinion. And he talked about mathematics as a means for us not only to attain truth through proof, but to discover what exactly we consider the criteria of truth to be.

In the conversation that followed, tutors and students asked earnest questions about the goings-on of the classrooms and about the nature of the program as a whole. A tutor addressed what he saw as a lack of Statistics and Probability in the curriculum of the math program, which gave rise to the ever-important question of what should be read and for how long, and instead of what. It was mentioned emphatically by an audience member that this education used to be (and still often is, I opine) understood as “only the beginning” of a life where learning and study are central. If a student wants to study Statistics and Probability, but the college doesn’t offer these themes adequately in the math program, there is always more time and more books.

The presence of both students’ and tutors’ voices in the conversation allowed both particular and institutional questions to be addressed.

More than anything else, it was just plain fun to hear so many serious and caring people discuss mathematics as a part of an education of which everyone at St. John’s is a part.