Strangely moved by math and science

"There's no such thing as suction" and other insights from my ongoing education in the Great Books

I recently finished a major project: I worked through the entirety of Newton’s Principia: The Central Argument, edited by the former St. John’s College faculty member Dana Densmore. Following Newton’s own suggestions as to what represents the main through-line of his sprawling and difficult work, Densmore presents a selection of Newton’s own text as well as a guide to expanding his often sketchy proofs as he might have completed them. This is more difficult than it may sound, because Newton relied on proportions rather than modern algebra and cast everything—even the proofs involving calculus—in geometric terms drawn from Euclid’s Elements and Apollonius’s Conic Sections.

Densmore recommends that the student attempt to develop their own proof before turning to the commentary, an activity she characterizes as “fun,” but I doubt that approach would be realistic outside the atmosphere of St. John’s rigorous Great Books program, where students learn math in part by working through Euclid and Apollonius. Indeed, the rare student with an easy mastery of high school geometry would struggle with the nuances of conic sections, which are curved figures generated by cutting a cone with a plane. (Even the great physicist Richard Feynman, in a famous lecture where he attempts his own proof of some of Newton’s propositions, confesses that he was defeated by the conic sections.)

Wikipedia’s diagram of the four basic conic sections. The visual imagination of ancient Greek thinkers never ceases to amaze me.

Hence I missed out on the “fun,” but working through Densmore’s rigorous proofs—sometimes, admittedly, with limited comprehension of anything but the overall gist—was incredibly challenging and rewarding. Even without constructing my own proofs, I still felt I was participating in the moment of discovery. In this extremely foreign and cumbersome format (reportedly chosen for its greater intuitive appeal!), I came to understand very basic and important things about calculus and about the motion of the planets in a way I never had before. I had always been puzzled by the inverse relationship between integration (finding the area under a curve) and differentiation (finding the tangent line to a local segment of a curve), but working through the lemmas and propositions where he lays out his methods showed me, more clearly than my distant memories of high school calculus, how the two activities related and relied on a similar style of thought.

Most rewarding, though, was Newton’s proof that the moon was perpetually falling toward the earth, affected by the same gravitational force as the familiar objects close to the surface. At Densmore’s suggestion, I crunched all the numbers myself—perhaps the first time I tried to do my own proof as recommended from the start—and went down several wrong paths, some of which I led commenters down as well after they questioned my initial presentation of the proof. When I finally did hit on the same answer as Newton, the knowledge felt much more meaningful than simply reading it as an isolated fact in a textbook. (Unfortunately, in the comments, it became clear that I didn’t retain the underlying logic as well as I would have hoped, but I promise that I grasped it in the moment and will very firmly grasp it before I have to present it to students!)

I didn’t just find these hard-won insights clarifying—I found them moving. Coming to understand something fundamental in a new way is not solely an intellectual experience for me, but an emotional one. And it’s one that I have had many times in recent years, as I have increasingly expanded my teaching in the Shimer Great Books program to embrace our math and science curriculum. My first venture into that field was as a student back at the independent Shimer College, where part of our training was to occasionally take courses outside of our area. I was thrown into Natural Sciences 1, which is a kind of history of chemistry that basically traces the breakdown of the traditional four elements into the periodic table. Guided by Jim Donovan, a master teacher who led generations of humanities-oriented Shimer students through the rough territory of natural science, I not only learned where the modern understanding of things like air pressure or fire came from, but also what made the traditional understanding plausible.

While I thrilled to see researchers stumbling to the idea of the periodic table—which they discovered by literally listing elements by atomic weights and noticing that similar properties came about at regular intervals, so that you could almost break them into rows that would line up…—my favorite insight was what I summarize to my students (now that I have graduated to teaching that course) as “there is not such thing as suction.” That is to say, our intuition about the “direction” of suction is wrong, because it is driven by differential air pressure. When we suck on a straw, for example, we are not “pulling” the liquid in, we are expanding our lungs and reducing the air pressure inside us, creating an opportunity for the greater air pressure outside to push it in as it seeks equilibrium.

When we moved to North Central College, we reorganized the Natural Sciences part of the curriculum, placing a course on cosmology at the beginning of the sequence in the hopes of framing the contrast between ancient and modern worldviews. Previously that material was in the senior capstone, and when I came to teach those materials in that context, my immediate thought was: “We should have had this the whole time!” Prepping for that course, which we entitled “The Shape of the World,” I got very invested in naked-eye stargazing. That presented unique challenges for me, since I live just south of downtown Chicago, the greatest source of light pollution for a thousand miles around. Yet that “filtering” effect produced some insights—like the fact that the planets are much brighter than most normal stars and actually appear first after sunset, meaning that their strange behavior would stand out much more to the ancients. I have succeeded in seeing all five “traditional” planets with my naked eye, even Mercury (which is a tough one!), and I’ve started a project of photographing the sun’s position in the sky at a certain time of day at weekly intervals, to use as a conversation piece for the first day of class the next time I teach the course. More than any of these courses, this one has affected my everyday experience of the world—I’m always on the lookout for the few stars and constellations we can see in Chicago, and most recently I have been annoying My Esteemed Partner by closely tracking the motions of Mercury in the hopes of “getting” it a second time.

My next venture was our course on Logic and Math, which is heavy on Aristotle and Euclid but also ventures into non-Euclidean geometry. Teaching Aristotle’s logical texts remains difficult and unrewarding, though I have gradually gained some insight into his approach in Prior Analytics. But the geometry is surprisingly fun. I remember when I spent the summer working through Euclid and flipped ahead to see a diagram of the Pythagorean Theorem—all of a sudden, my efforts felt newly worth it. I knew that, if the students got only one thing out of the class, it had to be the proof of the Pythagorean Theorem. Most students (including me, up to that point) probably view the old a² + b² = c² as a mysterious brute fact, but here they would be given the opportunity to finally understand it. (When I discussed this prospect with a colleague, he quipped, “It’s probably going to be the first time they truly understand anything about math!”) Non-Euclidean geometry was a greater challenge, but my second time around (aided by the discovery of this book) I felt I had some real grasp of odd realm—which presumably very few theology PhD’s wind up exploring.

Wikipedia really is such a great resource. We take it for granted.

Back at the independent Shimer, Natural Sciences 2—focused on biology and evolution—was treated as the one “easy reach” for us humanities and social sciences types. It was also one that felt personal to me, as a post-evangelical, and alongside the cosmology course, the one that most related to my background in theology. Here the most rewarding experience was simply sitting down and reading The Origin of Species and seeing how intuitively and irresistibly the argument unfolds. It is rare for someone to invent a whole new style of thought and simultaneously hit on the best way to present it to a broader audience (as Newton painfully illustrates), but Darwin did it. I have also found Konrad Lorenz’s On Aggression a challenging and rewarding text, both for illustrating the sometimes counterintuitive path of evolution and for forcing me to take very seriously the fact that humans are the product of evolution—something that “we all know,” of course, but that few of us really ponder in anything but a hand-wavy way.

---

These are not the only new experiences I have had as a result of teaching in a Great Books program. I have learned much more about the visual arts and classical music than I ever would have anticipated, and both are a much bigger part of my life as a result. We are regulars at the CSO, and we structure our vacations around art museums. I’ve also read and really studied so many great texts that I might have just read through and set aside, not only from the Western tradition but from many world cultures. My experience, in all those cases, has been encapsulated in my spontaneous reaction after our trip to Spain, where we saw Velázquez’s Las Meninas, Bosch’s Garden of Earthly Delights, Picasso’s Guernica, and Gaudí’s Sagrada Familia, among so many other amazing works: “my heart is full.”

My heart has been full after a beautiful concert, after finishing a great novel or work of philosophy (perhaps for the tenth time), after really sitting with a work of art—and it has also been full at the moment when I grasped the logic of Euclid’s proof of the Pythagorean Theorem, or understood intuitively that the stars appear to be spangled on a sphere rotating around the North Star, or saw how the undeniable fact of intentional breeding of domestic animals unfolded into the theory of evolution, or looked up at the moon with a clearer knowledge that the force keeping it in motion around the earth is the same force that keeps me on the ground.

There is a beauty and power in understanding, an emotional payoff for intellectual exertion. Every student deserves to experience that—to receive an education that is moving and beautiful, an education that lets them finally grasp the most familiar facts in a way that makes them realize they never really understood it before. It is good stuff, powerful stuff, human stuff. It is life.