Ancient authors tell us that, at the door to Plato’s Academy, there was a sign which read “Let no one who does not geometrize enter here.” Regardless of the literal truth of that story, the practice of geometry clearly held an important place in the Academy, and in the ancient Greek philosophical tradition more generally. Over the course of four weeks together, we’ll explore some of the reasons why.
We see geometric themes throughout Plato’s dialogues: from the divided line of the Republic; to the extended scene in the Meno where Socrates helps an uneducated servant solve challenging geometric problems; and throughout the Timaeus, which begins with Socrates asking “One, two, three; where’s the fourth?”, and proceeds to the harmony of the celestial spheres, and the construction of fire, air, water, and earth from special triangles!
This exploration will combine hands-on, practical, collaborative work in doing geometry, with philosophical reading and reflection on what we’re doing, and why it matters.
In order to take seriously the message on the sign, each and every week, we’ll combine four different elements:
- Practical exercises in geometry: constructing figures with a pencil, straightedge, and compass, in order to discover and demonstrate geometric and philosophic truths.
- Short readings from the dialogues of Plato, where we see the application of geometric methods and ideas in philosophy.
- Ideas drawn from other ancient philosophers like Aristotle, Euclid, Iamblichus, and Proclus, who explicitly reflect on the place of geometry within philosophy.
- Lively and active conversation and collaboration, as a community of learners, exploring together the interplay of these philosophical and geometrical themes.
In other words, we’ll geometrize, and we’ll reflect on why we’re doing so.
Some of our important themes and questions will include:
- The ethos of geometry: What is it like to geometrize, especially when we’re working together in a community? We’ll see that for the ancients, rather than being a competition, or something done by private individuals alone in their rooms, geometry was a community practice.
- Why is geometry so special, compared to all the other arts and sciences? We’ll see how this geometry has a special way of bridging the gap between other branches of knowledge.
- The relation between geometric truths and metaphysical truths—the fundamental realities that create, sustain, and explain the cosmos. How can the interplay between bound and infinite, definite and indefinite, which gives order and structure to geometry, also help us to explain the well-ordered world in which we live?
- The “flash of insight” that is so essential to both geometric discoveries, and to philosophical wisdom, as the Platonic tradition understands it. Think of Archimedes jumping out of his bathtub and running naked through the city, shouting “Eureka!” … but keeping our clothes on!
- The methods and approaches of geometry: How does a geometer approach a new problem, and how can similar methods be used more broadly by philosophers? How can doing geometry help us to notice and appreciate the harmony of nature?
Through all of this active, collaborative practice, we will first and foremost be trying to “think ourselves into” the mindset of the Platonic tradition of philosophy, for whom geometry was so highly valued. Some of the ideas we consider may seem foreign or counter-intuitive, while others may be quite familiar. We’ll be trying these ideas on, not as some final dogma that anyone should blindly accept, but as something worth exploring on its own terms, to see what unexpected vistas it might lead us to.
Our goal is that we will all come away with a new perspective and deeper appreciation for geometry and the ancient traditions of philosophy.