Yes, I'm on sabbatical, and yes, I'm teaching a class anyway. UF's Center for the Humanities and the Public Sphere has a team-teaching initiative. My friend and colleague Eric Kligerman and I submitted a proposal a year ago for a course with the above title; the selection committee liked it, and here we are. The title of course references Orwell's 1984 and Winston Smith's final submission to the state, but it also refers to this great Radiohead song. My plan is to blog about this weekly; maybe we'll turn it into an article. Maybe not.
Our first class was Thursday, January 8. We meet once a week for three hours. That's intense and I'm not used to it (math is usually done in smaller chunks). The class is not just about instances of mathematics in literature (like the coin flipping in Rosencrantz and Guilderstern are Dead), although we will point them out as they arise. The real focus is on various authors' use of mathematics as metaphor and structure in their works. Up first: Book VII of Plato's Republic, which contains the famous Allegory of the Cave. This is also the book in which Socrates is discussing which subjects are suitable for the education of his philosopher kings. The first subject, after gymnastics, is arithmetic. Socrates points out that Agamemnon was a horrible general, mostly because he didn't know his figures, but there's a bigger reason he's interested in it. Namely, he argues that rulers need to understand the higher logical functions that come along with learning about numbers (he argues for geometry after arithmetic). Indeed, there's a reason we still teach plane geometry in high school--it's not just its utility in describing things, but it's the first introduction to a rigorous logic system. The skills learned in geometry apply to other fields and make the king fit to rule (once he reaches 50, of course).
To the Greeks, "geometry" meant Euclidean geometry and so we spent some time discussing this. We introduced Euclid's five postulates, the first four of which are entirely obvious. The fifth, often called the Parallel Postulate, was the subject of some controversy, even to Euclid. Indeed, he avoided using it in proofs in the Elements until Proposition XXIX, which you can probably recite in its modern form: when parallel lines are cut by a transversal, alternate interior angles are congruent. For 2,000 years, mathematicians tried to prove that the Parallel Postulate is a consequence of the others, to no avail. It wasn't until the 1800s that someone asked the question of what happens if you negate it. (More accurately, it's easier to work with Playfair's Axiom, which is equivalent.) It turns out that it is possible to construct interesting, naturally occurring geometries in which the Parallel Postulate does not hold. The first of these should have been obvious, even to Euclid, since the Greeks knew the Earth is a sphere. On the surface of a sphere, given any "line" \(\ell\) and a point \(P\) not on the line, every line through \(P\) intersects \(\ell\). Of course, "line" here means a great circle (think of longitudes) since they are the shortest paths between points on the surface of a sphere. (Ever wonder why flights to Europe pass over Newfoundland and then swing by Iceland? They're following a great circle, more or less.) But let's be honest, it's a bit unfair to use our 21st Century hindsight to criticize the ancients for missing this one.
The other interesting non-Euclidean geometry is the hyperbolic plane. In hyperbolic space, there are infinitely many lines through \(P\) that miss \(\ell\). A model for this is the unit disc in the plane (not including the boundary circle) where "lines" are circular arcs orthogonal to the boundary circle, along with diameters. Here's a picture of a point and infinitely many lines missing another line:
You've seen this before. M.C. Escher famously used the hyperbolic plane to make pieces like this:
And, if you've ever eaten green leaf lettuce, then you've digested hyperbolic space thoroughly. In fact, hyperbolic structures show up when an object needs to curl up to conserve space. Coral reefs behave this way for example.
So, with some non-Euclidean ideas in hand we're ready to proceed. We ended class with this passage from Dostoyevsky's Brothers Karamazov:
I'll leave it to you to decide whether or not this argument is valid.
Up next: Tom Stoppard's Arcadia, which includes references to discrete dynamical systems, Fermat's Last Theorem, and the second law of thermodynamics. Tune in next time.