Here I'll try to keep a list of the various extraneous topics I have a tendency to bring up in lecture that aren't directly relevant to the course, but which are interesting in and of themselves. Wikipedia is usually a good resource for math-related topics (though less so for political or controversial topics), so if you're interested in anything, that might be a good place to look. I'll also try to provide links to other resources where appropriate. All of this stuff lies outside the scope of the class, so don't feel like you're obligated to read it.
Leibniz and Newton — The inventors of calculus. They had an ongoing dispute not only about which "invented" calculus first (Newton seems to have come up with it first, but Leibniz was the first to publish his results), but also about epistemology, metaphysics, and pretty much everything else. This dispute is one of the central themes of Neal Stephenson's excellent Baroque Cycle, consisting of Quicksilver, The Confusion and The System of the World (note: in case it's not clear from the context, Stephenson's books are fiction).
Different Infinities and the Continuum Hypothesis — The Continuum Hypothesis was first formulated by Georg Cantor, who was the first person to really figure out what infinity is all about. An interesting and accessible biography of Cantor is Amir D. Aczel's The Mystery of the Aleph. Also worth checking out (in my opinion; many people hate his style and/or his approach to math) is David Foster Wallace's Everything and More: A Compact History of ∞ (on that note, and entirely unrelated from math, I strongly recommend Wallace's Infinite Jest).
The Banach-Tarski Paradox — Which states that a marble can be cut up into five pieces and then the pieces can be reassembled into a solid ball the size of a planet. Originally intended to show that the Axiom of Choice was false, most mathematicians now think that the Axiom of Choice is true and that it just enables us to prove bizarre results. Many (see Dr. Weinstein in the philosophy department, for example) believe that Banach-Tarski is only paradoxical because we don't understand it well enough to state it properly in "normal" language, not because anything fishy is going on. On the topic of language and math, I ought to mention Wittgenstein's interesting Lectures on the Foundations of Mathematics. More generally, Wittgenstein's Tractatus Logico-Philosophicus and Philosophical Investigations are worth reading for those interested in philosophy, epistemology, language and how they relate. Speaking of which...
Gödel's Incompleteness Theorem — Actually, Gödel proved two incompleteness theorems, the more well-known of which says, more or less, that any logical system powerful enough to be useful allows us to make statements which are demonstrably true but which cannot be proved within that system. We shouldn't necessarily see this as a blow against rationalism (Gödel certainly didn't; in fact he saw it as an affirmation of rationalism), though such has been argued quite a bit in the last 70 years, mostly by people who don't understand what the theorem really says very well (note: there are other, stronger cases to be made against rationalism). Though I haven't read it yet, Rebecca Goldstein's Incompleteness: The Proof and Paradox of Kurt Gödel promises to be interesting (and, of course, one shouldn't overlook Newman and Nagel's Gödel's Proof).
Archimedes — Most famous among non-mathematicians for Archimedes' screw and for running naked through the streets shouting "Eureka!", he is also generally considered one of the greatest mathematicians of all time. Much of his work anticipated the use of infinitesimals that serves as the foundations for calculus.
John Napier — Pioneered the use of logarithms to facilitate computations, especially those being done by astronomers. Those techniques underpin the utility of the slide rule, an omnipresent accessory of mathematicians and engineers until the introduction of hand-held electronic calculators. Like Archimedes, Napier is (possibly apocryphally) credited with the invention of a wide variety of sophisticated military equipment, including a super-accurate artillery piece that could supposedly hit a cow from a mile away, but most such accounts suggest he was afraid of the destructive potential of these weapons and so suppressed their design.