Luke Wolcott's post on ODEs and the meaning of life ponders how philosophical we can be in a math class. I have thought about this too -- has every teacher?
I have taught a lot of calculus, of course. I treat a terminal freshman calculus course as a liberal arts experience as much about philosophy as anything else. The ideas wrapped up in the concepts of limit and infinity are amazing and can blow an 18-year-old's mind. (It's not limited to 18 year olds, but they are the most fun to watch!) As Luke says (although he's talking about modern algebra) we study structure for its own sake. The amazing thing, to me, is that this abstract structure then gives us magical knowledge about the real world. How amazing is that? Why should it work? When does it fail?
I'm very pragmatic in other respects. The terminal freshman calculus course is a life skills course. It's hard, harder than many other courses students take, and students always seem surprised. Just yesterday a student was remarking that it's amazing and disturbing that my class is harder than her AP high school class was. Goodness gracious, I'd certainly hope so! The class is about seeing hard scary things and then doing them in a supportive environment that nevertheless has certain incontrovertible truths. It's like a ropes course except that intellectually you're not wearing a harness and you could actually fall off the rope ladder from one tree to the next. It's like lifting a heavy weight: you can move it, or you can't, but you can always build up to that weight through training and concentration. The math is out there in the world and doesn't care about your feelings. It's like gravity. My students have often come from backgrounds that lead them to believe that trying really does count for everything, and math is a great place for them to learn that trying is only one ingredient of accomplishing something.
The great part of math is that you don't end up with a broken bone if you make that mistake. You just smile and start again. My new students often can't believe it: if you make a mistake, doesn't it hurt? or mean that you're not good at math and so you shouldn't do it?
In a higher-level class like ODEs many students are perfectionists but they aren't so afraid. Math is not such a mystery to them. Here's where my philosophies disagree with Luke's: I found no determinism in the course! I guess I taught ODEs from a "dynamical systems" point of view. We did a lot of modeling. While we did work on finding the recipes for solving various types of ODEs, we spent a lot of time on numerical and qualitative analysis. Every time we encountered a type of ODE we could solve analytically I introduced an ODE that we couldn't solve analytically. In every section we used visualization tools to consider the kinds of behavior these ODEs exhibited. We spent a lot of time on fisheries models...
Sensitive dependence on initial conditions got a lot of attention, and students at the end of the semester even came up with their own chaotic models. We talked about approximation in the computer and looked at nice intro examples like y' = e^t cos t --- this goes nuts pretty fast if you use Euler's method to approximate a solution. You can use the existence and uniqueness theorem to show that your computer estimate is wrong, wrong, wrong. I think this bore some real fruit: one of the end-of-semester projects included a student implementation of a three-body-problem modeling algorithm and the student showed the whole class how rounding error in the computer lead to drastically different outcomes.
This is philosophy, and political science! We talked a lot in a qualitative way about how to think about error and long-term behavior of systems. These have real implications for a lot of topics. I think students gained something beyond mathematics when I made them model a fish population, then add harvesting, then add a disease, then add an oil spill, then change the reproductive rate... I presented the SZR model of the zombie apocalypse and students were picking apart every assumption!
Anyhow. Life lessons (practice, work hard, don't be afraid of challenges) and abstract ideas (limits, infinity, sensitive dependence on initial conditions, strange attractors, the unreasonable effectiveness of mathematics). The idea of truth outside of ourselves that is unmoved by our emotional pleas but will yield to patient and persistent examination. Different kinds of philosophies than Luke's, appearing in some set of classrooms in middle America.