Like I suspected, this is rife with confusion-of-levels.
There are a lot of phenomena -- in mathematics, in the cosmos, and in everyday experience -- that you cannot understand without knowing something about differential equations. There are hardly any phenomena that you can't understand without knowing the difference between a cardinal and an ordinal number. That's all I mean by "fundamental."
That's like saying that you can get through life without knowing about atoms more easily than you can without knowing about animals, and so biology must be more fundamental than physics. Completely the wrong sense of the word "fundamental".
Differential equations govern most of our everyday experiences, including the experience of writing out the axioms for set theory and deducing theorems from them.
This is a classic confusion of levels. It's the same mistake Eliezer makes when he allows himself to talk about "seeing" cardinal numbers, and when people say that special relativity disproves Euclidean geometry, or that quantum mechanics disproves classical logic.
And we can model differential equations in a first order theory of real numbers, which requires no set theory
Your conception of "differential equations" is probably too narrow for this to be true. Consider where set theory came from: Cantor was studying Fourier series, which are important in differential equations.
But likewise, the standard reduction to set theory does not illuminate differential equations
...and nor does the reduction of biology to physics "illuminate" human behavior. That just isn't the point!
And we can model differential equations in a first order theory of real numbers, which requires no set theory
Your conception of "differential equations" is probably too narrow for this to be true.
Nope. It is literally possible to reduce the theory of Turing machines to real analytic ODEs. These can be modeled without set theory.
Did computer programming make you a clearer, more precise thinker? How about mathematics? If so, what kind? Set theory? Probability theory?
Microeconomics? Poker? English? Civil Engineering? Underwater Basket Weaving? (For adding... depth.)
Anything I missed?
Context: I have a palette of courses to dab onto my university schedule, and I don't know which ones to chose. This much is for certain: I want to come out of university as a problem solving beast. If there are fields of inquiry whose methods easily transfer to other fields, it is those fields that I want to learn in, at least initially.
Rip apart, Less Wrong!