In my experience, mathematics can be roughly divided in two: topics that use real numbers all the time, and topics that rarely need them. In the former, almost

In my experience, mathematics can be roughly divided in two: topics that use real numbers all the time, and topics that rarely need them. In the former, almost everything builds on calculus and/or linear algebra. In the latter, they are just two topics among many others. The true foundational classes in the typical undergraduate mathematics curriculum are logic and abstract algebra. People rarely start with them, because the usual way of teaching mathematics is applications before foundations. You learn linear algebra before abstract algebra, proofs before formal logic, and axiomatic probability before measure theory. And there is definitely such thing as too much linear algebra. Once upon a time, I wanted a decent mathematical background for theoretical CS and continued (at least) until the first graduate-level class in most major topics. Graduate linear algebra was "foundations without applications" for me, as I've never worked on anything building on it.