You write: "With Calculus, the mapping of math to real-world objects leads to all sorts of messy realities. With group theory, you dodge that mess by simply sta

You write: "With Calculus, the mapping of math to real-world objects leads to all sorts of messy realities. With group theory, you dodge that mess by simply stating rules upfront." (Prologue: I encountered group theory first by drawing pictures of pegboards and strings to illustrate permutations (before I knew the word!), not by reading the rules up front.) You are describing a schism between pure/formal and applied mathematics (and between formalism and intuition, to some extent). It's completely cool for you to have an interest in pure math completely separated from applied math, real-world physics, programming etc. It is also cool for someone to pursue applied math,physics, etc without any pure math, but that is sad because a lot of beautiful symmetry and cross-disciplinary value would be last. (Goodbye, encryption!) I personally strive to connect pure and applied mathematics. After getting burned (in an emotional/psychological way) by chasing pure math study beyond my ability to intuit and apply it, I now commit myself to learning theory and application in tandem. (I'll certainly appreciate the fact that pure theorists such as your ideal have gone several steps ahead and I can study their results without trying to discover them from scratch.) In fact, my most recent flight of fancy / big dream is to write math/CS tutorials that provide such a tight integration of theory and application, abstract and concrete, general and specific. And I want to use modern web tools (hyperlink, animation, multi-dimensional page layout) to do so. I'd love to talk to anyone interested in working with me on that :-)