> Given my situation, how would you try to convince me that I should learn math

Learning math has been the only time that I have ever been truly humbled by the aesthetic beauty of an idea. I didn't intend on studying math, I wanted to be an english major, but the woman I was dating at the time was somewhat of a math prodigy and I wanted to understand her world more // impress her. This may seem silly to some of the more mathematically mature people here, but the moment I fully understood Cantor's theorem it felt like my mind was dunked into this sublime understanding that left me in a daze for a week. Russell sums this up pretty well,

""" Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry. """

> how would I teach myself the actually interesting parts without getting bogged down by doing lengthy calculations that a computer could do. The standard curriculum is so heavy on the latter and mostly absent of the former.

This wont be much of a problem if you're interested in pure math as numbers bigger than 10 are rare in that domain. I would recommend picking up an intro to proofs type book like [1]. The reviews are mixed but I would take that with a grain of salt as I suspect many of them were written by bitter students.

[1] https://www.amazon.com/Chapter-Zero-Fundamental-Abstract-Mat...

I do not know if it is exactly what you want, but Chapter Zero ( http://www.amazon.com/Chapter-Zero-Fundamental-Abstract-Math... )comes close. It talks about a lot of the foundational mathematics that and symbolism that is often never taught to non-math majors, but it does so in an easily understood way.

Proofs and Refutations by Lakatos also touches on this topic, though it is not the primary or sole discussion there.

This Chapter Zero,

http://www.amazon.com/Chapter-Zero-Fundamental-Abstract-Math...

right? I have not read this particular book, but in general most students who desire to study math should take a look at a "transitions course" textbook or two, to fill in the gaps left by studying only in the fast lane to calculus.

http://www.amazon.com/Mathematical-Proof/lm/R3LUNI80ZOUYK4/

http://www.amazon.com/Mathematical-logic-and-foundations/lm/...

http://www.amazon.com/How-solve-prove-books-useful/lm/31B4QV...