Thanks for that! I needed it after a rough week.

It sounds like you are learning math perfectly. Besides being logic, math is like climbing stairs. What you learn today builds into tomorrow, etc.

I have been trying to help a friend learn math because he wants to go into mechanical engineering and he keeps trying to learn trig before he has a solid grasp of algebra and predictably stumbles. He can get through the first few chapters fairly easily but triangles disappear pretty quickly and turns into circles and waves and he gets lost because he doesn't have the background yet. I tell him that, but it falls on deaf ears. He seems to think I have some magic incantation that will make him understand it without putting in the prerequisite work.

Keep up the great work, it is doubly impressive that you are doing this using distance courses. I doubt I could have done that when I first started out. I needed the sweet security blanket of being able to march into the professors office.

A lot of the axioms used have to be taken on faith and some can't really be proven, at least not in the system that they exist in. Another fun math thing - Godel's Incompleteness Theorem. I really love math, I wanted to double major in it but my health prevented it. I completed a minor in it which is enough for 90% of my major, a lot of people miss the fact that computer science and programming are applied mathematics, but you really don't need a full math major for most of it unless you really want to dive deep into the theoretical pool.

I do self-study now and again to learn new things. I finished Godel Escher and Bach: An Eternal Golden Braid last summer. It was the second toughest math based book I have read and I am not sure I understood half of it. I will probably need to read it about 10 more times and it is a long read.

I am not sure if I ever had to write a formal proof about why a negative times a negative is a positive or even if one exists. The closest to a proof I have seen that if it wasn't so, lots of things would break and that would send mathematicians into a tizzy-fit, and trust me you don't want that.

I agree, algebra was the most fun part for me, I took three algebra courses, not counting linear algebra which isn't as much fun, IMO. Later courses suck some of the fun by rigorously defining things like what is a function. The vertical line test worked just fine and dandy for me but they had to ruin the party.