What am I doing here?

Hello, and welcome to the first real posting on Triangular Tunings, a blog about math topics I wish to blather about.

I'm a computer programmer who has an interest in math, and desire to learn as much about it as I can. While in high school, I didn't take many math courses -- not because I wasn't good at it, but because I was too good. My State had a requirement that all high school students had to take two years of high school math to graduate, and could take a third year to complete a "sequence" in math. I managed to fulfill the State's requirement for a math sequence before I even entered high school. While in high school, I was given the opportunity to take college-level courses in Calculus -- and did so, until integration turned into a major mental blocker for me. I understand what it's about, I just can't do it reliably.

In college, I met the requirements for my degree: courses in probability, statistics, as little calculus as needed, discrete maths, et.c. I had to take a 400-level elective, so I chose a course in "category theory", which (surprising for a 400-level course) had no prerequisites. The two main examples of categories used in that course were straight from abstract algebra and linear algebra. So the next year I took abstract and linear algebra courses.

I love physics, and wish to have an understanding, at the mathematical structural level, of the underlying physical theories of quantum mechanics and relativity (both special and general). Unfortunately, these theories involve math concepts a touch beyond what I know -- Hilbert spaces, hermitian operators, tensors, Chrisofel symbols, Lie groups and Lie algebras, Eigenvectors, Eigenvalues, Eigenstates and Eigenspaces. These are all things I ran into while looking at physics, and had to study up on my own. Dover is a wonderful source of cheap math books, and I have a lot of them.

Of course, many books on these topics assume prerequisites which I might not already have: Tensors are typically defined using Einstein notation and by how they transform when coordinate system transformations are performed. If you don't know Einstein notation, and don't have a grounding in multivariate calculus, then you either give up or pull open the Dover catalog again. Did I mention I have a lot of Dover math books?

As you can tell, my math education has been very eclectic. It's exposed me to a lot of math of various types. As a result, I believe that math can be very broad in scope, and very beautiful.

This isn't apparent to many people as a result of how they are educated about math. Starting in primary school, they learn arithmetic: how to do sums and products of numbers of various sorts. In high school, they are introduced to algebra, or how to manipulate equations involving variables, then geometry, where they might get a ming-numbing introduction to formal proofs. If they are going for "higher math", they take high school courses in trigonometry, where they learn to memorize and regurgitate identity formulas, and finally, at the pinnacle of their high school math career, they get to wade into the shallow end of calculus, starting with the tedium of epsilon-delta proofs of limits. It's no wonder most high school kids don't bother with the whole thing; it's no wonder people say they are no good at math almost as if they are proud of it. Math to them is tedious, boring, rote manipulation of figures by half-remembered rules which make no sense to them.

Obviously, I disagree. There is a lot of mathematics out there beyond what is learned in the quick route to calculus. Not all of math involved numbers, or variables, or calculus, or... well, just about anything you can think of. What they do all involve is critical thinking, intensely logical reasoning, and a willingness to accept some pretty remarkable results.

So what is this blog about? My goal is to provide a forum for me to help improve my own math skills, and to perhaps provide an education for others as well.

And so, with the take-home message that math can be fun and isn't all tedium and numbers, our first topic will be: numbers, and tediously defining and building them up... Um, yeah.