Why Am I Posting This Crap?

Working with physical models inevitably leads to mathematical problems. In physics education, many professors build up enough tools and techniques to shove these problems aside, and then quickly move on to the real physics. That's simply a reflection of the fact that they care deeply about the physics itself, and I appreciate their efforts to keep students from getting bogged down in details. But taken to the extreme, this attitude produces calculating-maniacs with tons of knowledge and little mathematical intuition. Take at look at the canonical book on math methods for physics (Arfken), and you'll see that it's simply an encyclopedic dictionary of methods.

I'm only now becoming mature enough to go back and re-examine many of the things I learned earlier with an eye towards appreciation rather than computation. This practice intrigues me. The purpose of a blog is ostensibly to create content for public consumption. That's not what I'm doing. I'm working through things for myself. Not because I feel a dire need to understand. Because I can get enraptured by the absurd power of a little solid thought to unearth something I never saw before, and yet later becomes laughably obvious.

According to Google Analytics, in the last month, 58 total people have visited the site in 172 separate visits, and viewed 268 pages. They've spent a total of 5 hours, 41 minutes reading the posts. That's only a small fraction of the cumulative time it took me to write the posts. Also, a lot of the statistics I quoted are just me, logging in from various computers, trying to crank up the scores so I'll feel more important.

Nikita, Julia, and Kangway have actually been participating and giving me feedback, which is pretty much incredible, so thank you for doing that. But looking at the recent posts I've written, I realized a few things:

• not that many people care about things such as the geometric interpretation of a vector product rule
  
• people who do care were better students the first time around than I was, and mostly already know
• people who would care, but don't already know this stuff, most likely don't have the assumed background to learn much from my posts.
  
• even if the content of the posts is "just right" for you, I skip a bunch of steps and gloss over the parts of the arguments that are less interesting to me, because they're too tedious to draw/ for the blog. Hence, following the posts from scratch would be a rough road. It's not the sort of reading people are looking for when browsing through blog posts. They want quick, to-the-point, easily-comprehensible and entertaining content. This blog is the antithesis.
  

My goal is simply to explore nuances that interest me, however mundane. That explains a few of the points above. When I start writing a post, I have to think about how much background knowledge to assume the reader has. I want to write about the parts that I think are interesting, which means essentially the parts (and only the parts) that I didn't completely understand before I started working on the post. I generally assume that everything I am totally comfortable with, the reader is as well. That means people will frequently be flabbergasted by the strange logical jumps I take and the way I write certain details off as trivial. On the other hand, everything that I had to work out and think about for a while, I feel the need to explain in more detail than many readers would be interested in wading through.

Today I finished a post about vector products and determinants. I was aware of the geometric interpretations from a few sources, but I had never put the pieces together for myself. By the end, I was amazed. Not only had I built up the idea of a determinant, motivated geometrically, but I had also shown how to use the determinant to solve a system of equations. I didn't even know that would pop out of the argument. If you had asked me to demonstrate how a determinant works before I started writing, I probably wouldn't even have known. Additionally, the path that led me there was remarkably short, and since I had worked through it myself, it walks through a logical (to my mind) problem-solving process.

Math books tend to define and develop the determinant as a number characterizing a linear operator on an n-dimensional vector space. The treatment is more abstract. The proofs showing how to calculate it and how it solves systems of equations frequently rely on manipulating sums of indices and leave me feeling unsatisfied. I believe them, and see that they're correct, but I have difficulty visualizing them as well as the author of the book. I don't knock their approach. I think it's probably the best way. But this supplementary cogitation on the blog has bolstered my appreciation.

Physics books, on the other hand, will normally skip over the entire journey and roll the answer out right away, then tell you to start inverting some matrices. Save the math questions for your linear algebra class!

Writing this blog is my way of taking control of my own educational process. I think about whatever I want, at my own pace, in my own way. Why then post it on the internet? Anything I write has probably been written somewhere else, and probably with more insight and clarity than I'm providing. If my goals are personal education, why paste it all over this blog, advertising it for people to see?

If I were to keep these developments to myself, I wouldn't think them through fully. I'd work until I saw the gist of the argument, then say to myself, "the rest is just mopping up details from here. better move on to something more important."

If I don't take the time to mop up the details, the bits of understanding I did build flit away. So by assuming the responsibility to clean things up to the point where someone could potentially read through the post and work out the results for themself, I force myself to clean up the ideas in my own head. That's what I'm after.