It occurs to me that I haven’t really explained what I’m doing here. My reasons for writing this blog aren’t the same as they were when I first started it, so I’ll discuss both the old and the new ones.
When I started this blog in the summer of 2006, I was a sophomore in high school fresh out of the Program in Research for Young Scientists, an experience that convinced me that I wanted to be a mathematician. The name and subtitle of the blog are a reference to Professor Glenn Stevens, the director of the program. At the time, I found a lot of truth in what he said. One of the biggest lessons of the program is a strong emphasis on rigor; that is, on annoying precision.
Anyway, I found that some of the fascinating tools I learned about at PROMYS (factorization in the Gaussian integers, continued fractions) were useful in solving problems on the Art of Problem Solving forums, but often they required some explanation or the setting up of some machinery. So I decided that instead of explaining these things over and over I would explain them once in a blog post and link to the explanation thereafter. So I started writing about some of the interesting things I learned and how they made a lot of problems easier. One of the early examples I saw was that a lot of problem solvers who understood modular arithmetic didn’t realize that they could apply the same principles to polynomials. So Annoying Precision became a blog about interesting problem solving techniques directed at students preparing for the AMC or Olympiads or similar.
But at some point my posts became more contemplative. Instead of just solving problems, I wanted to discuss the “deeper reasons” certain techniques worked or certain theorems had the forms they did. Of course, it’s likely I ended up saying some very stupid things as a result, but all the same I started to realize that having a solution to a problem isn’t the same thing as understanding it.
A little after that, I started reading math blogs, which was probably the best thing that happened to my mathematical education all of last year. It started with the master expositors Terence Tao and Tim Gowers. As I read through their archives, I marveled at how they were able to summarize and generalize technical arguments in non-technical but still enlightening ways. Once I learned that there’s more to mathematics than rigor, I realized that what Tao and Gowers do mentally is something like an enormous feat of compression. Rather than memorize the details of the proofs of the important results in their areas, it is both more efficient and ultimately more enlightening to compress an argument into a few important ideas, and provided you understand the subject well enough, you can (in principle) rewrite the entire argument from these big ideas. And the great thing about focusing on these big ideas rather than on the details of certain proofs is that you can apply these ideas to other situations where the details are different but the big ideas are the same. I would be more specific, but anyone who has read the above two blogs already knows what I’m talking about.
So I stopped focusing on problem-solving techniques. Instead, I started trying to consolidate what I already knew into a more efficient framework. I developed a habit of giving multiple proofs of a result and then trying to relate them, such as in the sums of squares post.
At some point last semester I learned that I was being linked to by other math blogs. Someone told me that the Secret Blogging Seminar had a link to Annoying Precision, although I then learned that Todd and Vishal’s blog had been linking to me for awhile before then. And then I discovered Mathlog, which unfortunately is in German. And then I realized that I had a chance to actually participate in the math blogosphere (the blathosphere?)! Of course, being on WordPress would make that a lot easier, so eventually I moved here.
I think the blathosphere has endless potential. There’s probably nothing I can say about it that hasn’t already been said, but reading the informal writing of masters like Tao and Gowers has done more to inform my mathematical worldview (so to speak) than any of the actual mathematics I’ve learned in the past year. Seeing their thought processes laid out on paper is a lot more valuable than seeing a polished paper with all the motivation and intuition removed, and I hope the rest of the blathosphere continues to follow this model. Perhaps one day the blathosphere will become an indispensable part of mathematics education and research (if it isn’t already). That having been said, this is some of what I want to do in my little corner:
- Try to explain “big ideas” with examples from high-school or undergraduate contests (when appropriate).
- Write down an idea and hope someone reading can point me to the generalization.
- Toss out questions and hope someone reading knows the answer. (David Speyer calls this a “bleg.”)
It should be a fun ride.