One of my pet peeves is the lack of deep understanding I have regarding mathematics. The feeling of confusion is very similar to the way I feel about long-running television soap operas with a cast of characters spanning generations. I guess what I’m looking for is the story of mathematics, where the characters are not mathematicians but the concepts themselves. In other words: how were these concepts born, and how do they relate to each other?
A few days ago I started to browse the Internet to cobble together an overview of what current math education is like these days; I figured grabbing the curricula from K-12 would give me some insights into what students were being prepared for and give me an idea of what the overall pattern and intention was behind it. As far as I could tell, the intention was to introduce the right concepts at the right time, but the original motivation for doing so was inaccessible to me.
It occurred to me that I actually do have some preconceptions about math, and it’s this: the systems of thought behind the concepts should make sense. However, the systems of thought are buried beneath hundreds of years of forgone assumptions, and it’s these assumptions that I think I need to categorize and lay out in a systematic fashion. Some of these assumptions are so basic that it’s only through digging into the philosophy of mathematic foundation that their distinguishing properties are illuminated. For example, I have been starting with what I think would be the basics: natural numbers. Natural numbers, as I have just re-learned, are the “counting numbers”, and there are two main areas of interest. In pure mathematics, there’s the exploration of the properties and “behavior patterns” of such numbers, which is called Number Theory. The second area is the applied side with regards to counting and ordering numbers; this is called Combinatorics. What’s common to both sides is the huge number of properties, operations, and relationships that have been discovered, conjectured, invented, or derived. I’m thinking that there can’t be THAT many of them in foundational math up to the college level, and yet I’ve never seen a really good diagram of how it all hangs together.
It could also be that I am just missing the point, or am not being sufficiently focused in which point I want to focus on. Frankly, I have no idea what my choices are, so I’m leveraging Wikipedia to build a holistic understanding of what mathematicians talk about. In the way that calculus is the study of functions, I’m attempting to develop a road map of mathematic based on studying the conversation itself. This wasn’t possible before the Internet without a lot of back-and-forth to the library, and so I am optimistic.
The first insight I had was that the development of mathematics is just like any other field of endeavor. It starts with a need: in this case, it was the need to be able to record what you had (counting), and figure out ways of manipulating the counts to make projections (accounting). What followed was the need to solve certain classes of problems that related what you had with some other desire, which leads to the development of algebra. Simultaneously, the need to be able to construct things with some level of accuracy gives rise to the study of shapes (geometry) and how to construct and measure them. Eventually, the two fields merge together and start to share similar methods for solving the problems using inventions of manipulation that are shown to be self-consistent (and therefore reliable). This is not unlike craft. A few people can’t help but notice that some of these things they’re doing are pretty nifty in their own right, possessed of a certain aesthetic unto themselves, and fields of study devoted to this (essentially, what I might call art). Over time, specialized fields of study develop, each with their own language and set of important principles.
I think what I’m looking for is to develop at least a little mastery of the fundamental principles behind the big questions and answers that Mathematics has given us. I don’t know exactly what I mean by that, but I am thinking it might be analogous to my understanding of computers and computer programming. To me, a computer is a bunch of on/off switches that have been combined in remarkable ways to create ever-more sophisticated behaviors. Computer programming is the same thing to me: raw boolean operations simulated by taming the wild electronic forces that scamper between semiconducting substrates that gate them into a semblance of order. They have been shaped by a few generations of scientists and engineers into being able to do some pretty cool things, but underneath it all I sense its original nature and see the lineage. It’s like perceiving the grain in a piece of wood, and being able to tell when that grain is obeyed and when it is violated. The hardest part about learning how to program a computer these days is understanding the ways which people have imposed their own logical systems on the grain of the bare metal underlying it all, and most of this understanding is locked away in the heads of the programming architect or expressed as inadequate documentation. There are a lot of programmers these days who code by invoking magic words in the order they learned, which is possible now because the true wizards have packaged the magic into a form that CAN be deployed. It makes my life a lot easier, but I’m very grateful that I was born in a time where I could witness the advent of computing. Can I learn to “read the grain” of mathematics also? Is the trail too cold? I sense it’s there to be rediscovered, but this could be wishful thinking.