This project is an exploration of mathematics from the perspective of someone who has learned some, forgotten most, and wants to learn more. I'm starting from the most basic concepts of pre-algebra, and I'm hoping to work through the typical American curriculum through geometry and then to cover some advanced areas that are related to my other projects. I won't set out a more detailed roadmap until I get closer to those waypoints.
I'm not a mathematician, and I'm not writing a textbook, even though I call the sections chapters. These are just my notes and reflections on aspects of math that I think will help me understand it. I'm hoping they'll help other people as well or that they'll at least be interesting. I'll also link to other resources that seem helpful. Think of this project as a commentary on other people's math education materials.
My major assumption in this project is that mathematical concepts relate to each other and to the world, and that knowing these relationships can help us understand, remember, and use these concepts. It can also help us appreciate them as objects of beauty, which can itself help us remember them.
What is mathematics?
Because of the different ways we use the term, there's a certain ambiguity in the question of what mathematics is. Am I asking about the academic activity of discovering mathematical concepts? the body of mathematical knowledge we've amassed? the abstract, timeless mathematical realities our knowledge is describing? the mathematical language we use to represent and communicate about those realities? I think to a certain degree the ambiguity can hang there without causing much trouble, but primarily I mean the activity of mathematical discovery.
Like many people, I started out thinking of math as the study of numbers and the formulas for working with them. But there was one wrinkle in this assumption--geometry. Why was geometry a part of math? Shapes aren't numbers. Did somebody make a mistake?
Poking around on Wikipedia made me aware that math is about much more than number. Numbers are the arithmetic, number theory, and maybe algebra part of math, and they are only one type of mathematical object. Math also covers space (geometry), change (calculus), and structure, whatever that is.
Expanding the scope of my understanding of math is good, but simply listing its concerns doesn't entirely help me. I look for coherence in my definitions, so I want to know what it is these areas have in common that draws mathematicians' interest and what distinguishes them from topics outside mathematics. Since space is involved, I'm especially unclear about the boundary between math and physics. These are issues I'll address as I learn. Category theory seems like a promising avenue for exploring the idea of mathematical objects, so I'm hoping to explore that area at some point.
Mathematicians sometimes characterize their field as the science of patterns. I'm not sure what to make of that yet, but it also sounds promising. I can see mathematicians studying patterns and looking for them, and I can imagine there are non-obvious patterns lying behind other mathematical concepts. But I'd like more clarity on what kinds of patterns interest mathematicians, what features of patterns they like to study, and how. I'm pretty sure, for example, they wouldn't publish papers on the emotional effects of color choices in flower-themed wallpaper patterns. Mathematical patterns seem to be related to logic.
For now I'll define math as doing mathematical things with mathematical objects, and as I learn I'll refine my definition. As possibly part of that definition, I'll also keep an eye out for how math's concepts can be characterized as patterns.
But for all that, math does seem preoccupied with numbers. No matter what area I'm glancing at, there's something numeric involved. As I've done my initial thinking about math, I've come up with a speculation I'd like to investigate as I learn, that math is concerned with quantity and the concepts that branch out from it in particular ways. I'll call it the Quantity Relatedness Conjecture (QRC).
- Wikipedia, s.v. “Mathematics,” last modified July 14, 2014, http://en.wikipedia.org/wiki/Mathematics; Wikipedia, s.v. “Areas of mathematics,” last modified June 18, 2014, http://en.wikipedia.org/wiki/Areas_of_mathematics
- Wikipedia, s.v. “Mathematical object,” last modified March 14, 2014, http://en.wikipedia.org/wiki/Mathematical_object