As usual, I’d like to pick up where I left off. Last time I said I’d want to try to learn category theory. I did some research and was absolutely bombarded by prerequisites. It came to my attention that the notion of trying to learn “category theory” from my standpoint is incredibly naive. There are many seemingly simple concepts or standards that are never discussed in school and never explicitly stated in educational texts. I honestly had no idea how to become familiar with these without being in an undergraduate or graduate university course.
To try to get my bearings I figured that the thing that will immediately throw you off if you don’t have familiarity is notation. Notation is one of the most fundamental parts of mathematics, the symbolic language, and one of the most arbitrary parts of it as well. There is no way of getting a grasp of, say, any single symbol unless the text tells you what it is. Especially since symbols are used for so many different things: functions, operations, and properties to name a few. For example take this (probably bad example since it’s from quantum mechanics): (1)
The point is there’s a lot of abstraction. Written out in words — this is much easier to understand, but if you don’t understand a single symbol then you don’t understand the whole expression. Plus, how do you go about googling symbols you can’t type? Obviously, you need some educational text to teach you about a symbol explicitly.
One notation that I see in almost every paper that really annoys me is the use of ellipses and indexing numbers, such as in the definition of a polynomial: This is overly hard to read for a concept taught in middle school.
There is also the annoyingly specific mathematical lingo that comes from math being so incredibly formal. For example a math paper might require the exact knowledge of each term in this (fake) example: infinitely differentiable and continuous smooth analytic hairy manifold. This creates a high knowledge ceiling for any one math paper, as well as a huge knowledge gap between papers in different areas. This is well satirized in the random math research paper generator, Mathgen (2), which I recommend taking a look at to see where I’m coming from.
To familiarize myself I took a look at a few sources which list out expressions and their meaning (3, 4), skimmed a paper on notation history (5) (Interesting fact from that: instead of using parenthesis to indicate order of operations, old mathematicians used a vertical line over the grouped sections called a vinculum), and read the Wikipedia articles on a few seemingly simple concepts (6, 7). Reading these may have helped a little bit, but what I really needed to do was find a source that does introduce common concepts.
I looked for recommendations online and discovered that, instead of category theory, what I was really trying to learn about was abstract algebra. I found a book on it by a famous expositor, Serge Lang’s Algebra (8). This is a book intended as a curriculum for a graduate course so I’m not expecting to be able to grasp it completely; indeed, I’ve ran into the problems I mentioned above many times on the first few pages. The author also has an undergraduate version (9), and a book on high school mathematics that I’m sure would fill in any gaps in my knowledge and prepare me to read his other texts (10). Unfortunately, neither of these are in the public domain so it will be hard to read them legally for free, one of the big failings of self-teaching online.
A big area of applied math (and probably the only area of applied math I care about) is computer graphics. In my first post I mentioned the aesthetic of fractals. But I would like to bring your attention to something much prettier: 3D fractals.
I also had a lot fun wasting time playing with this 2D light transport simulation (13). If you scroll down you can see how complicated (and full of annoying, esoteric notation) the math gets.
This led me to a neat discovery called the steridian (14), which is a generalization of the 2D radian to 3D. Generalizations have become a more important aspect of math than they used to, and especially to me, because they often lead to interesting logistics and better intuition. Often times in math class I’ll question how a concept can be carried into higher dimensions or something similar e.g. How does trigonometry work in 3D? Is there a 3D sine wave? How do polar coordinates work in 3D space? (answer: cylindrical or spherical coordinates). What happens if you do repeated exponentiation? (answer: hyperoperations, which I talked about in my last post). What happens if you add more foci to a conic? I am often disappointed by the lack of available information on these (15). These questions are researched well in various papers, but they remain inaccessible.
I think high school math education would be superior if information on generalizations were more incorporated, as understanding the whole of a concept leads to better intuition. Indeed, the whole of high school education often treats certain topics like forbidden knowledge, to be taught later on or not at all, in order to avoid overwhelming the students or create a natural progression. But I find this very annoying because I like to play with a concept and see where it progresses to by myself, and being forbidden from certain parts of a concept make this more difficult. I rarely find myself overwhelmed when a teacher decides to leak a little bit of this information, and I don’t think others are either. That is why I don’t like the pace of high school and instead try to teach myself when my time’s not completely consumed by high school.