Independent learning: mathematical notation
Studying math and physics, one of my ongoing difficulties has been with mathematical notation— it varies a lot. For one thing, the math and physics communities sometimes have different ideas on the right notation for a given concept. But also, within both of those communities, there’s a lot of variation.
On top of that, the notations are often ambiguous. By that, I mean that sometimes a given notation means one thing, and sometimes something quite different. Not to mention that the same concept often has multiple names. For example, using an asterisk (‘*’) as a superscript on a matrix can denote a conjugate transpose (also called a Hermitian matrix), or it can denote a Hermitian adjoint (also called Hermitian conjugate, adjoint matrix, or simply Hermitian). These two concepts are quite different. And both of those concepts can also be denoted by a dagger character superscript (‘†’)!
So in my notes, I’ve addressed this problem in a couple of ways. First, I wrote a section that summarized all these various uses and conflicts (currently over seven pages). Second, I made some (sometimes hard) decisions on which notation I would standardize on for each of these conflicting uses within the rest of my notes.
This means that when I’m adding information into my notes, I often have to convert the notation used by my source materials into my standard notation. And when I’m reviewing my notes against related material I’ve found from other sources, I often have to translate back and forth in my head.
The complexity doesn’t end there. I often have to figure out from context what my sources mean by the notation they use.
Even discovering that such conflicts in usage exists has been difficult. Trying to sort such things out is complicated by the fact that asking questions about such things is really painful with the usual web search engines.
The fact that the more popular search engines now have AI `overviews’ has been a big help in this regard. But before that came along, I started using ChatGPT, and later some other AI other chatbots to sort this out. I quickly learned that these tools understood LaTeX notation for expressing math, which means I can also use LaTeX notation for asking questions! So that was another huge, unforseen benefit of using LaTeX for creating and maintaining my notes. Synergy!
Using AI chatbots has been hugely beneficial in another way. I can ask very detailed and specific questions (and get useful answers!) that would never have been possible with web search engines. And if I don’t understand the answers, I can ask followup questions that take advantage of the previous response as context. Some of the chatbots are even capable of doing some fairly complex derivations. They are the next best thing to having my own personal professor.