Math Learning Project

Michael Hartl • Jul 20, 2023

After selling my company last year and completing my final Learn Enough tutorial, I decided to take a sabbatical to reconnect with my roots in theoretical physics. Somewhat to my surprise, this quickly mutated into a full-blown math learning project. I thought I’d share some of the considerations that went into this as well as some of the goals, strategies, and resources I’m using to guide my learning. It’s my hope that this may be useful to others interested in undertaking similar projects, or even in following along with this one.

The general direction I’m heading is toward active physics research, especially the relationship between classical and quantum physics. This relationship concerns what I consider to be some of the most important outstanding problems in science, including subjects like wave function collapse and quantum gravity. I have no particular pretensions of making original contributions to these fields (though I’m certainly open to the possibility); at this point, my main goal is just to understand them well enough to appreciate where the principal challenges lie.

I initially started down the path of this rather long journey by reviewing parts of the elementary physics curriculum, including rewatching The Mechanical Universe (a lecture series based on a course at Caltech) and rereading around half of Volume I of The Feynman Lectures on Physics. But in the back of my mind, I felt a little tug—although my knowledge base in physics is pretty solid (even if my skills are a little rusty), I realized my knowledge base in mathematics was shaky at best.

Now, anyone who’s familiar with my background or who’s read The Tau Manifesto can tell that I know at least a little math. Having gone through postdoctoral research in theoretical physics, it would be surprising if I didn’t. But many people don’t realize how little “pure” mathematics appears in the standard physics curriculum, which is heavily weighted toward applications.

In all my years of schooling, I’ve taken only one math course (real analysis) that could be considered truly rigorous. This means I’ve had essentially no exposure to rigorous treatments of subjects like number theory, abstract algebra, topology, differential geometry, algebraic geometry, or complex analysis. And even with real analysis, although I did fine in the class I took, I didn’t really get it, due in large part to a lack of experience with rigorous proofs.

One reason this is relevant is that, when looking (even tentatively) toward some of the frontiers of physics, I distinctly feel my mathematical limitations. My strong suspicion is that a better-than-average background in rigorous mathematics would be a significant advantage in understanding the problems that most interest me. It was this feeling that initially got me moving in this direction—what I’ve sometimes referred to as “taking the scenic route back to physics through math.”

Even if my assessment of pure math’s utility for physics is wrong or overstated, I also realized that the subject is of sufficient intrinsic interest to me that it’s worth studying for its own sake. Indeed, I asked myself, when was the last time I could take the time to really learn mathematics, even if it might take a few years? The answer is, basically never. I was too busy in high school and college with distribution requirements and my physics major. In grad school, I couldn’t exactly tell my advisor I wanted to go off and learn math instead of working on my dissertation. And as an entrepreneur, I had an obligation to my cofounders and to our customers to, you know, actually work on the company.

Now that I’m free of such obligations, at least for a while, I feel immensely fortunate to have the luxury of being able to take the time to do this project. It’s still a big challenge, though, and there are many questions that arise. What are the best resources to use for learning math? What learning strategies should I apply? Which goals might I use to guide my studies and determine in which direction to proceed?

These are some of the questions I aim to explore (and maybe even answer!) in upcoming posts. I also expect to share some of the math lessons I learn along the way. I hope you’ll join me!