Backstory
In college, I pursued both math and philosophy majors.1 From the outset, I frequently said that I’d see how far I could do both, but would likely end up choosing. And choose I did: in my junior year I got a zero on an abstract algebra exam, dropped the class, and that was that. Nor was this my first failure in the study of undergraduate mathematics - at the end of my emotionally turbulent freshman year, one kindly professor let me retake both her finals, without which mercy I’d have lost all my scholarship money.
I never felt quite done with abstract algebra, though. As a little kid I was fascinated by what I would later learn to be algebra: I inferred that negative numbers should exist as a little kid, and as a young teen I was excited to figure out (and prove!) that the difference between the squares of any two primes greater than 3, was always divisible by 24. The point is, I was famously a math kid for a long time, and abstract algebra was precisely up my alley.
I kept the textbook, and a few times over the years took another crack at it. I never got that far. The same thing happened in self-study as had happened in the course: I dug the vibes, but I couldn’t keep all the finicky bits straight, my head would hurt, and I’d lose interest.
Then, last November, I decided to try again. I’d already built myself a habit tracker, and was doing various tasks daily. So I added “study math” to my routine, started the textbook from a chapter near the beginning, and kept at it.
Yesterday, I finished the last exercise in the book. I didn’t do every single exercise, but I did many of them from every chapter. My habit tracker keeps track of total days spent on any given habit, and the current streak. To actually get through the book took 151 days of study, the last 120 of which were in a row. Most of those days I only spent half an hour or so, but actually doing it every day made a big difference.
Tradeoffs
Physical exercise can be pretty difficult, but at least it’s intuitive how big an exertion any given activity will take. An hour straight of pushups is obviously ridiculous, while an hour-long walk is totally reasonable. But cognitive labor is a lot harder to gauge. Abstract algebra felt closer to push ups than a stroll, but mental exhaustion is different from physical exhaustion. I would often find myself getting really agitated when doing my daily algebra, even when it was fun. Music in the background, or even the sight of my wife fidgeting out of the corner of my eye, would break my concentration and make me feel terrible, then feel terrible about feeling terrible. Even when I wasn’t doing math, the cognitive overload could make me feel a little useless for a while afterward. In some sense, it was cool to know that I was pushing myself, that I really was challenging the frontiers of my capacity. But sometimes it sucked.
Beyond the basic tradeoff of pushing myself making me tired and grumpy, there was the tradeoff of regularity versus inspiration. When I’d attempted abstract algebra before, either in college or after, I’d only studied when I was really excited. It felt great to do math when I felt like it, and my mind was abuzz with energy. But in practice, the plodding regularity of doing it every day, even when I was slow and stupid and made tons of embarrassing mistakes, worked way better. Many days I felt like I was getting nowhere, but the sheer momentum carried me forward. And having that incremental progress helped me build the muscles I needed to get to insights (like the existence of quaternions!) that were very exciting.
On a micro level, a common tradeoff was whether to forge ahead with a problem that seemed hard, to cut my losses and move on to the next section, or to seek guidance. The most durable (and most fun) learning came from actually figuring things out myself, but if I wasn’t close to figuring something out, getting a leg up could help me get to a later stage where I could actually figure something out. There was also the question of what was on the critical path - some problems might be an interesting bit of miscellany, while others might be foundational to what came next. Skipping some of the former was necessary to keep moving and hold my interest, but skipping the latter would make me end up confused. Near the very end, a few proofs in the actual text of the chapters felt like too much, and it was a tough call to keep at them or keep going. My choice varied.
Another complicating factor, though not exactly a tradeoff, was that the answers weren’t so easy to just look up. Near the very end of the book I did find a repo where someone had put their answers, but it wasn’t anything official, and I’m glad I didn’t lean on it sooner. A mainline theorem was fine to look up for a second perspective, but a random exercise wasn’t exactly Google search compliant.
My Tutors, My Pupils
Almost no one I know is familiar with abstract algebra. Some friends were willing to humor me when I ranted, but I didn’t really have a social scene to learn with. (Though having a semi-advanced math textbook at a bar is a great way to get math nerds to strike up conversation.)
However, I did have something on this attempt that I’d never had in previous ones: LLMs. I talked to GPT-4 and Claude Opus about abstract algebra on a daily basis. Sometimes when I was actually working through the day’s problems, but also often when I was bored in the middle of the night. I’d ask about some implication from the last thing I’d studied, or an edge case, or test my understanding of how two concepts fit together.
I’m not sure I could say which model was better between GPT-4 and Claude Opus, though GPT-4o has been (surprisingly) better than either. I may go into more detail on this in a future post, but at a high, reflective level, I’ll just say that none of the models were perfect. There were funny errors, like declaring that 2 does not equal 2 in a proof by contradiction, but also just lots of getting definitions wrong or making sloppy inferences. Which, I think, actually helped my learning! Because, without worrying about hurting their feelings (though I was reasonably polite), I could correct them.
By having access to multiple interlocutors, who I could both learn from (sometimes) and “teach” in turn, my motivation increased, and I was more rarely confused. Instead of just hand waving in my own mind, I could hash things out, and get to feel smart when I offered a correction to a model almost, but not quite, as good as I was at figuring out the inferences. For anyone trying to learn a technical subject, I highly recommend chatting with AI about it.
How Does It Feel?
Good! Complicated, but good. It’s hard to believe the book is done, and it’s hard to believe that I’ll break my algebra streak today. And sad - it’s always a little sad to break a good habit of four months, even though I think I need the rest.
There are emotional considerations; it’s nice to confirm that I did, in fact, have it in me, even if I don’t know whether my understanding is at the undergraduate level I sought a decade ago. In any case, I’m pretty confident I’ve got the basics, and serial flatterer Claude says that I’m great at it.
But also, abstract algebra is really cool! I’ve kept thinking about it, especially the last few weeks, just for fun. My picture of math, and to some degree my picture of the world, is a little more vivid now. And who knows? Maybe, once I get a rest, I’ll build on what I’ve learned.
Technically, at my fancy liberal arts college, areas of concentration.