On learning difficult things
I have been autodidacting quite a bit lately. You may have seen my reviews of books on the MIRI course list. I’ve been going for about ten weeks now. This post contains my notes about the experience thus far.
Much of this may seem obvious, and would have seemed obvious if somebody had told me in advance. But nobody told me in advance. As such, this is a collection of things that were somewhat surprising at the time.
Part of the reason I’m posting this is because I don’t know a lot of autodidacts, and I’m not sure how normal any of my experiences are. (Though on average, I’d guess they’re about average.) As always, keep in mind that I am only one person and that your mileage may vary.
When I began my quest for more knowledge, I figured that in this modern era, a well-written textbook and an account on math.stackexchange would be enough to get me through anything. And I was right… sort of.
But not really.
The problem is, most of the time that I get stuck, I get stuck on something incredibly stupid. I’ve either misread something somewhere or misremembered a concept from earlier in the book. Usually, someone looking over my shoulder could correct me in ten seconds with three words.
“Dude. Disjunction. Disjunction.”
These are the things that eat my days.
In principle, places like stackexchange can get me unstuck, but they’re an awkward tool for the job. First of all, my stupid mistakes are heavily contextualized. A full context dump is necessary before I can even ask my question, and this takes time. Furthermore, I feel dumb asking stupid questions on stackexchange-type sites. My questions are usually things that I can figure out with a close re-read (except, I’m not sure which part needs a re-read). I usually opt for a close re-read of everything rather than asking for help. This is even more time consuming.
The infuriating thing is that answering these questions usually doesn’t require someone who already knows the answers: it just requires someone who didn’t make exactly the same mistakes as me. I lose hours on little mistakes that could have been fixed within seconds if I was doing this with someone else.
That’s why my number one piece of advice for other people attempting to learn on their own is do it with a friend. They don’t need to be more knowledgeable than you to answer most of the questions that come up. They just need to make different misunderstandings, and you’ll be able to correct each other as you go along.
The thing I miss most about college is tight feedback loops while learning. When autodidacting, the feedback loop can be long.
I still haven’t managed to follow my own advice here. I’m writing this advice in part because it should motivate me to actually pair up. Unfortunately, there is nobody in my immediate circle who has the time or patience to read along with me, but there are a number of resources I have not yet explored (the LessWrong study hall, for example, or soliciting to actual mathematicians). It’s on my list of things to do.
Read, reread, rereread
Reading Model Theory was one of the hardest things I’ve done. Not necessarily because the content was hard, but because it was the first time I actually learned something that was way outside my comfort zone.
The short version is that Basic Category Theory and Naïve Set Theory left me somewhat overconfident, and that I should have read a formal logic textbook before diving in. I had basic familiarity with logic, but no practice. Turns out practice is important.
Anyway, it’s not like Model Theory was impossible just because I skipped my logic exercises. It was just hard. There are a number of little misconceptions you have when you’re familiar with something but you’ve never applied it, and I found myself having to clean those out just to understand what Model Theory was trying to say to me.
In retrospect, this was an efficient way to strengthen my understanding of mathematical logic and learn Model Theory at the same time. (I’ve moved on to a logic textbook, and it’s been a cakewalk.) That said, I wouldn’t wish the experience on others.
In the process, I learned how to learn things that are way outside my comfort zone. In the past, all the stuff I’ve learned has been either easy, or an extension of things that I was already interested in and experienced with. Reading Model Theory was the first time in my life where I read a chapter of a textbook and it made absolutely no sense. In fact, it took about three passes per chapter before they made sense.
The first pass was barely sufficient to understand all the words and symbols. I constantly had to go research a topic. I followed proofs one step at a time, able to verify the validity of each step but not really understand what was going on. I came out the other end believing the results, but not knowing them.
Another pass was required to figure out what the book was actually trying to say to me. Once all the words made sense and I was comfortable with their usage, the second pass allowed me to see what the theorems and proofs were actually saying. This was nice, but it still wasn’t sufficient: I understood the theorems, but they seemed like a random walk through theorem-space. I couldn’t yet understand why anyone would say those particular things on purpose.
The third pass was necessary to understand the greater theory. I’ve never been particularly good at memorizing things, and it’s not sufficient for me to believe and memorize a theorem. If it’s going to stick, I have to understand why it’s important. I have to understand why this theorem in particular is being stated, rather than another. I have to understand the problem that’s being solved. A third pass was necessary to figure out the context in which the text made sense.
After a third pass of any given chapter, the next chapter didn’t seem quite so random. When the upcoming content started feeling like a natural progression instead of a random walk, I knew I was making progress.
I note this because this is the first time that I had to read a math text more than once to understand what was going on. I’m not talking about individual sentences or paragraphs, I’m talking about finishing a chapter, feeling like “wat”, and then starting the whole chapter over. Twice.
I’m not sure if I’m being naïve (for never having needed to do this before) or slow (for having to do this for Model Theory), but I did not anticipate requiring three passes. Mostly, I didn’t anticipate gaining as much as I did from a re-read; I would have guessed that something opaque on the first pass would remain opaque on a second pass.
This, I’m pretty sure, was naïvety.
So take note: if you stumble upon something that feels very hard, it might be more useful than anticipated to re-read it.
Cognitive exchange rates
When reading Model Theory, I was only able to convert 30-50% of my allotted “study time” into actual study.
This is somewhat surprising, as I had no such troubles with Basic Category Theory or Naïve Set Theory.
(I often have the opposite problem when writing code; this is probably due to the different reward structure.)
I was somewhat frustrated with my inability to study as much as I would have liked. My usual time-into-studying conversion rate is much higher (I’d guess 80%ish, though I haven’t been measuring).
I’m not sure what factor made it harder for me to study model theory. I don’t think it was the difficulty directly, as I often tend to work harder in the face of a challenge. I’d guess that it was either the slower rate of rewards (caused by a slower pace of learning) or actual cognitive exhaustion.
In the vein of cognitive exhaustion, there were a few times while reading Model Theory where I seem to have become cognitively exhausted before becoming physically exhausted. This was a first for me. I’m not referring to those times when you’ve done a lot of mental work and you shy away from doing anything difficult, that’s happened to me plenty. Rather, in this case, I felt fully awake and ready to keep reading. And I did keep reading. It just… didn’t work. I’d have trouble following simple proofs. I’d fail at parsing sentences that were quite clear after resting.
I’m still not sure what to make of this, and I don’t have sufficient data to draw conclusions. However, it seems like there are mental states where my I feel awake and able to continue, but my mind is just not capable of doing the heavy lifting.
Again, the fact that I’m only just realizing this now is probably naïvety, but it’s something to remember before getting frustrated with yourself.
Explain it to someone
As I’ve said before, one of the best ways to learn something is to do the problem sets. For Model Theory, though, there were times when I finished reading through a chapter and was not capable of doing the problems.
Re-reading helped, as mentioned above. Another thing that helped was explaining the concepts.
I explained model theory pretty extensively to a text file on my computer. I sketched the proofs in my own words and stated their significance. I explained the syntax being used. I tried to motivate each idea. (The notes are still lying around somewhere; I haven’t posted them because they’re pretty much a derivative work at this point.)
I found that this went a long way towards helping me track down places where I’d thought I learned something, but actually hadn’t. If you’re having trouble, go explain the concept to somebody (or to a text file). This can bridge the gap between “I read it” and “I can do the problems” quite well. For me, this technique often took problems from “unapproachable” to “easy” in one fell swoop.
Don’t book yourself solid
I’m pretty good at avoiding stress. I have the (apparently rare) ability to drop all work-related concerns at the door when I leave. I don’t even know how to get stressed by bad luck, especially if I made good choices given the information I had at the time. I get normally tense in stressful situations with time constraints, but I’m adept at avoiding the permastress that I’ve seen plague friends and family — unless I’ve booked myself solid.
I’ve had a packed schedule these past few weeks. I try to move the needle on at least two projects a day (more on weekends). Even if it’s entirely reasonable to fit all these things into my schedule, I have not yet found a way to avoid the stress.
Even when I know that, if I push myself, I can read this much and write that much and code this feature all in one day, I haven’t found a good way to push myself without pressure-stress.
I’m still hoping that I’ll learn how to move quickly without stress as I learn my capabilities, but I’m not sure I’ve been adequately accounting for the cost of stress.
It’s worth remembering that doing less than you’re capable of on purpose might be a good strategy for maximizing long-term output.
There you go. Those are my notes gathered from trying to learn lots of things very quickly (and trying to learn one hard thing in particular). Comments are encouraged; I am by no means an expert.