I planned to publish this a few weeks after beginning my self-studying, but I failed to because I tried too hard to make it ‘big,’ rather than providing little notes, which might prove to be more substantial in their accumulation. Sorry for that! June came and went, and I was awarded full funding for my PhD. I’ve now started that, and am working with Caswell Barry and Tim Behrens at UCL on neural representations. Things are really exciting, and I’m already very grateful for my supervisors and the people I’ve met. Knowing that people believe in you is an amazing motivator.
My self-studying was totally essential in getting me to this point, as was the Gatsby Bridging Programme, which was one of the most important things I’ve ever done. I wish I could claim that my crazy self-studying period was nearly as fruitful, but working with mathematicians (crazy, enthusiastic, genius PhD students!) was way more useful than I could have imagined. There is a real ‘you don’t know what you don’t know’ effect with studying math. Or maybe more importantly, ‘you don’t know what you actually need.’ My work now would have been totally unrealizable without my self-studying or the Gatsby programme.
I think the biggest thing I gained was this: there is a sense now with which I operate when I work with math that I didn’t have before. It’s not an “Oh, I know this already,” but more often than not, it’s a feeling of “I think I could work this out…” That is, I have a real feeling that I could conquer a problem, even if it would take me weeks. I can’t recommend the course enough.
You might be reading this after seeing my first post, and expect some lessons and tips I’ve learned from teaching myself math, rather than being taught math. My first tip is to find a mentor and engage with them regularly. I do have someone I’d agreed on mentorship with, but our rule was, “If you get stuck on a problem, talk to me.” I never got stuck on a problem enough in my self-study period. If you have a relationship like that, you should do harder problems for shorter periods of time.
Some other things I learned:
- Four hours of doing math in one day is difficult. Six to eight will leave you delirious. My flatmates would regularly look at me funny when I slumped on the couch after my studying. If you don’t find this to be true, you’re probably not working that hard (or on hard enough problems for your level).
- If you can’t prove it, you don’t really understand it. That said, you need to have a strong understanding of the operations underlying methods of proof before you really prove stuff. Proving stuff is really hard. Proving stuff is really worthwhile.
- Don’t let yourself get too bored, or your inner child will start acting a fool. If you’re really bored, change the subject. Do some other kind of math. Watch 3Blue1Brown videos.
- Doing eight hours a day of math is possible—I did it. Most days were seriously painful, physically and emotionally. I got into running. I hate running, but the adrenaline rush was needed. The last hour was typically very painful. But if you need to do this, you should do it.
- Pomodoro timers are a gift from heaven. Back then I did 50-minute sessions with 10-minute breaks, then a 60-minute break after four. Two sets. Now I do 60 minutes followed by 20-minute breaks, with a 90-minute break after four. I find this more sustainable.
It’s hard to overstate how important this whole thing has been for me. I really feel that going through this intensity changed me as a person. I learned that I’m not just a spontaneous and in-the-moment, ADHD-type scientist, but that I could harness discipline, often through a real desire for intensity. Anger became a friend during this time—either anger at the problem, at my tiredness, or just used as raw fuel. It reminded me of the competitive spirit from doing sports as a kid. Maybe a better word is drive, but it often feels like anger.
From April to June, I did math for six to eight hours a day, every day, (most) weekends included. From June to August, I learned math for around six hours a day, and at the beginning would go home and do problem sets for a couple hours. By the end, I was totally, absolutely dead. I took basically three months off, at least from any real cognitive work—I went to Burning Man, moved from Paris to London, and spent a lot of very good time with a lot of people I love. It was really needed, but I don’t want to do it that way again. I was definitely burnt out, and the cognitive fatigue was kind of exhausting, emotionally.
I’m sure I have a lot more to say, but want to just get this update out there. I want to reinforce that if you’re thinking about doing any kind of ultralearning project, you should reorganize your life as aggressively as you can and do it. At least once. Fly to a new country and learn the language as fast as possible. Drop everything and learn an instrument for a month. Teach yourself math, physics, computer science. For more inspiration, see Scott Young’s MIT challenge, which greatly inspired me. His books are decent, but not really necessary.
Also, I totally did fail at my original goals—I wanted to finish the Math Academy sequence through probability/statistics and differential equations by the time the Gatsby programme started, but that did not come close to happening. There was probably a lot of material I could have skipped, and I wish I had been more aggressive with retaking the diagnostic tests as bits and bobs flooded back into my memory. I’m still working through these courses, and taking some courses in machine learning at UCL.
The (old) post #
I also want to share some of the thoughts that I had in the moment during the self-study period, before I started the Gatsby summer school. I’ll leave this unedited. A lot of it is just me saying nice things about Math Academy, which I still do use, just in combination now with textbooks.
The last few weeks have been going really well. That said, it’s worked out quite a bit differently than a) how I envisioned the process and b) the outputs that I wanted to create with it. The first thing to note is that Math Academy explicitly discourages taking notes (link to this). I do agree with their reasoning, that the content taught within is meant to be practiced, not thought about very much. I imagine that in more advanced courses, this will change pretty drastically, and I’ll need to take thorough notes to follow the material at all—that is, to prove things myself.
That poses an issue for now, which is that I don’t really have a way to prove to anyone that I’ve done anything. Math Academy has a ‘progress report’ feature, but that just isn’t very interesting for a blog, now is it? I had hoped to write an interesting diatribe or two on some theorem I found interesting. The truth is, I’m trying to move too fast for that right now. I think that day will come in the future, after the Gatsby programme. Until then, I’m trying to speed-run as much (linear) algebra, calculus, statistics, and probability as I can.
I was working through “Mathematical Foundations II,” MA’s course primarily composed of trigonometry, algebra, etc. I was originally placed in MF I, which was sort of right and sort of wrong—there were a lot of things that I forgot, down to basics like when the area of a triangle formula works and when it doesn’t. Once I remembered those things, I moved on very quickly, so I’ve used the ‘diagnostic tests’ feature to speed-run through bits of the course I just needed brief reminders on. For the second part of the course, I’ve tried to work through it sequentially, but this has grown quite tedious. I switched courses to “Mathematics for Machine Learning,” a course which covers most of what the Gatsby programme covers, and am hoping that MA’s remediation efforts are not lost on me as I move through this course.
The concern with this approach is that I begin to develop false intuitions for concepts in, e.g., linear algebra, that aren’t well founded in the underlying theorems that I no longer have a good grasp on, or simply have never been exposed to. We will see! If I begin to see that happening, I’ll jump down to “Mathematical Foundations III” and move up from there. According to the scheduling tool, 375 XP a day will have me finishing the course by mid-June. The most I’ve been able to manage is around 250 XP in a day, which was around five hours of active effort.
That’s the other thing—doing math all day is really hard. I also came down with a cold recently, which didn’t help very much. My original calculations were made by averaging the time spent on each lesson (~10 XP) for a number of lessons, but toward the beginning of my work session. That put 400 XP at around six hours of work—I’ve done a few eight-hour days (trying to up those numbers!)—and have gotten nowhere near there. We’ll see if I can do it.
The last thing I really like about Math Academy, now having worked with the programme for a couple weeks, is the gamification. It’s just a bit easier to muster up the energy when I a) know there are people working harder than me and b) I can see it—they have more XP than me. I’m doing this full time right now, so not being first on the leaderboard is, well, embarrassing. That’s not the ‘healthiest’ motivation, but as a secondary source of energy it works.