I very much appreciate your interest. I’m sympathetic to the points that you raise. The trouble is that it’s hard to even state my main thesis without presenting a lot of background information (!!). It’s as though someone wanted to know about monstrous moonshine, and I started explaining what a modular function is, and the person said “ok, rather than giving so much motivation, I’d prefer it if you just told me what monstrous moonshine is.” Then I state the theorem, and the person says “wait, what’s the modular j-function?”
I did consider the possibility when I wrote that that I was grossly underestimating the amount of background information necessary to understand your main thesis. After all, you’ve been tutoring for, 15 years, I believe you said? Presumably you’ve collected a ton of anecdata, besides the information you’ve gathered from actual research. That’s a lot to distill. But do you really think it’s comparable to having to understand advanced abstract algebra/group theory? If you’re really confident that stating your main thesis in your next post will only discredit you, then I don’t think you should do it. But I do think you should be open to the possibility that that’s not the case, and I think there’s a non-negligible risk that you’ll alienate the portion of your audience that strongly believes that mathematical ability is almost completely, if not completely, innate, if you take too much time to motivate.
But it’s much easier to speak to an individual’s situation than it is to speak to the general question of how people can get better at math. What’s your background and what are your goals? I may not be able to respond at length individually, but I’ll try to offer quick thoughts at least, and your comments will inform what I write about subsequently.
The rest of this comment is going to require considerable personal disclosure, so others should move on if they don’t like that or care. I hope that my comment doesn’t decrease my credibility with you or others; I think I’ve made valid points. My background is weak, far below what I expect to be the LessWrong norm. I’m going to explain my background, innate ability, interests, and the provisional goals that I’ve set based on my preliminary research. I understand that your time is limited, so I should say that any links are for elaboration and do not necessarily need to be read. I do apologize if this is too long, and I won’t be offended if you don’t offer a response.
I only have a high school diploma and I’m not a college student. I was trapped in the bowels of anti-epistemology until the end of last year. I have strong ugh fields around the traditional math curriculum because I always memorized passwords and my math classes took place very early in the newest part of my school, in which the floors were ceramic, the walls concrete, and the air conditioning diabolically effective; I literally associate the traditional curriculum with being exhausted and confused in a cold, hard, white room that is beset with motivational posters and otherwise featureless. A pressing need for mathematical logic (as you will see) segued nicely into my need for a subject in formal science that I had not been conditioned to hate and that required little or no prerequisite knowledge from the traditional curriculum. I started reading about logic in February, and I got up to soundness and completeness proofs a few days ago. My first book used semantic tableaux, and now I’m working my way from the bottom up again with Fitch-style natural deduction. I have other books for Gentzen-style, Hilbert-style, and sequent calculi.
As for innate ability, that timeline should give you an idea of my learning rate, although I should mention that my mother died on February 24th and it decreased my productivity considerably. I identified the pattern in the Raven matrix in Innate Mathematical Ability. I took the Stanford-Binet IV when I was 8 and scored 135, although I know that one test before age 16 is not a very reliable indication of my adult IQ. Presumably my writing and analysis on this post thus far should have given you an idea of my verbal reasoning ability. I detail my innate ability because I expect there’s a possibility that the benefits of your main thesis are diminishing as innate ability increases, just as rationality training is usually a monumental improvement for those with little initial rationality and not-so-great for those with considerable existing rationality.
As for interests, I’m curious about AGI (I won’t go so far as to say that I expect to become a researcher), but I’m lacking in tools to seriously understand the field. I’ve already read the popular introduction.
As for goals, I’ve been trying to construct a model of what I need to study because the MIRI Research Guide assumes more background knowledge than one would infer at a glance. Khan Academy is enough to solidify my traditional foundations once I get over the ugh fields. Peter Smith’s Teach Yourself Logic guide is basically all I need to give me an idea of how to work through first-order logic, model theory, proof theory, etc. I’ve found Halmos accessible for some set theory. The early relevant chapters in Jaynes require at least calculus, and I have Apostol and Spivak (to compare one another) for that. Axler seems good enough for linear algebra; I checked out the first chapter and it wasn’t too frightening. I’m aware of the Decision Theory FAQ, and have also found that accessible.
I infer that more reading recommendations, especially on subjects that I may have missed, and general learning techniques of the sort that I expect you will eventually detail and that I might not have picked up elsewhere, would be most useful to me, although I defer to your pedagogical experience in determining what advice to give me.
I did consider the possibility when I wrote that that I was grossly underestimating the amount of background information necessary to understand your main thesis. After all, you’ve been tutoring for, 15 years, I believe you said? Presumably you’ve collected a ton of anecdata, besides the information you’ve gathered from actual research. That’s a lot to distill. But do you really think it’s comparable to having to understand advanced abstract algebra/group theory? If you’re really confident that stating your main thesis in your next post will only discredit you, then I don’t think you should do it. But I do think you should be open to the possibility that that’s not the case, and I think there’s a non-negligible risk that you’ll alienate the portion of your audience that strongly believes that mathematical ability is almost completely, if not completely, innate, if you take too much time to motivate.
The rest of this comment is going to require considerable personal disclosure, so others should move on if they don’t like that or care. I hope that my comment doesn’t decrease my credibility with you or others; I think I’ve made valid points. My background is weak, far below what I expect to be the LessWrong norm. I’m going to explain my background, innate ability, interests, and the provisional goals that I’ve set based on my preliminary research. I understand that your time is limited, so I should say that any links are for elaboration and do not necessarily need to be read. I do apologize if this is too long, and I won’t be offended if you don’t offer a response.
I only have a high school diploma and I’m not a college student. I was trapped in the bowels of anti-epistemology until the end of last year. I have strong ugh fields around the traditional math curriculum because I always memorized passwords and my math classes took place very early in the newest part of my school, in which the floors were ceramic, the walls concrete, and the air conditioning diabolically effective; I literally associate the traditional curriculum with being exhausted and confused in a cold, hard, white room that is beset with motivational posters and otherwise featureless. A pressing need for mathematical logic (as you will see) segued nicely into my need for a subject in formal science that I had not been conditioned to hate and that required little or no prerequisite knowledge from the traditional curriculum. I started reading about logic in February, and I got up to soundness and completeness proofs a few days ago. My first book used semantic tableaux, and now I’m working my way from the bottom up again with Fitch-style natural deduction. I have other books for Gentzen-style, Hilbert-style, and sequent calculi.
As for innate ability, that timeline should give you an idea of my learning rate, although I should mention that my mother died on February 24th and it decreased my productivity considerably. I identified the pattern in the Raven matrix in Innate Mathematical Ability. I took the Stanford-Binet IV when I was 8 and scored 135, although I know that one test before age 16 is not a very reliable indication of my adult IQ. Presumably my writing and analysis on this post thus far should have given you an idea of my verbal reasoning ability. I detail my innate ability because I expect there’s a possibility that the benefits of your main thesis are diminishing as innate ability increases, just as rationality training is usually a monumental improvement for those with little initial rationality and not-so-great for those with considerable existing rationality.
As for interests, I’m curious about AGI (I won’t go so far as to say that I expect to become a researcher), but I’m lacking in tools to seriously understand the field. I’ve already read the popular introduction.
As for goals, I’ve been trying to construct a model of what I need to study because the MIRI Research Guide assumes more background knowledge than one would infer at a glance. Khan Academy is enough to solidify my traditional foundations once I get over the ugh fields. Peter Smith’s Teach Yourself Logic guide is basically all I need to give me an idea of how to work through first-order logic, model theory, proof theory, etc. I’ve found Halmos accessible for some set theory. The early relevant chapters in Jaynes require at least calculus, and I have Apostol and Spivak (to compare one another) for that. Axler seems good enough for linear algebra; I checked out the first chapter and it wasn’t too frightening. I’m aware of the Decision Theory FAQ, and have also found that accessible.
I infer that more reading recommendations, especially on subjects that I may have missed, and general learning techniques of the sort that I expect you will eventually detail and that I might not have picked up elsewhere, would be most useful to me, although I defer to your pedagogical experience in determining what advice to give me.