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eli_sennesh comments on Is Scott Alexander bad at math? - Less Wrong
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I say everything I'm about to say as a person who is more certain than not that you have something valuable to contribute through this sequence, and who eagerly awaits more.
All of your posts in this sequence have purportedly been written to motivate your main thesis, but it's not clear to me what that is. I think you should stop motivating and very clearly reveal your Big Secret. What can I do right now to improve my mathematical ability? That's what I want to know.
Consider these points:
Eliezer tried to explain his metaethics the first time and failed, but it was okay because he was able to write his epistemology sequence after that and clarify. His posts pretty much always motivate people to continue reading because they usually have such high insight density and people know that they do; but your posts in this sequence, as far as I can tell, have just been anecdotes, quotes, nonstandard definitions, references detailing your nonstandard definitions, and promises of elucidation in future posts, and in your case, there isn't common knowledge of high insight density to make people trust you even when they don't understand where you're going. Your second post had less karma than your first post, and I predict that this post will have even less. I think that people are getting antsy. If no one understands your Big Secret and it turns out that all of this motivation really was necessary, then you can just do what Eliezer did and try again.
It seems to me that you're getting strawmanned a lot. I want to argue against those things but I don't have a real man to point to, and if you wrote a clear thesis statement, then I would. Maybe someone would say that I'm defending a side rather than seeking the truth if I want to argue against another position when I have no clear position, but I think that amounts to claiming that your predicate's extension is empty rather than that it's vague. There are things that people have claimed about your posts that I can't consider an accurate statement of your (vague) position no matter how charitable my interpretation. The most obvious is that you're claiming that innate ability is irrelevant; you've explicitly claimed the opposite. Other possibilities in order of increasing plausibility include:
You've explicitly stated the fourth item, and it's the one I'm most sympathetic to, and it would be useful to me and many others if this were elucidated regardless of whether or not it means I can be a famous mathematician. If you're claiming more than one of the above, then maybe you could just refine your points about that particular point into public form, and substantiate your more radical assertions afterwards. I really want to see what you have to say about pedagogical techniques more than anything else. I want to be better at math. I'm open to the possibility that I'm misinterpreting everyone's attitudes about this and I'm the only one who wants to know what you have to say about that more than anything else.
I also think it's weird that I see a lot of people throwing around 'good at math' and 'bad at math' as if those terms mean the same thing to everyone. Some people mean Calculus III, some people mean Fields Medal or thereabouts, and some people mean somewhere in between. Whether someone is good or bad will depend on what people mean. It also surprises me because that's an amateur mistake here. It also doesn't help when you quote modest mathematicians but describe people who are characterized as 'bad at math' in your anecdotes. It makes it too easy to assume that you're claiming those people are secret Grothendiecks, and I don't think you are. If you are, then I want to know.
Well, I certainly find it plausible that most reasonably intelligent (ie: at the mean to one standard deviation above) people can learn math to the level of, say, a first-year undergrad, or an advanced high-schooler. For one thing, some countries have more advanced and rigorous math curricula for high-schoolers than others ("engineering-major calculus" and linear algebra are reasonably common in high school curricula), and yet their class-failure rate, to my very limited knowledge, seems to vary with the quality of the pedagogy rather than being uniformly higher.
Could most people of such an intelligence level also learn an entire undergraduate math major? I don't know: nobody is trying that experiment. I do think most of the people who already complete engineering, natural science, or computer-science degrees could probably complete undergraduate math -- but nobody is trying the experiment of controlling the "double major" variable, either. Instead we discourage physics, engineering, and comp sci majors who don't voluntarily double-major from doing additional theoretical math courses in favor of the applied math they need for their own domain (ie: algebra for the quantum physicist, differential equations and optimization for the engineer, logic and computability for the computer scientist). And that's when they take a theoretical track, instead of just cutting out to the applications ASAP!
Could most people who do PhD-level research work in other natural and formal sciences do work in mathematics? At the research level, the distinction has collapsed: they partly already do! I've actually heard it said that you're not really well-prepared for PhD-level CS or physics if you didn't double-major in math, or for PhD-level biology if you didn't take an undergrad major or minor in statistics, anyway. You certainly can't work in type theory or machine learning (to bang on my own interests) these days without using and doing research-level work in "math" as an inherent part of your own research field.
It's very hard to make the relevant inferences when we lack data on what happens when we try to teach people math instead of using math as a weed-out subject, and then jumping out from behind a bush at young researchers in the other sciences yelling, "SHOULDA LEARNED MORE MATH!".
I'm finding this discussion very interesting because of my personal background. The general population would describe me as "good at maths". I would describe myself (because of context) as "bad at maths". I was one of the best all the way through high school and then started an undergraduate maths course known for being challenging. After a few weeks I completely hit a wall and couldn't progress any further with it. (I changed course to music.) My sister, father and brother-in-law all completed a whole undergraduate course in maths - I couldn't finish the first year. So I think I am bad at maths.
Following on: I think I have a much deeper aesthetic understanding of music than my father and sister. They, the "better" mathematicians, are excellent musicians, but in a functional sense. I'd say that I, the "worse" mathematician, have a much more profound insight into music than they do.
And I failed my first go at Machine Learning, and nearly failed my first go at Intro to Statistics!
(Because I hadn't taken continuous probability first, and was depressed.)
What was it like from the inside, hitting that wall? Math is a particularly easy subject to hit a wall in, because if you're lacking a prerequisite of some sort, all of a sudden everything you're seeing turns to apparent nonsense.
Actually I found it very difficult to understand how it had happened. At school I was one of the best, I enjoyed maths, I understood the concepts and mostly found it easy. At University that all reversed: I was one of the worst, I couldn't do the assignments, I found the lectures boring, and I thoroughly disliked it. I found it very hard to comprehend how such a complete reversal had happened. And more than 15 years later, I still don't really get it... It's rather destabilising when you can't do the thing you expected to devote three years of your life to!