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IlyaShpitser 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.
Let me take a stab at my (not OPs) views on math:
(a) A single number model of intelligence is toxic and silly. IQ is a single number proxy for a complex multidimensional space.
(b) Effective test taking has very little to do with math ability. Many excellent mathematicians are bad test takers (e.g. do not think quickly on their feet): this means basically nothing.
(c) Brains are complicated, and there is a huge amount of heterogeneity in how people process information and think about mathematics (and indeed all topics, but it is clearer in mathematics perhaps). Some are very visual, some are big on calculation.
(d) There is no separate magisterium called "math," there is a gently sloping continuum from common sense to "novel math work." When someone says "I am bad at math," I am not sure if they mean "I can't think carefully at all," "math notation scares me," "I can't think abstractly," [something else].
(e) If you haven't engaged with math beyond high school, you probably don't have enough information to evaluate the counterfactual "would a hypothetical me that pursued a math education make a good mathematician?" A lot of people's current trajectories in life are based on contingent things like what sort of teachers you had, social situation, etc.
(f) There is a skill component to math that needs practice and repetition, as with anything else. I call it "taking the time to insert metal struts into your brain."
(g) I think a non-trivial % of college-educated population can have a non-trivial engagement with mathematical topics, if they get over their panic. A smaller % can do novel work, if they wanted to put in the time.
(h) There is an enormous overlap between doing math properly and doing rationality properly. Math gives you quite a bit of good rationality habits for free (e.g. "I am notice I am confused.."), and (?conversely?) you can't engage with rationality without a bit of math.
A data point for you: I am not particularly good at math. What this means is that at certain levels going forwards suddenly becomes much more difficult. I can continue, but slowly and only with a lot of effort. It's a slog. By comparison, I'm much better at logic/patterns and going deeper there is just easier. I do NOT mean that I can't think carefully or abstractly or that notation scares me.
Note that I'm using a fairly narrow definition of math here. In particular, I distinguish math and statistics and believe that they require two different propensities. People good at math are rarely good at statistics; people good at statistics are rarely good at math.
I am not sure what exactly going deeper at logic/patterns means if not getting into mathematical logic. It is incredibly easy to read mathematics you know and incredibly difficult to read mathematics that you don't due to how dense it is. It might be the case that your impression is due to comparing these two.
I am training to become a mathematician and I do not know of a single person for whom learning mathematics is not slowly and with a lot of effort, I do not think you are particularly exceptional in that but I know very little about your particular scenario.
I am not sure how to properly express the difference, but it has to do with math being more, um, hard-edged and rigid (in the sense of a rigid mechanical construction as opposed to a reed bending with the wind).
To use a quote from Conan Doyle, a logic/patterns ability would allow one to "from a drop of water ... infer the possibility of an Atlantic or a Niagara without having seen or heard of one or the other." Mathematical logic cannot do that.
So something more like intuitive pattern-recognition/completion than rigorous formalization?
Definitely less rigorous formalization and more gestalt pattern recognition.
In general, I think of math as dealing with well-defined "things" -- you may not know the shape/properties/characteristics at the moment, but they exist, they are precisely defined, and they are not going anywhere. In contrast to math, statistics deals with fuzzy amorphous "things" that you will likely never know in precise detail, that mutate as more data becomes available, and that usually require interpretation and/or some guessing to fill in the gaps.
Cutting edge math is actually mostly about converting fuzzy stuff, at least the parts of math I am interested in(Algebraic Geometry - Grothendieck/Weil for example). Both the mentioned mathematicians worked in a field where people had some stuff that worked but no foundations.
Also, the foundations of math have been changing for quite a long time and continue to do so. I think your reaction to mathematics might be to badly taught mathematics rather than mathematics as practiced. However, I don't see an easy way to fix it.
To teach mathematics well would require a high amount of mastery and we don't have enough people like that around.
I doubt it -- I generally teach myself things and just ignore bad instruction. The underlying cause is likely to be the curse of the gifted -- I'm lazy and when I run into walls I usually go around instead of starting a wall disassembly project. And I was never attracted to math sufficiently to apply a lot of effort.
Can you give an example of the level where things suddenly become more difficult?
As I said in another post, I struggled quite a bit with early calculus classes, but breezed through later "more difficult" classes that built on them.
Also, I disagree with the math and stats thing. Many of the best statisticians I know have strong grounding in mathematics, as do many of the best data scientists I know.
I hit a wall in my string theory course, after having to apply a lot more effort than expected in a QFT course the year before. Didn't have that issue with GR at all. Well, maybe with some finer points involving algebraic topology, but nothing insurmountable.
And importantly, brains are more heterogeneous at the extremes than at the means. A person of average intelligence might not have much difference between their verbal and calculation abilities; a person of great verbal or calculative intelligence may have a large gap between their stronger and (comparatively) weaker cognitive abilities.