(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.
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].
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 what exactly going deeper at logic/patterns means if not getting into mathematical logic.
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.
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 think your reaction to mathematics might be to badly taught mathematics rather than mathematics as practiced.
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.
Can you give an example of the level where things suddenly become more difficult?
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.
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.
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.
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.