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.
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.