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This approach eventually breaks down. Could you give a useful colorized explanation for Urysohn's metrization theorem? Or the Vitali covering lemma? Colors are

This approach eventually breaks down. Could you give a useful colorized explanation for Urysohn's metrization theorem? Or the Vitali covering lemma? Colors are useless. Just learn to think abstractly. This is what (supposedly) distinguishes you from lesser animals.

threatofrain 9y ago
Even if this is true, the way you deliver insults probably sets back the message.
- catnaroek 9y ago
  I'm not arguing against examples. I'm arguing against using colors to make up for the inability to understand abstract structures. In my experience, the people who argue for this are usually “math educators” who don't do much mathematics themselves.
gugagore 9y ago
I hope most people care about understanding something more than they do distinguishing themselves in your view. For many people, concrete examples can help them understand an abstract idea. Even if someone doesn't grasp the full abstract idea, at least they might have a concrete example that exemplifies the idea. I'm not familiar with your examples, but it is true that complex nuggets of math have evolved to be simpler, more elegant and sometimes more powerful. I try not to pride myself to much for understanding something complicated and abstract, because it would be even better if I had a way to conceptualize of it that made it simple and obvious. Finally, most _things_ breakdown. Doesn't mean that they aren't useful in a limited context.
- catnaroek 9y ago
  > For many people, concrete examples can help them understand an abstract idea. Sure enough. > Even if someone doesn't grasp the full abstract idea, at least they might have a concrete example that exemplifies the idea. Examples are only useful inasmuch as they pave the road to understanding abstract definitions. On their own, examples are pretty much useless. In any case, your comment has nothing to do with the original topic, namely, whether coloring different parts of mathematical definitions is particularly helpul. It seems to me that the real problem here is that a certain kind of people has great difficulty understanding the syntactic and semantic structure of mathematical definitions. Now: (0) The syntactic structure of mathematical definitions is the usual syntactic structure of English, or whatever your first language is. If it hasn't been drilled into your head in elementary school, sue your elementary school. (1) As for the semantic structure of mathematical definitions, well, I guess there is no way around thinking hard about it, annoying as though it may be. > but it is true that complex nuggets of math have evolved to be simpler, more elegant and sometimes more powerful. I try not to pride myself to much for understanding something complicated and abstract Simplicity usually comes from having better abstractions (e.g., how Stokes' theorem evolved to its modern version in terms of differential forms), not from rejecting abstractions altogether.
  - gugagore 9y ago
    > On their own, examples are pretty much useless. This sounds like being familiar with an example of an abstract idea is as ["pretty much" as] useless as not knowing anything about it. Furthermore, you don't really get to decide what other people find useful for their understanding. > Simplicity usually comes from having better abstractions Absolutely. And sometimes using better abstractions involves less abstract thinking. I am not persuaded that thinking concretely is akin to rejecting abstractions, as you seem to be. Notice that in the course of your argument, you keep bringing concrete math examples to make your point. You are not rejecting abstraction and yet you are not purely speaking abstractly either. I would say that school teaches the (negative) integers is an abstraction over "what they actually are", the construction of the integers from natural numbers. Yet, thinking about that construction, or even knowing what the word "equivalence class" means requires abstract thinking. In any case, I think it's a pretty trivial idea to show which parts of a natural language phrase correspond to which parts of something written in math. It's a gloss. You can do it with colors, or gesturing, or whatever. Finally, I believe you are thinking too mechanistically about humans, especially humans learning about math. Elsewhere you suggest that (ask about whether) recognizing Greek letters should be as easy as recognizing letters from the Roman alphabet you are familiar with... Yeah, I think if you haven't used Greek letters enough, it's going to be harder to visually parse something with Greek letters. So now some more of your cognition is being spent on something that is wholly unrelated to the task at hand. Apparently you can even measure that German people are a little slower at some arithmetic problems because their names for quantities don't always list the digits in the same order of significance. Your WORD for a number causes you to be faster or slower at arithmetic. So I think using unfamiliar symbols might also make a difference. Here you are suggesting that the syntactic structure of math definitions is the same as the syntactic structure of English. Right! They are written in English! However, even in the realm of natural language, humans will have a hard time understanding utterances with nested quantifiers, negation, and clauses. The notion of syntax is an abstraction, and from that abstract point of view, there is no difference in the expressiveness of English and Mathlish. But humans apparently don't process not even natural language syntax as mechanistically as you suggest they do. Even though shallow and deep trees can be generated by the same grammar, more complex parse trees are harder to understand. You have to be careful with an abstraction because sometimes the details (like how many words are in a sentence, or how many layers of quantifiers there are) matter.

threatofrain 9y ago

Even if this is true, the way you deliver insults probably sets back the message.

catnaroek 9y ago
I'm not arguing against examples. I'm arguing against using colors to make up for the inability to understand abstract structures. In my experience, the people who argue for this are usually “math educators” who don't do much mathematics themselves.

gugagore 9y ago

I hope most people care about understanding something more than they do distinguishing themselves in your view. For many people, concrete examples can help them understand an abstract idea. Even if someone doesn't grasp the full abstract idea, at least they might have a concrete example that exemplifies the idea. I'm not familiar with your examples, but it is true that complex nuggets of math have evolved to be simpler, more elegant and sometimes more powerful. I try not to pride myself to much for understanding something complicated and abstract, because it would be even better if I had a way to conceptualize of it that made it simple and obvious. Finally, most _things_ breakdown. Doesn't mean that they aren't useful in a limited context.

catnaroek 9y ago
> For many people, concrete examples can help them understand an abstract idea. Sure enough. > Even if someone doesn't grasp the full abstract idea, at least they might have a concrete example that exemplifies the idea. Examples are only useful inasmuch as they pave the road to understanding abstract definitions. On their own, examples are pretty much useless. In any case, your comment has nothing to do with the original topic, namely, whether coloring different parts of mathematical definitions is particularly helpul. It seems to me that the real problem here is that a certain kind of people has great difficulty understanding the syntactic and semantic structure of mathematical definitions. Now: (0) The syntactic structure of mathematical definitions is the usual syntactic structure of English, or whatever your first language is. If it hasn't been drilled into your head in elementary school, sue your elementary school. (1) As for the semantic structure of mathematical definitions, well, I guess there is no way around thinking hard about it, annoying as though it may be. > but it is true that complex nuggets of math have evolved to be simpler, more elegant and sometimes more powerful. I try not to pride myself to much for understanding something complicated and abstract Simplicity usually comes from having better abstractions (e.g., how Stokes' theorem evolved to its modern version in terms of differential forms), not from rejecting abstractions altogether.
- gugagore 9y ago
  > On their own, examples are pretty much useless. This sounds like being familiar with an example of an abstract idea is as ["pretty much" as] useless as not knowing anything about it. Furthermore, you don't really get to decide what other people find useful for their understanding. > Simplicity usually comes from having better abstractions Absolutely. And sometimes using better abstractions involves less abstract thinking. I am not persuaded that thinking concretely is akin to rejecting abstractions, as you seem to be. Notice that in the course of your argument, you keep bringing concrete math examples to make your point. You are not rejecting abstraction and yet you are not purely speaking abstractly either. I would say that school teaches the (negative) integers is an abstraction over "what they actually are", the construction of the integers from natural numbers. Yet, thinking about that construction, or even knowing what the word "equivalence class" means requires abstract thinking. In any case, I think it's a pretty trivial idea to show which parts of a natural language phrase correspond to which parts of something written in math. It's a gloss. You can do it with colors, or gesturing, or whatever. Finally, I believe you are thinking too mechanistically about humans, especially humans learning about math. Elsewhere you suggest that (ask about whether) recognizing Greek letters should be as easy as recognizing letters from the Roman alphabet you are familiar with... Yeah, I think if you haven't used Greek letters enough, it's going to be harder to visually parse something with Greek letters. So now some more of your cognition is being spent on something that is wholly unrelated to the task at hand. Apparently you can even measure that German people are a little slower at some arithmetic problems because their names for quantities don't always list the digits in the same order of significance. Your WORD for a number causes you to be faster or slower at arithmetic. So I think using unfamiliar symbols might also make a difference. Here you are suggesting that the syntactic structure of math definitions is the same as the syntactic structure of English. Right! They are written in English! However, even in the realm of natural language, humans will have a hard time understanding utterances with nested quantifiers, negation, and clauses. The notion of syntax is an abstraction, and from that abstract point of view, there is no difference in the expressiveness of English and Mathlish. But humans apparently don't process not even natural language syntax as mechanistically as you suggest they do. Even though shallow and deep trees can be generated by the same grammar, more complex parse trees are harder to understand. You have to be careful with an abstraction because sometimes the details (like how many words are in a sentence, or how many layers of quantifiers there are) matter.