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> 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

> 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.
- catnaroek 9y ago
  > This sounds like being familiar with an example of an abstract idea is as ["pretty much" as] useless as not knowing anything about it Right! > I am not persuaded that thinking concretely is akin to rejecting abstractions, as you seem to be. I'm not arguing against thinking concretely. I'm arguing against highlighting the syntax of English sentences and math formulas just because some people couldn't parse them, let alone semantically analyze them, otherwise. > Notice that in the course of your argument, you keep bringing concrete math examples to make your point. Of course. As I previously said, examples serve to illustrate a general concept. > it's a pretty trivial idea to show which parts of a natural language phrase correspond to (...) Indeed. > You can do it with colors, or gesturing, or whatever. You can do it with your brain. > 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. Um, wat. Operating on a number doesn't require thinking of the word you use for that number. > However, even in the realm of natural language, humans will have a hard time understanding utterances with nested quantifiers, negation, and clauses. And they systematically let people graduate from high school in this pathetic state? Maybe I got it wrong: Don't sue your elementary school. Sue your country's education ministry.
  - gugagore 9y ago
    > And they systematically let people graduate from high school in this pathetic state? You're being nasty. There's nothing pathetic about recognizing that almost everyone experiences limits in their cognition. If you don't, then that must be very nice. Perhaps you misunderstood. People can understand a sentence with a few levels of negation, quantification, embedded clauses, but not too many. And sentences in math are often more complex than sentences in casual speech. You made a bold claim that math syntax is equivalent to natural language syntax, and I am demonstrating why that claim is wrong, or at best not relevant to coloring math expressions. I understand you are claiming they are formally be the same (who knows, since no one has completely formalized "The Math Language" or probably any natural language, but I understand your point in the context of baby first order logic syntax). And I am claiming that that doesn't matter. > I'm arguing against highlighting the syntax of English sentences and math formulas just because some people... No one is saying you should colorize every textbook or every anything, so I don't understand what you are arguing. > You can do it with your brain. Yeah, and you need to feed your brain some input. There are lots of ways you can encode the same information. If you suddenly no longer had vision, someone could read to you unambiguously every math expression you needed, but I bet you're going to suffer a hit in your math performance. Despite using your brain. > Um, wat. Operating on a number doesn't require thinking of the word you use for that number. Obviously. A computer operates on a number without having a "word" for a number. (Well...). And some humans are better at being computers than other humans. That's not responding to my point that shows that people must be thinking about words, because you can measure delays when the words aren't very good, like the number words in German or French. Yeah, I agree it's "Um, wat" in an interesting way, but I fear you're just dismissing it.

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.

catnaroek 9y ago
> This sounds like being familiar with an example of an abstract idea is as ["pretty much" as] useless as not knowing anything about it Right! > I am not persuaded that thinking concretely is akin to rejecting abstractions, as you seem to be. I'm not arguing against thinking concretely. I'm arguing against highlighting the syntax of English sentences and math formulas just because some people couldn't parse them, let alone semantically analyze them, otherwise. > Notice that in the course of your argument, you keep bringing concrete math examples to make your point. Of course. As I previously said, examples serve to illustrate a general concept. > it's a pretty trivial idea to show which parts of a natural language phrase correspond to (...) Indeed. > You can do it with colors, or gesturing, or whatever. You can do it with your brain. > 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. Um, wat. Operating on a number doesn't require thinking of the word you use for that number. > However, even in the realm of natural language, humans will have a hard time understanding utterances with nested quantifiers, negation, and clauses. And they systematically let people graduate from high school in this pathetic state? Maybe I got it wrong: Don't sue your elementary school. Sue your country's education ministry.
- gugagore 9y ago
  > And they systematically let people graduate from high school in this pathetic state? You're being nasty. There's nothing pathetic about recognizing that almost everyone experiences limits in their cognition. If you don't, then that must be very nice. Perhaps you misunderstood. People can understand a sentence with a few levels of negation, quantification, embedded clauses, but not too many. And sentences in math are often more complex than sentences in casual speech. You made a bold claim that math syntax is equivalent to natural language syntax, and I am demonstrating why that claim is wrong, or at best not relevant to coloring math expressions. I understand you are claiming they are formally be the same (who knows, since no one has completely formalized "The Math Language" or probably any natural language, but I understand your point in the context of baby first order logic syntax). And I am claiming that that doesn't matter. > I'm arguing against highlighting the syntax of English sentences and math formulas just because some people... No one is saying you should colorize every textbook or every anything, so I don't understand what you are arguing. > You can do it with your brain. Yeah, and you need to feed your brain some input. There are lots of ways you can encode the same information. If you suddenly no longer had vision, someone could read to you unambiguously every math expression you needed, but I bet you're going to suffer a hit in your math performance. Despite using your brain. > Um, wat. Operating on a number doesn't require thinking of the word you use for that number. Obviously. A computer operates on a number without having a "word" for a number. (Well...). And some humans are better at being computers than other humans. That's not responding to my point that shows that people must be thinking about words, because you can measure delays when the words aren't very good, like the number words in German or French. Yeah, I agree it's "Um, wat" in an interesting way, but I fear you're just dismissing it.