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> You'd be wasting your time trying to get a deep understanding of the Russian phrase, but there's a reason to do it with the mathematical phrase. I think you'

> You'd be wasting your time trying to get a deep understanding of the Russian phrase, but there's a reason to do it with the mathematical phrase. I think you're simply wrong here. You're not talking about the amount of conceptual depth underlying it, you're talking about the relative delta in the conceptual structures between languages you already know and the language you're learning. On the whole, the Russian phrase has greater conceptual depth; however, Russian is closer to English than mathematics is, and thus you get greater conceptual transfer between your knowledge of English and Russian than your knowledge of English and mathematics, so more of your past experience remains relevant. In essence, I have something like 36-72k hours considering semantically equivalent statements to the Russian one in English, while I have something like 5-10k hours of studying mathematical statements at all. Your comment just amounts to saying that having 6x as much practice at task A as task B makes task A relatively easier, which, while true, isn't perhaps as profound as you'd meant it. Consider this: the entire field of academic inquiry containing Euler's identity is fundamentally an effort to fully understand "I went to the store".

westoncb 10y ago
> you're talking about the relative delta in the conceptual structures between languages you already know and the language you're learning I don't think so. In both of my examples, the one from mathematics and the one from Russian, language is used as a way of representing some concepts—now, the language itself also has concepts underlying it, but these are about grammatical structures, lexicon, etc.—not about the subject the language is used to talk about. To be clear, I'll call the first category of concepts 'subject concepts' and the second 'language concepts.' When you learn a new foreign language you must learn new language concepts, but not new subject concepts. You aren't re-learning what it means for one to visit a store, you are only learning a new scheme for representing that concept, which isn't nearly as deep as the initial concept itself. In learning mathematics you're having to pick up the language concepts and the subject concepts, and the subject concepts (as in the example I gave) can be quite deep.
- SomeStupidPoint 10y ago
  You've outlined my point exactly: you're comparing a case of learning language concepts to a case of learning language and subject concepts. Of course one of those is easier, but it doesn't mean that the two processes are fundamentally dissimilar, which was your initial point. Your comment is no more insightful than to say that it's easier to learn set theory knowing category theory than it is to learn English while knowing no languages, since it's purely acquisition of language rather than subject concepts. (This is actually untrue -- which is why it's easier to say, switch to romance languages than going to an Asian one from English; there's a bit of subject conceptualization in the nature of the language concepts.) It's not comparing apples to apples, which sort of reduces the point about the relative complexity of statements and the way that you learn the underlying concepts -- both language and subject. You're comparing the complexity differential of two encodings on the one hand and the total complexity involved in the other. Nonsense comparison. If you want to talk about learning Russian while knowing English, why not contrast it with learning set theory while knowing category theory? Because the language differential between spoken languages (eg, you do learn new subject concepts if you learn Japanese versus English) is comparable to the difference in mathematical underpinnings, eg, the switch from set theory to category theory.
  - westoncb 10y ago
    > ... it doesn't mean that the two processes are fundamentally dissimilar, which was your initial point. That was not my initial point. You've read the 'fundamentally' part into it. > If you want to talk about learning Russian while knowing English, why not contrast it with learning set theory while knowing category theory? That is what I was doing. I'm not saying learning math versus foreign language is fundamentally different, I'm talking about practical differences in actually learning one or the other: who in the audience here doesn't already speak one natural language? So, with any foreign language they will be in the position of somebody knowing category theory and attempting to learn set theory. That is by definition not, however, the case for someone trying to break into mathematics for the first time. So you always have language concepts only for a foreign language and language concepts + subject concepts for mathematics—practically speaking.

westoncb 10y ago

> you're talking about the relative delta in the conceptual structures between languages you already know and the language you're learning I don't think so. In both of my examples, the one from mathematics and the one from Russian, language is used as a way of representing some concepts—now, the language itself also has concepts underlying it, but these are about grammatical structures, lexicon, etc.—not about the subject the language is used to talk about. To be clear, I'll call the first category of concepts 'subject concepts' and the second 'language concepts.' When you learn a new foreign language you must learn new language concepts, but not new subject concepts. You aren't re-learning what it means for one to visit a store, you are only learning a new scheme for representing that concept, which isn't nearly as deep as the initial concept itself. In learning mathematics you're having to pick up the language concepts and the subject concepts, and the subject concepts (as in the example I gave) can be quite deep.

SomeStupidPoint 10y ago
You've outlined my point exactly: you're comparing a case of learning language concepts to a case of learning language and subject concepts. Of course one of those is easier, but it doesn't mean that the two processes are fundamentally dissimilar, which was your initial point. Your comment is no more insightful than to say that it's easier to learn set theory knowing category theory than it is to learn English while knowing no languages, since it's purely acquisition of language rather than subject concepts. (This is actually untrue -- which is why it's easier to say, switch to romance languages than going to an Asian one from English; there's a bit of subject conceptualization in the nature of the language concepts.) It's not comparing apples to apples, which sort of reduces the point about the relative complexity of statements and the way that you learn the underlying concepts -- both language and subject. You're comparing the complexity differential of two encodings on the one hand and the total complexity involved in the other. Nonsense comparison. If you want to talk about learning Russian while knowing English, why not contrast it with learning set theory while knowing category theory? Because the language differential between spoken languages (eg, you do learn new subject concepts if you learn Japanese versus English) is comparable to the difference in mathematical underpinnings, eg, the switch from set theory to category theory.
- westoncb 10y ago
  > ... it doesn't mean that the two processes are fundamentally dissimilar, which was your initial point. That was not my initial point. You've read the 'fundamentally' part into it. > If you want to talk about learning Russian while knowing English, why not contrast it with learning set theory while knowing category theory? That is what I was doing. I'm not saying learning math versus foreign language is fundamentally different, I'm talking about practical differences in actually learning one or the other: who in the audience here doesn't already speak one natural language? So, with any foreign language they will be in the position of somebody knowing category theory and attempting to learn set theory. That is by definition not, however, the case for someone trying to break into mathematics for the first time. So you always have language concepts only for a foreign language and language concepts + subject concepts for mathematics—practically speaking.