Language is sometimes confused for thinking. That's a category error. Language is a tool to express, organize, store, and recall thoughts. It works for most types of thought, but not all. And there are other ways to express thoughts -- gestures, facial expressions, drawings, etc.
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You are so close, yet so far. No, language doesn't do that by itself. Writing systems do that.
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It's unfortunate that they do require a high level of mastery to accurately render the details of most underlying thoughts. And some of those systems have encoding limitations too, especially when it comes to real-time rendering.
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Oh. <Click> Did I not already know that? If I did, why have I assumed the recursive feature of human thought arose from tool use? But perhaps tools and language both drove recursive thought.
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I constantly think if the structure of language has some impact on the way people are. More structured languages make the society more organized maybe?
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Yes, and we should to this: language is the interface and the way we can communicate and program others how to think by reading the thoughts that we express. (Only for addressees that recognize them, of course).
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The most telling examples are qualias: when you talk about "pain" for instance, this word bears no meaning in itself if you have not felt pain once yourself. It is in stark opposition to mathematical sentences for instance in which meaning is carried by the syntax.
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In another perspective language is the portal between the world of ideas and the world of objects. Writing allows you to turn ideas in a tangible object, dually reading makes the world of objects modify your thoughts.
End of conversation
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