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> Which fields of human endeavour cannot be rendered symbolically? All of them other than math. > do you consider geometric proofs math? Yes, because diagram

by lisper 5y ago

> Which fields of human endeavour cannot be rendered symbolically? All of them other than math. > do you consider geometric proofs math? Yes, because diagrams are symbols. They happen to be rendered in 2-D but they are symbols nonetheless. What makes something a symbol is that it is subject to a convention that allows it to fall into one of a finite number of equivalence classes, so that small changes in its physical details don't change its function within the context of the symbol-manipulation activity. Geometric diagrams have this property, so they qualify as symbols.

bawolff 5y ago
I don't think i agree with that definition of symbol. I think that's closer to the definition of an abstraction, which i would consider to be different. Regardless, for the sake of argument lets accept it. Words also have this property. The word "Dog" refers to an equivalence class of a bunch of different physical manifestations which are considered equivalent in the context the word is used. Lets take a first aid textbook. This usually consists of a bunch of words describing situations, and a bunch of 2-D diagrams. This meets your definition of symbolic as far as i can tell. Is first-aid a sub-discipline of mathematics in your view?
- lisper 5y ago
  > I don't think i agree with that definition of symbol. Really? How would you define it? > The word "Dog" refers to an equivalence class Yes, and if you wanted to, you could use dogs (actual physical dogs, not the word "dog") as symbols. It would be kind of a silly thing to do, but you could. (For the sake of completeness, because I'm not 100% sure this is what you were referring to, the word "dog" is unambiguously a symbol, one which denotes the equivalence calls of actual physical dogs. The equivalence class of the word dog includes things like "Dog" and "DOG" and "perro" and "hund".) > Is first-aid a sub-discipline of mathematics in your view? No, because the business of first aid is not the business of manipulating those symbols, it is the business of manipulating actual physical bodies. First-aid isn't math for the same reason that (say) building computers isn't math. This is not to say that math can't be a useful part of either of those activities, but they aren't equivalent.
  - bawolff 5y ago
    > Really? How would you define it? Well first off, i'm not sure why you're requiring finiteness or what that means in context. But more to the point, i would define it as something that represents (denotes) something else. The thing it represents may be concrete, it might be abstract (represent a group of all things that some property), it might not exist at all, etc. I suppose i'm actually a bit confused by your definition, i find it hard to parse. "What makes something a symbol is that it is subject to a convention that allows it to fall into one of a finite number of equivalence classes,": it sounds like what you're saying is that a symbol is defined as being a representative member of one or more equivalence classes, which seems obviously wrong, as symbols typically aren't members of the class of objects they represent. I was previously interpreting your statement as missing the word "denote". But denote is kind of the core of the definition of a symbol. If what you meant by your definition was that a symbol is something that denotes an equivalence class, you could probably drop the equivalence class bit, as all concepts can be thought of as equivalence classes with zero or more members, so it doesn't add anything to the definition. > No, because the business of first aid is not the business of manipulating those symbols, it is the business of manipulating actual physical bodies. Sure (although what about the activity of writing first-aid textbooks?), but that's a pretty dramatic shift in goal posts from "precisely at the point where you can render those images symbolically". Being able to "Render" something symbolically is very different from having the end goal of the activity being the manipulation of the symbols. I agree having symbol manipulation be the chief end, feels closer to a definition of math, but it still feels kind of lacking. Mathematicians don't just arbitrarily manipulate symbols, they choose what to do based on a variety of criteria, such as real world applications, beauty, connectedness to other areas, etc. The ultimate end for actual mathematicians isn't solely symbol manipulation. On the other side, poets manipulate symbols (words) following certain rules. They do so for a variety of reasons, but usually to evoke emotion or perhaps beauty in the minds of their readers. The final goal is a series of symbols manipulated just the right way. Are they mathematicians?
    - lisper 5y ago
      > symbols typically aren't members of the class of objects they represent That's right. The essential characteristic of a symbol is that it retains its essential character through certain kinds of changes. It's hard to illustrate via ascii text because all of the instances of a given symbol render in exactly the same way, but if I could hand-draw two letter A's I could make them look slightly different but they would still be indentifiable as A's. In some contexts, an upper and lower case character can be the "same symbol", so for example, "DOG" and "dog" are the "same word" (which is to say, the same symbol, because words are symbols). The reason the number of equivalence classes has to be finite is because the things doing the manipulating are physical entities (like humans or computers) they are only capable of dealing with a finite number of different kinds of things. > what about the activity of writing first-aid textbooks? You posed that as a parenthetical but it's a fair question. Let me reflect it back at you: imagine that AI technology has advanced to the point where it is capable of producing original first-aid textbooks. Is that math? > Mathematicians don't just arbitrarily manipulate symbols, they choose what to do based on a variety of criteria, such as real world applications, beauty, connectedness to other areas, etc. The ultimate end for actual mathematicians isn't solely symbol manipulation. Of course. That's why part of my definition was that there had to be interesting or useful results. > poets manipulate symbols (words) following certain rules Really? What are those rules?

by lisper 5y ago

bawolff 5y ago

I don't think i agree with that definition of symbol. I think that's closer to the definition of an abstraction, which i would consider to be different. Regardless, for the sake of argument lets accept it. Words also have this property. The word "Dog" refers to an equivalence class of a bunch of different physical manifestations which are considered equivalent in the context the word is used. Lets take a first aid textbook. This usually consists of a bunch of words describing situations, and a bunch of 2-D diagrams. This meets your definition of symbolic as far as i can tell. Is first-aid a sub-discipline of mathematics in your view?

lisper 5y ago
> I don't think i agree with that definition of symbol. Really? How would you define it? > The word "Dog" refers to an equivalence class Yes, and if you wanted to, you could use dogs (actual physical dogs, not the word "dog") as symbols. It would be kind of a silly thing to do, but you could. (For the sake of completeness, because I'm not 100% sure this is what you were referring to, the word "dog" is unambiguously a symbol, one which denotes the equivalence calls of actual physical dogs. The equivalence class of the word dog includes things like "Dog" and "DOG" and "perro" and "hund".) > Is first-aid a sub-discipline of mathematics in your view? No, because the business of first aid is not the business of manipulating those symbols, it is the business of manipulating actual physical bodies. First-aid isn't math for the same reason that (say) building computers isn't math. This is not to say that math can't be a useful part of either of those activities, but they aren't equivalent.
- bawolff 5y ago
  > Really? How would you define it? Well first off, i'm not sure why you're requiring finiteness or what that means in context. But more to the point, i would define it as something that represents (denotes) something else. The thing it represents may be concrete, it might be abstract (represent a group of all things that some property), it might not exist at all, etc. I suppose i'm actually a bit confused by your definition, i find it hard to parse. "What makes something a symbol is that it is subject to a convention that allows it to fall into one of a finite number of equivalence classes,": it sounds like what you're saying is that a symbol is defined as being a representative member of one or more equivalence classes, which seems obviously wrong, as symbols typically aren't members of the class of objects they represent. I was previously interpreting your statement as missing the word "denote". But denote is kind of the core of the definition of a symbol. If what you meant by your definition was that a symbol is something that denotes an equivalence class, you could probably drop the equivalence class bit, as all concepts can be thought of as equivalence classes with zero or more members, so it doesn't add anything to the definition. > No, because the business of first aid is not the business of manipulating those symbols, it is the business of manipulating actual physical bodies. Sure (although what about the activity of writing first-aid textbooks?), but that's a pretty dramatic shift in goal posts from "precisely at the point where you can render those images symbolically". Being able to "Render" something symbolically is very different from having the end goal of the activity being the manipulation of the symbols. I agree having symbol manipulation be the chief end, feels closer to a definition of math, but it still feels kind of lacking. Mathematicians don't just arbitrarily manipulate symbols, they choose what to do based on a variety of criteria, such as real world applications, beauty, connectedness to other areas, etc. The ultimate end for actual mathematicians isn't solely symbol manipulation. On the other side, poets manipulate symbols (words) following certain rules. They do so for a variety of reasons, but usually to evoke emotion or perhaps beauty in the minds of their readers. The final goal is a series of symbols manipulated just the right way. Are they mathematicians?
  - lisper 5y ago
    > symbols typically aren't members of the class of objects they represent That's right. The essential characteristic of a symbol is that it retains its essential character through certain kinds of changes. It's hard to illustrate via ascii text because all of the instances of a given symbol render in exactly the same way, but if I could hand-draw two letter A's I could make them look slightly different but they would still be indentifiable as A's. In some contexts, an upper and lower case character can be the "same symbol", so for example, "DOG" and "dog" are the "same word" (which is to say, the same symbol, because words are symbols). The reason the number of equivalence classes has to be finite is because the things doing the manipulating are physical entities (like humans or computers) they are only capable of dealing with a finite number of different kinds of things. > what about the activity of writing first-aid textbooks? You posed that as a parenthetical but it's a fair question. Let me reflect it back at you: imagine that AI technology has advanced to the point where it is capable of producing original first-aid textbooks. Is that math? > Mathematicians don't just arbitrarily manipulate symbols, they choose what to do based on a variety of criteria, such as real world applications, beauty, connectedness to other areas, etc. The ultimate end for actual mathematicians isn't solely symbol manipulation. Of course. That's why part of my definition was that there had to be interesting or useful results. > poets manipulate symbols (words) following certain rules Really? What are those rules?