4 ms·

> 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 essent

by 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?

dwohnitmok 5y ago
> Really? What are those rules? This is too fun so I can't help myself. A stark example here is of haikus, which generally have few requirements by way of grammar, and only require syllabic rules. To parallel propositional logic, we can define an English haiku as a logic through the following: The set of symbols are the words of the English language, with punctuation marks and newline symbols. The language is defined as the free language generated by the symbols. Transformation (inference) rules consist of the following: If we have no sentences we may add (infer) an axiom. If the first sentence consists of five syllables, we may infer a new sentence of free words consisting of seven syllables. If the second sentence consists of seven syllables, we may infer a new sentence of free words consisting of five syllables. If the third sentence consists of five syllables, we may infer top. Otherwise if none of these rules are productive, we may infer bottom (we never infer bottom). Our axiom set consists of all five syllable sets of English words. That is the formal system of haikus. Now you can argue that this system doesn't have interesting or useful results, but it's a bit difficult to argue that purely syntactically. You probably need some notion of semantics, and it becomes quite difficult to explain semantics purely using symbols and rules. In particular it can get a bit thorny to describe what a "result" is purely syntactically, i.e. only as the consequence of symbols and manipulation rules.
- lisper 5y ago
  > haikus I must confess I had to think about this one for a while. Yes, haikus adhere to strict rules (as do sonnets). But here is something that adheres to the rules of haiku: foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo which I think you would agree is not a haiku, or at least not a very good one (unless you're a Dadaist). There is something essential to a haiku, and to a poem in general, that is not captured by the rules. But there are situations where the boundary between math and poetry can be very fuzzy indeed.
  - bawolff 5y ago
    I think i would say the same about math. You can manipulate logical statements in an infinite number of ways, but only a subset are interesting. What makes an interesting/good poem is just as subjective as what makes an interesting/good theorem.
    - lisper 5y ago
      That's not quite true. Some math is objectively useful in ways that poems are not because the math corresponds to -- and hence allows you to accurately predict -- the behavior of physical systems.
bawolff 5y ago
> 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). Ok, I see where you're coming from with that. I think I would conceptualize this differently: ¼+¼=½ and 1/4+1/4=1/2 Are both two separate "symbols". These symbols both however denote another symbol, the abstract idea of the equation. This abstract idea is what mathematicians are manipulating. The scratches on a piece of paper are just a symbol that represents the symbol being manipulated. This abstract symbol in turn is a symbol which represents different concrete realizations (e.g. it might represent drinking a quarter cup of water, and then another one, for a total of half a cup. Or whatever). To summarize, People add "numbers" together. They don't add pieces of writing together, those just represent what they add together. Each numeral written on a piece of paper is unique, but they represent the same number. I suppose that view might be more philosophical, where its not definitively right or wrong but more a matter of taste. On this view, you could certainly create an equivalence class of all the symbols that denote the same thing, but its hardly necessary and not core to the definition. In my view, symbols don't have to be physically realizable, so finiteness is not a requirement. > 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? I have a different definition of math than you. While i think i disagree with your definition, its not exactly obvious to me what the correct one would be. I think I would go with something along the lines of: Math is the process of discovering new truths and consequences of abstract formal systems using logical deduction. Good math, is math that discovers truths which are interesting from a practical or aesthetic standpoint. On my definition, I would say no. While writing a book does involve the manipulation of abstract symbols as an end (Which i think makes it math in your definition), it doesn't involve discovering any consequences of a formal system. I don't think whether an AI or a human writes it makes much difference. > Really? What are those rules? Sibling post discussed haikus which are an obvious example. Other examples involve meter or rhyming scheme. If nothing else, there's rules about which sets of characters make words in english and what not. The key thing I'm trying to get at is that poetry involves putting symbols in a specific order following certain rules to get some affect. That seems to match your definition of math afaict. I don't think it is math, because it doesn't capture any truths about abstract systems (Maybe you could argue it captures emotional essence of what it is to be human in a similar way as how math captures the essence of what it is to be a triangle and what not, but that's kind of a tangent).

by lisper 5y ago

dwohnitmok 5y ago

> Really? What are those rules? This is too fun so I can't help myself. A stark example here is of haikus, which generally have few requirements by way of grammar, and only require syllabic rules. To parallel propositional logic, we can define an English haiku as a logic through the following: The set of symbols are the words of the English language, with punctuation marks and newline symbols. The language is defined as the free language generated by the symbols. Transformation (inference) rules consist of the following: If we have no sentences we may add (infer) an axiom. If the first sentence consists of five syllables, we may infer a new sentence of free words consisting of seven syllables. If the second sentence consists of seven syllables, we may infer a new sentence of free words consisting of five syllables. If the third sentence consists of five syllables, we may infer top. Otherwise if none of these rules are productive, we may infer bottom (we never infer bottom). Our axiom set consists of all five syllable sets of English words. That is the formal system of haikus. Now you can argue that this system doesn't have interesting or useful results, but it's a bit difficult to argue that purely syntactically. You probably need some notion of semantics, and it becomes quite difficult to explain semantics purely using symbols and rules. In particular it can get a bit thorny to describe what a "result" is purely syntactically, i.e. only as the consequence of symbols and manipulation rules.

lisper 5y ago
> haikus I must confess I had to think about this one for a while. Yes, haikus adhere to strict rules (as do sonnets). But here is something that adheres to the rules of haiku: foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo which I think you would agree is not a haiku, or at least not a very good one (unless you're a Dadaist). There is something essential to a haiku, and to a poem in general, that is not captured by the rules. But there are situations where the boundary between math and poetry can be very fuzzy indeed.
- bawolff 5y ago
  I think i would say the same about math. You can manipulate logical statements in an infinite number of ways, but only a subset are interesting. What makes an interesting/good poem is just as subjective as what makes an interesting/good theorem.
  - lisper 5y ago
    That's not quite true. Some math is objectively useful in ways that poems are not because the math corresponds to -- and hence allows you to accurately predict -- the behavior of physical systems.

bawolff 5y ago

> 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). Ok, I see where you're coming from with that. I think I would conceptualize this differently: ¼+¼=½ and 1/4+1/4=1/2 Are both two separate "symbols". These symbols both however denote another symbol, the abstract idea of the equation. This abstract idea is what mathematicians are manipulating. The scratches on a piece of paper are just a symbol that represents the symbol being manipulated. This abstract symbol in turn is a symbol which represents different concrete realizations (e.g. it might represent drinking a quarter cup of water, and then another one, for a total of half a cup. Or whatever). To summarize, People add "numbers" together. They don't add pieces of writing together, those just represent what they add together. Each numeral written on a piece of paper is unique, but they represent the same number. I suppose that view might be more philosophical, where its not definitively right or wrong but more a matter of taste. On this view, you could certainly create an equivalence class of all the symbols that denote the same thing, but its hardly necessary and not core to the definition. In my view, symbols don't have to be physically realizable, so finiteness is not a requirement. > 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? I have a different definition of math than you. While i think i disagree with your definition, its not exactly obvious to me what the correct one would be. I think I would go with something along the lines of: Math is the process of discovering new truths and consequences of abstract formal systems using logical deduction. Good math, is math that discovers truths which are interesting from a practical or aesthetic standpoint. On my definition, I would say no. While writing a book does involve the manipulation of abstract symbols as an end (Which i think makes it math in your definition), it doesn't involve discovering any consequences of a formal system. I don't think whether an AI or a human writes it makes much difference. > Really? What are those rules? Sibling post discussed haikus which are an obvious example. Other examples involve meter or rhyming scheme. If nothing else, there's rules about which sets of characters make words in english and what not. The key thing I'm trying to get at is that poetry involves putting symbols in a specific order following certain rules to get some affect. That seems to match your definition of math afaict. I don't think it is math, because it doesn't capture any truths about abstract systems (Maybe you could argue it captures emotional essence of what it is to be human in a similar way as how math captures the essence of what it is to be a triangle and what not, but that's kind of a tangent).