> 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).