Math is just a language. I say “just” not to discount its power, but because it really doesn’t exist outside of our conception of it, just as English doesn’t exist outside of our conception of it. It’s a convention.
The key difference between math and spoken language is that it’s unambiguous enough to extrapolate on fairly consistently. If English were that precise we might be able to find truth in the far reaches of the language, just like greek philosophers tried to do. With math, such a thing is actually possible.
So, 2+3=5 corresponds to your dots or sheep, and that’s the whole fact of the matter. Cats are called cats because that’s what we feel like calling them and calling them dogs won’t change their cat-ness.
It FEELS like there should be more because of the way we are accustomed to extrapolating math. There is no additional fact to account for, though.
The only time this isn’t really the case is with exotic math which corresponds to a basically “counterfactual” world like “What if the world were made of city blocks?” (Taxi Cab Geometry). It’s true that we can imagine false worlds and invent precise language to describe those worlds, but such a description does not make them less false, just more vivid fiction.
Math is just a language. I say “just” not to discount its power, but because it really doesn’t exist outside of our conception of it, just as English doesn’t exist outside of our conception of it. It’s a convention.
The key difference between math and spoken language is that it’s unambiguous enough to extrapolate on fairly consistently. If English were that precise we might be able to find truth in the far reaches of the language, just like greek philosophers tried to do. With math, such a thing is actually possible.
So, 2+3=5 corresponds to your dots or sheep, and that’s the whole fact of the matter. Cats are called cats because that’s what we feel like calling them and calling them dogs won’t change their cat-ness.
It FEELS like there should be more because of the way we are accustomed to extrapolating math. There is no additional fact to account for, though.
The only time this isn’t really the case is with exotic math which corresponds to a basically “counterfactual” world like “What if the world were made of city blocks?” (Taxi Cab Geometry). It’s true that we can imagine false worlds and invent precise language to describe those worlds, but such a description does not make them less false, just more vivid fiction.