If you're playing dungeons and dragons, then unicorns "exist" within the context of that game, and there are rules for interacting with them. If you're reading a fantasy novel, then unicorns exist within the context of that fictional universe.
And because people enjoy playing Dungeons and Dragons and reading fantasy novels, that existence has real value both for those people and for the people creating those novels and games. And you can say that certain things are true about unicorns within the context of those games and fictional worlds. (For example, it is true that unicorns have horns).
If it makes you feel better to put infinity into the same category of reality as "unicorns", then I think that's fine? It doesn't necessarily need to be the case that anything in mathematics corresponds to something that is physically manifested, and probably _most_ of mathematics does not -- not just infinities.
Yes. Infinity doesn't exist in the same way unicorns don't exist. The difference is that some people religiously believe in the existence of infinities and try to defend their existence even though nobody has ever seen one: see other replies to this thread.
No, you haven't. You've seen some symbols that represent the ludicrous physical notion of a negative amount of an object. Look outside, count negative 3 sheep. You can't. It's a _fiction_ which happens to be useful for accounting for debt.
Sure I can. I start with the number of sheep I expect: 30, and then count backwards. When I count to -3 I know that I have too many sheep. There is no rule that says counting backwards isn't useful. Any positive number can be represented as a negative number in relation with a constant and remain a perfectly isomorphic model of reality.
Infinities are quite different, given that it is by definition impossible to measure whether something in reality is infinite or not unless you have infinite time.
> Infinities are quite different, given that it is by definition impossible to measure whether something in reality is infinite or not unless you have infinite time.
It's impossible to count a negative number of things. It's impossible to count an imaginary number of things, it's impossible to count a matrix of things. It's impossible to count an irrational number of things. These are all fictions that people use because they're useful. Infinity is exactly the same as any of those other concepts.
No, infinity is a different concept to all of those quite distinct concepts. There are also clear and observable uses for all of those things that make them hard to replace, whereas infinity is unique in that if you drop it as an axiom then everything still works and your mathematical system becomes more rigorous.
Also I clearly just demonstrated that you can count negative numbers easily so I don't understand why you find it impossible.
If it makes you feel better to put infinity into the same category of reality as "unicorns", then I think that's fine? It doesn't necessarily need to be the case that anything in mathematics corresponds to something that is physically manifested, and probably _most_ of mathematics does not -- not just infinities.