Comment by empath-nirvana | Hacker News Reader

Comment by empath-nirvana | Hacker News Reader

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Mathematics is just an exercise in symbol manipulation according to a particular rule set. Mathematics is a _game_. Park Place exists in the context of a game of Monopoly and also exists in Atlantic City. The tokens in a game of Reversi don't correspond to anything in the real world. You might be playing it on a physical board of a particular size, but you could just as easily play it on an imaginary board, or on a piece of paper, or a computer, or in some imaginary board of infinite size. Whether or not either exists in "reality" doesn't matter in the context of the game, though.

If the rules of mathematics allow for infinities, then they exist in the context of mathematics. Whether they exist in reality doesn't matter. You can still say things that are true about these imaginary objects in the context of the game that everyone is playing with these symbols, just like you can say things that are true about the game of Reversi, even one played entirely in the minds of two players on an imaginary, infinite board.

If you use "exist" to mean "is a symbol defined within some system" rather than the typical "has isomorphism with reality" then you can say literally anything exists. Unicorns exist because English has a word for them.

So be it, but it is a useless definition of exists then isn't it?

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.

They don't religiously defend their existence, they're defending the second form of exist in which Unicorns are said to exist.

Have you ever seen a negative number of things?

Yes. I have seen debt before.

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.

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