> Now, there might be a way to weasel out of it by talking > about "possible universes": Hercules exists in one [logically] possible > universe, and Heavystone exists in a different [logically] possible > universe. But then you're waffling about the meaning of the word "exists".
The meaning of the term "mathematically exists" is indeed at issue here.
> Do you mean exists within *one* universe, or exists within *any* universe?
I really am having difficulty understand what is being asked, so I don't know how to answer it.
Hercules exists in one logically possible universe, Heavystone exists in a different logically possible universe. Both possible universes exist within the universe of logically possible universes -- I don't see the problem.
It's like this -- suppose we find in a text on number theory that there "exists" a unique prime decomposition for every natural number n > 1.
If we have some doubt about the nature of the existence of the naturals themselves, how much more so about these more complicated objects, prime decompositions of naturals? What sort of "existence" do they have?
But really, in ordinary use, there's no mystery at all. For the purpose of doing number theory, we accept the existence of naturals with their (axiomatically?) given properties, whether as hypothetical entities or as eternal Platonic objects, or whatever, and it then follows that for these entities, whatever their nature, there will be a unique prime decompostion for each one greater than one. That's all that we mean or need to mean about "existence" to do number theory.
There is no question, unless we're pretty confused, that the existence of these "prime number decompositions" is like the material existence of the Empire State building, having a definite position in space and time, etc. It's just that given that one thing "exists", it follows that another thing "exists". This is an utterly ordinary usage: nothing deep is going on here. When we study number theory, we don't worry about the ontology of naturals -- their "existence" is simply accepted as posited.
And that's really all there is to it. We can posit anything we want (e.g. Conway's "surreal" numbers") and reason about what properties these posited objects "have" -- that is, what properties of these objects "exist".
It's just a way of talking -- we suspend disbelief, and take them as having "real" existence of a sort. And until a contradiction turns up, the game is a good game. You never know, it might usefully model some phenomena in the physical world -- it's happened before.
That's what I mean when I say that mathematical existence is logically possible existence. It's really very ordinary.