
Re: Mathematics as a language
Posted:
Nov 6, 2010 3:11 AM


in article <4CD4EFCA.B10CF99C@gmail.com>, herbzet <herbzet@cox.net> wrote:
I guess the problem people are having with my thesis is that they are willing to accept (a) the mathematical existence of S, similar to that of the object 6, and they are willing to accept (b) the mathematical existence of a bijection f from w_1 to P(N), similar to that of the object 6, but they are not willing to accept both (a) and (b), because the object 6 is special  it really and truly exists in some sense, and that property of really existing cannot be shared by contradictory objects like S and f.  One thinks of sets as collections of objects, and one supposes that the collection P(N) either has an uncountable subset S not equipollent to P(N), or it doesn't  there's a fact of the matter about collections of objects. One of the objects S or f has an existence at least as factual as that of the object 6, and the other doesn't  though the other may enjoy some sort of hypothetical existence, not as real as the object 6.  I'm thinking that this is the reason for the caviling about the mathematical existence of logically possible objects: some mathematical objects are accepted as really existing in some sense  they compose the actual mathematical universe.  Whereas, though we may reason about hypothetical, logically possible, but nonexistent objects, they are not part of the actual mathematical universe; not like integers and real numbers, quaternions and the square root of 1, and so on. And sets.
no, it's much simpler than that. you're pointing to an actor playing the tooth fairy and you're saying "i guess that you're telling me that this actor doesn't exist.". no, we're telling you that the tooth fairy doesn't exist.

jdolan@math.ucr.edu

