herb z
Posts:
1,187
Registered:
8/26/06


Re: Mathematics as a language
Posted:
Nov 6, 2010 2:03 AM


Herman Jurjus wrote: > Marshall wrote: > > herbzet wrote: > >> Herman Jurjus wrote: > >>> Marshall wrote: > >>>> Herman Jurjus wrote: > >>>>> herbzet wrote: > >>>>>> Aatu Koskensilta wrote: > >>>>>>> herbzet writes: > >>>>>>>> Bill Taylor wrote: > >> > >>>>>>>>> Or whether the number 6 really exists. Does it? > >> > >>>>>>>> It *could* exist  therefore, mathematically, it *does* exist. > >> > >>>>>>> This is a traditional and appealing idea. But just what is meant by > >>>>>>> "could" here? What sort of possibility is involved? > >> > >>>>>> For rhetorical punch, I purposely left out the modifier, which is "logical". > >> > >>>>>> What logically could exist  that is, what is not inherently self > >>>>>> contradictory  has mathematical existence. > >> > >>>>> Corollary: CH is false. > >>>>> Proof: Since Cohen 1963 we know that it is logically consistent to > >>>>> assume that there exists S, subset of P(N), equipollent neither to N nor > >>>>> to P(N). > >> > >>>> Consistent with what? In what theory? > >> > >>> Coconsistent with ordinary mathematics, of course. > >>> (I.e. with ZFC, and then also with any weaker theory.) > >> > >> Right  the assumption here is that ordinary mathematics > >> (i.e. ZFC, more or less) is itself consistent  the ordinary > >> and unremarkable gentleman's agreement. > > > > Ok. But again, all the "counterexamples" just amount to saying > > that (most) theories have undecidable sentences, right? > > It may be the case that the remark 'most theories have undecidable > sentences' is used early on in the argument, but it's not the crux of > what's being said. > > Here's the same argument in terms of 'real' mathematics: > > If you start a paper or discourse with > "Let S be an uncountable subset of P(N), not equipollent to P(N)" (*) > and you proceed from there, then, as long as you stick to ordinary > mathematics, you'll never reach a contradiction (unless the ordinary > mathematics that you use is in itself already inconsistent). > > So, if mere absence of contradiction (using ordinary mathematical > reasoning alone) is argument enough for mathematical existence, then you > would be compelled to accept the mathematical existence of S, similar to > that of the object '6'.
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.
 hz
> (*) And likewise, of course, with > "Let f be a bijection from w_1 to P(N)"

