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From: Math1723 on 13 Jan 2010 11:40 On Jan 13, 10:28 am, Leonid Lenov <leonidle...(a)gmail.com> wrote: > > Now I have to have this book :). It will also give me more arguments > with which to defend Platonism form all those "heretics". Why are > there some people against Platonism will never be clear to me... I am certainly not a critic of Platonism, but the danger (I think) that might arise is when a mathematician chases what he thinks is "real" as opposed to relying on truths of mathematics. I think of myself probably in between. For example, I believe in the Axiom of Choice. It seems ridiculous to disallow one from, given a collection of sets, being able to choose one element out of each set. For me, AC is absolutely and fundamentally "true". However, that does not prevent me from seeing what theorems come out of ZF + ~AC. There is a rich and interesting collection of objects that result from the denial of A.C. (for example, sets that are infinite but Dedekind-finite). Certain very interesting large cardinals are inconsistent with A.C. As a Formalist, I can enjoy all of these "what if" objects. But in the end, I see them as fictional. I believe in A.C. Strangely though, I don't "believe in" the truth or falsity of Euclid's Parallel Postulate. Like a game, I can decide up front whether or not I agree to PP (or ~PP) and investigate the rich results from either decision. Sorry for that long winded response, and a sincere thank you for your input.
From: scattered on 13 Jan 2010 11:46 [snip] > > Again, I am curious as to how you apply this principle to my analogy > with Geometry. > [snip] The difference is that standard axiomizations of the real numbers are categorical - meaning that they have only one model (e.g. Dedekind- complete ordered fields are unique up to a field isomporhpism). The first four postulates of Euclid do *not* specify a categorical theory. What is happenening with Euclidean vs. NonEuclidean geometries is much more akin to what is happenening with Abelian vs. nonAbelian groups. It doesn't make sense to ask if *the* group is really abelian, since there is no "the group". Similarly it does not make sense to say if "the geometry" is really Euclidean - since there isn't any unique geometry. But there *is* a unique set of real numbers (up to isomorphism - which preserves cardinality) hence it makes perfect sense to ask if the real numbers really satisfies CH. The fact that ZFC can't answer that question doesn't detract from the cogency of the question. -scattered
From: Leonid Lenov on 13 Jan 2010 12:01 On Jan 13, 5:40 pm, Math1723 <anonym1...(a)aol.com> wrote: > However, that does not prevent me from seeing what theorems come out > of ZF + ~AC. There is a rich and interesting collection of objects > that result from the denial of A.C. (for example, sets that are > infinite but Dedekind-finite). Certain very interesting large > cardinals are inconsistent with A.C. As a Formalist, I can enjoy all > of these "what if" objects. And as a Platonist you cannot? I think that consistency implies existence. That is, if a set of axioms is consistent then there exist objects that those axioms talk about. For example if ZFC is consistent than the sets of ZFC exist. However that does not imply that you cannot enjoy all other "what if" objects as you call them... They all exist provided there is a consistent set of axioms that describe them.
From: Math1723 on 13 Jan 2010 12:44 On Jan 13, 11:46 am, scattered <still.scatte...(a)gmail.com> wrote: > [snip] > > > Again, I am curious as to how you apply this principle to my analogy > > with Geometry. > > [snip] > > The difference is that standard axiomizations of the real numbers are > categorical - meaning that they have only one model (e.g. Dedekind- > complete ordered fields are unique up to a field isomporhpism). Yes, but "Dedekind-completeness" is not itself a model-independent concept. You can have two models of R, say M1 and M2, with reals R1 and R2 respectively, in which R1 and R2 are not isomorphic. In M1, R1 is necessarily M1-Dedekind-complete and in M2, and R2 is necessarily M2-Dedekind-complete. Yet (depending on M1 and M2), neither would be "Dedekind complete" in the other model. It seems that all you have done here is replace the concept of a "real" R with the concept of "real" Dedekind Completeness. Nothing you have said persuades me out of thinking it's *possible* that there are more than one concepts of real numbers, each equally valid and real, and each with differing values for 2^Aleph_0. > The first four postulates of Euclid do *not* specify a categorical > theory. What is happenening with Euclidean vs. NonEuclidean > geometries is much more akin to what is happenening with Abelian > vs. nonAbelian groups. It doesn't make sense to ask if *the* group > is really abelian, since there is no "the group". Similarly it does > not make sense to say if "the geometry" is really Euclidean - since > there isn't any unique geometry. I admit my ignorance of Category Theory, so I cannot respond to your point on that. However, a century or two ago, there were people arguing passionately that there is indeed a unique geometry, in much the same way you argue that there is a unique concept of R, and saying rather ugly things about proponents of Non-Euclidean Geometry. You say now that there isn't a unique Geometry. How can you be certain that a hundred years from now, mathematicians will agree that R isn't unique either. > But there *is* a unique set of real numbers (up to > isomorphism - which preserves cardinality) hence it makes perfect > sense to ask if the real numbers really satisfies CH. The very fact that there are models of R with differing answers to CH seems to me to make such an assertion a bit hasty. Thanks for continuing the discussion.
From: Math1723 on 13 Jan 2010 12:51
On Jan 13, 12:01 pm, Leonid Lenov <leonidle...(a)gmail.com> wrote: > > And as a Platonist you cannot? I think that consistency implies > existence. That is, if a set of axioms is consistent then there exist > objects that those axioms talk about. For example if ZFC is consistent > than the sets of ZFC exist. However that does not imply that you > cannot enjoy all other "what if" objects as you call them... They all > exist provided there is a consistent set of axioms that describe them. Fair point. Then would you not concede then that since ZFC + CH and ZFC + ~CH are each equally consistent, that therefore asking about the "actual" truth of CH is meaningless? What does it mean for CH to be "actually" true (or "actually" false) when each possibility equally "exists" (as per your "consistency implies existence")? |