Ralf Bader wrote: > Lester Zick wrote: > > > On 14 Jul 2006 12:30:47 -0700, "Gene Ward Smith" > > <firstname.lastname@example.org> wrote: > > > >> > >>Lester Zick wrote: > >>> On 13 Jul 2006 19:35:42 -0700, "Gene Ward Smith" > >>> <email@example.com> wrote: > >> > >>> >I'm hardly in a position to play Miss Manners, but I think one > >>> >could reasonably complain in both cases. One thing I am very tired > >>> >of reading on this list are the constant attempts to denigrate the > >>> >mathematical work of one distinguished mathematician or another, or > >>> >one school of mathematicians or another. I think ripping in to the > >>> >positions people take on the philosophy of mathematics is one > >>> >thing, but treating the mathematical work of the people involved is > >>> >quite something else. > >>> > >>> And how exactly does one distinguish one from the other? > >> > >>If they prove theorems, they are doing mathematics. In which case, > >>keep your greasy paws off of them in terms of philosophy, and look at > >>the proofs and theorems. > > > > If one reasons in formal a priori terms one is doing math? > > > >> People who > >>> call themselves mathematicians invariably work within the context of > >>> one mathematical paradigm or another and one set of axiomatic > >>> beliefs or another. > >> > >>This is quite false. Mathematicians can, and do, differ sharply on > >>philsophical issues, but can readily agree nonetheless on mathematical > >>issues. > > > > How about the axiomatic assumptions they reason from? Are they math or > > greasy mathematical philosophy? > > > >>> Their math is no more correct than the paradigm. > >> > >>This makes no sense. Two people prove the same theorem independently, > >>and send it in for publication. One gets rejected because he holds the > >>wrong philosophical beliefs about mathematics, and hence his proof is > >>invalid. The other, who used the same proof, has his paper accepted > >>because his philosophy of mathematics is correct. What is wrong with > >>this picture? > > > > How does one know whether any philosophy of math is correct or not? It > > may or may not be conventional but whether or not it's correct in the > > sense of being true is something axiomatic reasoners are not able to > > address or demonstrate conclusively. If your complaint is that greasy > > mathematical birds flock together,it's undoubtedly only because people > > who deal in speculative assumptions of truth for axioms have no choice > > because they have no better basis for demonstrating the truth of what > > they assume to be true than their assumptions of truth. > > Why do you think that mathematical axioms, and therefore the theories > derived from them, should be true? Can a piece of music be true? An > architect's plan for a building? A computer program? I don't see any > sense in asserting that the axioms of Euclidean geometry or of group > theory are true or wrong.
This is a situation where we typically consider several different models and it's natural to say what's true depends on the model. In the case of set theory, we have a canonical semantics and we are interested in what is true with respect to this semantics. It is perfectly reasonable to believe that the axioms of set theory are true with respect to this semantics.
> And this does not mean that axioms of this > kind are arbitrary assumptions. There seem to be very few axiom systems > that lead to interesting theories. > > > Ralf