>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?
Because the assumption of problematic axioms is trivial.
> Can a piece of music be true? An >architect's plan for a building? A computer program?
Are any of these things constructed theoremetrically from demonstrably true assumptions?
> I don't see any >sense in asserting that the axioms of Euclidean geometry or of group >theory are true or wrong.
But you see plenty of sense in assuming the truth of what you can't demonstrate true or false?
> 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.