On 17 Jul 2006 16:22:23 -0700, "Karl Malbrain" <firstname.lastname@example.org> wrote:
> >Lester Zick wrote: >> On 17 Jul 2006 14:15:19 -0700, "Karl Malbrain" <email@example.com> >> wrote: >> >> > >> >firstname.lastname@example.org wrote: >> > >> >> I don't find that to be the case for me; at least for the usual >> >> meanings of the terms "believe" and "truth" /in this context/. >> > >> >truth: >> > 1. The quality or being true; as: >> > (a) Conformity to fact or reality; exact accordance with >> > that which is, or has been; or shall be. >> > (b) Conformity to rule; exactness; close correspondence >> > with an example, mood, object of imitation, or the >> > like. >> > >> >If a mathematical result conforms to the axioms you agree to opoerate >> >within, you get mathematical truth. >> >> Not exactly. All you've got at that point is axiomatic truth, self >> consistency between axiomatic assumptions and extraplated theorems. > >And an expansion of the truth of the system.
I'm inclined to disagree because the axioms already subsume the truth of the system whatever it may be.
>(...) > >> >> On the other hand, I do believe in the truth of the statement "if we >> >> assume Euclidean geometry with the parallel postulate, then all >> >> triangles have angles summing to 180 degrees"; but that is a belief in >> >> the truth of the fact that it logically follows from the given axioms - >> >> not a belief in the truth of the axioms themselves. >> > >> >Exactly. The axioms are not things in themselves. >> >> Sez who? Axioms are as much things in themselves as any other >> assumptions. > >As part of their system they go beyond that and become >things-for-themselves.
I don't quite see what you mean by "things in themselves" versus "things for themselves". Axioms are assumptions and like all assumption are things whether part of a system or just plain ole assumptions. The utility and value of various assumptions can be justified by theorems extrapolated within their framework. But they remain assumptions.