Aatu Koskensilta wrote: > firstname.lastname@example.org wrote: > > Karl Malbrain wrote: > >>Belief means you agree with the axioms. > > > > Consider: > > > > http://en.wikipedia.org/wiki/Elementary_group_theory > > > > I don't "agree" or "disagree" with the axioms that define a group. What > > could it even mean to "disagree" with the axiom "for every element, > > there exists an inverse"? > > Obviously it makes no sense to agree or disagree with the group axioms. > It does make perfect sense to agree or disagree, or remain agnostic > about, that there exists a measurable cardinal, or that a particular > primitive recursive ordering of naturals has no primitive recursive > infinite descending sequences. >
Well, this is a bit out of my depth, but my understanding is that, say, regarding the Continuum Hypothesis, one can have no opinion, one can assert it, or one can assert its negation.
In each case, there are a different set of theorems which can be proven under these assumptions. And a different universe of sets which can be said to exist or not exist.
Let us simply restrict ourselves to the cases where one makes no assumption regarding CH, and where one asserts CH.
How is this conceptually different for you than considering on the one hand the usual axioms of group theory, and on the other assuming in addition commutativity, so that all groups are Abelian?
> > I happen to agree; a common definition of "mathematical truth" is > > "follows logically from a particular set of axioms". > > That's a very silly definition, even if one does in fact run into it > from time to time in the "philosophical" outpourings of some mathematicians. >
Yes, well, it does assert that true->provable; which isn't right.