Allow me please, one more remark: my ID for an axiom is *a "ground-rule" derived to facilitate the acceptance of a theory.* I suspect the axioms were invented AFTER the theoretical considerations to make them acceptable. They are called axioms because we cannot justify their acceptability. I am not ready to defend this. JM On Wed, Feb 27, 2013 at 2:40 PM, meekerdb <meeke...@verizon.net> wrote:
> On 2/27/2013 2:59 AM, Bruno Marchal wrote: > > > On 26 Feb 2013, at 21:40, meekerdb wrote: > > On 2/26/2013 1:24 AM, Bruno Marchal wrote: > > > How did number arise? We don't know that, but we can show that if we > don't assume them, or equivalent (basically anything Turing Universal), > then we cannot derive them. > > > I'm not sure how you mean that? > > > I meant that you cannot build a theory, simpler than arithmetic in > appearance, from which you can derive the existence of the numbers. All > theories which want talk about the numbers have to be turing universal. > So I meant this in the concrete sense that if you write your axioms, and > want talk about numbers, you need to postulate them, or equivalent. You can > derive the numbers from the equational theory: > > Kxy = x > Sxyz = xz(yz) > > + few equality rules, > > But that theory is already Turing universal, and assume as much the > number than elementary arithmetic. > > > > > We know that we experience individual objects and so we can count them > by putting them in one-to-one relation with fingers or notches or marks. > So what are you calling an "assumption" in this? > > > A theory is supposed to abstract from the experiences. The experiences > motivates the theory, but does not justify it logically. > > > But "justify logically" seems like a bizarre concept to me. We just make > up rules of logic so that inferences from some axioms, which we also make > up, preserve 'true'. To say that it is "justified logically" seems to mean > no more than "we have avoided inconsistency insofar as we know." Sure it's > important that our model of the world not have inconsistencies (at least if > our rules of inference include ex contradictione sequitur quodlibet) but > mere consistency doesn't justify anything. > > Brent > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to everything-list+unsubscr...@googlegroups.com. > To post to this group, send email to everything-list@googlegroups.com. > Visit this group at http://groups.google.com/group/everything-list?hl=en. > For more options, visit https://groups.google.com/groups/opt_out. > > > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list?hl=en. For more options, visit https://groups.google.com/groups/opt_out.
