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
I am not ready to defend this.
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.
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