John,


Allow me please, one more remark:

I allow you an infinity of remarks. But not one more :)


my ID for an axiom is a "ground-rule" derived to facilitate the acceptance of a theory.

Hmm... That is not the standard idea. An axiom is simply an hypothesis. Like the hypothesis that there is a moon, or that 0 + x = x, etc. It is what we accept to proceed.



I suspect the axioms were invented AFTER the theoretical considerations to make them acceptable.

That is true, but they are useful to communicate ideas and beliefs to others. When formalized, the axioms and theorems don't depend on the many interpretations that they can have. In applied science, we cannot use such axiom, and so must do some semi-axiomatization, with implicit hypotheses, like the existence of the domain of application, like when we send persons or robots to the moon.


They are called axioms because we cannot justify their acceptability.

Yes.



I am not ready to defend this.

Without (semi)-axioms, we remain unclear and non refutable, so we can't so easily progress.

Bruno





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

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