On 27 Feb 2013, at 20:40, meekerdb 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'.


We have intuition and/or evidences for accepting the axiom. Here the question is really: do you accept the axiom of Peano Arithmetic (for example). And with comp we don't need more. Then we prove theorems, which can be quite non trivial.



To say that it is "justified logically" seems to mean no more than "we have avoided inconsistency insofar as we know."

Yes. We can't hope for more. But in case of arithmetic, we do have intuition. Well, in physics too.




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.

Consistency justifies our existence. I would say. We are ourself hypothetical with comp. We are divine hypotheses, somehow. I think you ask too much for a justification. I don't think that what you ask is possible, even if I am pretty sure that x + 0 = x, x + s(y) = s(x + y), etc.

Bruno



http://iridia.ulb.ac.be/~marchal/



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