On 27 Feb 2013, at 20:40, meekerdb wrote:

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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 ifwe don't assume them, or equivalent (basically anything TuringUniversal), then we cannot derive them.I'm not sure how you mean that?I meant that you cannot build a theory, simpler than arithmetic inappearance, from which you can derive the existence of the numbers.All theories which want talk about the numbers have to be turinguniversal.So I meant this in the concrete sense that if you write youraxioms, and want talk about numbers, you need to postulate them, orequivalent. 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 thenumber than elementary arithmetic.We know that we experience individual objects and so we can countthem by putting them in one-to-one relation with fingers ornotches or marks. So what are you calling an "assumption" in this?A theory is supposed to abstract from the experiences. Theexperiences motivates the theory, but does not justify it logically.But "justify logically" seems like a bizarre concept to me. We justmake up rules of logic so that inferences from some axioms, which wealso 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 haveinconsistencies (at least if our rules of inference include excontradictione sequitur quodlibet) but mere consistency doesn'tjustify 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/ -- 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.