On 28 Feb 2013, at 19:03, meekerdb wrote:

On 2/28/2013 4:46 AM, Bruno Marchal wrote:

On 28 Feb 2013, at 05:04, meekerdb wrote:

On 2/27/2013 7:17 PM, Bruno Marchal wrote:

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.

You are assuming that justification comes from logic; and indeed it is too much to expect from such a weak source. I look for such "justification" as can be found from experience, which you demoted to mere "motivation".

Where did I say "motivation"?

"The experiences motivates the theory, but does not justify it logically."

Indeed. It is the inductive inference part of the learning process. It is very important. All theories, including "brains" comes from this.






I use the term "intuition", and I demote nothing, as it correspond to to the first person (the hero of comp, the inner God, the third hypostase, Bp & p; S4Grz1, etc.).
But justification for me invokes "proof", formal or informal.

Logical proof is relative to axioms. So the justification can be no stronger than the axioms.

Sure.









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.

I'm not at all sure that there is successor for every x.

Then you adopt ultrafinitism, and indeed comp does not make sense with such hypothesis, and UDA1-7 suggests that ultrafinitism might save physicalism, but step 8 put a doubt on this. The axiom that all natural numbers have a successor is used in basically all scientific paper though.

It is assumed, but I'm not sure it is used in an essential way. I recognize it difficult to do mathematics without it, but still it may be only a convenience.

OK. I don't see the problem with this. Convenience is a fuzzy notion. A brain too is convenient. Universes can be convenient. I am not sure to see your point.




You need it, or equivalent, to define "machine", "formal systems", "programs", "Church's thesis", "string theory", "eigenvector", "trigonometry", etc.




My intuition doesn't reach to infinity. It seems like an hypothesis of convenience.

I propose a theory, that's all. You don't need to believe in infinity, unlike in set theory (yet also used by many). You need just to believe (assume) that 0 ≠ s(x), and that x ≠ y entails s(x) ≠ s(y). Notion like provability and computability are based on this.

I think you need to accept that every number has a successor in order to prove things like Godel's theorems.

That follows from the axiom above. It is easy to prove this, and Gödel's incompleteness, *in* the theory, if the theory has the induction axioms. If the theory has no induction axioms, you can still prove this at the meta-level, *about* the numbers in the theory.

You have the terms 0, s(0), s(s(0)), etc. When you say that you are not sure all numbers have a successor, you are telling me that there is a number x = to some s(s(s(...(0)...) for which has no successor. this can be true for some weird notion of numbers, but non sensical in all models of Robinson Arithmetic, Peano Arithmetic, of the arithmetic in all model of set theories like ZF, etc. I am not sure you why you want things like that, unless you are searching for some ultrafinitist theory, which is your right, and is a mean to escape the UDA1-7 conclusion. But still not the step 8 (on which I will come back soon on the FOAR list of Russell Standish).

Bruno


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



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