On Thursday, February 28, 2013 1:03:40 PM UTC-5, Brent 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."
>
>
>  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.
>

And axioms are no stronger than the capacity to make sense of them.
 

>
>
>  
>  
>  
>  
>  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.
>
>  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.
>
> Brent
>
>  
>  Bruno
>
>  
>   http://iridia.ulb.ac.be/~marchal/
>
>  
>  
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