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. >

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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/ > > > > No virus found in this message. > Checked by AVG - www.avg.com > Version: 2013.0.2899 / Virus Database: 2641/6133 - Release Date: 02/25/13 > -- > 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-li...@googlegroups.com <javascript:>. > To post to this group, send email to everyth...@googlegroups.com<javascript:> > . > Visit this group at http://groups.google.com/group/everything-list?hl=en. > For more options, visit https://groups.google.com/groups/opt_out. > > > > > -- 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.