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
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
A theory is supposed to abstract from the experiences. The
experiences motivates the theory, but does not justify it
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
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"? 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.
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. 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
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.
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