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