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
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
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. 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 +
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 post to this group, send email to firstname.lastname@example.org.
Visit this group at http://groups.google.com/group/everything-list?hl=en.
For more options, visit https://groups.google.com/groups/opt_out.