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
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'. To say
that it is "justified logically" seems to mean no more than "we have avoided inconsistency
insofar as we know." 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.
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