On 27 May 2012, at 20:59, meekerdb wrote:
On 5/27/2012 5:02 AM, Bruno Marchal wrote:
As Bruno said, "Provable is always relative to some axioms and
rules of inference. It is quite independent of "true of
reality". Which is why I'm highly suspicious of ideas like
deriving all of reality from arithmetic, which we know only from
axioms and inferences.
We don't give axioms and inference rule when teaching arithmetic in
high school. We start from simple examples, like fingers, days of
the week, candies in a bag, etc. Children understand "anniversary"
before "successor", and the finite/infinite distinction is as old
In fact it can be shown that the intuition of numbers, addition and
multiplication included, is *needed* to even understand what axioms
and inference can be, making arithmetic necessarily known before
any formal machinery is posited.
But only a small finite part of arithmetic.
I don't think so. Our arithmetical intuition is already not
formalizable. If it was, we would be able to capture it by a finite
number of principle, but then we would be persuade that such finite
theory is consistent, and that intuition is not in the theory.
I suspect that our intuition is full second order arithmetic, which is
not axiomatizable. In fact it is the very distinction between finite
and infinite that we cannot formalize. Like consciousness, we know
very well what finite/infinite means, but we cannot defined it,
without using implicitly that distinction. The natural numbers are
*the* mystery, and it has to be like that: no machine will ever been
able to define what they are. Assuming comp, neither will we.
Arithmetical truth per se, as no corresponding complete TOE. It is
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