On 5/28/2012 12:36 AM, Bruno Marchal wrote:
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
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 as humanity.
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 inexhaustible.
Our intuition is that space is euclidean, the earth is stationary and flat, and that there
is only one world. It seems to me that the infinity of arithmetic is just the intuition
we should always be able to add one more. But intuition fails us in precisely in
questions like Hilbert's hotel. Why should you be so trusting of your intuition is just
this particular instance.
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