On 28 May 2012, at 18:02, meekerdb wrote:
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 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 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.
Bruno
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.
Not really. I think there are complete theories of (N, successor). But
we have an intuition of adding and multiplying and this makes that
intuition inexhaustible.
Intuition is not entirely a given, it is something which develop with
the familiarity and life working. It is different for all of us, so it
nice that we can share some big initial segment of the arithmetical
truth. Comp does not need more than the sigma_1 intuition, at the
ontic level.
But intuition fails us in precisely in questions like Hilbert's hotel.
Why? Not sure, but it does not concern us, as comp builds on the
intuition of the finite things.
Why should you be so trusting of your intuition is just this
particular instance.
Do you doubt elementary arithmetic?
Bruno
http://iridia.ulb.ac.be/~marchal/
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