On 29 May 2012, at 08:46, meekerdb wrote:
On 5/28/2012 1:13 PM, Bruno Marchal wrote:
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?
I doubt infinities.
I can doubt actual infinities. Not potential infinities, which gives
sense to any non stooping program notion.
Comp is ontologically finitist. As long as you don't claim that there
is a biggest prime number, there should be no problem with the comp
hyp. Infinities can be put in the epistemology, or at the meta-level:
they are mind tool, souls attractor etc.
Bruno
Brent
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