On 11 Feb 2017, at 15:04, Telmo Menezes wrote:
Hi Bruno,
On Fri, Feb 10, 2017 at 6:44 PM, Bruno Marchal <marc...@ulb.ac.be>
wrote:
On 10 Feb 2017, at 04:27, Telmo Menezes wrote:
https://www.quantamagazine.org/read-offline/4739/20130524-is-nature-unnatural.print
It critics also the many universe as non testable. But all the
following
theories are not testable:
there is 0 universe,
This one seems refutable to me. I know there is at least one universe
because I know that I am aware of some reality.
OK. I should have said "physical universe", or "reality independent of
me".
But even just "universe" (or reality), you cannot prove that there is
one. You can know (in some sense) but you still cannot prove it, in
the usual sense of convincing some other peer. You know that you are
aware, and I bet you are, but you cannot prove this. OK?
there is 1 universe,
there is 2 universes,
there is 3 universes,
there is 4 universes,
there is 5 universes,
there is 6 universes,
there is 7 universes,
...
There is aleph_0 universes,
There is aleph_1 universes,
There is aleph_2 universes,
There is aleph_3 universes,
There is aleph_4 universes,
...
There is aleph_aleph_aleph_aleph_aleph_aleph_1004 universes,
...
If mechanism is true, then, as long as we are correct, the theories
like
"There is a universe", or "there is a reality" are somehow (up to
some
annoying and long to made slight nuances) absolutely undecidable
(and true,
hopefully).
With computationalism, even the belief in the "standard model of
Peano
arithmetic" requires some faith, which all scientists have, but not
always
with the awareness of the faith, which requires the
computationalist theory
to be explained. That kind of faith is cabled probably through
evolution.
Could you elaborate? Why is faith required to believe in Peano
arithmetic?
To believe in the standard model, or to believe in any model of Peano
Arithmetic is equivalent, (by Gödel's completeness theorem) to believe
in the consistency of Peano Arithmetic, which requires, by Gödel's
incompleteness theorem, some other theory to prove it (consistency of
PA), which will be as much doubtable than Peano Arithmetic.
Now, most of us have *that* faith, and usually, we are completely
convinced by many simple proofs of the consistency of PA, like the
usual one, which is done implicitly in set theory, or in second-order
logic. In fact most of the math used everyday requires a bit more than
PA, like analysis, physics, real numbers, etc. That explains why we
are almost unaware that we use some faith there, and it is only
through the work of the logicians that we can become conscious of that
faith.
Keep also in mind that I use "proof" in a sense closer to the
technical sense, than the sense based on evidences. Smullyan (who died
recently, RIP Raymond) explains that the incompleteness does not throw
doubt on PA's consistency, and he is quite right on this. But some
people infers from this that we can prove PA's consistency, and people
mention often Gentzen's quite sophisticated proof of it, but that is
incorrect, at least in the context of the computationalist hypothesis.
We cannot prove the consistency of arithmetic, even if we have no
reason at all to doubt it.
Well, even that is not entirely true, and when some (rare) logicians,
like Nelson, did claim one year ago, to have found a contradiction in
PA, logicians took him seriously, and took the time (long) to find
precisely where Nelson was ... wrong. They did find the mistake, and
Nelson agreed that it was a mistake, but he will continue to doubt its
consistency (as he refuses the impredicative definitions latent in the
induction axioms).
Bruno
Telmo.
Bruno
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