On 07 Mar 2016, at 20:28, Brent Meeker wrote:
On 3/7/2016 5:09 AM, Bruno Marchal wrote:
the glass-of-beer created the natural numbers (with their law of
addition and multiplication), and (and that is the new thing) all
the rest emerges from them, or a construction by them.
But you insist that the numbers exist because they provably satisfy
predicates like "successor of 0". So there is no role for the gob.
I have never said that.
I say that arithmetical relation are true because they are true in the
standard interpretation in which we all believe. The relation between
the theories/machines and the model (the structure (N, 0, +, *) have
been studied.
The realism I use is not bigger than the one you need to not decide to
take your children out of school in case they taught your children
that 2+2=5 is not correct.
It has nothing to do with the fact that this or that theory or machine
believes or prove such theorem. I have often insisted that realism
means that 2+2=4 is independent of you or me, or PA or RA, or ZF. But
then the talk of PA, RA, exists also in the arithmetical relation, and
we must be careful not to mix the level of talk.
If you don't find Riemann hypothesis senseless, you are enough realist
to make sense of what the machine will and will not been able to
justify rationally from its primitive beliefs and knowledge, whatever
the means are used to encapsulate them in a finitely, relative to some
universal number, describable way. And that exists by * by
assumption* of the digital mechanist hypothesis.
I don't do any 'metaphysics' on the nature of what the number can be.
I like NBG(*) definition of the natural numbers: what belong to the
intersection of all inductive classes.
In all sciences, including theology, we must avoid ontological
commitment. We can only agree on things that we do have an intuition,
but cannot defined.
What is a number? Do they exist in some metaphysical sense? I don't
talk about that. I juste hope you accept classical predicate calculus,
and that you are OK with some axioms on them, like that any number n,
when added to the number 0 gives itself, n. Plus some more.
We need to accept such laws, not for doing metaphysics, but for
talking about the digitalization of your brain in the experience of
step zero, the definition of digital mechanism.
Then the reasoning shows that matter (sensible appearances) prediction
is given by some statistics on sigma_1 sentences, and indeed []p & p,
[]p & <>t and []p & <>t & p ([] = beweisbar of Gödel 1931, <>p =
~[]~p) gives interesting 'quantizations' when p is restricted on the
sigma_1 sentences.
I am a conservative. The laws of (platonist) thought (platonist means
the belief in the truth of the law A v ~A. The laws of mind when
assuming mechanism is determined by what a machine can say about that
itself, and there we have all the nuances brought by incompleteness,
in many forms, not all related.
I show that there is a big problem with the mechanist hypothesis, and
then I show that the universal machine has already solve it, and in a
testable way. And here QM illustrates that an evidence for a collapse
would be a refutation of digital mechanism. Physics is already like
the universal machine seems to say.
I don't pretend the universal machine is true, only that what it says
is testable/refutable. With all the technical nuances which would be
long to described, but I have explain the main one which is that we
test not comp but comp or perverse emulation in a normal world, it is
the re-entry of the dream argument at the metalevel.
Bruno
(*) I recommend the very nice book on set theory (a dover book) by
Smullyan and Fitting. They also treats the Cohen forcing method
through a use of a "quantization" ([]<>p) on a S4 modal logic, which
provides a nice illustration of a "quantization", in a context of
building models.
http://www.amazon.com/Theory-Continuum-Problem-Dover-Mathematics/dp/0486474844
Brent
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