On 31 May 2012, at 17:03, John Clark wrote:
On Wed, May 30, 2012 Bruno Marchal <[email protected]> wrote:
> The axiom of choice just asserts that an arbitrary product of a
family of non empty set is non empty.
True, but my dictionary says "arbitrary" means "based on a random
choice or personal whim".
It math, if P(x) is true for arbitrary x, it just means that P(x) is
true for all x.
> There is no clue of direct relationship with physics
If modern physics said randomness does not exist then there would be
a conflict with the Axiom of Choice,
The axiom of choice has nothing to with randomness, a priori. I can
imagine the existence of theories bringing relation, for divine (non
turing emulable) entities. But then you have to present those theories.
they could not both be true; but physics says randomness DOES exist
so they are compatible.
Comp says randomness does exist, and physics confirms that, OK. But
again, this has nothing to do directly with the axiom of choice which
concerns set theory.
There are evidence that 'mathematical" physics can live in little
constructive toposes, and if I remember well the reading of papers
some time ago, I think that the axiom of choice makes those toposes,
or topoi, boolean, that is still obeying classical logic, which is not
so much liked by the constructive people. Physics lives in very short
initial segment of ZF, it is not clear if the axiom of choice says
anything about the physical reality, nor even of the math, except by
making true some nice completing property, like having *all* Hilbet
space having orthonormal base, in physics, or like all consistent set
of sentences having unique consistent extensions. But, with comp, this
concerns the epistemology, and things are very difficult.
Consciousness surfs on coherent dreams, and it is just an open
question if that converges to a unique physical universe, or a unique
multiverse, or a unique multi-multiverse, or and this on all ordinals
(in which theory? With AC?.
> It has a priori nothing to do with free will
Of course it doesn't, nothing real can have anything to do with
"free will" because "free will" is gibberish.
That is *your* theory, and to be honest, I don't find it so much
interesting. I do agree that some definition of free will are
"gibberish", that is either inconsistent, or empty, but some are not.
I suggest that free-will is the machine awareness of the possibility
of hesitating in front of a spectrum of possibilities.
Butterflies are close to free will, imo, because of the spectrum of
flowers and nectars, but I have no evidence that butterfly have free
will because I have no evidence that butterfly can infer and reflect.
They might be mainly attracted. But I gave evidence that jumping
spiders and octopi have free will in the sense that they do infer the
possibilities, and reflect on it.
Relatively to their cognitive abilities, they have as much free will
than you, me and PA (with the definition above).
But the Axiom of Choice does have something to do with cause and
effect and randomness because those things are not gibberish,
We could make that true if we would formalize physics in set theory.
But there are conceptual reason why such an enterprise is doomed at
the start.
ZF is the "fortran" of the mathematical theories. Just an altar for
category theory and "natural transformations" (Eilenberg and MacLane).
I love ZF, but as a very imaginative Löbian machine.
To say that the axiom of choice has something to do with the notion of
cause and effect, without saying in which theory you work is confusing.
it even has something to do with intelligence. When Alan Turing
designed the first stored program electronic digital computer, the
Manchester Mark 1, he insisted it have a hardware random number
generator incorporated in it because he felt that pseudo-random
numbers being produced by a numerical process could not be truly
random. He thought that if a machine could sometimes make purely
random guesses and then use logic to examine the validity of those
guesses it might be able to overcome some of the limitations he
himself had found in pure Turing Machines (although he never used
that name for them), and then you could make what he called a
"Learning Machine. He thought that in this way the limitations all
deterministic processes have that he and Godel had found might be
overcome, at least in part.
For problem solving this in vindicated by the result that Random
Oracle can enlarged classes of problem solving. Those are given by
necessary non constructive proofs. This does not overcome
Incompleteness or insolubility, but can reduce complexities in
relative way. That might play a role in the first person indeterminacy
comp measure problem, as it gives freely a first person "random
Oracle" a priori, relativized by their many computational extensions.
Bruno
http://iridia.ulb.ac.be/~marchal/
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