On 31 May 2012, at 17:03, John Clark wrote:

On Wed, May 30, 2012  Bruno Marchal <marc...@ulb.ac.be> 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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