On 17 Jan 2012, at 17:26, John Clark wrote:

On Mon, Jan 16, 2012 at 4:56 AM, Bruno Marchal <marc...@ulb.ac.be> wrote:

" But in computability theory we have only natural numbers. A real number like PI or e is modeled by a total computable function from N to N."

Yes, but real numbers like PI or e are very much the exception, they are rare, quite literally infinitely rare oddball real numbers, because nearly all the numbers on the number line are not computable so there is no way for a Turing Machine, or anything else, to come arbitrarily close to one like you can for PI or e.

You don't need to assume them. They already exists at the natural numbers' epistemological level. But you can assume them if you want. Just rememeber that when I use the term "number", I mean a natural number.

 " By "number" I always mean natural number."

Then numbers can not be the only thing that is fundamental.

Together with the laws of addition and multiplication, they are. The rest is numbers dreams (themselves recovered by number relations, definable with addition and multiplication, like I explained briefly in a recent post to David). This comes from the fact that elementary arithmetic (on integers) is Turing universal. Elementary (first order logical) analysis is NOT Turing universal. Real numbers are, computationally, to much simple. You need to postulate the trigonometrical function to recover the natural numbers from the real. With the natural numbers, you recover all constructive reals with only degree four polynomes (by a famous result by Matiyasevitch). Fermat formula is trivial on the reals, but it took 300 hundred years to handle the case for the natural numbers. Arithmetical truth is not axiomatisable. *all* effective theories miss infinitely many truth about them.

" By mechanism I mean the idea that the brain (or whatever needed for consciousness) is Turing emulable."

OK. Then mechanism has not been proven and will never be proven it is just assumed,

Exact. It can even be justified by mechanism, that mechanism cannot be proved (even taken as axiom). It is a necessary meta principle. It is even a theological assumption. A belief in a form of reincarnation, obeying to theological laws already intuited by the Platonists and the neo-platonist.
Now, *all* theories are assumption.
I am a theoretician. I don't want to argue for truth or falsity. Mechanism is just my working assumption.

and the ground that assumption is built on is exactly as strong or as weak as the assumption that you are not the only conscious being in the universe.

It is not that stronger. Someone doubting mechanism is not necessarily solipsist.

" we live in a non deterministic reality."

That has been known for nearly a century.

We don't know that. We infer it from QM-without collapse, itself inferred from observation. Now, I deduce it from simple mechanism. It is always a success when we prove something, especially something contentious, in a simpler theory.

" Non determinism is a simple consequence of mechanism"

Determinism or non-determinism has nothing to do with consciousness, its irrelevant.

You have to study my sane04 paper, or my explanation to Elliot Temple in the FOR list, or my recurring explanations, on this list. The first person indeterminacy is based on the fact that mechanism supposes that there is a level of substitution of my parts such that my consciousness remains invariant (it is much weaker than most version of comp in the literature). Then indeterminacy is explained by self- duplication, as seen by conscious (first) person. You have just to distinguish carefully first and third person points of view notions.

" Universal machine can always been optimized by change of software only, and one way to do that is allowing the machine to believe in non provable propositions."

Yes that makes sense but I don't see what it has to do with consciousness, that's true for any axiomatic system including Euclid's geometry. And there is a danger, the reason the proposition is non-provable may have nothing to do with Godel, it may simply be plain ordinary false. If it's false you'd better hope it's non- provable in your logical system.

Right. And consciousness will be a result of integrating a non conscious bet in such a self-consistency (the idea that we don't prove false sentence). This is equivalent with a bet in the existence of a reality. (By Gödel's completeness theorem).

"BTW I tend to use "competence" for what you call "intelligence". "Intelligence" requires consciousness"

If what you call competence and Intelligence can both produce the same behavior then you might as well say that Intelligence and consciousness are synonyms because they are both equally unobservable and untestable. In common usage intelligence is simply what intelligent behavior implies, and redefining familial words in unfamiliar ways is not the path to clarity or enlightenment.

In all circumstances, I use the terms the most used by the experts in the fields. That's the only option in serious interdisciplinary researchs. Technical approaches forces to redefine many terms having more vague usages in everyday life. That's always the case.

But I have seen that when people understand the nuance between intelligence (needed to create some competence) and competence itself, it helps them in everyday learning process.

" Competence needs some amount of intelligence, but it has a negative feedback on intelligence."

I don't know what that means.

It means that very competent people can begin to believe that their are intelligent, and that's leads to stupidity. A little genius is someone saying little stupidities in his colleagues' fields. A big genius is someone telling big stupidities in his colleagues' fields.

Competence is more like the derivative of intelligence, with a minus sign. Somehow. This is an image.



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