On 14 Jan 2012, at 19:00, John Clark wrote:

On Sat, Jan 14, 2012  Bruno Marchal <marc...@ulb.ac.be> wrote:

> OK, but today we avoid the expression "computable number".

Why? Seems to me that quite a large number of people still use the term. A computable number is a real number that can be computed to any finite amount of digits by a Turing Machine, however most irrational numbers, nearly all in fact, are NOT computable . So the sort of numbers computers or the human mind deals in can not be the only thing that is fundamental because most numbers can not be derived from them.

> All natural number are computable

Yes, but very few numbers are natural numbers.

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. It makes things simpler. there is no real theory of computability for the real numbers. There are no equivalent to the Church Turing thesis for them. And with comp we don't need any ontological numbers other than the natural numbers. The whole of analysis and physics is eventually made espistemological ("number's ideas").

> With mechanism it is absolutely indifferent which fundamental finite object we admit.

If by "mechanism" you mean determinism then your remarks are irrelevant because we don't live in a deterministic universe, and even the natural numbers are not finite.

No. By mechanism I mean the idea that the brain (or whatever needed for consciousness) is Turing emulable. This shows immediately (UDA1-3) that we live in a non deterministic reality. Non determinism is a simple consequence of mechanism, which arise from self-duplication.

>> There is no way consciousness can have a direct Darwinian advantage so it must be a byproduct of something that does have that virtue, and the obvious candidate is intelligence.

> I disagree. Consciousness has a "darwinian role" in the very origin of the physical realm.

If Evolution can't see something then it can't select for it, and it can't see consciousness in others any better than we can, just like us all it can see is behavior.

I am talking on the Evolution of the physical laws. You have to follow the whole UDA to understand the special and crucial role of consciousness. Physical reality arise from the communicable first plural part of the consciousness flux existing in elementary arithmetic as a whole. I know this is not obvious at all. That's why it is a non trivial discovery. It makes physics a branch of mathematical computer science (alias number theory). By "number" I always mean natural number.

>>>    like relative universal self-speedin

>>  I don't know what that means.

> It means making your faculty of decision, with respect to your most probable environment, more quick.

In other words thinking fast. The fastest signals in the human brain move at a about 100 meters per second and many are far slower, the fastest signals in a computer move at 300,000,000 meters per second.

That's why consciousness plays a key role. Any slow universal machine can be arbitrarily speed up, on almost all its inputs, by change of software. This is Blum speed-up theorem. 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. That's why biological evolution selected conscious machine. They know much more than what they can communicate, and eventually get puzzled by such knowledge. BTW I tend to use "competence" for what you call "intelligence". "Intelligence" requires consciousness in my approach and definitions. Competence needs some amount of intelligence, but it has a negative feedback on intelligence.



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