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.
Bruno
http://iridia.ulb.ac.be/~marchal/
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