On 24 Oct 2014, at 18:58, John Clark wrote:
On Thu, Oct 23, 2014 at 7:10 PM, meekerdb <[email protected]>
wrote:
> They are non-computable by a Turing machine - which is already
assumed to have unlimited tape and time. It is likely that in the
real world almost all integers are not computable too.
Any integer can be calculated with a Turing machine
That is ambiguous. You can give an intensional definition of the
natural numbers by listing the
phi_0, phi_1, phi_2, phi_3, phi_4, phi_5, ..., with the phi_i
enumerating effectively the programs with zero argument.
All natural numbers will have a definition among those phi_i, and thus
are computable *in that sense*, but not all definition of a natural
number of that kind will define such a natural number, as some
programs will not stop.
that has unlimited tape and time, and even with a finite tape and
finite time good approximations can be found for the rational
numbers and some irrational numbers, even a few transcendental
numbers,
Yes, all constructive reals can be defined in arithmetic, and even
many non constructive one.
but for nearly all real numbers not even approximations can be
calculated, not even with a infinite tape and infinite time. They're
just not computable.
In classical analysis, that is correct. In intuitionist analysis, many
models satisfy the Brouwer axiom: all functions are continuous, or all
functions are computable. In some models: computable = continuous. It
is interesting, and even useful for the engineers, but it brings
conceptual difficulties in cognitive science (typically it leads to
different sorts of solipsisms: it cannot solve the "other mind"
problem, and it is a form of consciousness of the other eliminativism).
And if a mechanical process like a Turing Machine can't produce them
can the Real numbers have anything to do with physics? I don't claim
to have a answer I'm just asking a question.
Take the infinite WM self-duplication, it predicts that almost all
2^n, with n large, observers coming from such an experience, and who
bet white noise, or non computable sequences, win the bets. So in a
classical deterministic context we can justify the presence of non
computable randomness in the subjective discourses (the diaries) of
the observers involved. Now in front of the whole arithmetical truth,
it is an harder question, but we can already ask non trivial question
to the machines about this, and get some results. Normally, we should
get indeterminacy below our substitution level, and quantum mechanics,
when taken literally might confirm this, or refute it..
The computationalist hypothesis, taken with the classical definition
of knowledge (the modal logic S4) in the cognitive science is testable.
Bruno
John K Clark
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