On 11 Dec 2015, at 07:27, Dan wrote:
If we could harness the undecidability of the spectral gap to
implement infinite real weights for Hava Siegelmann's analog
recurrent neural network model, then I suppose we could in theory
achieve hypercomputation.
The undecidability of the spectral gap reflects only the Turing-
universality of physics. I am not sure you can use that to *harness*
hypercomputation.
To show that a simple problem is undecidable is more impressive, like
showing that there is no algorithm for deciding if a diophantine
polynomial equation has a solution or not (Matiyazevitch), that is
impressive, as it shows that a tiny part of the arithmetical reality
is already Turing-Universal, and that it emulates all computations (by
the intensional Church Thesis, which follows from the usual Church's
thesis).
Evolution (Darwin) can be said to make an hypercomputation, as time
can play the oracle for the halting problem in the limit.
Hava Sielgelmann still consider programs, that you can implement with
serial digital computer. Those are interesting from an engineering
perspective, like quantum computation, but they do not, without
further ado, make possible to escape the consequence in physics of
computationalism (in the cognitive science).
This would have many incredible consequences. We wouldn't be able to
develop an algorithm to reproduce consciousness, but perhaps by
trying all possibilities (perhaps guided by intuition, as Goertzel
suggests) we might then actually stumble upon a conscious being.
Maybe our brains are analog recurrent neural networks that exploit
this undecidability of the spectral gap...
I think we have met algorithms which can implement consciousness: any
universal machine/program/number.
Then the Löbian machines/programs/numbers, like Peano Arithmetic, and
which are the universal machines capable of knowing that they are
universal, are as much conscious and self-conscious that you and me,
but with much less prejudices, illusions and other contingent built-in
theories/beliefs. See my papers for the theology that such machine can
discover when looking inward (in a precise sense due to Gödel, Kleene,
etc.)
We *have* stumbled on the conscious being, since Gödel 1931. It is
just our own dumbness/spitual-sleepiness, which makes us slow in
realizing this. It still needs the study of some amount of mathematics
and logic to understand what is happening.
Bruno
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