On 9/15/2012 4:11 AM, Russell Standish wrote:
On Sat, Sep 15, 2012 at 02:55:17AM -0400, Stephen P. King wrote:
Could you elaborate on what your definition of "a digital
Anything Turing emulable.
OK. But you do understand that this assumes an unnecessary
restrictive definition of computation. I define computation as "any
transformation of information" and Information is defined as "the
difference between a pair that makes a difference to a third".
That is far too inclusive a definition of computation.
Not really, it only requires some way of representing the
information such that it can be transformed. The integers are not the
only kind of number that we can represent numbers (or any other
mathematical object) with. IMHO, we are naive to think that Nature is
hobbled to only use integers to perform her Computations. We must never
project our deficiencies on Nature.
A map from i in
N to the ith decimal place of Chaitin's number Omega would satisfy you
definition of transformation of information, yet the posession of such
an "algorithm" would render oneself omniscient.
That is exactly my point! I am forcing the issue of the implication
of Universal Turing Machines, they are implicitly omniscient unless they
are restricted in some way. Turing et al, considered the case of
computations via NxN -> N functions but abstracted away the resource
requirements and we get very smart people, like Bruno, taking this as to
means that we can completely ignore the possibility of actually
implementing a computation and not jsut reasoning about some abstract
object in our minds. The Ultrafinitists and Intuitionist (like Normal
Wildberger <http://en.wikipedia.org/wiki/Rational_trigonometry> for
instance) have a valid critique but forget that they too are fallible
and project their limitations on Nature. I am trying very hard to to do
You can answer any
question posable in a formal language by means of running this
algorithm for the correct decimal place. See Li and Vitanyi, page 218
for a discussion, or the reference they give:
Bennett & Gardiner, (1979) Scientific American, 241, 20-34.
Sure, but you are missing the point that I am trying to make.
Unless there is at least the possibility in principle for a given
computation to be implemented somehow, even if it is in the form of some
pattern of chalk marks on a board or pattern of neurons firing in a
brain, there is no "reality" to a abstraction such as a Universal Turing
Machine. I am arguing against Immaterialism (and Materialism!) of any
kind and for a dual aspect monism (like that which David Chalmers
discusses and argues for in his book <http://consc.net/book/tcm.html>).
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