On 9/15/2012 4:11 AM, Russell Standish wrote:
On Sat, Sep 15, 2012 at 02:55:17AM -0400, Stephen P. King wrote:
Dear Bruno,

    Could you elaborate on what your definition of "a digital
machine" is?
Anything Turing emulable.
Dear Bruno,

     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".

Hi Russell,

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 that!

  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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