On 20 May 2014, at 22:07, Craig Weinberg wrote:
On Sunday, May 18, 2014 9:59:10 PM UTC-4, Liz R wrote:
On 19 May 2014 07:37, Craig Weinberg <[email protected]> wrote:
You did not provide evidence that they cannot do that.
His evidence was the negative answer to Hilbert's 10th problem.
To be exact, it's claimed to be how he arrived at that answer. The
extract says that he arrived at a proof that "no algorithm could
have found". How did he find it?
From what I can gather, Matijasevich proved that the already proven
unsolvable Halting Problem can be represented as a Diophantine
equation, so that there is at least one Diophantine equation that
can't be solved by a Turing machine. I'm sure its more complicated
than that, but at this point, that's what I'm getting as a general
overview.
Not bad. He proves in particular that diophantine equations are turing
universal.
Not only we can translate any computational processes/proof in
arithmetical sentences/programs (by the intensional Church thesis,
which is a consequence of the usual classical Church thesis), but we
can limit the arithmetical languages to the degree four diophantine
polynomial, like the one I sent in my recent post.
The paper is far too high powered for my little brain, so I am
hoping for an answer for dummies. Did he decide that the answer
might have some particular form using intuition, say, tried it, and
found it worked? How did he (or anyone) then show there was no
algorithm for finding it?
(This is reminiscent of "The Emperor's New Mind", which IIRC
attempts to prove that some gifted mathematicians are not machines!)
Well that contains the main error we can do by using Gödel to defend
non-comp. Penrose corrected it in his second book, but does not seem
to realize the consequence.
You cannot derive from Gödel that we are not machine, but you can
derive from Gödel that IF we are machine, THEN we cannot know nor
justify which machine we are, nor which universal computations
supports us. It is the arithmetical reason for the machine first
person indeterminacy.
Bruno
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