On Tue, Jun 17, 2008 at 5:58 PM, Abram Demski <[EMAIL PROTECTED]> wrote:
>
> Hector Zenil,
>
> I do not think I understand you. Your argument seems similar to the following:
>
> "I do not see why Turing machines are necessary. If we can compute a
> function f(x) by some Turing machine, then we could compute it up to
> some value x=n. But we could construct a lookup table of all values
> f(0), f(1), f(2),... , f(n) which contains just as much information."
>
> Obviously the above is a silly argument, but I don't know how else to
> interpret you. A Turing machine can capture a finite number of the
> outputs of a hypercomputer. Does that in any way make the
> hypercomputer reducible to the Turing machine?
>
This nicely boils the fallacy down from 20 pages to a few lines.
Merely providing the lookup table or adding more states is not
sufficient to turn a Turing machine into a hypercomputer as it would
follow from the paper main argument: that humans can always find
bb(n+1) once bb(n) calculated, therefore humans are capable of
hypercomputing (modulo other strong assumptions).
In fact the paper acknowledges that more information is needed at each
jump, so eventually one would reach either a physical or a feasible
limit unless the brain/mind is infinite in capabilities, falling into
the traditional claims on hypercomputation, and not necessarily a new
one.
I recall that my suggestion was (reductio ad absurdum) to encode (or
provide the program) a n-state Turing machine T_n after knowing bb(n)
so at every moment when people is working on bb(n+1) there is always a
T_n behind able to calculate bb(n). Once the hyperhuman finds bb(n+1)
then he encodes T_{n+1} to compute bb(n+1) while the hyperhuman H
computes bb(n+2) but one knows that at the next step one will be able
to code T_{n+2} to calculate bb(n+2), just as H does. Following their
argument, if there is always a machine able to calculate bb(n+1) for
any n when bb(n) is calculated (as there is a hyperhuman according to
their claim), therefore T (the universal Turing machine that emulates
all those T_i for all i) would turn into a hypercomputer (absurd since
it would collapse the classes of computability!).
Notice that my use of hypercomputer is the traditional use of a
computer: a machine able to compute at a Turing degree other than the
first.
I still might be missing something, but hope this clarifies my objection.
People might be also interested in the work of Kevin Kelly:
"Uncomputability: The Problem of Induction Internalized," Theoretical
Computer Science, pp. 317: 2004, 227-249.
as an epistemological approach to traditional computability, as some
have suggested in this thread induction as evidence for
hypercomputability.
--
Hector Zenil http://zenil.mathrix.org
> On Tue, Jun 17, 2008 at 4:35 PM, Vladimir Nesov <[EMAIL PROTECTED]> wrote:
>> On Tue, Jun 17, 2008 at 11:38 PM, Abram Demski <[EMAIL PROTECTED]> wrote:
>>> V. N.,
>>> What is inhuman to me, is to claim that the halting problem is no
>>> problem on such a basis: that the statement "Turing machine X does not
>>> halt" only is true of Turing machines that are *provably* non-halting.
>>> And this is the view we are forced into if we abandon the reality of
>>> the uncomputable.
>>>
>>
>> Why, you can also mark up the remaining territory by "true" and
>> "false", these labels just won't mean anything there. Set up to sets,
>> T and F, place all true things in T, all false things in F, and all
>> unknown things however you like, but don't tell anybody how. Some
>> people like to place all unknown things in F, their call.
>> Mathematically it can be convenient, but really, even of "computable"
>> things you can't really compute that much, so the argument is void for
>> all practical concerns anyway.
>>
>> --
>> Vladimir Nesov
>> [EMAIL PROTECTED]
>>
>>
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