Re: Analog = digital?

Very interesting! Thanks very much,

John

<tchow@xxxxxxxxxxxxx> wrote in message
news:458ecb38$0$572$b45e6eb0@xxxxxxxxxxxxxxxxxxxxxxxxxxxx
In article <458d93c3$0$16152$4fafbaef@xxxxxxxxxxxxxxxxxxx>,
JMF <jfavaro@xxxxxx> wrote:
friend thought he remembered that way back then when the pro-digital folks
were pushing for going completely digital and abandoning analog computers,
they appealed to this proof somebody had supposedly made that digital
computers could compute anything analog computers could -- and that
therefore by going the digital route they wouldn't be giving up
computational possibilities. But as I said, he's begun to wonder whether
that supposed proof exists.

It would be highly misleading to use the word "proof" to denote any such
argument. At best, it would be something along the lines of the arguments
for the Church-Turing thesis (that Turing machines can compute anything
that can be algorithmically computed by a person or any other physical
system)---i.e., a plausible heuristic argument based on experience as well
as on mathematical argumentation. (The situation is different with "GOTO"
since there you have a well-defined mathematical question that can be
settled with a rigorous argument.) There might be some plausible models
of analogue computation for which you can prove the equivalence to digital
computers, but you could never prove that there aren't some actual
analogue
computers with powers not captured by your model.

In fact, nowadays there is a considerable flurry of interest in the
concept
of "hypercomputation." While I personally feel that much of this work is
seriously flawed, there is certainly no mathematical *proof* that
hypercomputation is absurd. If you look up some of that work, you'll see
that it has the flavor of "analogue computers that can do things that
digital computers can't."
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great,
will
never exceed four of those miles of which as many thousand separate us
from
the center of the earth. ---Galileo, Dialogues Concerning Two New
Sciences


.

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