Hal Finney wrote:
> > M. Pour-El and I. Richards [Computability in Analysis and Physics, Springer,
> > Berlin, 1989] have proven that ......................
> I haven't seen this result, but it is well known that Newtonian physics
> has similar properties. In fact you can do an infinite amount of
> computation in finite time using Newtonian point sources. I wouldn't be
> surprised if this QM result relied on similar unrealistic idealizations.
Freeman Dyson wrote something about that result.
<The superiority of analog-life is not so surprising if you are familiar
with the mathematical theory of computable numbers and computable functions.
Marian Pour-El and Ian Richards, two mathematicians at the University of Minnesota,
proved a theorem twenty years ago that says, in a mathematically precise way,
that analog computers are more powerful than digital computers.
They give examples of numbers that are proved to be non-computable
with digital computers but are computable with a simple kind of analog computer.
The essential difference between analog and digital computers is that an analog
computer deals directly with continuous variables while a digital computer deals
only with discrete variables. Our modern digital computers deal only with zeroes and ones.
Their analog computer is a classical field propagating though space and time and
obeying a linear wave equation. The classical electromagnetic field obeying the Maxwell
equations would do the job. Pour-El and Richards show that the field can be focussed
on a point in such a way that the strength of the field at that point is not computable
by any digital computer, but it can be measured by a simple analog device.
The imaginary situation that they consider has nothing to do with biological information.
The Pour-El-Richards theorem does not prove that analog-life will survive better in a cold.>
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