[FOM] Physical Theories and Hypercomputation
pratt at cs.stanford.edu
Sat Mar 10 11:12:25 EST 2012
In response to Dmytro Taranovsky I would say simply that any speculation
about the computational capability of nature that ignores the
implications of quantum mechanics is dead on arrival.
The prospect of anything approaching an analytic universe, or even
hypercomputation, is undermined by the arithmeticity of action as the
product of conjugate variables, for example time and energy, position
and momentum, and the angular momenta about any two orthogonal axes, in
combination with Heisenberg uncertainty which can be seen as a principle
that makes ostensibly analytic variables arithmetic when considered
jointly with their conjugate partners.
Every bit of precision you gain in time, for example, must be taken from
energy. Any sufficiently precise clock must be therefore prepared to
deal with unpredictable gusts of energy. Long before your clock has
gained enough bits even just to think of itself as the most accurate
clock in the universe, let alone one capable of approaching the
physically fictitious (phyctitious?) Cauchy-Dedekind domain of
analyticity, its neighborhood will have been torn to shreds by an
unfortunately large such gust.
By the same token any sufficiently precise ruler will be torn apart by
gusts of momentum. Likewise for a cog in a sufficiently precise
mechanical clock, which will fall victim to gusts of angular momentum
about random axes in the cog's equatorial plane.
The logarithm of the reciprocal of Planck's constant is about 76.4
nepers (natural digits), or in other bases 110 bits or 33.2 decimal
digits. I take this as meaning that the portion of the universe in that
neighborhood of spacetime to which our current understanding of it
grants us access is given by nature at every point to a precision of
that many digits in your preferred radix.
Note that I am not saying nature has a preferred radix (Wheeler's "It
from Bit" should be understood only as "It from arithmetic"), nor that
she is finite, nor that Planck's constant binds every natural
observable. I claim only that Planck's constant constrains the
observables currently known to physics, in concert with our instruments
for measuring them, to a known finite precision.
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