[FOM] Physical Theories and Hypercomputation

Vaughan Pratt 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.

Vaughan Pratt

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