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I would like to ask if any of the fundamental physical quantities like the speed of light or plancks constant (all measured according to a common standard of of units) can be classified as computable or non-computable in the sense of of Turing? I don't know if the question makes sense, becuase I have not been training in either theoretical physics or computability theory. But still, the question is relevant in comparing the power of analog vs digital computers, the engineering design of which is the prime concern of EE techies.

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  • $\begingroup$ See this $\endgroup$ Feb 8, 2014 at 16:53
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    $\begingroup$ The the speed of light is integer by definition (en.wikipedia.org/wiki/Speed_of_light). $\endgroup$ Feb 8, 2014 at 17:02
  • $\begingroup$ 1 kg is the mass of a body of a specified cylindric shape made of a specific platinum/iridium alloy. The Avogadro constant is the number of atoms in 0.012 kg of C$^{12}$. So with a suitably good model of quantum physcs you might compute the Avogadro constant from first principles. But it is a really long way to go (and requires trust in the models being accurate enouhg) $\endgroup$ Feb 8, 2014 at 17:05

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Some of the physical constants are measured, some are defined, and some are computed. The measured ones are just that and have some error attached. The best measured value is a terminating decimal, which is certainly computable. There are (infinitely many) values within the uncertainty range that are not computable. The defined ones again are fixed values with a terminating decimal. For example, the speed of light is defined to be $299 792 458 m/s$ The anomalous magnetic moment of the muon can (in theory) be computed as an infinite sum over all the interaction processes of interest. It can then be compared with experiment. We can imagine computing as many decimals as desired.

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  • $\begingroup$ I am primarily interested in physical constants that are not computable. The idea is to obtain say, an "analog" computer out of some mechanical contraption that uses a rod for basic copmutation and whos length is turing non-computable. So the first thing is, how to construct such a "non-computably long" rod. One could imagine accelerating to a speed such that relativistic effects would contract a "computable" rod to some "noncomputable" one. Obviously, there has to be some physical constant that is noncomputable that enters into the transformation from one frame to the other. $\endgroup$
    – Iconoclast
    Feb 8, 2014 at 19:19
  • $\begingroup$ On some more reflection, the actual length of any (most) rods would be non-computable right? $\endgroup$
    – Iconoclast
    Feb 8, 2014 at 19:23
  • $\begingroup$ The problem is that the computable and noncomputable numbers are both dense in the reals. You can't measure any physical quantity and say whether it is computable or not because the uncertainty covers numbers of both sorts. On the other hand, some argue that space is granular, say at the Planck length. Then all physical objects have a length that is computable because it is an integer multiple of the Planck length. I don't think there is anything useful in linking the computable numbers and the physical world. $\endgroup$ Feb 8, 2014 at 19:26

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