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Is there anything to be said about the computability of some important physical constants? Specifically, the gravitational constant turns out to be very hard to approximate to even its third significant figure. Can this be considered evidence towards a negative answer? What about other constants?

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 Your question doesn't make sense. $G$ can only hope to make sense as a real number relative to a model of gravitation. You haven't specified which model you're using. Moreover, in some models $G$ isn't a constant. For example, in Wun-Yi Shu's model: arxiv.org/abs/1007.1750 -- this is a particularly nice model because it appears to remove the need for "dark energy" – Ryan Budney Aug 18 2010 at 8:09
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 Note that $G$ is not a dimensionless constant, so the numerical part of the value will depend upon the units chosen. – Rhubbarb Aug 18 2010 at 8:09
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 Also the difficulty in determining G to more significant figures with our current technology has a lot to do with the fact that gravity is much weaker than the other forces. That being said, I'm not sure this is a question for MO. – J. M. Aug 18 2010 at 8:12
Such a physical constant is only as meaningful as the units it is expressed in. Else you can use en.wikipedia.org/wiki/Planck_units in which G and some another constants equal 1. I am more curious about understanding the ratio of the proton mass and the electron mass. – Dan Brumleve Aug 18 2010 at 8:15
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 Dirac had some numerological argument that the fine structure constant is exactly 1/137 ... but subsequently more accurate measurements showed him wrong. – Gerald Edgar Aug 18 2010 at 8:25
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closed as not a real question by Ryan Budney, Robin Chapman, Yemon Choi, Franz Lemmermeyer, Charles Siegel Aug 18 2010 at 10:02

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