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I see there are different accounts on the number of physical constants. Is there any theory predicting what the number of physical constants is? I agree there should be at least one but I have no justification why there should be more than one constant.

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"Should"?!? Really? Seems like philosophy to me. –  dmckee Jan 21 '13 at 0:51
Higgs had a theory in which there was an unobserved particle and which was highly consistent with observations. –  dmckee Jan 21 '13 at 1:04
You can always encode n constants in one constant by eg. interleaving the digits. –  user1708 Jan 21 '13 at 1:14
Generically adding in more parameters allows you to more easily fit the data. Generally this is considered a bad thing because of overfitting en.wikipedia.org/wiki/Overfitting. An ideal theory would be one with no free parameters that nevertheless matched all observations. –  Michael Brown Jan 21 '13 at 2:15
@Nick: What do you define as a physical constant? Or what I mean to ask is, how do you define and then count the set of all physical constants? For instance, do you count $h$ and $\hbar$ as separate constants? If you include $c$, $\hbar$, and $G$ as physical constants, then do you also include the Planck length, time, and mass? Without a clear definition of what constitutes an independent physical constant, this question is not really answerable. –  David Z Jan 21 '13 at 3:56
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