I am wondering whether the five$^1$ units of the natural unit system really is dictated by nature, or invented to satisfy the limited mind of man? Is the number of linearly independent units a property of the nature, or can we use any number we like? If it truly is a property of nature, what is the number? -five? can we prove that there not is more?
In the answers please do not consider cgs units, as extensions really is needed to cover all phenomenon. - or atomic units where units only disappear out of convenience. - or SI units where e.g. the mole for substance amount is just as crazy as say some invented unit to measure amount of money.
$^1$length, time, mass, electric charge, and temperature (or/and other linearly independent units spanning the same space).
Answer
Even seasoned professionals disagree on this one. Trialogue on the number of fundamental constants by M. J. Duff, L. B. Okun, G. Veneziano, 2002:
This paper consists of three separate articles on the number of fundamental dimensionful constants in physics. We started our debate in summer 1992 on the terrace of the famous CERN cafeteria. In the summer of 2001 we returned to the subject to find that our views still diverged and decided to explain our current positions. LBO develops the traditional approach with three constants, GV argues in favor of at most two (within superstring theory), while MJD advocates zero.
Okun's thesis is that 3 units (e.g. $c$, $\hbar$, $G$) are necessary for measurements to be meaningful. This is in part a semantic argument.
Veneziano says that 2 units are necessary: action $\hbar$ and some mass $m_{fund}$ in QFT+GR; or a length $\lambda_s$ and time $c$ in string theory; and no more than 2 in M-theory although he's not sure.
Finally, Duff says there is no need for units at all, all quantities are fundamentally subject to some symmetry, and units are merely conventions for measurement.
This is a very fun paper and answers your question thoroughly.
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