The definition of temperature is: 1/T = ∂S/∂U but mathematically this is a bit weird. Because S = klnW, where W denotes the multiplicity. And the multiplicity certainly does not form a continous basis, but rather a set of spread out integers. So S must be a discrete function! Therefore my question is: How come it still works? I have sometimes used the stirlingsapproximation and differentiated the expressions to find the maximum multiplicity. And indeed it did work, but mathematically it's just nonsense to me, that you can ever differentiate S or W. And another question (If you can bear over with all of them): The above definition of S = klnW: Is this JUST A DEFINITION because of the niceness of the additivity of the ln or is there more to it - i.e. is the ln somehow incorporated in nature? Because I get different answers for this, most saying it's a pure definition. So could you actually just as well have defined: 1/T = ∂W/∂U (maybe multiplied by k) and got the same result? Obviously the formulas would need to be revised such that they fit with W not being additive.