The first and second laws of thermodynamics allow you to recover the inequality between the arithmetic and the geometric means:  Bring together n identical heat reservoirs with heat capacity $C$ and temperatures $T_1,\ldots,T_n$ and allow them to reach a final temperature $T$.  The first law of thermodynamics tells you that $T$ is the arithmetic mean of the $T_i$.  The second law of thermodynamics demands the non-negativity of the change in entropy, which is  

$$ C_n \, log(T/G) $$

where $G$ is the geometric mean.  It follows that $T > G$.

I believe this argument was first made by P.T. Landsberg (no relation!).