The "paradox" arises because any detectable constitutive distinction, no matter how slight, can lead to a considerably large change in amount of entropy as a result of mixing. Though a continuous change in the properties of the materials that are mixed might make the degree of constitutive difference tend continuously to zero, the entropy change would nonetheless vanish discontinuously when the difference reached zero. From a general physical viewpoint, this discontinuity is paradoxical. But from a specifically thermodynamic viewpoint, it is not paradoxical, because in that discipline the degree of constitutive difference is not questioned; it is either there or not there. Gibbs himself did not see it as paradoxical. Distinguishability of two materials is a constitutive, not a thermodynamic, difference, for the laws of thermodynamics are the same for every material, while their constitutive characteristics are diverse. Though one might imagine a continuous decrease of the constitutive difference between any two chemical substances, physically it cannot be continuously decreased till it actually vanishes. It is hard to think of a smaller difference than that between ortho- and para-hydrogen. Yet they differ by a finite amount. The hypothesis, that the distinction might tend continuously to zero, is unphysical. This is neither examined nor explained by thermodynamics. Differences of constitution are explained by quantum mechanics, which postulates discontinuity of physical processes. For a detectable distinction, some means should be physically available. One theoretical means would be through an ideal semi-permeable membrane. It should allow passage, backwards and forwards, of one species, while passage of the other is prevented entirely. The entirety of prevention should include perfect efficacy over a practically infinite time, in view of the nature of thermodynamic equilibrium. Even the slightest departure from ideality, as assessed over a finite time, would extend to utter non-ideality, as assessed over a practically infinite time. Such quantum phenomena as tunneling ensure that nature does not allow such membrane ideality as would support the theoretically demanded continuous decrease, to zero, of detectable distinction. The decrease to zero detectable distinction must be discontinuous. For ideal gases, the entropy of mixing does not depend on the degree of difference between the distinct molecular species, but only on the fact that they are distinct; for non-ideal gases, the entropy of mixing can depend on the degree of difference of the distinct molecular species. The suggested or putative "mixing" of identical molecular species is not in thermodynamic