This paper is about Gibbs’ paradox in thermodynamics. There is also a well known and much discussed paradox in statistical mechanics which concerns justifying a factor of N! in the expression for the entropy in order to make it extensive. This paradox has attracted and continues to attract much more attention in the philosophy and physics literature than the thermodynamic version. The consensus is that the thermodynamic paradox has been satisfactorily solved. This paper pushes against this consensus through a combination of philosophical analysis of the mathematical and physical foundations of thermodynamics and historical interpretation and reconstruction of Gibbs’ 1875-1878 argument, thereby analysing the paradox in thermodynamics in more detail than has so far been done. The first aim of the paper is to point out that there are three distinct versions of the paradox in thermodynamics in the literature. This is important because all discussions and solutions so far are targeted at only one of the versions and thus no treatment to date can claim to provide the full picture. Furthermore, I show that they all follow from two premises. The second aim is to argue that the paradoxes disappear when we derive from first principles the equation used to calculate entropy changes. This exercise also supplies a definition of thermodynamic distinguishability. The third aim is to show that this analysis and definition reconstructs and clarifies Gibbs’ reasoning concerning gas mixing.