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References
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Strangely enough, all the above formulations exhaust the undecidable problems of constructive analysis: equality between differentials, ordering, extrema, equality between numbers (being any thermodynamical magnitude a function of two others, all said for numbers holds as well as for functions).
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A very good example is: Truesdell, C.: The tragicomedy of classical thermodynamics. Berlin, Heidelberg, New York: Springer 1971.
An historical enquiry reveals that in the XIXth century all students following a “thermodynamical” point of view used a very simple mathematics, and all those following a “mechanical” point of view used intensively analysis and differential equation (the only exceptions where P. Duhem and M. Planck). An essentially “mechanical” point of view was followed by Boltzmann for formulating statistical mechanics; for this reason we don't examine it in this first paper about thermodynamics and mathematics.
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Drago, A. Constructive analysis and thermodynamics formulations. Arch math Logik 24, 17–21 (1984). https://doi.org/10.1007/BF02007138
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DOI: https://doi.org/10.1007/BF02007138