I have had some experience in using mathematical theory to formulate physical theory. For example, this was done in special and general relativity [1]. When Minkowski formulated Einstein special relativity (SR) in terms of spacetime, he changed the basis of the discussion from Lorentz transformation (LT) to the invariance of the interval ds2 to any observer. The LT became a consequence [1], not a cause. GR was created on this new SR, not the old SR, by Einstein,and depends on this insight.

Mutatis mutandis, one can explore the consequences to thermodynamics when Caratheodory changed the basis of the discussion, by focusing on Pfaffian differential forms and adiabatic processes (without transfer of heat or mass).

In this case, James Joule determined the value of the mechanical equivalent of heat in 1843. For many years, well into the twentieth century, the result of Joule's fundamental experiment was only physically understood and explained [2].

One can adopt the interpretation of Max Born and his co-worker Caratheodory in 1909 [2]. Caratheodory's formulation of thermodynamics was studied by J. B. Boyling [3], as well.

REFERENCES:

[1] Edgardo V. Gerck. Special Relativity Applied to Arbitrary Motion. 2018. Preprint at Preprint Special Relativity Applied to Arbitrary Motion

[2] J. Kestin. A Course In Thermodynamics. Blaisdell Publishing Company, 1966.

[3] J. B. Boyling. An Axiomatic Approach to Classical Thermodynamics. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, vol. 329, no. 1576, pp. 3570, 1972.

Comparison of New and Old Thermodynamics

1. Logic of the Second Law of Thermodynamics: Subjectivism, Logical Jump, Interdisciplinary Argumentation.

2. New thermodynamics pursues universality, two theoretical cornerstones:

2.1 Boltzmann formula: ro=A*exp(-Mgh/RT) - Isotope centrifugal separation experiments show that it is suitable for gases and liquids.

2.2. Hydrostatic equilibrium: applicable to gases and liquids.

3. The second and third sonic virial coefficients of R143a derived from the new thermodynamics are in agreement with the experimental results.

3.1. The third velocity Virial coefficient derived is in agreement with the experimental data, which shows that the theory is still correct when the critical density is reached.

4. See Appendix Pictures and Documents for details.

I explained elsewhere on the danger, such as confirmation bias, of following one's opinion, or what one can imagine -- that it is better to focus on questions than (supposed) facts or (outdated) quotes. I give an example that seems far removed, but harbors the same issue.

It comes from the theory and the practice of tri-state logic design and as taught in EE in good universities. And, in Physics, let us remind ourselves that there is a notion of right and wrong --- it it agrees with Nature, it is right; but wrong otherwise. Opinions and voting do not matter in Physics, Maths, EE, and son on.

Everyone should know this, and it can change lives, can bring in new sound ideas, did influence logic, all as explained below, in different logic levels, as well.

In RG, for example, we can use it. Some answers are YES, some NO, and some are DON'T CARE. You have three logical states, but RG gives a point for any answer!

Is RG one state? Always YES? No, it's a multi-state computation, where each answer is counted according to different values, and any answer has more than three states it can be found, but RG shows only one, a point.

Go get your point, now, carelessly -- one might think, any anwer is a point -- and a disagreement gives me more points, more answers, more traffic. But, in reality, RG is part of life and life is a school. Being an RG-whale does not make one better, it actually can disqualify -- as all one can do... in life, evaluation follows several logic states, not just Boolean YES and NO.

Same with a tri-state IC. The final state of each chip IC is not just YES and NO, it can also be X. And all is computed in tri-state, one can't mix-in two-state chips, and just imagine that the end is somehow tri-state because the last chip is tri-state.

If you are paying attention, the array of tri-state chips can connect to another tristate device, compatibly. In EE, one can use even 12 states, or more, consult the IEEE for information and standards.

Silicon Valley would not exist otherwise, literally, and tri-state is a TM, it has commercial value for others -- it is not just maths. It changes lives, and give sound ideas one can follow, and trade-mark, and advance technology along a more sure path -- that Physics also follows -- that of the unknown.

Boolean logic is two centuries behind, cannot compete? No, it just doesn't agree with Nature, as already measured in Physics, EE, Maths, and so on. Time to catch-up, why it makes sense in Physics, for example, to follow Minkowski SR --- that it agrees with Nature more completely, and avoids "paradoxes." A similar thing happens with Caratheodory in thermodynamics.

The second law of thermodynamics is incorrect. See figure below for details.

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This is in opposition to the geometric view, which includes a model a priori. Mathematics and Geometry are all made up, made by humans, granted. Further, there is no Euclidean geometry (a geometry in which Euclid's fifth postulate holds, sometimes called parabolic geometry) or non-euclidean geometry (the complement, sometimes called hyperbolic geometry). There is just ageometry -- meaning, the absence of geometry -- from the Greek suffix "a-". The insistence that it must obey geometry has been a straight-jacket for ideas since the times of Euclid and Nikolai Lobachevsky. Caratheodory, for example, did not follow into this trap, but provided us with a global framework in differential forms.