> it is quite surprising that these two sets of axioms that look nothing alike produce the same self-consistent physics
Why is that surprising? This sort of thing seems to be ubiquitous in our universe. F=ma is equivalent to the principle of least action. Turing machines are equivalent to the lambda calculus. Electrodynamics is equivalent to the existence of a finite reference velocity.
Also:
> You need to define temperature as an associative observable
Did you mean transitive? My understanding is that the required axiom is that if system A is in thermal equilibrium with system B, and B is in equilibrium with C, then A will be in equilibrium with C. That's transitivity, not associativity.
Indeed, this type of "surprises" is how you know you are probably on the right track with your choice of abstraction. But I still find the word "surprise" reasonable for the emotion I feel when I observe something like that. Very pleasing reassuring intellectually satisfying surprise. A "sufficiently smart armchair theorist" might discover these "surprises" in advance simply from consistency and beauty and simplicity considerations, but these armchair theorists are usually just a made up thing we imagine when teaching.
I corrected the wrong use of the word associative, thanks!
Why is that surprising? This sort of thing seems to be ubiquitous in our universe. F=ma is equivalent to the principle of least action. Turing machines are equivalent to the lambda calculus. Electrodynamics is equivalent to the existence of a finite reference velocity.
Also:
> You need to define temperature as an associative observable
Did you mean transitive? My understanding is that the required axiom is that if system A is in thermal equilibrium with system B, and B is in equilibrium with C, then A will be in equilibrium with C. That's transitivity, not associativity.