A friendly introduction to Hilbert's sixth problem with special focus in its meaning and issues

Question

While I was reading the Wikipedia's article dedicated to Hilbert's problems I've known the so-called Hilbert's sixth problem from this page and the corresponding Wikipedia's entry dedicated to this problem that I've read.

Question. I would like to know a friendly introduction to Hilbert's sixth problem, in which you can do more focus about the meaning of the problem and the issues (I think about the axiomatization of the more relevant theories in physics); isn't required the history of the problem, nor status. Your introduction can be done from an informative point of view but adding those remarkable facts, formulas and reasonings if you want to exemplify some point. Many thanks.

Thus I am asking about an explanation from your words and knowledges, in a post as your answer.

If you need/want to enrich your explanation with some reference (preferably also with an informative spirit) add such references and I am going to search and read what I can from the literature.

Answer 1

A poster child would be special relativity and the Lorenz transformation. Your axioms are that the universe is homogeneous and isotropic, which says the laws of physics are the same in all reference frames, and that the speed of light is independent of the motion of the source. From those you can derive all of special relativity. This is done in many texts.

The maximization of entropy can link statistical mechanics to may other subjects. You can get the classical black body radiation curve, then introduce photons and get the Planck curve. You can get the ideal gas law. You can get a lot of information about phase changes.

Answer 2

A key reference fot this is

Corry, Leo. On the origins of Hilbert's sixth problem: physics and the empiricist approach to axiomatization. International Congress of Mathematicians. Vol. III, 1697–1718, Eur. Math. Soc., Zürich, 2006.

The book notes that

"The sixth of Hilbert's famous 1900 list of twenty-three problems is a programmatic call for the axiomatization of physical sciences. Contrary to a prevalent view this problem was naturally rooted at the core of Hilbert's conception of what axiomatization is all about. The axiomatic method embodied in his work on geometry at the turn of the twentieth-century originated in a preoccupation with foundational questions related with empirical science, including geometry and other physical disciplines at a similar level. From all the problems in the list, the sixth is the only one that continually engaged his efforts over a very long period, at least between 1894 and 1932.''

Answer 3

I suppose I should recommend my own book, https://www.amazon.com/Hilberts-Sixth-Problem-Axiomatization-Probability/dp/3330072938 which is "friendly" only in the sense that you should have been either a math major or a physics major.
In particular, I cannot really agree that entropy is "foundational", it is merely a useful approximation under certain limitations, but plays no role in the axioms of the foundations of physics.