Formal systems for physical theory

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Are there any attempt to express a part of physical theory, for example, Newtonian mechanics, in the form of a formal system such as first-order or higher-order predicate logic?

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This is, essentially, Hilbert's sixth problem:

Mathematical Treatment of the Axioms of Physics. The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which already today mathematics plays an important part; in the first rank are the theory of probabilities and mechanics.