David Hilbert himself devoted much of his research to the sixth problem;[3] in particular, he worked in those fields of physics that arose after he stated the problem.
In the 1910s, celestial mechanics evolved into general relativity. Hilbert and Emmy Noether corresponded extensively with Albert Einstein on the formulation of the theory.[4]
In the 1920s, mechanics of microscopic systems evolved into quantum mechanics. Hilbert, with the assistance of John von Neumann, L. Nordheim, and E. P. Wigner, worked on the axiomatic basis of quantum mechanics (see Hilbert space).[5] At the same time, but independently, Dirac formulated quantum mechanics in a way that is close to an axiomatic system, as did Hermann Weyl with the assistance of Erwin Schrödinger.
In the 1930s, probability theory was put on an axiomatic basis by Andrey Kolmogorov, using measure theory.
Since the 1960s, following the work of Arthur Wightman and Rudolf Haag, modern quantum field theory can also be considered close to an axiomatic description.
In the 1990s-2000s the problem of "the limiting processes, there merely indicated, which lead from the atomistic view to the laws of motion of continua" was approached by many groups of mathematicians. Main recent results are summarized by Laure Saint-Raymond,[6] Marshall Slemrod,[7] Alexander N. Gorban and Ilya Karlin.[8]
Hilbert’s sixth problem was a proposal to expand the axiomatic method outside the existing mathematical disciplines, to physics and beyond. This expansion requires development of semantics of physics with formal analysis of the notion of physical reality that should be done.[9] Two fundamental theories capture the majority of the fundamental phenomena of physics:
Hilbert considered general relativity as an essential part of the foundation of physics.[11][12] However, quantum field theory is not logically consistent with general relativity, indicating the need for a still-unknown theory of quantum gravity, where the semantics of physics is expected to play a central role. Hilbert's sixth problem thus remains open,[13] Nevertheless, in recent years it has fostered research regarding the foundations of physics with a particular emphasis on the role of logic and precision of language, leading to some interesting results viz. a direct realization of uncertainty principle from Cauchy's definition of `derivative' and the unravelling of a semantic obstacle in the path of any theory of quantum gravity from the axiomatic perspective,[14] unravelling of a logical tautology in the quantum tests of equivalence principle[15] and formal unprovability of the first Maxwell's equation.[16]