Hilbert's sixth problem

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Hilbert's sixth problem is to axiomatize those branches of science in which mathematics is prevalent. It occurs on the list of Hilbert's problems given out in 1900.

The explicit statement reads

6. 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.¹

In the decade that followed, new foundational physics in the form of quantum theory and special relativity arose. These, clearly, could not have been anticipated when Hilbert formulated the problem. He himself subsequently worked on the axiomatic approach to more classical parts of physics.

When it came to formulating general relativity, Hilbert had an influence. The abstract approach of Dirac to the developed quantum mechanics of the 1920s resembles an axiomatic study; but would not be considered to be a complete axiomatisation in mathematical terms. Efforts have been made to put quantum field theory on some axiomatic basis. While the programme suggested by Hilbert has had some influence, therefore, it cannot be said to have been fulfilled along the lines suggested. In fact, fundamental physics still eludes any precise description.

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