Hilbert’s Sixth Problem
In the year 1900 Hilbert presented his problems to the International Congress of Mathematicians (he presented 10 problems at the talk; the full list of 23 problems was published later). The list of 23 Hilbert’s problems was very influential for 20th century mathematics. The sixth problem concerns the axiomatization of those parts of physics which are ready for a rigorous mathematical approach
The original Hilbert’s formulation (in English translation) was:
“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.” This is definitely “a programmatic call” for the axiomatization of existent physical theories.
In a further explanation Hilbert proposed two specific problems: (i) axiomatic treatment of probability with limit theorems for the foundation of statistical physics, and (ii) the rigorous theory of limiting processes “which lead from the atomistic view to the laws of motion of continua”:
"As to the axioms of the theory of probabilities, it seems to me desirable that their logical investigation should be accompanied by a rigorous and satisfactory development of the method of mean values in mathematical physics, and in particular in the kinetic theory of gases. ... Boltzmann's work on the principles of mechanics suggests the problem of developing mathematically the limiting processes, there merely indicated, which lead from the atomistic view to the laws of motion of continua."
The sixths problem has inspired several waves of research. Its mathematical content changes in time what is very natural for a “programmatic call” . In 1930s, the axiomatic foundation of probability seemed to be finalized on the basis of measure theory . Nevertheless, Kolmogorov and Solomonoff in 1960s stimulated new interest to the foundation of probability (Algorithmic Probability) .
Hilbert, Chapman and Enskog created the asymptotic expansions for the hydrodynamic limit of the Boltzmann equations [4,5]. The higher terms of the Chapman-Enskog expansion are singular and truncation of this expansion does not have rigorous sense (Bobylev ). Golse, Bardos, Levermore and Saint-Raymond proved rigorously the Euler limit of the Boltzmann equation in the scaling limit of very smooth flows [7,8], but recently Slemrod used the exact results of Karlin and Gorban [9,10] and proposed a new, Korteweg asymptotic of the Boltzmann equation . These works attracted much attention and a special issue of BAMS was published with papers of L. Saint-Raymond , A. Gorban and I. Karlin . There was a media reaction: QuantaMagazine published on July 21, 2015 a paper “Famous Fluid Equations Are Incomplete: A 115-year effort to bridge the particle and fluid descriptions of nature has led mathematicians to an unexpected answer.”
It seems that Hilbert presumed the kinetic level of description (the “Boltzmann level”) as an intermediate step between the microscopic mechanical description and the continuum mechanics. Nevertheless, this intermediate description may be omitted. Now, L. Saint-Raymond with co-authors is developing a new approach to the problem “from the atomistic view to the laws of motion of continua” without intermediate kinetic equations .
Quantum mechanics was invented after the Hilbert problems were stated. When, twenty-five years after his Paris address, quantum mechanics came of age with the almost simultaneous papers of Heisenberg and Scrhödinger, it was immediately clear to Hilbert that this theory was as much a new mechanics as a new probability calculus and that these features implied a change of perspective on his sixth Paris problem. This awareness is proved by the fact that, almost simultaneously with the birth of quantum theory (in 1925), he devoted a seminar to the description of its mathematical structure, in which the non-classical features of the new probabilistic calculus were clearly described. The notes of this seminar, collected by von Neumann and Nordheim (Hilbert's assistants at that time), were later published in a joint paper . In this paper the authors declare that the goal of an axiomatization of a physical discipline consists in: “… formulating the physical requirements so clearly, that the mathematical model becomes uniquely determined by them ...” The main idea is that a physical theory consists of three, sharply distinguishable parts: (i) physical axioms, (ii) analytic machinery (also called “formalism”), and (iii) physical interpretation. The first attempt to formalization of quantum mechanics was performed by von Neumann .
The Kolmogorov formalization of the classical probabilistic model came to light in a time (1933) when already quantum mechanics was making extensive use of a completely different probabilistic formalism. In the following 20 years each of the two disciplines was strongly concentrated on its own inner development and this is probably the reason why we have to wait until the second Berkley Symposium on Probability and Statistics (1950) when Feynman, in his communication, for the first time explained to a large and qualified international audience of probabilists, the scientific challenge posed by the existence of two, apparently mutually incompatible, mathematical models of probability theory .
After that, the axiomatic approach to the quantum probability was developed by many researchers  and there are many versions of its axiomatization [19,20]. The fast development of this area of research has involved ideas from algebra, functional analysis, and fuzzy set theory. There has even developed a programming language based on non-commutative logic . Ideas and methods of quantum computing [22,23] and quantum cryptography  transform research in the foundation of quantum mechanics into an applied discipline with a perspective of engineering applications. Many new mathematical structures and methods have been invented (see, for example, [25,26]). Applications of quantum probability are now much wider than just mechanics of subatomic particles .
Work on Hilbert’s sixth problem involves many areas of mathematics: mathematical logic, algebra, functional analysis, differential equations, geometry, probability theory and random processes, theory of algorithms and computational complexity, and many others. It remains one of the most seminal area of interdisciplinary dialog in mathematics and mathematical physics.
1. L. Corry, David Hilbert and the axiomatization of physics (1894–1905), Arch. Hist. Exact Sci. 51 (2) (1997), 83–198,
2. A. Kolmogoroff, Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer-Verlag, Berlin, 1977 (German). Reprint of the 1933 original.
3. A.K. Zvonkin and L.A. Levin, The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms, Russian Mathematical Surveys 25 (6) (1970), 83-124.
4. S. Chapman and T. G. Cowling, The mathematical theory of non-uniform gases. An account of the kinetic theory of viscosity, thermal conduction and diffusion in gases, Cambridge University Press, London, 1970.
5. P. Degond, Macroscopic limits of the Boltzmann equation: a review, In Modeling and Computational Methods for Kinetic Equations, pp. 3-57. Birkhäuser Boston, 2004.
6. A. V. Bobylev, On the Chapman-Enskog and Grad methods for solving the Boltzmann equation, Dokl. Akad. Nauk SSSR 262 (1) (1982), 71–75.
7. F. Golse and L. Saint-Raymond, The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels, Invent. Math. 155 (1) (2004), 81–161.
8. L. Saint-Raymond, Hydrodynamic limits of the Boltzmann equation, Lecture Notes in Mathematics, vol. 1971, Springer, Berlin, 2009.
9. A. N. Gorban and I. V. Karlin, Short-wave limit of hydrodynamics: a soluble example, Phys. Rev. Lett. 77 (1996), 282–285.
10. I.V. Karlin, Exact summation of the Chapman-Enskog expansion from moment equations, J. Phys. A 33 (45) (2000), 8037–8046,
11. M. Slemrod, From Boltzmann to Euler: Hilbert’s 6th problem revisited, Comput. Math. Appl. 65 (10) (2013), 1497–1501.
12. L. Saint-Raymond, A mathematical PDE perspective on the Chapman–Enskog expansion, Bull. Amer. Math. Soc., 51 (2) (2014), 247–275.
13. A.N. Gorban, I. Karlin, Hilbert's 6th Problem: exact and approximate hydrodynamic manifolds for kinetic equations, Bull. Amer. Math. Soc., 51(2), 2014, 186-246.
14. T. Bodineau, I. Gallagher, L. Saint-Raymond, The Brownian motion as the limit of a deterministic system of hard-spheres, Invent. Math. (2013), 1-61.
15. D. Hilbert, L. Nordheim, J.von Neumann, Über die Grundlagen der Quantenmechanik. Mathematische Annalen, 98 (1927), 1-30.
16. J. von Neumann, Mathematical foundations of quantum mechanics, Investigations in Physics, Princeton, NJ: Princeton University Press, 1955.
17. R. P. Feynman, The Concept of Probability in Quantum Mechanics, in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, ed. by J. Neyman, (University of California Press, Berkeley, 1951), 533–541.
18. L. Accardi, Topics in quantum probability, Physics Reports 77 (3) (1981), 169-192.
19. A. Khrennikov, Contextual approach to quantum formalism. Springer, 2009.
20. G. Chiribella, G. M. D’Ariano, P. Perinotti, Informational derivation of quantum theory, Physical Review A 84 (1) (2011), 012311.
21. R. Baudot, Non-commutative logic programming language NoClog. In Symposium LICS, Santa Barbara, 2000, 3-9.
22. D. Deutsch, Quantum theory, the Church-Turing principle and the universal quantum computer, Proceedings of the Royal Society of London A 400 (818) (1985), 97-117.
23. H. Nishimura, M. Ozawa, Computational complexity of uniform quantum circuit families and quantum Turing machines, Theoretical Computer Science 276 (1) (2002), 147-181.
24. C.H. Bennett, G. Brassard, A. Ekert, Quantum cryptography, In Progress in Atomic physics Neutrinos and Gravitation, proceedings of the XXVIIth Rencontre de Moriond Series: Moriond Workshops, held January, p. 371. 1992.
25. M Evans, RL Hudson, Multidimensional quantum diffusions, Quantum Probability and Applications III, Springer Berlin Heidelberg, 1988, 69-88.
26. S. Majid, Foundations of quantum group theory. Cambridge University Press, 2000.
27. L Accardi, Non Kolmogorov Probability Models Outside Quantum Mechanics, Foundations of Probability and Physics-5 1101 (1) (2009), 3-7.