Editing Hilbert's sixth problem

{{Short description|Axiomatization of probability and physics}}
'''Hilbert's sixth problem''' is to [[axiom]]atize those branches of [[physics]] in which [[mathematics]] is prevalent. It occurs on the widely cited list of [[Hilbert's problems]] in mathematics that he presented in the year 1900.<ref>{{cite journal |first=David  |last=Hilbert |title=Mathematical Problems |journal=Bulletin of the American Mathematical Society |volume=8 |issue=10 |pages=437–479 |year=1902 |doi=10.1090/S0002-9904-1902-00923-3 |mr=1557926|doi-access=free }} Earlier publications (in the original German) appeared in ''Göttinger Nachrichten'', 1900, pp. 253–297, and ''Archiv der Mathematik und Physik'', 3rd series, vol. 1 (1901), pp. 44-63, 213–237.</ref> In its common English translation, the explicit statement reads:
[[File:StairsOfReduction.png|thumb|200px|Stairs of model reduction from microscopic dynamics (''the atomistic view'') to macroscopic continuum dynamics (''the laws of motion of continua'')  (Illustration to the content of the book<ref>{{cite book |last1=Gorban |first1= Alexander N.|last2= Karlin |first2= Ilya V. |date=2005 |title= Invariant Manifolds for Physical and Chemical Kinetics| url= https://www.academia.edu/17378865| location= Berlin, Heidelberg |publisher= Springer|series= Lecture Notes in Physics (LNP, vol. 660)| isbn= 978-3-540-22684-0|doi= 10.1007/b98103| archive-url= https://web.archive.org/web/20200819052923/https://www.academia.edu/17378865/Invariant_Manifolds_for_Physical_and_Chemical_Kinetics|archive-date= 2020-08-19}} [https://archive.org/details/gorban-karlin-lnp-2005 Alt URL]</ref>)]]
: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.''

Hilbert gave the further explanation of this problem and its possible specific forms:

:"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."

==History==
[[David Hilbert]] himself devoted much of his research to the sixth problem;<ref>{{cite journal |first=L. |last=Corry |title=David Hilbert and the axiomatization of physics (1894–1905) |journal=[[Archive for History of Exact Sciences]] |volume=51 |issue=2 |pages=83–198 |year=1997 |doi=10.1007/BF00375141 }}</ref>  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.{{r|Sauer}}

In the 1920s, mechanics of microscopic systems evolved into [[quantum mechanics]]. Hilbert, with the assistance of [[John von Neumann]], [[Lothar Wolfgang Nordheim|L. Nordheim]], and [[Eugene Wigner|E. P. Wigner]], worked on the axiomatic basis of quantum mechanics (see [[Hilbert space]]).<ref>{{cite journal | title=Von Neumann's contributions to quantum theory | first=Léon | last=van Hove | journal=Bull. Amer. Math. Soc. | volume=64 | issue=3 | year=1958 | pages=95–99 | mr=0092587 | zbl=0080.00416 | doi=10.1090/s0002-9904-1958-10206-2| doi-access=free }}</ref>  At the same time, but independently, [[Paul Dirac|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]],<ref>{{cite book |first=L. |last=Saint-Raymond |title=Hydrodynamic Limits of the Boltzmann Equation |publisher=Springer-Verlag |series=Lecture Notes in Mathematics |volume=1971 |year=2009 |isbn=978-3-540-92847-8  |doi=10.1007/978-3-540-92847-8 }}</ref> Marshall Slemrod,<ref>{{cite journal |first=M. |last=Slemrod |title=From Boltzmann to Euler: Hilbert's 6th problem revisited |journal=Comput. Math. Appl. |volume=65 |issue=10 |pages=1497–1501 |year=2013 |doi=10.1016/j.camwa.2012.08.016 |mr=3061719|doi-access=free }}</ref> [[Alexander Nikolaevich Gorban|Alexander N. Gorban]] and [[Ilya Karlin]].<ref>{{cite journal |first1=A.N. |last1=Gorban |first2=I. |last2=Karlin |title=Hilbert's 6th Problem: exact and approximate hydrodynamic manifolds for kinetic equations |journal=Bull. Amer. Math. Soc. |volume=51 |issue=2 |pages=186–246 |year=2014 |doi=10.1090/S0273-0979-2013-01439-3|doi-access= free |arxiv=1310.0406 }}</ref>

In 2025, a group of mathematicians made the claim that they had derived the full set of fluid equations, including the [[Compressible Euler equations|compressible Euler]] and [[incompressible Navier-Stokes-Fourier equations]], directly from Newton's laws. {{As of|2025|05}} their work is being examined by other mathematicians.<ref>{{cite arXiv |eprint=2503.01800 |last1=Deng |first1=Yu |last2=Hani |first2=Zaher |last3=Ma |first3=Xiao |title=Hilbert's sixth problem: Derivation of fluid equations via Boltzmann's kinetic theory |date=2025 |class=math.AP }}</ref><ref>{{Cite web |last=Murtagh |first=Jack |title=Mathematicians Crack 125-Year-Old Problem, Unite Three Physics Theories |url=https://www.scientificamerican.com/article/lofty-math-problem-called-hilberts-sixth-closer-to-being-solved/ |access-date=2025-04-26 |website=Scientific American |language=en}}</ref>

==Status==
Hilbert’s sixth problem was a proposal to expand the [[Axiomatic system|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.<ref>{{cite journal |first=A.N. |last=Gorban |title=Hilbert's sixth problem: the endless road to rigour |journal=Phil. Trans. R. Soc. A |volume=376 |issue=2118 |pages= 20170238|year=2018 |doi=10.1098/rsta.2017.0238 |pmid=29555808 |doi-access= free|pmc=5869544 |arxiv=1803.03599 |bibcode=2018RSPTA.37670238G }}</ref> Two fundamental theories capture the majority of the fundamental phenomena of physics:
* [[Quantum field theory]],<ref>{{cite book | editor=Felix E. Browder | editor-link= Felix Browder | title= Mathematical Developments Arising from Hilbert Problems | series=[[Proceedings of Symposia in Pure Mathematics]] | volume= XXVIII | year=1976 | publisher= [[American Mathematical Society]] | isbn=0-8218-1428-1 | first=A.S. | last= Wightman | author-link= Arthur Wightman | chapter= Hilbert's sixth problem: Mathematical treatment of the axioms of physics | pages=147–240 }}</ref> which provides the mathematical framework for the [[Standard Model]];
* [[General relativity]], which describes space-time and gravity at macroscopic scale.
Hilbert considered general relativity as an essential part of the foundation of physics.<ref>{{cite journal |first=David |last=Hilbert |title=Die Grundlagen der Physik. (Erste Mitteilung) |journal=Nahrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-physikalische Klasse |volume=1915 |pages=395–407 |year=1915 |url=https://eudml.org/doc/58946}}</ref><ref>{{harvnb|Sauer|1999}}</ref> 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.<ref>Theme issue {{cite journal |title=Hilbert's sixth problem |journal=Phil. Trans. R. Soc. A |volume=376 |issue=2118 |year=2018 |doi=10.1098/rsta/376/2118 |doi-access=free }}</ref> 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,<ref>{{Cite journal |author=A. Majhi |title=Cauchy's Logico-Linguistic Slip, the Heisenberg Uncertainty Principle and a Semantic Dilemma Concerning "Quantum Gravity" |journal=International Journal of Theoretical Physics |volume=61|issue=3 |year=2022 |page=55 |doi=10.1007/s10773-022-05051-8|arxiv=2204.00418|bibcode=2022IJTP...61...55M }}</ref> unravelling of a logical tautology in the quantum tests of [[equivalence principle]]<ref>{{cite journal |first1=A. |last1=Majhi |first2=G. |last2=Sardar |title=Scientific value of the quantum tests of equivalence principle in light of Hilbert's sixth problem |journal= Pramana - J Phys |volume=97 |issue=1 |year=2023 |page=26 |doi=10.1007/s12043-022-02504-x|arxiv=2301.06327 |bibcode=2023Prama..97...26M }}</ref>  and formal unprovability of the first Maxwell's equation.<ref>{{Cite journal|author=A. Majhi |title=Unprovability of first Maxwell's equation in light of EPR's completeness condition: a computational approach from logico-linguistic perspective|journal= Pramana - J Phys|volume=61|issue=4 |year=2023 |page=163 |doi=10.1007/s12043-023-02594-1|url=https://hal.science/hal-03682283v2|arxiv=2310.14930|bibcode=2023Prama..97..163M }}</ref>
Regarding the problem of "developing mathematically the limiting processes [...] which lead from the atomistic view to the laws of motion of continua." an active area of research is focused on deriving the continuum equations of fluid motion and of elastic solids starting from atomistic particle-based descriptions. For example, a derivation of the equations of laminar viscous flow and of viscoelasticity has been achieved starting all the way from an atomistic particle-based microscopically reversible Hamiltonian<ref>{{Cite journal|author=A. Zaccone |title=General theory of the viscosity of liquids and solids from nonaffine particle motions|journal= Physical Review E|volume=108|pages= 044101 |year=2023|issue=4 |doi=10.1103/PhysRevE.108.044101|pmid=37978701 |url=https://journals.aps.org/pre/abstract/10.1103/PhysRevE.108.044101|arxiv=2306.05771|bibcode=2023PhRvE.108d4101Z |hdl=2434/1093228 }}</ref> and subsequently generalized from classical mechanics to relativistic mechanics.

==See also==
*[[Wightman axioms]]
*[[Constructive quantum field theory]]

==Notes==
{{Reflist|refs=
<ref name=Sauer>{{harvnb|Sauer|1999|p=6}}</ref>
}}

==References==
* {{cite journal | last=Sauer | first=Tilman | year=1999 | title=The relativity of discovery: Hilbert's first note on the foundations of physics | journal=Arch. Hist. Exact Sci. | volume=53 | number=6 | pages=529–575 | arxiv=physics/9811050 | zbl=0926.01004  | bibcode=1998physics..11050S }}
* {{cite book | editor=Felix E. Browder | editor-link= Felix Browder | title= Mathematical Developments Arising from Hilbert Problems | series=[[Proceedings of Symposia in Pure Mathematics]] | volume= XXVIII | year=1976 | publisher= [[American Mathematical Society]] | isbn=0-8218-1428-1 | first=A.S. | last= Wightman | author-link= Arthur Wightman | chapter= Hilbert's sixth problem: Mathematical treatment of the axioms of physics | pages=147–240 }}

==External links==
* [http://aleph0.clarku.edu/~djoyce/hilbert/problems.html#prob6 David Hilbert, Mathematical Problems, Problem 6, in English translation].

{{Hilbert's problems}}
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[[Category:Hilbert's problems|#06]]
[[Category:Unsolved problems in physics]]