Hilbert's sixth problem

{{Short description|Axiomatization of probability and physics}} [[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. | year = 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>)]] '''Hilbert's sixth problem''' is to axiomatize 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. Quoted in its common English translation, the explicit statement reads:<ref name="hilbert">{{cite journal | last = Hilbert | first = David | author-link = David 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> {{blockquote|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 a further explanation of this problem and its possible specific forms:<ref name="hilbert"/> {{blockquote|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, L. Nordheim, and 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, 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.

In 1932 John von Neumann, basing on his earlier works, made an attempt to place quantum mechanics on a rigorous mathematical basis in his book ''Mathematical Foundations of Quantum Mechanics'',<ref>{{Cite book |last=Von Neumann |first=John |author-link=John von Neumann |url=https://press.princeton.edu/titles/11352.html |title=Mathematical foundations of quantum mechanics |date=2018 |publisher=Princeton University Press |isbn=978-0-691-17856-1 |editor-last=Wheeler |editor-first=Nicholas A. |location=Princeton Oxford |translator-last=Beyer |translator-first=Robert T.}}</ref>. While in the time of its release this was considered the most complete book about quantum mechanics<ref>{{cite journal|doi=10.1090/S0002-9904-1933-05665-3|title=Book Review: ''Mathematische Grundlagen der Quantenmechanik''|year=1933|last1=Margenau|first1=Henry|author-link=Henry Margenau|journal=Bulletin of the American Mathematical Society|volume=39|issue=7|pages=493–495|mr=1562667|doi-access=free}}</ref>, further developments made it insufficient.

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 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 and incompressible Navier-Stokes-Fourier equations, directly from Newton's laws.{{clarification needed|reason=Currently, there are no articles about those two. It might be helpful to explain what are these two.|date=January 2026}} {{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 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 |article-number= 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>

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

==References== * {{cite journal | last=Sauer | first=Tilman |author-link=Tilman Sauer | 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].

#06 Category:Unsolved problems in physics