
David Hilbert (1862–1943) concluded his retirement address to the Society of German Scientists and Physicians on 1930-09-08, with Wir müssen wissen. Wir werden wissen. = We must know. We shall know. The words were given in response to the Latin: Ignoramus et ignorabimus = We do not know and we shall not know. His epitaph on his tombstone in Göttingen repeats his response. This weblog post is being published on he 95th anniversary of this address.
Hilbert was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries. He discovered and developed a broad range of fundamental ideas including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics, particularly proof theory.
Hilbert revealed a list of 10 problems on 1900-08-08, during the International Congress of Mathematicians held at the Sorbonne in Paris. These were problems later numbered 1, 2, 6, 7, 8, 13, 16, 19, 21 and 22, when the list was expanded to 23 problems. Sources for these are listed at the end of this post. This weblog post is only concerned about the sixth problem, often described as: Mathematical treatment of the axioms of physics: (a) axiomatic treatment of probability with limit theorems for foundation of statistical physics; (b) the rigorous theory of limiting processes “which lead from the atomistic view to the laws of motion of continua.”
Until recently, that status of this problem was unresolved, or partially resolved, depending on how the original statement is interpreted. Items (a) and (b) were two specific problems given by Hilbert in a later explanation. Kolmogorov’s axiomatics (1933) is now accepted as standard for the foundations of probability theory. There is some success on the way from the “atomistic view to the laws of motion of continua”, but the transition from classical to quantum physics means that there would have to be two axiomatic formulations, with a clear link between them. John von Neumann made an early attempt to place Quantum Mechanics on a rigorous mathematical basis in his book Mathematical Foundations of Quantum Mechanics (1932), but subsequent developments have occurred, further challenging the axiomatic foundations of quantum physics.
The sixth problem called for axiomatizing physics = determining the bare minimum of mathematical assumptions behind all its theories. Unfortunately, it’s not clear that mathematical physicists could ever know if they had resolved this challenge. However in the intervening 125 years, researchers have refined Hilbert’s vision into concrete steps needed to meet its solution.
In 2025-03 mathematicians Yu Deng (ca 1987 – ), Zaher Hani (ca 1986 – ) and Xiao Ma (1972 – ) posted a new paper to the preprint server arXiv.org that claims to have cracked one of these goals. If their work withstands scrutiny, it will mark a major stride toward grounding physics in mathematics and may open the door to analogous breakthroughs in other areas of physics.
These researchers may have figured out how to unify three physical theories that explain the motion of fluids. These theories govern a range of engineering applications that rested on assumptions that hadn’t been rigorously proven. This breakthrough won’t change the theories themselves, but it mathematically justifies them, which means that it strengthens mankind’s confidence that the equations will be useful.
Each theory differs in scale. At the microscopic level, fluids are composed of particles, which can be regarded as balls zooming around, sometimes colliding. Here, Newton’s laws of motion work well to describe their trajectories.
At a mesoscopic level a model cannot effectively model each particle individually. It needs to model the collective behavior of vast numbers of particles. This was originally developed in 1872 by Austrian theoretical physicist Ludwig Boltzmann, who addressed this when he developed the Boltzmann equation, that considers the likely behaviour of a typical particle, rather the the individual behaviour of every particle. This dismisses low-level details in favor of higher-level trends. The equation allows physicists to calculate how quantities such as momentum and thermal conductivity in the fluid evolve, without considering every microscopic collision.
More cosmology
In 1929, astronomer Edwin Hubble published a paper demonstrating that the universe is expanding. It gave rise to the Hubble constant, the number that describes how fast the universe is expanding. This created a puzzle, the Hubble tension, because this cosmic expansion differs depending on what cosmic objects are used to measure it. However, a new mathematical model could resolve the Hubble tension by assuming the universe rotates. New research, published in 2025-03 in Monthly Notices of the Royal Astronomical Society, suggests that our universe completes one revolution every 500 billion years. This rotation could resolve the discrepancy between different measurements of the Hubble constant.
Note
I am not a mathematician. In my studies, I learned some mathematics, including calculus and mathematical logic. However, at irregular intervals I felt I should add some extra synapses to my brain, by spending time working with physics and mathematics problems. In particular, I remember: 1. the first of these was devoting energy attempting to understand Kurt Gödel’s incompleteness theorems, as found in Ernest Nagel (1901-1985), James Roy Newman (1907–1966) Gödel’s Proof (1958). This was in the late 1960s. This proof demonstrates that in any sufficiently complex mathematical system, there are true statements that cannot be proven within that system. This means that no set of axioms can be both complete and consistent, fundamentally changing mankind’s understanding of mathematics. 2. I have also worked my way through George Polya’s (1887 – 1985) How to Solve It: A New Aspect of Mathematical Method, second edition (1957) [My copy was purchased 1983-04-19, 42 years ago, and I studied it shorly after that time.] 3. Alexander Woodcock and Monte Davis, Catastrophe Theory (1978). [My copy is undated, but states it was bought in Bodø, where we lived from 1985 – 1988.] 4. On a daily basis, working my way through Joseph Kane (1938 – 2024) and Morton Sternheim (1933 – 2023), Physics, 3rd edition (1988) [purchased 1997-05-31]. After that, I felt capable of taking university physics courses. Much later, I took some applied physics at the Andøya Rocket Range, now Andøya Spaceport, on Andøya (a municipality and an island) in northern Norway.
Source materials:
These are for a detailed study, only suitable for mathematicians and historians of mathematics and physics.
The original ten problems can be found in: Mathematische Probleme. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse = News of the Society of Sciences at Göttingen, Mathematical-Physical Class: 253–297. The larger list of 23 problems was published in 1901 in the original German in Archiv der Mathematik und Physik in Mathematische Probleme. Archiv der Mathematik und Physik. 3rd series (in German). 1: 44–63, 213–237. from which an English translation was made in 1902 by Mary Frances Winston Newson and published in the Bulletin of the American Mathematical Society, as Mathematical Problems. Bulletin of the American Mathematical Society. 8 (10): 437–479. doi:10.1090/S0002-9904-1902-00923-3. The authorship of these works is attributed to David Hilbert.
Problem 6, is examined in detail in: Corry, L. (1997). “David Hilbert and the axiomatization of physics (1894–1905)”. Arch. Hist. Exact Sci. 51 (2): 83–198. doi:10.1007/BF00375141. S2CID 122709777, and in: Gorban, A. N.; Karlin, I. (2014). “Hilbert’s 6th Problem: Exact and approximate hydrodynamic manifolds for kinetic equations”. Bulletin of the American Mathematical Society. 51 (2): 186–246. arXiv:1310.0406. doi:10.1090/S0273-0979-2013-01439-3
Kolmogorov’s axiomatics were initially formulated in Andrey Kolmogorov: Foundations of the theory of probability (1933). New York, US: Chelsea Publishing Company.
One source of John Von Neumann’s approach is found in, John Von Neumann and Nicholas A. Wheeler (editor). Mathematical foundations of quantum mechanics (2018). Translated by Robert Beyer: Princeton University Press. ISBN 978-0-691-17856-1
New updated information about the sixth problem, can be found in Scientific American 2025-04-14.







