Hilbert's Sixth Problem Partially Solved After 125 Years: A 2025 Mathematical Breakthrough Illuminating the Arrow of Time
Introduction: 2025 Was a Historic Year for Mathematics
Quanta Magazine declared that "2025 marked a historic year in mathematics," highlighting the partial solution of Hilbert's sixth problem as one of three major mathematical breakthroughs of the year. After 125 years of remaining unsolved, this formidable challenge has finally seen significant progress.
What Is Hilbert's Sixth Problem?
The 23 Problems That Shaped 20th Century Mathematics
In 1900, German mathematician David Hilbert presented 23 unsolved problems at the Second International Congress of Mathematicians in Paris. These problems became the guiding compass for mathematical research throughout the 20th century, and solving any one of them was considered the highest honor for mathematicians.
The sixth problem concerned the "axiomatization of physics." Simply put, it asked whether the laws of physics could be reconstructed as a rigorous logical system, similar to mathematics.
From the World of Particles to the World of Fluids
This problem specifically addresses the behavior of everyday substances like air and water.
Air and water are composed of incredibly small particles (molecules) invisible to the naked eye. The motion of individual particles can be explained by Newton's simple laws of motion. However, the behavior of a "fluid" consisting of vast numbers of particles is described by completely different, complex equations called the Navier-Stokes equations.
Hilbert's sixth problem asks: "Can the laws describing macroscopic fluid motion be rigorously derived mathematically from the laws describing microscopic particle motion?"
The 2025 Breakthrough
Three Mathematicians Crack the Code
In March 2025, Yu Deng of the University of Chicago, along with Zaher Hani and Xiao Ma of the University of Michigan, published a paper presenting a solution to this 125-year-old problem.
Their approach consisted of two stages:
In the first stage, they rigorously derived the Boltzmann equation—which describes the statistical behavior of particles—from a system of countless small spheres undergoing repeated collisions. The key concept here is the "Boltzmann-Grad limit," which shows that collision frequency is appropriately maintained as the number of particles increases and their size decreases.
In the second stage, they derived the fundamental equations of fluid dynamics—the Euler equations and Navier-Stokes equations—from the Boltzmann equation.
Why Was This So Difficult?
Previous research had only proven the first stage derivation for extremely short time periods. Because countless particles in a fluid undergo repeated collisions, tracking all possible collision patterns was mathematically extremely challenging.
The research team developed new mathematical techniques that extended proofs previously valid only for shorter than "the blink of an eye" to work for arbitrarily long time periods. This completed, for the first time, the full mathematical pathway from microscopic particle motion to macroscopic fluid behavior.
New Light on the Mystery of the "Arrow of Time"
Reversibility and Irreversibility of Time
This research also provides new perspectives on one of physics' fundamental mysteries: the "arrow of time."
In the microscopic world governed by Newtonian mechanics, time is reversible. Physical laws hold even if time runs backward. For instance, playing a billiard ball collision in reverse presents no physical contradictions.
However, in the macroscopic world, time is irreversible. We age but don't grow younger. Ink dropped into water disperses but doesn't naturally reconcentrate. Hot things cool down but don't spontaneously heat up. Why can time run backward at the microscopic level but only in one direction at the macroscopic level?
Statistical Inevitability
This research provides a mathematical explanation for this mystery. Even though individual particle motions are time-reversible, statistically speaking, nearly all collision patterns proceed in the direction following the "arrow of time." Gas dispersal is natural, but spontaneous contraction has virtually zero probability.
Potential Impact on Our Lives
Advances in Fluid Dynamics
If this research gains formal recognition, it could promote new theoretical developments in fluid dynamics and statistical mechanics. This might contribute to improved weather forecasting accuracy, optimization of aircraft and ship design, and better understanding of biological fluid behavior such as blood flow.
Applications to Other Unsolved Problems
The new mathematical techniques developed by the research team may be applicable to other unsolved problems. For example, they might contribute to solving one of the Millennium Prize Problems: "the existence and smoothness of Navier-Stokes equation solutions."
Impact on Fundamental Physics
Deeper understanding of time's irreversibility could affect fundamental physics concepts, including reinterpretation of the second law of thermodynamics and reassessment of the law of entropy increase. Furthermore, developments in cosmology and bridging quantum mechanics with classical mechanics are anticipated.
Conclusion: Realizing a 125-Year-Old Dream
The question Hilbert posed in 1900 has finally received a significant answer after 125 years. Of course, the paper is still under peer review, and rigorous verification by the entire mathematical community is necessary. However, there is no doubt that this research represents a groundbreaking step in the fusion of mathematics and physics.
In Japan, reactions to this mathematical breakthrough include comments like "I can't understand it as a layperson, but something amazing has happened" and "The explanation about the arrow of time is fascinating." Even without being experts in mathematics or physics, learning that such deep mathematical structures lie hidden behind the flow of water and movement of air we observe daily might change how we see the world.
How has this major mathematical discovery been reported in your country? What are your thoughts on the relationship between physics and mathematics? We'd love to hear from you!
References
- https://www.quantamagazine.org/epic-effort-to-ground-physics-in-math-opens-up-the-secrets-of-time-20250611/
- https://www.quantamagazine.org/the-year-in-mathematics-20251218/
- https://arxiv.org/abs/2503.01800
- https://www.itmedia.co.jp/news/articles/2503/19/news080.html
- https://gigazine.net/news/20250510-hilberts-6th-problem-solved/
- https://gigazine.net/news/20251230-breakthroughs-in-mathematics-2025/
Reactions in Japan
I saw the news that Hilbert's sixth problem may have been solved. It's amazing that a problem unsolved for 125 years might finally have an answer. This is a moment when mathematical history moves. I want to watch whether it passes peer review.
It's wonderful that part of the grand goal of axiomatizing physics has been achieved. However, since the proof is under the limited conditions of dilute gas, extension to more general situations will be the next challenge.
Honestly, I don't understand the content at all, but I'm moved that there was a problem that took 125 years to solve. Mathematicians are amazing.
It's fascinating that this provides a mathematical explanation for the 'arrow of time' problem. Research approaching the mystery of why time can reverse at the micro level but is irreversible at the macro level is philosophically interesting too.
Since it's still at the preprint stage, we should be cautious until it passes peer review. There have been cases in the past where 'solutions' were announced and later retracted. But I do have high expectations.
I became interested when I heard this might lead to improved fluid dynamics simulation accuracy in the future. If applied to weather forecasting or aircraft design, it would be relevant to daily life.
When students ask me 'What is math good for?', I want to introduce research like this. I think it's a great example showing the power of mathematics supporting the foundations of physics.
A problem Hilbert posed in 1900 is now being solved more than 100 years later. The story of intellectual challenge continuing from Boltzmann's era is truly inspiring.
I heard a rebuttal paper came out in April. There are also criticisms that the dilute gas condition is unrealistic. Whether this can be called a complete solution seems debatable.
Even if it's a mathematically rigorous proof, applicability to real physical systems is a separate issue. But applied research is born from such basic research. I want to pay respect to these steady efforts.
The explanation that time's irreversibility emerges from statistical inevitability seems interesting as it might relate to debates about determinism and free will.
I finally understood the overview after reading the GIGAZINE explanation article. I'm grateful for media that explains specialized papers for general audiences.
If theoretical understanding of the Navier-Stokes equations deepens, it could lead to improvements in numerical simulation methods. This is research with practical significance too.
The fact that Quanta Magazine selected this as one of the three major mathematical breakthroughs of 2025 means it's internationally recognized. I hope it gets more coverage in Japan too.
A problem that was a distant dream during my active years is now being solved. Living long has its rewards. I'm happy to see young mathematicians thriving.
This is a monumental moment in mathematical history. I can't believe that a question Hilbert posed in 1900 is finally getting an answer 125 years later. Congratulations to the research team.
As a physics major, I feel the importance of this research is immeasurable. With the bridge between micro and macro mathematically proven, the foundations of physics have become more solid.
While I'm proud that German mathematician Hilbert's problem was solved by an American team, I also feel this is a result of international scientific collaboration.
The explanation about the 'arrow of time' is particularly fascinating. I was moved that mathematics can provide an answer to the philosophical question of why time only flows in one direction.
A wonderful achievement, but we should evaluate it cautiously until it passes peer review. In mathematics, there have been cases where claimed 'solutions' were later found to have errors.
As a regular person, it's hard to understand the importance of such basic research, but hearing it could lead to improved weather forecasting accuracy makes it feel more relatable.
As a fluid dynamics engineer, I have high expectations for this research. If it leads to improved simulation technology, it could revolutionize aircraft and automobile design.
This has become a hot topic in China's mathematical research community as well. It's a great example showing that investment in basic science produces such results.
Hilbert's 23 problems have guided the development of mathematics. Some problems still remain unsolved, but I hope the day will come when they're all solved.
Looking at it critically, a proof under the limited conditions of dilute gas may have limitations in real-world application. I expect a more general proof.
As someone from Japan doing research in America, I'm proud of such international research achievements. Mathematics has no borders.
As a science journalist, I feel the difficulty of communicating such complex research to the general public. However, continuing to convey the importance of basic science is our mission.
A problem scientists have worked on since Boltzmann's era is being solved with modern technology and knowledge. I really felt the power of accumulated scientific progress.
As a former Korean Mathematical Olympiad participant, I always dreamed of becoming a mathematician who solves such historic problems. The research team are my heroes.
I hope this research will also develop into bridging quantum mechanics and classical mechanics. It might be a step toward a unified theory of physics.