Quanta Magazine called 2025 a historic year in mathematics and named the partial solution of Hilbert's sixth problem one of the year's three major breakthroughs. The problem had been open since 1900.

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

On March 3, 2025, Yu Deng of the University of Chicago, with Zaher Hani and Xiao Ma of the University of Michigan, posted a preprint to arXiv 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 team built new techniques that extended proofs previously valid for less than the blink of an eye to arbitrarily long time spans. For the first time, the mathematical path from microscopic particle motion to macroscopic fluid behavior runs end to end.

The setting, though, is a dilute gas of elastically colliding hard spheres, taken in the Boltzmann-Grad limit. More realistic interaction potentials and dense fluids remain untouched. That is why this is a partial solution rather than the whole of Hilbert's sixth problem.

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 work supplies a mathematical account of that asymmetry. Individual particle motions are time-reversible, but statistically almost every collision pattern runs in the direction the arrow of time points. A gas spreading out is ordinary; a gas spontaneously collecting itself into a corner is not impossible, merely so improbable that it never happens.

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.

Realizing a 125-Year-Old Dream

The question Hilbert posed in 1900 has, 125 years on, received a substantial answer. It is a bridge thrown across the border between mathematics and physics.

Update: On July 23, 2026, at the International Congress of Mathematicians in Philadelphia, Yu Deng was awarded a 2026 Fields Medal. The citation includes this work, the rigorous derivation of the Boltzmann equation from hard-sphere dynamics for rarefied gases. Deng also delivered an invited ICM lecture titled "Hilbert's Sixth Problem: Particles and Waves."

You do not need to be a mathematician or a physicist for this to land. Knowing that this much structure sits under the flow of water and the movement of air changes how those things look.

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!

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