After the Fields Medals were announced, a great deal of public discussion developed around the winners. I was not especially interested in that discussion. I was much more curious about what Yu Deng had actually proved. His work concerns the way reversible microscopic motion can produce irreversible macroscopic behaviour, a question closely connected with how I have thought about entropy for many years.
The proof is far beyond my mathematical ability. I cannot follow its technical development in any complete way. That does not prevent me from considering the conclusion and its possible significance. The first requirement is to establish what the research proves and the conditions under which it holds. The mathematical result can then be separated from the philosophical interpretation built upon it. The interpretation remains a hypothesis; it should not be presented as part of the theorem.
I have long tended to understand mathematics through a broadly Kantian framework. Human beings cannot know an object independently of the a priori forms and rational conditions through which it becomes intelligible to us. Mathematics removes as much particular empirical content as possible and retains quantity, relation, structure and rules of inference. It is not nature itself, but it is the most stable and rigorous form through which human reason can grasp nature.
Mathematics should not be reduced to formal logic alone. For Kant, mathematics consisted of synthetic a priori judgements and also depended on space, time and the possibility of construction. A more accurate formulation would be that mathematics is the necessary development of possible experience under the a priori forms, logical rules and constructive relations available to human reason.
Mathematical conclusions are always conditional. No theorem exists independently of its axioms, definitions and model. Once those conditions have been accepted, however, the conclusion is no longer a matter of preference. We may question whether a model adequately corresponds to reality, but we cannot accept its premises and then reject what follows from them. Mathematics shows us which conclusions cannot be avoided within a rigorously defined rational framework.
I have never understood entropy simply as an increase in “disorder”. That word is too vague. Entropy increase is better understood as a process in which differences that were once concentrated, identifiable and available for use become dispersed, while simple relations pass into increasingly complex higher-order correlations. Information may not disappear at the microscopic level, yet a macroscopic description becomes less able to recover or use it. The structure may still exist, but no longer in its former accessible form.
Boltzmann’s H-theorem has long established that physical entropy does not decrease under the relevant conditions. A foundational difficulty nevertheless remained. The Boltzmann equation is irreversible, while the Newtonian mechanics beneath it is reversible. If every particle velocity could be reversed exactly, the particles should in principle retrace their paths. How irreversible macroscopic behaviour emerges from reversible microscopic motion has therefore remained a central problem in statistical mechanics.
Yu Deng, Zaher Hani and Xiao Ma studied the elastic collisions of a large number of hard spheres in a rarefied gas. Under specified statistical initial conditions and in the Boltzmann–Grad limit, they rigorously derived the Boltzmann equation from Newtonian hard-sphere dynamics. Their result remains valid for as long as the corresponding strong solution of the Boltzmann equation exists. A connection that had previously been established only over very short periods was thereby placed on a much stronger mathematical foundation.
This work does not prove, without qualification, that entropy must increase everywhere in the universe. It concerns a particular model and depends on its statistical initial state, scaling limit and assumptions about the existence of solutions. Nor does the complete microscopic information simply vanish. Correlations created by particle collisions move into higher-order structures that are no longer tracked by one-particle distributions and macroscopic variables. Irreversibility appears through this change of scale.
That is already enough to raise a philosophical question. If mathematics expresses the basic form of human reason, what does it mean when reason begins with reversible local motion, follows strict rules, and arrives at irreversible behaviour at the collective level? Entropy may be more than an empirical tendency observed in external nature. It may also represent a structural resistance that reason encounters as it advances according to its own rules.
“Resistance” here does not refer to a natural force acting against reason. It is closer to a statistical direction. States that remain ordered, identifiable and directly accessible occupy only a small region among the available possibilities, while many different microscopic states can produce the same macroscopic result. Concentrated differences require conditions. Relations require constraints if they are to remain accessible. An established structure does not preserve itself automatically.
Human reason encounters a related problem when it deals with complex objects. It can define local rules, but it cannot continue to preserve and use every detail as the scope of the object expands. It must abstract, classify, compress and use statistical descriptions. Some differences remain visible while others recede into the background. The further reason moves from individual events towards complex wholes, the harder it becomes to recover information that has passed into higher-order correlations. Entropy may therefore reveal not only a macroscopic direction in physical systems, but also a limit that finite reason cannot entirely remove when it attempts to comprehend complex wholes.
Today, AI is increasingly used in physics, chemistry, and biology to discover patterns, generate hypotheses, perform calculations, and derive results, which can ultimately be tested through observation and experiment. Mathematics is different. AI can propose conjectures, explore proof strategies, and even generate lengthy proofs, but for a result to become mathematical knowledge, its reasoning must still be open to human verification. Human scrutiny remains a necessary step in the acceptance of a mathematical theorem.
Deng’s work clearly carries philosophical significance. It does not settle the whole relationship between entropy and reason, but it greatly strengthens the connection between reversible microscopic motion and irreversible macroscopic behaviour. Mathematics is the most rigorous development of human reason, and here reason, following its own rules, arrives at a directionality that cannot simply be removed at the macroscopic level. This gives my earlier thoughts about mathematics, entropy and reason a new point of connection. The circle is not yet complete, but one of its most important conditions now rests on a substantial mathematical foundation.
Further reading
International Mathematical Union, Fields Medals 2026
https://www.mathunion.org/imu-awards/fields-medal/fields-medals-2026
Yu Deng, Zaher Hani and Xiao Ma, Long Time Derivation of the Boltzmann Equation from Hard Sphere Dynamics
https://arxiv.org/abs/2408.07818
Yu Deng, Zaher Hani and Xiao Ma, Hilbert’s Sixth Problem: Derivation of Fluid Equations via Boltzmann’s Kinetic Theory
https://arxiv.org/abs/2503.01800
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