t the turn of the 20th century, the renowned mathematician David Hilbert had a grand ambition to bring a more rigorous, mathematical way of thinking into the world of physics. At the time, physicists were still plagued by debates about basic definitions — what is heat? how are molecules structured? — and Hilbert hoped that the formal logic of mathematics could provide guidance.

On the morning of August 8, 1900, he delivered a list of 23 key math problems to the International Congress of Mathematicians. Number six: Produce airtight proofs of the laws of physics.

The scope of Hilbert’s sixth problem was enormous. He asked “to treat in the same manner [as geometry], by means of axioms, those physical sciences in which mathematics plays an important part.”

His challenge to axiomatize physics was “really a program,” said Dave Levermore (opens a new tab), a mathematician at the University of Maryland. “The way the sixth problem is actually stated, it’s never going to be solved.”

But Hilbert provided a starting point. To study different properties of a gas — say, the speed of its molecules, or its average temperature — physicists use different equations. In particular, they use one set of equations to describe how individual molecules in a gas move, and another to describe the behavior of the gas as a whole. Was it possible, Hilbert wondered, to show that one set of equations implied the other — that these equations were, as physicists had assumed but hadn’t rigorously proved, simply different ways of modeling the same reality?

For 125 years, even axiomatizing this small corner of physics seemed impossible. Mathematicians made partial progress, coming up with proofs that only worked when they considered the behavior of gases on extremely short timescales or in other contrived situations. But these fell short of the kind of result that Hilbert had imagined.

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Now, three mathematicians have finally provided such a result. Their work not only represents a major advance in Hilbert’s program, but also taps into questions about the irreversible nature of time.

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