At 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?
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