Hilbert's Sixth Problem & the Cost of Infinite Information

In 1900, David Hilbert challenged mathematicians to axiomatize physics with the same rigor as geometry. The sixth of his famous problems specifically called for a mathematical treatment of probability in mechanics. Over a century later, the problem remains formally open. This essay argues that Hilbert's question reduces to an information-theoretic constraint: does nature require infinite precision to specify its state? The ultraviolet catastrophe demonstrated what happens when classical physics requests infinite computational channels from finite substrate. Planck's resolution introduced discreteness, but the deeper question persisted—why must there be a cutoff at all? Modern constraints provide the answer. Landauer's principle establishes minimum energy per bit operation. The Margolus-Levitin theorem bounds processing speed. Holographic principles limit information to surface area. Together, these constraints cap the universe at approximately 10^123 bits—finite, not infinite. Statistical mechanics works because nature cannot track individual trajectories with infinite precision. What appears as randomness is resolution limits of the computational substrate. This reframing does not resolve Hilbert's problem in the formal axiomatic sense. It provides explanatory closure: the reason physics admits statistical description is that infinite-precision determinism exceeds the universe's computational capacity. The constraint is physical, not epistemic.