CONCEPT
Hilbert’s Program
David Hilbert’s early-twentieth-century project to prove all of mathematics complete, consistent, and decidable by finite mechanical means—the most ambitious attempt to mechanize truth ever made, whose refutation by Gödel and Turing simultaneously destroyed the dream and, in the wreckage, created the theory of computation and every digital computer that followed.
Hilbert’s Program is the clearest statement ever made of the belief that reason could be fully mechanized, and its refutation is the most productive failure in the history of thought.
David Hilbert proposed in the 1920s that the entire edifice of mathematics could be placed on a foundation with three properties: completeness (every true statement provable), consistency (no contradiction derivable), and decidability (a mechanical procedure to settle any question). The third demand, the Entscheidungsproblem, was the most far-reaching: it asked whether there existed an algorithm for logical truth itself—a procedure that, given any mathematical claim, would determine in finitely many steps whether it followed from the axioms. If such a procedure existed, reasoning would in a precise sense be
solved: reduced to bookkeeping any patient clerk, or machine, could carry out. This is, almost exactly, the dream that animates the strongest claims made for artificial