David Hilbert’s contributions to logical theory

Charles Sanders Peirce famously declared that “no two things could be more directly opposite than the cast of mind of the logician and that of the mathematician” (Peirce 1976, p. 595), and one who would take his word for it could only ascribe to David Hilbert that mindset opposed to the thought of his contemporaries, Frege, Gentzen, Gödel, Heyting, Łukasiewicz, and Skolem. They were the logicians par excellence of a generation that saw Hilbert seated at the helm of German mathematical research. Of Hilbert’s numerous scientific achievements, not one properly belongs to the domain of logic. In fact several of the great logical discoveries of the 20th century revealed deep errors in Hilbert’s intuitions—exemplifying, one might say, Peirce’s bald generalization. Yet to Peirce’s addendum that “[i]t is almost inconceivable that a man should be great in both ways” (Ibid.), Hilbert stands as perhaps history’s principle counter-example. It is to Hilbert that we owe the fundamental ideas and goals (indeed, even the name) of proof theory, the first systematic development and application of the methods (even if the field would be named only half a century later) of model theory, and the statement of the first definitive problem in recursion theory. And he did more. Beyond giving shape to the various sub-disciplines of modern logic, Hilbert brought them each under the umbrella of mainstream mathematical activity, so that for the first time in history teams of researchers shared a common sense of logic’s open problems, key concepts, and central techniques. It is not possible to reduce Hilbert’s contributions to logical theory to questions of authorship and discovery, for the work of numerous colleagues was made possible precisely by Hilbert’s influence as Europe’s preeminent mathematician together with his insistence that various logical conundra easily relegated to the margins of scientific activity belonged at the center of the attention of the mathematical community. In the following examination of how model theory, proof theory, and the modern concept of logical completeness each emerged from Hilbert’s thought, one theme recurs as a unifying motif: Hilbert everywhere wished to supplant philosophical musings with definite mathematical problems and in doing so made choices, not evidently necessitated by the questions themselves, about how to frame investigations “so that,” as he emphasized in 1922, “an unambiguous answer must result.” This motif and the wild success it enjoyed are what make Hilbert the chief architect of mathematical logic as well as what continue to inspire misgivings from several philosophical camps about logic’s modern guise.