Hilbert's Program Revisited

Abstract

ABSTRACT. After sketching the main lines of Hilbert's program, certain well-known and influential interpretations of the program are critically evaluated, and an alternative interpretation is presented. Finally, some recent developments in logic related to Hilbert's program are reviewed.In its heyday in the 1920s, Hilbert's program was arguably the most sophisticated and progressive research program in the foundations of mathematics. However, after Gödel's celebrated incompleteness results it became an almost universally held opinion that Hilbert's program was dead and buried, and consequently interest in it diminished and the received picture of it became somewhat caricatured and unfair. But more recently, there has been lots of new serious interest in Hilbert's program. Consequently, there now exists some illuminating historical work on Hilbert's thought. Moreover, there are also new systematic interpretations of Hilbert's program, which argue -in various ways -that there is a sound core in the program which was not affected by Gödel's results.2 My aim in this paper is to critically evaluate these recent influential interpretations in the light of both historical (textual) and systematic logical facts and to hopefully settle some of the controversies. I shall first give a brief and relatively uncontroversial description of Hilbert's program, and then proceed to more detailed and controversial issues of interpretation.On the one hand, the roots of Hilbert's program go back to the foundational debates in the late 19th century, especially to Kroenecker's attack on Cantorian set theory and the abstract analysis just developed. This debate affected Hilbert permanently. On the other hand, Hilbert's own thought went through various important changes, and it would be an error to simply equate Hilbert's views in, say, 1900, and his mature program, which was formulated only in the early 1920s.Hilbert spoke about a consistency proof for arithmetic, or analysis, already in his famous 1900 talk on the open problems in mathematics (Hilbert 1900). This may give the wrong impression that Hilbert's program was already there. However, in 1900 Hilbert thought that this consistency Synthese 137: 157-177, 2003.