The Philosophical Significance of Hilbert’s Problems

William Ewald

Hilbert is often regarded, in Otto Blumenthal’s phrase, as ‘the man of problems’ —as a brilliant technician who moved from problem to problem, picking up new problems and discarding the old as his interests shifted. This aspect of Hilbert’s practice has created the impression that the various aspects of his mathematical work —in number theory, in geometry, in mathematical physics, in foundations— were not related in any systematic manner. In particular his search for a consistency proof tends to be viewed as one problem among many, and specifically as an attempt to overcome the difficulties occasioned by the discovery of the paradoxes. It is argued here that Hilbert’s search for consistency proofs was not motivated by the discovery of the paradoxes but had independent grounds, and that a recognition of this fact makes it possible to view Hilbert as engaged in a more systematic enterprise than is commonly recognized.

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