Hilbert On Consistency As a Guide to Mathematical Reality

Fiona Doherty


In his early work Hilbert puts forward the principle that in mathematics consistency is enough for existence. The standard understanding of Hilbert’s contention is that he is assuming the completeness of his system. I look at the evidence for this interpretation and conclude that at the time he made this claim Hilbert had not yet developed a sophisticated conception of meta-mathematical concepts like consistency and completeness to allow him to formulate the completeness theorem. I then consider how we should understand Hilbert’s contention in light of this and suggest that, for Hilbert, consistency is conceptually prior to existence. On the basis of this I present a new reading of Hilbert’s Principle which recovers Hilbert’s actual contention, and along with it the philosophical significance of Hilbert’s early work which, in particular, provide a new approach to questions of ontology in mathematics.


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