The Philosophy of Mathematics Today

By Matthias Schirn | Go to book overview

11
Hilbert's Finitism and the Notion of Infinity

KARL-GEORG NIEBERGALL AND MATTHIAS SCHIRN

An ihren Früchten sollt ihr sie erkennen,
heiβt es auch für die Theorien
.
(By their fruits ye shall know them--
that applies also to theories.)

Hilbert, "'Über das Unendliche'", 98 [384]

When Hilbert set about developing his foundational programme in the 1920s, proof theory was still enjoying a state of pre-Gödelian innocence. The principal objective of the programme was to establish once and for all the entire heritage of classical mathematics by means of a finitist metamathematical consistency proof. Hilbert thus aimed at justifying, by using only finitist methods, what he took to be the disputable non-finitist modes

____________________
Versions of this material (some in Portuguese or Spanish or German) were presented at the 3rd International Congress of Analytical Philosophy held in September 1995 at the Federal University of Santa Catarina (UFSC) in Florianópolis, the Center of Logic, Epistemology and History of Science of the State University of Campinas (CLE, UNICAMP), the Catholic University of Rio de Janeiro (PUC-RJ), the Federal University of Pernambuco (UFPE) in Recife, the Catholic University of São Paulo (PUC-SP), the Institute of Philosophical Investigations of the National Autonomous University of Mexico (IIF, UNAM), the National Autonomous University of Costa Rica, the Argentinean Society of Philosophical Analysis (SADAF) in Buenos Aires, the International Congress "'Truth: Logic, Representation and World'" held in January 1996 at the University of Santiago de Compostela, the University of Sevilla (Dept. of Mathematical Analysis), the History of Logic Conference held in April 1996 at the University of Texas at Austin, the University of Paris, the University of Regensburg, and at the University of Munich. Some topics of Hilbert's programme were also discussed at the University of São Paulo (USP). Thanks to the audiences for interesting discussion. We are grateful to Ulrike Ritter, Robert Bublak, Manfred Harth, Stefan Iwan, and Roberto Torretti for reading an earlier version of this paper. Special thanks are due to Nicholas Jacob-Flynn and Uwe Lück.

The idea of writing a paper on Hilbert's finitism grew out of our joint seminar on his prooftheoretic programme at the University of Munich in the winter term 1993-4. Hilbert's classical papers on his proof theory, stimulating as they are and transparent as they may appear at first glance, are in fact not free from ambiguity and vagueness. We do not blame him for that, of course, nor do we mean to apologize for any remaining shortcomings in our account. Nevertheless, we feel inclined to spell this out here. With the exception of Hilbert ( 1926) and (1928a) the translations of Hilbert's writings are our own. For the most part, we have slightly modified the existing translations. As to the translation of the world 'inhaltlich', we have adopted the neologism 'contentual', coined by the translator of Hilbert ( 1926), Stefan Bauer- Mengelberg. The references to the English translations of Hilbert ( 1926), (1928a) and Tarski ( 1935) appear in square brackets.

-271-

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