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The Philosophy of Mathematical Practice$
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Paolo Mancosu

Print publication date: 2008

Print ISBN-13: 9780199296453

Published to Oxford Scholarship Online: February 2010

DOI: 10.1093/acprof:oso/9780199296453.001.0001

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Reflections on the Purity of Method in Hilbert's Grundlagen der Geometrie

Reflections on the Purity of Method in Hilbert's Grundlagen der Geometrie

(p.198) 8 Reflections on the Purity of Method in Hilbert's Grundlagen der Geometrie
The Philosophy of Mathematical Practice

Michael Hallett

Oxford University Press

In the Conclusion to his Grundlagen der Geometrie of 1899, Hilbert stated that the concern with ‘purity of method’ is nothing more than a ‘subjective interpretation’ of the demand for a careful examination of central mathematical propositions, the search either for rigorous proofs from clearly specified axioms, or the proof of the impossibility of such a proof. This chapter examines Hilbert's treatment of purity in the lecture notes surrounding the Grundlagen. In particular, it presents three important case studies, concerning Desargues's Theorem, the Euclidean Isosceles Triangle Theorem, and the Three Chord Theorem. These examples show how important ‘higher’ mathematical knowledge is for Hilbert, and how this can often shape and instruct the intuitive level, which is often where a ‘purity’ question about geometry first arises; in the examples examined here, this forces a reassessment of what is ‘appropriate’ (or ‘pure’).

Keywords:   Hilbert, foundations of geometry, purity of method, impossibility proofs, geometrical intuition, Desargues's Theorem, Isoceles Triangle Theorem, Three Chord Theorem

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