"A Mathematical Proof Must Be Surveyable" What Wittgenstein Meant by This and What It Implies

Author: Mühlhölzer, Felix

Source: Grazer Philosophische Studien, Deepening our Understanding of Wittgenstein. Edited by Michael Kober , pp. 57-86(30)

Publisher: Rodopi

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Abstract:

In Part III of his Remarks on the Foundations of Mathematics Wittgenstein deals with what he calls the surveyability of proofs. By this he means that mathematical proofs can be reproduced with certainty and in the manner in which we reproduce pictures. There are remarkable similarities between Wittgenstein's view of proofs and Hilbert's, but Wittgenstein, unlike Hilbert, uses his view mainly in critical intent. He tries to undermine foundational systems in mathematics, like logicist or set theoretic ones, by stressing the unsurveyability of the proof-patterns occurring in them. Wittgenstein presents two main arguments against foundational endeavours of this sort. First, he shows that there are problems with the criteria of identity for the unsurveyable proof-patterns, and second, he points out that by making these patterns surveyable, we rely on concepts and procedures which go beyond the foundational frameworks. When we take these concepts and procedures seriously, mathematics does not appear as a uniform system, but as a mixture of different techniques.

Document Type: Research Article

Publication date: April 24, 2006

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