Abstract
The paper provides an interpretation of Wittgenstein’s claim that a mathematical proof must be surveyable. It will be argued that this claim specifies a precondition for the applicability of the word ‘proof’. Accordingly, the latter is applicable to a proof-pattern only if we can come to agree by mere observation whether or not the pattern possesses the relevant structural features. The claim is problematic. It does not imply any questionable finitist doctrine. But it cannot be said to articulate a feature of our actual usage of the word ‘proof’. The claim can be dissociated, however, from two tenable doctrines of Wittgenstein, namely that proofs can be used as paradigms for corresponding proof concepts and that a proof is not an experiment.
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Notes
The point actually seems to be acknowledged by Wittgenstein (cf. 1978, III, §10).
In this connection, Addis has argued that a ‘strict finitist’ in Dummett’s sense need not rely on such a vague notion as surveyability and would indeed be better advised to develop a sufficiently strict notion of a feasible operation (Addis 1995, p. 165).
Therefore, Shanker is right to maintain that, according to Wittgenstein, it would make no sense, for instance, ‘to speak of Appel and Haken “proving” the “four-colour theorem”’, since their ‘solution of the four-colour problem is empirical rather than mathematical’ (Shanker 1987, p. 157).
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Büttner, K.M. Surveyability and Mathematical Certainty. Axiomathes 27, 113–128 (2017). https://doi.org/10.1007/s10516-016-9292-4
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DOI: https://doi.org/10.1007/s10516-016-9292-4