Reading Ernest Gellner on logic.
Some people at the turn of the 20th century hoped: “if only the whole of logic and mathematics could be deduced from a limited set of safe premisses, the whole edifice would thereby be made trustworthy.” But now “the ideal itself has lost its appeal and authority.” W. V. O. Quine, for example, says (Gellner’s paraphrase) that “even in logic, our premisses cannot insure us against future surprises and the need to retrace steps” and that we “haggle and weigh advantages, risks, and convenience, rather than move rectilinearly in marble halls along pathways of clear and indubitable truth.” Logic, too, will “retrace steps, make opportunist and revocable choices, use anything that happens to be close to hand,” just like Lévi-Strauss’ prehistoric bricoleur, that opportunist neolithic handyman who ‘makes arrows from any timber that is close to hand’.” (p. 6)
Meanwhile, the philosopher of mathematics Imre Lakatos maintains, in Gellner's paraphrase, that
“definitions follow rather than precede proofs. Objections to proofs can only be raised and understood after the proof has been formulated; the refinement of the proof in the light of the objection leads to a reformulation of the initial concepts, with a view to weathering the objection or modifying the conclusion. The fully refined and powerful definitions of the initial concepts, and hence the axioms formulated through them, are only the fruits of many, many repetitions of this process. The definition, the official premiss, comes one at the end, not at the beginning — and even this end is only an interim one. The process can never be known to be complete, and indeed is unlikely ever to be terminated. Moreover, the order in which these sequences occur — proof, objection, redefinition — is not something contingent and accidental, something that only happens in the rough work and can be left out in the clean copy; it is, on the contrary, essential to and inherent in the very nature of mathematical discovery. The messy, insecure, step-retracing procedure is of the essence of the subject.” (p. 6–7)
Ernest Gellner, Legitimation of Belief Cambridge University Press, 1975.
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