Item description for Hilbert's Program: An Essay on Mathematical Instrumentalism (Synthese Library) by M. Detlefsen...
Hilbert's Program: An Essay on Mathematical Instrumentalism (Synthese Library) by M. Detlefsen
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Est. Packaging Dimensions: Length: 9.54" Width: 6.5" Height: 1.18" Weight: 1.05 lbs.
Release Date Apr 30, 1986
ISBN 9027721513 ISBN13 9789027721518
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Hilbert defended against Gödel Jul 23, 2009
Three arguments are raised against the standard view that Gödel's theorem proves the unfeasibility of Hilbert's programme:
(1) Gödel proved the nonprovability of a particular sentence (Con(T)) expressing the consistency of the system (T), but this does nothing to exclude the possibility that "there might still be some formula other than Con(T), expressing the same proposition that Con(T) expresses, that is provable in T." "Were this the case, the unprovability-in-T of Con(T) and its expression of T's consistency would best be taken as sheer coincidence." (p. 81).
(2) Gödel's result shows only that the consistency proof cannot be carried out *within the system itself,* but Hilbert is not committed to this. As Gödel himself said: his results "do not contradict Hilbert's formalistic viewpoint. For this viewpoint presupposes only the existence of a consistency proof in which nothing but finitary proof is used, and it is conceivable that there exist finitary proofs that cannot be expressed in the formalism [in question]" (p. 91). (Detlefsen tries to put his own spin on this argument by motivating it in terms of his "instrumentalist" interpretation of Hilbert, but it comes to the same thing.)
(3) Gödel's result shows that any system containing number theory will be unable to prove its own consistency. But to Hilbert formulas are largely "ideal" (as in "ideal numbers," "point at infinity," etc.), i.e., they are instruments invented to aid human thought. Therefore the consistency of this "ideal" formalism is of no interest in itself. The part of the formalism that is of any value excludes, for example, "all ideal proofs of real formulae that are to long or complex to be of any human epistemic utility" (p. 89). And since "'elementary number theory' ... designates an infinite system of proofs," "not every appreciable system of ideal proofs contains elementary number theory" (p. 87). "Thus, despite the Gödelian challenge, it may still prove possible to give both a finitary and a feasible demonstration of the soundness of the useful ideal methods." (p. 90). This is a weak argument, since it ignores the fact that Hilbert's programme is a mathematical research programme, and although Detlefsen's move may be philosophically safe it is obviously disastrous from this point of view.