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Gödel's Theorem in Focus (Philosophers in Focus)

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The most interesting part of the book is the discussion of Gödel's "Platonism." One is "struck by the great contrast between the deep platonist convictions Gödel held concerning the objective basis of mathematics and the special caution he exercised in revealing these convictions" (p. 96). In fact, Gödel made only "one public statement of a philosophical character in the period directly following his thesis work," namely: "Gödel openly criticized the view which took consistence as the criterion for existence ... '[Assuming the consistence of classical mathematics] one can even give examples of propositions ... which are really contentually true ... but are unprovable in the formal system of classical mathematics.' Hence, the negations of such statements could be adjoined to form a consistent system containing false statements; thus consistency would not guarantee existence in the intended sense." (pp. 102-103). "it should be noted that the heuristic principle of my construction of undecidable number theoretic propositions in the formal systems of mathematics is the highly transfinite concept of 'objective mathematics truth' as opposed to that of 'demonstrability'" (p. 107). However, "in consequence of the philosophical prejudices of our times ... a concept of objective mathematical truth as opposed to demonstrability was viewed with greatest suspicion and widely rejected as meaningless" (p. 107).

The book's main objective, however, is "to divest [Gödel's theorem] of its false metaphysical pretensions" (p. 235), as witnessed, for example, by the fact that the article on Gödel's own convictions is shorter than that by his hardcore enemy Detlefsen, who proclaims that all is fine and well with Hilbert's Programme. Speaking of page count, Shanker abuses his editor's privilege in inserting a hopelessly rambling 100-page article by himself on "Wittgenstein's remarks on the significance of Gödel's theorem," despite the fact that these "remarks" are little more than incoherent scraps (as Shanker himself pretty much admits, e.g.: "Of the many problems posed by Wittgenstein's fragmentary remarks on Gödel's theorem perhaps the most frustrating is that it is not even clear how these fit into the overall context of Wittgenstein's proposed resolution of the foundations crisis"; p. 177).