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Logik der Antinomien. (Logic of antinomies). (German) Zbl 0724.03003
Bern etc.: Peter Lang. 142 S. DM 36.00 (1990).

The author introduces his treatise as a logico-philosophical critique of Gödel’s incompleteness theorem and the Cantorian diagonal procedure used in proof of the theorem. The thesis is that if we properly understand the assumptions underlying Gödel’s and Cantor’s formulation of what they proved, we will find that there is nothing philosophically startling about these results. To establish this thesis he uses insights of Wittgenstein, Brouwer and Lorenzen to examine notions such as: The antinomies in general, the liar paradox, the set theoretic paradoxes, truth and self-reference. By themselves these examinations constitute a series of philosophical essays scrutinizing several mathematical procedures which, according to the author, have misled philosophers to think that something mysterious or wonderful was accomplished. For instance, the author argues at length that the true metalinguistic reading of a Gödelian formula is not a true, but unprovable formula, within an axiomatic arithmetic.

The treatise warrants serious consideration by philosophers of mathematics. The argumentation is careful and non-technical. However, some familiarity with the technical details of the mathematics discussed is important. To a large extent, this treatise is a systematic development of Wittgenstein’s unsystematic remarks that a careful description of mathematical activity will eliminate philosophically misleading interpretations of what has been accomplished.

MSC:
03A05Philosophical and critical
03-02Research monographs (mathematical logic)