Motivating Wittgenstein’s perspective on mathematical sentences as norms. (English) Zbl 1271.00015

Although Wittgenstein’s later philosophy of mathematics and Hilbert conception of axiomatic systems differ significantly, their views about the nature of mathematical sentences can be regarded as compatible to each other. The author admits that contrary to Hilbert, Wittgenstein regarded an axiom as self-evident truth, but “this is irrelevant for the role it performs in mathematical practice” (p.7). For the author there is rather a natural way from Wittgenstein’s understanding of mathematical sentences as conceptual norms to Hilbert’s conception of axioms as implicit definitions. The latter can be “employed as conceptual norms in the sense that they function as standards of what counts as employment of the concepts defined through them” (p.17). This normative mode also applies to the theorems derived from them. The author uses Zorn’s Lemma and the Axiom of Choice as examples.


00A30 Philosophy of mathematics
03-03 History of mathematical logic and foundations
03A05 Philosophical and critical aspects of logic and foundations
01A60 History of mathematics in the 20th century
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