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The status of set-theoretic axioms in empirical theories. (English) Zbl 0847.03002

Echeverria, Javier (ed.) et al., The space of mathematics. Philosophical, epistemological, and historical explorations. Revised papers from a symposium on structures in mathematical theories, Donostia/San Sebastian, Basque Country, Spain, September 1990. Berlin: Walter de Gruyter. Grundlagen der Kommunikation und Kognition. 156-167 (1992).
In an essay concerning the foundations of mathematics, I. Lakatos [Br. J. Philos. Sci. 27, 201-223 (1976; Zbl 0364.00028)] argued that (meta-)mathematics should be conceived as a “quasi-empirical” theory. He even raises the following question:
“Are we going to arrive, tracing back problemshifts through informal mathematical theories to empirical theories, so that mathematics will turn out in the end to be indirectly empirical, thus justifying Weyl’s, von Neumann’s and – in a certain sense – Mostowski’s and Kalmar’s position?”
In my paper I will try to explore a particular aspect of this question: in which sense and to which extent could a part of (meta-)mathematics, namely the set-theoretical axioms, obtain an empirical meaning? Unfortunately, I cannot present a definite answer, but I will arrive at a distinction between the empirical and the non-empirical part of set theory which probably proves right if some strong restrictions concerning the language of empirical theories are imposed.
For the entire collection see [Zbl 0839.00019].

MSC:

03A05 Philosophical and critical aspects of logic and foundations
00A30 Philosophy of mathematics

Citations:

Zbl 0364.00028
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