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From closed to open systems. (English) Zbl 0877.03006

Czermak, Johannes (ed.), Philosophy of mathematics. Proceedings of the 15th international Wittgenstein-Symposium, August 16-23, 1992, Kirchberg am Wechsel, Austria. Part I. Wien: Hölder-Pichler-Tempsky. Schriftenreihe der Wittgenstein-Gesellschaft. 20/I, 206-220 (1993).
The author proposes “computational logic” as a new paradigm for logic. Computational logic corresponds to the features of the programming language Prolog. The author maintains that this logic could serve as “the theory of communicating inference processes”. He sees the main difference to mathematical logic and to its mathematical corollary, the axiomatic method, in the analytical strength of computational logic. The analytical method, going back to Plato’s contributions to the history of logic (cf. pp. 207-213), is presented as a trial and error procedure which is in any case perfectible, because there are no ultimate hypotheses (p. 208). Computational logic stands for an open view on mathematics as opposed to the closed view represented by the axiomatic method. But this disjunction between open and closed systems is no complete disjunction as the author suggests. It would be complete if axiomatic systems were really fixed once for all. Isn’t the analytical method an appropriate method for finding the axioms of an axiomatic system? The author’s answer seems to be “no”. He fails to apply his pragmatic approach to the axiomatic method itself. If this is done, closed, static and timeless axiomatic proofs could easily be combined with the open, dynamic and time-dependent analytical method.
For the entire collection see [Zbl 0836.00022].

MSC:

03A05 Philosophical and critical aspects of logic and foundations
03B70 Logic in computer science
68N17 Logic programming
70A05 Axiomatics, foundations
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