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Is Mathias an ontologist? (English) Zbl 0786.03001

Set theory of the continuum, Pap. Math. Sci. Res. Inst. Workshop, Berkeley/CA (USA) 1989, Math. Sci. Res. Inst. Publ. 26, 119-122 (1992).
[For the entire collection see Zbl 0758.00014.]
[This article is reviewed together with the following one (see Zbl 0786.03002).]
Mathias in his above paper expresses some objections against positions which Mac Lane has formulated at different occasions and which, essentially, neglect the value of a set-theoretic foundation for mathematics. Mathias argues mainly in favour of two positions: (i) Mac Lane’s position is wrong among others in neglecting the procedural, i.e. algorithmic, recursive aspect of mathematics; (ii) there is some big universe of sets with lots of inner models which explain the huge amount of different independence results.
Mac Lane’s answer, besides some remarks in favour of the basic role of categorical understanding of mathematics, essentially is that he declares to have a protean view of mathematics not in need of any fixed ontological basis.
Of course, the whole – philosophical – problem behind this debate is not solved by thee two short remarks.

MSC:

03A05 Philosophical and critical aspects of logic and foundations
00A30 Philosophy of mathematics
03E99 Set theory
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