A set-theoretical reductionist holds that sets are the only abstract objects, and that (e.g.) numbers are identical to sets. (Which sets? A reductionist is a relativist if she is (e.g.) indifferent among von Neumann, Zermelo, etc. ordinals, an absolutist if she makes an argument for a priviledged reduction, such as identifying cardinal numbers with a equivalence classes under equipollence). Contrasting views: classical platonism, which holds that (e.g.) numbers exist independently of sets; and nominalism, which holds that there are no abstract particulars.
I'm interested in the relationship between "structuralism" as its understood by philosophers of science and mathematics and the structuralist methodology in mathematics for which Bourbaki is well known. A small point that I'm hung up on is the place of set theory in Bourbaki structuralism. I'm weighing two readings.
- (1) conventionalism: Bourbaki used set theory as a convenient "foundation", a setting in which models of structures may be freely constructed, but "structure" as understood in later chapters is not essentially dependent on the formal theory of structure developed in Theory of Sets,
- (2) reductionism: sets provide a ground floor ontology for mathematics; mathematicians study structures in the realm of sets.
In favor of conventionalism:
- (a) Leo Corry's arguments in “Nicolas Bourbaki and the Concept of Mathematical Structure” that the formal structures of Theory of Sets are to be distinguished from and play only a marginal role in the subsequent investigation of mathematica
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