In Objects, Structures, and Logics, FilMat Studies in the Philosophy of Mathematics (forthcoming)

Neil Barton
Universität Konstanz
Informal rigour is the process by which we come to understand particular mathematical structures and then manifest this rigour through axiomatisations. Structural relativity is the idea that the kinds of structures we isolate are dependent upon the logic we employ. We bring together these ideas by considering the level of informal rigour exhibited by our set-theoretic discourse, and argue that different foundational programmes should countenance different underlying logics (intermediate between first- and second-order) for formulating set theory. By bringing considerations of perturbations in modal space to bear on the debate, we will suggest that a promising option for representing current set-theoretic thought is given by formulating set theory using quasi-weak second-order logic. These observations indicate that the usual division of structures into \particular (e.g. the natural number structure) and general (e.g. the group structure) is perhaps too coarse grained; we should also make a distinction between intentionally and unintentionally general structures.
Keywords structuralism  set theory  second-order logic
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Mathematics as a Science of Patterns.Michael David Resnik - 1997 - Oxford, England: New York ;Oxford University Press.
Elements of Intuitionism.Michael Dummett - 1977 - Oxford University Press.
Philosophy and Model Theory.Sean Walsh & Tim Button - 2018 - Oxford, UK: Oxford University Press.

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