| Open-endedness, schemas and ontological commitment [penultimate version] |
| Marcus Rossberg, Nikolaj Jang Pedersen |
| forthcoming in Noûs |
| Area 1 Philosophy of Mathematics |
| None |
| Keywords schema schemata schemas open-endedness open-ended open ended endedness ontological commitment mathematics arithmetic Vann McGee Shaughan Lavine categoricity determinacy second-order logic |
| http://nikolaj.bol.ucla.edu/OpenendednessSchemasOntologicalCommitmentNous.pdf |
| http://homepages.uconn.edu/~mar08022/papers/PedersenRossbergSchemas.pdf |
| Second-order axiomatizations of certain important mathematical theories – such as arithmetic and real analysis – can be shown to be categorical. Categoricity implies semantic completeness, and semantic completeness in turn implies determinacy of truth-value. Second-order axiomatizations are thus appealing to realists as they sometimes seem to offer support for the realist thesis that mathematical statements have determinate truth-values. The status of second-order logic is a controversial issue, however. Worries about ontological commitment have been influential in the debate. Recently, Vann McGee has argued that one can get some of the technical advantages of second-order axiomatizations – categoricity, in particular – while walking free of worries about ontological commitment. In so arguing he appeals to the notion of an open-ended schema – a schema that holds no matter how the language of the relevant theory is extended. Contra McGee, we argue that second-order quantification and open-ended schemas are on a par when it comes to ontological commitment. |
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