Open-endedness, Schemas and Ontological Commitment
Article first published online: 25 MAY 2010
© 2010 Wiley Periodicals, Inc.
Volume 44, Issue 2, pages 329–339, June 2010
How to Cite
Pedersen, N. J. L. L. and Rossberg, M. (2010), Open-endedness, Schemas and Ontological Commitment. Noûs, 44: 329–339. doi: 10.1111/j.1468-0068.2010.00742.x
- Issue published online: 25 MAY 2010
- Article first published online: 25 MAY 2010
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.