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This chapter advocates and defends a broadly neo-Carnapian approach to the philosophy of mathematics more broadly.This approach combines potentialist set theory with platonism about non-set theoretic mathematical objects.
In many ways set theory lies at the heart of modern mathematics, and it does powerful work both philosophical and mathematical – as a foundation for the subject. However, certain philosophical problems raise serious doubts about our acceptance of the axioms of set theory. In a detailed and original reassessment of these axioms, Sharon Berry uses a potentialist (as opposed to actualist) approach to develop a unified determinate conception of set-theoretic truth that vindicates many of our intuitive expectations regarding set theory. Berry further defends her approach against a number of possible objections, and she shows how a notion of logical possibility that is useful in formulating Potentialist set theory connects in important ways with philosophy of language, metametaphysics and philosophy of science. Her book will appeal to readers with interests in the philosophy of set theory, modal logic, and the role of mathematics in the sciences.
How ontologically committal is common sense? Is the common-sense philosopher beholden to a florid ontology in which all manner of objects, substances, and processes exist and are as they appear to be to common sense, or can she remain neutral on questions about the existence and nature of many things because common sense is largely non-committal? This chapter explores and tentatively evaluates three different approaches to answering these questions. The first applies standard accounts of ontological commitment to common-sense claims. This leads to the surprising and counter-intuitive result that common sense has metaphysically heavyweight commitments. The second approach emphasizes the superficiality and locality of common-sense claims. On this approach, however, common sense comes out as almost entirely non-committal. The third approach questions the seriousness of ontological commitment as such. If ontological commitment is cheap, it becomes possible both to accept the commitments of common sense at face value and to avoid the counter-intuitive consequences of heavyweight metaphysical commitments.
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