What has Chihara's mathematical nominalism gained over mathematical realism?

Abstract

The indispensability argument, which claims that science requires beliefs in mathematical entities, gives a strong motivation for mathematical realism. However, mathematical realism bears Benacerrafian ontological and epistemological problems. Although recent accounts of mathematical realism have attempted to cope with these problems, it seems that, at least, a satisfactory account of epistemology of mathematics has not been presented. For instance, Maddy's realism with perceivable sets and Resnik's and Shapiro's structuralism have their own epistemological problems. This fact has been a reason to rebut the indispensability argument and adopt mathematical nominalism. Since mathematical nominalism purports to be committed only to concretia, it seems that mathematical nominalism is epistemically friendlier than mathematical realism. However, when it comes to modal mathematical nominalism, this claim is not trivial. There is a reason for doubting the modal primitives that it invokes. In this thesis, this doubt is investigated through Chihara's Constructibility Theory. Chihara's Constructibility Theory purports not to be committed to abstracta by replacing existential assertions of the standard mathematics with ones of constructibility. However, the epistemological status of the primitives in Chihara's system can be doubted. Chihara might try to argue that the problem would dissolve by using possible world semantics as a didactic device to capture the primitive notions. Nonetheless, his analysis of possible world semantic is not plausible, when considered as a part of the project of nominalizing mathematics in terms of the Constructibility Theory.

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Tomohiro Hoshi
Stanford University

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