Abstracta and Possibilia: Modal Foundations of Mathematical Platonism

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Abstract
This paper aims to provide modal foundations for mathematical platonism. I examine Hale and Wright's (2009) objections to the merits and need, in the defense of mathematical platonism and its epistemology, of the thesis of Necessitism. In response to Hale and Wright's objections to the role of epistemic and metaphysical modalities in providing justification for both the truth of abstraction principles and the success of mathematical predicate reference, I examine the Necessitist commitments of the abundant conception of properties endorsed by Hale and Wright and examined in Hale (2013); examine cardinality issues which arise depending on whether Necessitism is accepted at first- and higher-order; and demonstrate how a two-dimensional intensional approach to the epistemology of mathematics, augmented with Necessitism, is consistent with Hale and Wright's notion of there being epistemic entitlement rationally to trust that abstraction principles are true. A choice point that I flag is that between availing of intensional or hyperintensional semantics. The hyperintensional semantic approach that I advance is an epistemic two-dimensional truthmaker semantics. Epistemic and metaphysical states and possibilities may thus be shown to play a constitutive role in vindicating the reality of mathematical objects and truth, and in explaining our possible knowledge thereof.
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First archival date: 2017-06-28
Latest version: 35 (2021-11-26)
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