Indispensable infinities: an aspect-pluralist approach to mathematical idealization and ontology

Abstract

I investigate some epistemic and metaontological implications of infinite idealizations in mathematical physics. I situate the discussion within the framework of indispensability arguments for mathematical Platonism, which typically assume that mathematics is indispensable to science in virtue of a single distinctive explanatory role. In chapter two, through case studies of limit-taking techniques and infinitely idealized models in statistical and quantum mechanics, I argue that scientific practice reveals instead a plurality of genuinely epistemic (as opposed to merely pragmatic) reasons for their prevalence. These reasons correspond to a plurality of productive epistemic roles mathematics might play in fundamental physics. To make sense of this plurality, I develop a neo-Wittgensteinian contextualist framework, scientific aspect-realism. On this view, mathematical models remain truth-apt without requiring de-idealization or strict structural correspondence as necessary conditions on interpretation. Building on these conclusions, chapter three defends indispensability pluralism, the claim that mathematics is genuinely indispensable to science in multiple unique, irreducible ways. Finally, I argue that this observation neither straightforwardly supports traditional Platonist realism nor undermines indispensability arguments altogether. Instead, it motivates a novel form of ontological pluralism as applied to the mathematical. According to the resulting view, aspect-pluralism, the ontological commitments warranted by scientific practice depend directly on the epistemic roles mathematical structures perform, thereby licensing commitment to multiple kinds of mathematical entities while staving off metaphysical overinflation.