Alan Baker's enhanced indispensability argument demonstrates that mathematical objects do genuine explanatory work irreducible to nominalist metalinguistic or modal reformulations.
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Reformulations(as the alternate versions of math that nominalists propose)
Different ways of expressing or restating the same idea using different words or concepts.
enhanced indispensability argument(Philosophy of mathematics; debated by Lyon, Colyvan, and Saatsi)
A strengthened form of the indispensability argument that appeals not merely to the indispensability of mathematics in science, but to the genuine explanatory role of mathematical entities in scientific explanations, aiming to justify Platonism about those entities.
indispensability argument(Lewis uses a modified form: not strict indispensability, but sufficient theoretical benefit)
An argument that we ought to believe in the existence of entities that are required (or highly useful) for our best theories
irreducible(Personalist anthropology; distinguishes personhood from mere biological individuality)
That which is unique and unrepeatable in each human being, by virtue of which a person is not merely an individual of a species but a personal subject.
modal(in logic and metaphysics)
Dealing with possibility and necessity—questions about what could be true, what must be true, and what's merely contingent (could go either way).
nominalism(Metaphysics; opposed to realism about universals)
The view that abstract entities such as properties or universals do not exist, and that predicative facts must be explained without appealing to such entities.