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It appears that one important motivation for being a (Quinean) nominalist about mathematical objects is ontological parsimony: mathematical objects are ontologically extravagant; so if we can rewrite our scientific theories without quantifying over mathematical objects, then such theories should be preferred to theories that do quantify over numbers, functions, etc. But it is a mystery to me why a Quinean (who subscribes to naturalized epistemology) should concern herself with ontological parsimony when formulating the ontology of mathematics. Mathematicians, unlike physicists, don't seem bothered by postulating new objects - in fact that is what they do most of the time (contra physicists who are disturbed by multiverses or dark energy). So why is ontological parsimony a criterion for adjudicating between two ontological theories about mathematics, from the Quinean naturalistic perspective?

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    Actually, Quine does not subscribe to mathematical parsimony. The so-called Quine-Putnam indispensability argument for platonism advocates admitting mathematical objects (sets and numbers) into ontology because they are indispensable in scientific theories, and we should generally accept what our best theories posit as existent. Also, the main argument against platonism is not parsimony but Benacerraf's dilemma: there is no non-mystical explanation as to how we interact with ideal realm. – Conifold Jan 29 '18 at 20:47
  • thank you! correct me if I am wrong: Quine-Putnam indispensability argument presupposes that it is better to not admit mathematical objects into our ontology (due to ontological parsimony), and we are forced to because they are indispensable. The nominalist project (e.g. Field's) appears to be working under the assumption that dispensability of math objects in science is sufficient for their nonexistence. And this is what I don't understand from the Quinean perspective. – bluether Jan 30 '18 at 1:55
  • Perhaps a better way to put my question is this: Quine thinks i) we should evaluate theories by scientific standards, and ii) ontological parsimony is a virtue of all scientific theories. But if we acknowledge that iii) mathematics as a science, then i) and iii) falsifies ii) since ontological parsimony is not a virtue for mathematical theories. – bluether Jan 30 '18 at 2:01
  • p.s. a word on Benacerraf's dilemma: I also find it mysterious why a Quinean naturalist should find Benacerraf's dilemma persuasive since we do modal reasoning in science all the time (e.g. counterfactual for causal relation) and it is unclear that we have non-mystical access to modal knowledge either. – bluether Jan 30 '18 at 2:01
  • Young Quine was a nominalist, so he might have felt this way, but the QP argument itself is independent of such sentiment. The same goes for Field's challenge, one does not need generalities about existence to ask for non-mystical epistemology of platonism. Also, Quine has little use for "modal reasoning":"We modify a sentence with the adverb 'necessarily' when it is a sentence presumed acceptable to our interlocutor and stated only as a step toward the consideration of moot ones. Or... something that follows from generalities already expounded, as over against new conjectures or hypotheses" – Conifold Jan 30 '18 at 3:59

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