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- Cian Dorr, Of Numbers and Electrons.According to a tradition stemming from Quine and Putnam, certain theories that entail the existence of mathematical entities are better, qua explanations of our evidence, than any theories that do not, and thus we have the same broadly inductive reason for believing in numbers as we have for believing in electrons. In this paper I consider how the existence of nominalistic modal theories of the form 'Possibly, the concrete world is just as it in fact is and T' and 'Necessarily, if standard mathematics is true and the concrete world is just as it in fact is, then T' bears on this claim. I conclude that, while analogies with theories that attempt to eliminate unobservable concrete entities provide good reason to regard theories of the former kind as explanatorily bad, this reason does not apply to theories of the latter kind, which are not relevantly analogous to anything available to eliminativists about electrons.In my reading list | Discuss this article | Edit | Categorize |My bibliography
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According to a tradition stemming from Quine and Putnam, certain theories that entail the existence of mathematical entities are better, <em>qua</em> explanations of our evidence, than any theories that do not, and thus we have the same broadly inductive reason for believing in numbers as we have for believing in electrons. In this paper I consider how the existence of nominalistic modal theories of the form 'Possibly, the concrete world is just as it in fact is and <em>T</em>' and 'Necessarily, (...)
According to the indispensability argument, the fact that we quantify over numbers, sets and functions in our best scientific theories gives us reason for believing that such objects exist. I examine a strategy to dispense with such quantification by simply replacing any given platonistic theory by the set of sentences in the nominalist vocabulary it logically entails. I argue that, as a strategy, this response fails: for there is no guarantee that the nominalist world that go beyond the set of (...)
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Platonism about mathematics (or mathematical platonism as I will mostly call it) is typically defined as the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. Just as electrons and planets exist independently of us, so do numbers and sets. And just as statements about electrons and planets are made true or false by the objects with which they are concerned and these objects’ perfectly objective properties, so are statements (...)
According to Quine’s indispensability argument, we ought to believe in just those mathematical entities that we quantify over in our best scientific theories. Quine’s criterion of ontological commitment is part of the standard indispensability argument. However, we suggest that a new indispensability argument can be run using Armstrong’s criterion of ontological commitment rather than Quine’s. According to Armstrong’s criterion, ‘to be is to be a truthmaker (or part of one)’. We supplement this criterion with our own brand of metaphysics, 'Aristotelian (...)
In this essay, I defend a view I call “Robust Realism” about normativity. According to this view, there are irreducibly, perfectly objective, normative truths, that when successful in our normative inquiries we discover rather than create or construct. My argument in support of Robust Realism is modeled after arguments from explanatory indispensability common in the philosophy of science and the philosophy of mathematics. I argue that irreducibly normative truths, though not explanatorily indispensable, are nevertheless deliberatively indispensable, and that this kind (...)
The Quine-Putnam Indispensability argument is the argument for treating mathematical entities on a par with other theoretical entities of our best scientific theories. This argument is usually taken to be an argument for mathematical realism. In this chapter I will argue that the proper way to understand this argument is as putting pressure on the viability of the marriage of scientific realism and mathematical nominalism. Although such a marriage is a popular option amongst philosophers of science and mathematics, in light (...)
Indispensability arguments for realism about mathematical entities have come under serious attack in recent years. To my mind the most profound attack has come from Penelope Maddy, who argues that scientific/mathematical practice doesn't support the key premise of the indispensability argument, that is, that we ought to have ontological commitment to those entities that are indispensable to our best scientific theories. In this paper I defend the Quine/Putnam indispensability argument against Maddy's objections.
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| | Other links: philmat.oxfordjournals.org | Scholar | More.. I explicate and defend the claim that, fundamentally speaking, there are no numbers, sets, properties or relations. The clarification consists in some remarks on the relevant sense of ‘fundamentally speaking’ and the contrasting sense of ‘superficially speaking’. The defence consists in an attempt to rebut two arguments for the existence of such entities. The first is a version of the indispensability argument, which purports to show that certain mathematical entities are required for good scientific explanations. The second is a speculative (...)
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| | Other links: phonline.org | Scholar | More.. The Quine-Putnam indispensability argument in the philosophy of mathematics urges us to place mathematical entities on the same ontological footing as other theoretical entities essential to our best scientific theories. Recently, the argument has come under serious scrutiny, with many influential philosophers unconvinced of its cogency. This book not only outlines the indispensability argument in considerable detail but also defends it against various challenges.
I follow standard mathematical practice and theory to argue that the natural numbers are the finite von Neumann ordinals. I present the reasons standardly given for identifying the natural numbers with the finite von Neumann's (e.g., recursiveness; well-ordering principles; continuity at transfinite limits; minimality; and identification of n with the set of all numbers less than n). I give a detailed mathematical demonstration that 0 is { } and for every natural number n, n is the set of all natural (...)
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