21,264 research outputs found

    Indispensability Without Platonism

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    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 (...) realism', in order to identify the truthmakers of mathematics. We consider in particular as a case study the indispensability to physics of real analysis (the theory of the real numbers). We conclude that it is possible to run an indispensability argument without Quinean baggage

    Mathematical Representation: Playing a Role

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    The primary justification for mathematical structuralism is its capacity to explain two observations about mathematical objects, typically natural numbers. Non-eliminative structuralism attributes these features to the particular ontology of mathematics. I argue that attributing the features to an ontology of structural objects conflicts with claims often made by structuralists to the effect that their structuralist theses are versions of Quine's ontological relativity or Putnam's internal realism. I describe and argue for an alternative explanation for these features which instead explains the attributes them to the mathematical practice of representing numbers using more concrete tokens, such as sets, strokes and so on

    Categoricity, Open-Ended Schemas and Peano Arithmetic

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    One of the philosophical uses of Dedekind’s categoricity theorem for Peano Arithmetic is to provide support for semantic realism. To this end, the logical framework in which the proof of the theorem is conducted becomes highly significant. I examine different proposals regarding these logical frameworks and focus on the philosophical benefits of adopting open-ended schemas in contrast to second order logic as the logical medium of the proof. I investigate Pederson and Rossberg’s critique of the ontological advantages of open-ended arithmetic when it comes to establishing the categoricity of Peano Arithmetic and show that the critique is highly problematic. I argue that Pederson and Rossberg’s ontological criterion deliver the bizarre result that certain first order subsystems of Peano Arithmetic have a second order ontology. As a consequence, the application of the ontological criterion proposed by Pederson and Rossberg assigns a certain type of ontology to a theory, and a different, richer, ontology to one of its subtheories

    What we talk about when we talk about numbers

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    In this paper, I describe and motivate a new species of mathematical structuralism, which I call Instrumental Nominalism about Set-Theoretic Structuralism. As the name suggests, this approach takes standard Set-Theoretic Structuralism of the sort championed by Bourbaki and removes its ontological commitments by taking an instrumental nominalist approach to that ontology of the sort described by Joseph Melia and Gideon Rosen. I argue that this avoids all of the problems that plague other versions of structuralism

    Aristotle, The Pythagoreans, and Structural Realism

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    Aristotle’s main objection to Pythagorean number ontology is that it posits as a basic subject what can exist only as inherent in a subject. I then show how contemporary structural realists posit an ontology much like that of Aristotle’s Pythagoreans. Both take the objects of knowledge to be structure, not the subject of structure. I discuss both how pancomputationalists such as Edward Fredkin approach the Pythagorean account insofar as on their account all reality can in principle be expressed as one (very big) number, made up of discrete units, and even more moderate varieties of structural realism, like that of Floridi, share with pancomputationalism the aspect of “Pythagorean” ontology that Aristotle finds so objectionable: positing structure or form with no substrate. I conclude by arguing that Aristotle himself is drawn to something close or (identical) to a structural realist ontology in Metaphysics 7.3

    Grounding Concepts: the Problem of Composition

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    In a recent book C.S. Jenkins proposes a theory of arithmetical knowledge which reconciles realism about arithmetic with the a priori character of our knowledge of it. Her basic idea is that arithmetical concepts are grounded in experience and it is through experience that they are connected to reality. I argue that the account fails because Jenkins’s central concept, the concept for grounding, is inadequate. Grounding as she defines it does not suffice for realism, and by revising the definition we would abandon the idea that grounding is experiential. Her account falls prey to a problem of which Locke, whom she regards as a source of inspiration, was aware and which he avoided by choosing anti-realism about mathematics

    Is There a Compelling Argument for Ontic Structural Realism?

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    Structural realism first emerged as an epistemological thesis aimed to avoid the so-called pessimistic meta-induction on the history of science. Some authors, however, suggested that the preservation of structure across theory-change is best explained by endorsing the metaphysical thesis that structure is all there is. While the possibility of this latter, „ontic‟ form of structural realism has been extensively debated, though, not much has been said concerning its justification. In this paper, I distinguish between two arguments in favour of ontic structural realism that can be reconstructed from the literature, and find both of them wanting

    Rejecting Mathematical Realism while Accepting Interactive Realism

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    Indispensablists contend that accepting scientific realism while rejecting mathematical realism involves a double standard. I refute this contention by developing an enhanced version of scientific realism, which I call interactive realism. It holds that interactively successful theories are typically approximately true, and that the interactive unobservable entities posited by them are likely to exist. It is immune to the pessimistic induction while mathematical realism is susceptible to it
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