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Switch to: References
Citations of:
A reductio ad surdum? Field on the contingency of mathematical objects
Mind 103 (410):169-184 (1994)
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Let us distinguish two kinds of contingentism: entity contingentism and metaphysical contingentism. Here, I use ‘entity’ very broadly to include anything over which we can quantify—objects (abstract and concrete), properties, and relations. Then entity contingentism about some entity, E, is the view that E exists contingently: that is, that E exists in some possible worlds and not in others. By contrast, entity necessitarianism about E is the view that E exists of necessity: that is, that E exists in all possible (...) |
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In this paper I argue against Divers and Miller's 'Lightness of Being' objection to Hale and Wright's neo-Fregean Platonism. According to the 'Lightness of Being' objection, the neo-Fregean Platonist makes existence too cheap: the same principles which allow her to argue that numbers exist also allow her to claim that fictional objects exist. I claim that this is no objection at all" the neo-Fregean Platonist should think that fictional characters exist. However, the pluralist approach to truth developed by WQright in (...) No categories |
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Crispin Wright and Bob Hale have defended the strategy of defining the natural numbers contextually against the objection which led Frege himself to reject it, namely the so-called ‘Julius Caesar problem’. To do this they have formulated principles (called sortal inclusion principles) designed to ensure that numbers are distinct from any objects, such as persons, a proper grasp of which could not be afforded by the contextual definition. We discuss whether either Hale or Wright has provided independent motivation for a (...) |
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In Being Realistic About Reasons T. M. Scanlon argues that particular fact about reasons are explained by contingent non-normative facts together with pure normative principles. A question then arises about the modal status of these pure principles. Scanlon maintains that they are necessary in a sense, and suggests that they are ‘metaphysically’ necessary. I argue that the best view for Scanlon to take, given his other commitments, is that these pure normative principles are metaphysically contingent in some cases and necessary (...) No categories |
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It is often claimed that nominalistic programmes to reconstruct mathematics fail, since they will at some point involve the notion of logical consequence which is unavailable to the nominalist. In this paper we use an idea of Goodman and Quine to develop a nominalistically acceptable explication of logical consequence. |
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In a lot of domains in metaphysics the tacit assumption has been that whichever metaphysical principles turn out to be true, these will be necessarily true. Let us call necessitarianism about some domain the thesis that the right metaphysics of that domain is necessary. Necessitarianism has flourished. In the philosophy of maths we find it held that if mathematical objects exist, then they do of necessity. Mathematical Platonists affirm the necessary existence of mathematical objects (see for instance Hale and Wright (...) |
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Mathematical apriorism holds that mathematical truths must be established using a priori processes. Against this, it has been argued that apparently a priori mathematical processes can, under certain circumstances, fail to warrant the beliefs they produce; this shows that these warrants depend on contingent features of the contexts in which they are used. They thus cannot be a priori. -/- In this paper I develop a position that combines a reliabilist version of mathematical apriorism with a platonistic view of mathematical (...) |
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According to the species of neo-logicism advanced by Hale and Wright, mathematical knowledge is essentially logical knowledge. Their view is found to be best understood as a set of related though independent theses: (1) neo-fregeanism-a general conception of the relation between language and reality; (2) the method of abstraction-a particular method for introducing concepts into language; (3) the scope of logic-second-order logic is logic. The criticisms of Boolos, Dummett, Field and Quine (amongst others) of these theses are explicated and assessed. (...) |
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According to the species of neo‐logicism advanced by Hale and Wright, mathematical knowledge is essentially logical knowledge. Their view is found to be best understood as a set of related though independent theses: (1) neo‐fregeanism—a general conception of the relation between language and reality; (2) the method of abstraction—a particular method for introducing concepts into language; (3) the scope of logic—second‐order logic is logic. The criticisms of Boolos, Dummett, Field and Quine (amongst others) of these theses are explicated and assessed. (...) |
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One cannot escape the feeling that these mathematical formulae have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers. |
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One kind of structuralism holds that mathematics is about structures, conceived as a type of abstract entity. Another denies that it is about any distinctively mathematical entities at all—even abstract structures; rather it gives purely general information about what holds of any collection of entities conforming to the axioms of the theory. Of these, pure structuralism is most plausibly taken to enjoy significant advantages over platonism. But in what appears to be its most plausible—modalised—version, even restricted to elementary arithmetic, it (...) |
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We discuss abstraction principles in the context of modal and temporal logic. It is argued that abstractionism conflicts with both serious presentism and serious actualism. No categories |
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Recently, nominalists have made a case against the Quine–Putnam indispensability argument for mathematical Platonism by taking issue with Quine’s criterion of ontological commitment. In this paper I propose and defend an indispensability argument founded on an alternative criterion of ontological commitment: that advocated by David Armstrong. By defending such an argument I place the burden back onto the nominalist to defend her favourite criterion of ontological commitment and, furthermore, show that criterion cannot be used to formulate a plausible form of (...) |
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