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150Taking it Easy: A Response to ColyvanMind 121 (484): 983-995. 2012.This discussion note responds to Mark Colyvan’s claim that there is no easy road to nominalism. While Colyvan is right to note that the existence of mathematical explanations presents a more serious challenge to nominalists than is often thought, it is argued that nominalist accounts do have the resources to account for the existence of mathematical explanations whose explanatory role resides elsewhere than in their nominalistic content.
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178Platonism and anti‐Platonism: Why worry?International Studies in the Philosophy of Science 19 (1). 2005.This paper argues that it is scientific realists who should be most concerned about the issue of Platonism and anti-Platonism in mathematics. If one is merely interested in accounting for the practice of pure mathematics, it is unlikely that a story about the ontology of mathematical theories will be essential to such an account. The question of mathematical ontology comes to the fore, however, once one considers our scientific theories. Given that those theories include amongst their laws asser…Read more
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230Conventionalism, by Yemima Ben-MenahemMind 118 (472): 1111-1115. 2009.(No abstract is available for this citation)
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199Revolutionary Fictionalism: A Call to ArmsPhilosophia Mathematica 13 (3): 277-293. 2005.This paper responds to John Burgess's ‘Mathematics and _Bleak House_’. While Burgess's rejection of hermeneutic fictionalism is accepted, it is argued that his two main attacks on revolutionary fictionalism fail to meet their target. Firstly, ‘philosophical modesty’ should not prevent philosophers from questioning the truth of claims made within successful practices, provided that the utility of those practices as they stand can be explained. Secondly, Carnapian scepticism concerning the meaning…Read more
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59What's there to know? A Fictionalist Approach to Mathematical KnowledgeIn Mary Leng, Alexander Paseau & Michael Potter (eds.), Mathematical Knowledge, Oxford University Press. 2007.Defends an account of mathematical knowledge in which mathematical knowledge is a kind of modal knowledge. Leng argues that nominalists should take mathematical knowledge to consist in knowledge of the consistency of mathematical axiomatic systems, and knowledge of what necessarily follows from those axioms. She defends this view against objections that modal knowledge requires knowledge of abstract objects, and argues that we should understand possibility and necessity in a primative way.
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19Platonism and anti-platonism in mathematics (review)Bulletin of Symbolic Logic 8 (4): 516-517. 2002.
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14Creation and Discovery in MathematicsIn John Polkinghorne (ed.), Meaning in mathematics, Oxford University Press. 2011.
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140Structuralism, Fictionalism, and Applied MathematicsIn Clark Glymour, Wei Wang & Dag Westerståhl (eds.), Logic, Methodology and Philosophy of Science: Proceedings of the Thirteenth International Congress, College Publications. pp. 377-389. 2009.
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52Mathematics and RealityOxford University Press. 2010.This book offers a defence of mathematical fictionalism, according to which we have no reason to believe that there are any mathematical objects. Perhaps the most pressing challenge to mathematical fictionalism is the indispensability argument for the truth of our mathematical theories (and therefore for the existence of the mathematical objects posited by those theories). According to this argument, if we have reason to believe anything, we have reason to believe that the claims of our best emp…Read more
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194What's wrong with indispensability?Synthese 131 (3). 2002.For many philosophers not automatically inclined to Platonism, the indispensability argument for the existence of mathematical objectshas provided the best (and perhaps only) evidence for mathematicalrealism. Recently, however, this argument has been subject to attack, most notably by Penelope Maddy (1992, 1997),on the grounds that its conclusions do not sit well with mathematical practice. I offer a diagnosis of what has gone wrong with the indispensability argument (I claim that mathematics is…Read more
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74Phenomenology and mathematical practicePhilosophia Mathematica 10 (1): 3-14. 2002.A phenomenological approach to mathematical practice is sketched out, and some problems with this sort of approach are considered. The approach outlined takes mathematical practices as its data, and seeks to provide an empirically adequate philosophy of mathematics based on observation of these practices. Some observations are presented, based on two case studies of some research into the classification of C*-algebras. It is suggested that an anti-realist account of mathematics could be develope…Read more
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5Imre Lakatos and Paul Feyerabend, For and Against Method Reviewed byPhilosophy in Review 20 (2): 115-117. 2000.