Arithmetic and Ontology: A Non-realist Philosophy of Arithmetic

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Rodopi, 2006 - Mathematics - 393 pages
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This volume documents a lively exchange between five philosophers of mathematics. It also introduces a new voice in one central debate in the philosophy of mathematics. Non-realism, i.e., the view supported by Hugly and Sayward in their monograph, is an original position distinct from the widely known realism and anti-realism. Non-realism is characterized by the rejection of a central assumption shared by many realists and anti-realists, i.e., the assumption that mathematical statements purport to refer to objects. The defense of their main argument for the thesis that arithmetic lacks ontology brings the authors to discuss also the controversial contrast between pure and empirical arithmetical discourse. Colin Cheyne, Sanford Shieh, and Jean Paul Van Bendegem, each coming from a different perspective, test the genuine originality of non-realism and raise objections to it. Novel interpretations of well-known arguments, e.g., the indispensability argument, and historical views, e.g. Frege, are interwoven with the development of the authors account. The discussion of the often neglected views of Wittgenstein and Prior provide an interesting and much needed contribution to the current debate in the philosophy of mathematics."
 

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Contents

Introduction
35
Notes to Grundlagen
45
Objectivism and Realism in Freges Philosophy
73
The Peano Axioms
105
Existence Number and Realism
129
Arithmetic and Necessity
159
Arithmetic and Rules
183
Thesis One
215
Thesis Three
254
References
285
Colin Cheyne Numbers Reference and Abstraction
291
Sanford Shieh What Is NonRealism about Arithmetic?
317
Jean Paul Van Bendegem NonRealism Nominalism and Strict
343
Philip Hugly and Charles Sayward Replies to Commentaries
369
About the Contributors
387
Copyright

Thesis Two
241

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