Arithmetic and Ontology: A Non-Realist Philosophy of Arithmetic

Arithmetic and Ontology: A Non-Realist Philosophy of Arithmetic

Arithmetic and Ontology: A Non-Realist Philosophy of Arithmetic

Arithmetic and Ontology: A Non-Realist Philosophy of Arithmetic

Synopsis

"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."

Excerpt

Philosophy at its best is a stimulating dialogue between creative and articulate reasoners who present and defend different points of view on an issue. This insight is at the origin of the Monograph-in-Debate Series of the Poznań Studies in the Philosophy of the Sciences and the Humanities. This volume keeps faith to this insight. It documents a debate between five philosophers on one central issue in the philosophy of mathematics, i.e., the ontological status of arithmetic. The monograph Arithmetic and Ontology. A Non-Realist Philosophy of Arithmetic by Phil Hugly and Charles Sayward provides the basis for the development of this discussion. In the papers following the monograph, three commentators react to Hugly’s and Sayward’s proposal. Finally, the authors’ replies address some of the issues raised by the commentators. As it is often the case in a good debate, more ground is covered than what was at first promised. Although the ontological status of arithmetic is the core issue of the debate, novel interpretations of well known arguments and historical views, such as those of Frege, Wittgenstein, and Prior are interwoven with the development of the authors’ account and provide a valuable contribution on their own.

At the origin of Arithmetic and Ontology is a certain uneasiness with two widely spread practices within the philosophy of mathematics. The first is the appeal to non-spatially or temporarily located abstract objects; the second is the acceptance of Quine’s criterion of ontological commitment “to be is to be the value of a variable.” Hugly and Sayward have much to say about both practices. While their discussion of the former may aptly remind some of Wittgestein’s queries about linguistic uses, with regard to the latter, the authors explicitly acknowledge the influence of Prior’s criticism of Quine’s criterion. In either case, it is their doubts about these practices that motivate these authors’ search for an alternative to a realist ontological account of arithmetic.

Hugly’s and Sayward’s non-realist philosophy of arithmetic is based on two main sets of ideas, i.e., “(i) that pure arithmetic, taken in isolation from the use of arithmetical signs in empirical judgment, is an activity for which a formalist account (roughly one on which arithmetic is strictly a system of signs) is about right, and (ii) that arithmetical signs nonetheless have meanings, but only in and through belonging to a system of signs with empirical application” (p. 35). At the basis of either . . .

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