Arithmetic and Ontology

Arithmetic and Ontology

Author: Philip Hugly

Publisher: BRILL

Published: 2016-08-09

Total Pages: 397

ISBN-13: 9004333681

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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.


Book Synopsis Arithmetic and Ontology by : Philip Hugly

Download or read book Arithmetic and Ontology written by Philip Hugly and published by BRILL. This book was released on 2016-08-09 with total page 397 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Semantics and the Ontology of Number

Semantics and the Ontology of Number

Author: Eric Snyder

Publisher: Cambridge University Press

Published: 2021-05-20

Total Pages: 144

ISBN-13: 1108653057

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What are the meanings of number expressions, and what can they tell us about questions of central importance to the philosophy of mathematics, specifically 'Do numbers exist?' This Element attempts to shed light on this question by outlining a recent debate between substantivalists and adjectivalists regarding the semantic function of number words in numerical statements. After highlighting their motivations and challenges, I develop a comprehensive polymorphic semantics for number expressions. I argue that accounting for the numerous meanings and how they are related leads to a strengthened argument for realism, one which renders familiar forms of nominalism highly implausible.


Book Synopsis Semantics and the Ontology of Number by : Eric Snyder

Download or read book Semantics and the Ontology of Number written by Eric Snyder and published by Cambridge University Press. This book was released on 2021-05-20 with total page 144 pages. Available in PDF, EPUB and Kindle. Book excerpt: What are the meanings of number expressions, and what can they tell us about questions of central importance to the philosophy of mathematics, specifically 'Do numbers exist?' This Element attempts to shed light on this question by outlining a recent debate between substantivalists and adjectivalists regarding the semantic function of number words in numerical statements. After highlighting their motivations and challenges, I develop a comprehensive polymorphic semantics for number expressions. I argue that accounting for the numerous meanings and how they are related leads to a strengthened argument for realism, one which renders familiar forms of nominalism highly implausible.

Philosophy of Mathematics

Philosophy of Mathematics

Author: Stewart Shapiro

Publisher: Oxford University Press

Published: 1997-08-07

Total Pages: 290

ISBN-13: 0190282525

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Do numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic. As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests realism, but realism in this context suggests seemingly intractable epistemic problems. As a way out of this dilemma, Shapiro articulates a structuralist approach. On this view, the subject matter of arithmetic, for example, is not a fixed domain of numbers independent of each other, but rather is the natural number structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle. Using this framework, realism in mathematics can be preserved without troublesome epistemic consequences. Shapiro concludes by showing how a structuralist approach can be applied to wider philosophical questions such as the nature of an "object" and the Quinean nature of ontological commitment. Clear, compelling, and tautly argued, Shapiro's work, noteworthy both in its attempt to develop a full-length structuralist approach to mathematics and to trace its emergence in the history of mathematics, will be of deep interest to both philosophers and mathematicians.


Book Synopsis Philosophy of Mathematics by : Stewart Shapiro

Download or read book Philosophy of Mathematics written by Stewart Shapiro and published by Oxford University Press. This book was released on 1997-08-07 with total page 290 pages. Available in PDF, EPUB and Kindle. Book excerpt: Do numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic. As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests realism, but realism in this context suggests seemingly intractable epistemic problems. As a way out of this dilemma, Shapiro articulates a structuralist approach. On this view, the subject matter of arithmetic, for example, is not a fixed domain of numbers independent of each other, but rather is the natural number structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle. Using this framework, realism in mathematics can be preserved without troublesome epistemic consequences. Shapiro concludes by showing how a structuralist approach can be applied to wider philosophical questions such as the nature of an "object" and the Quinean nature of ontological commitment. Clear, compelling, and tautly argued, Shapiro's work, noteworthy both in its attempt to develop a full-length structuralist approach to mathematics and to trace its emergence in the history of mathematics, will be of deep interest to both philosophers and mathematicians.

Epistemology versus Ontology

Epistemology versus Ontology

Author: P. Dybjer

Publisher: Springer Science & Business Media

Published: 2012-07-10

Total Pages: 399

ISBN-13: 9400744358

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This book brings together philosophers, mathematicians and logicians to penetrate important problems in the philosophy and foundations of mathematics. In philosophy, one has been concerned with the opposition between constructivism and classical mathematics and the different ontological and epistemological views that are reflected in this opposition. The dominant foundational framework for current mathematics is classical logic and set theory with the axiom of choice (ZFC). This framework is, however, laden with philosophical difficulties. One important alternative foundational programme that is actively pursued today is predicativistic constructivism based on Martin-Löf type theory. Associated philosophical foundations are meaning theories in the tradition of Wittgenstein, Dummett, Prawitz and Martin-Löf. What is the relation between proof-theoretical semantics in the tradition of Gentzen, Prawitz, and Martin-Löf and Wittgensteinian or other accounts of meaning-as-use? What can proof-theoretical analyses tell us about the scope and limits of constructive and predicative mathematics?


Book Synopsis Epistemology versus Ontology by : P. Dybjer

Download or read book Epistemology versus Ontology written by P. Dybjer and published by Springer Science & Business Media. This book was released on 2012-07-10 with total page 399 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book brings together philosophers, mathematicians and logicians to penetrate important problems in the philosophy and foundations of mathematics. In philosophy, one has been concerned with the opposition between constructivism and classical mathematics and the different ontological and epistemological views that are reflected in this opposition. The dominant foundational framework for current mathematics is classical logic and set theory with the axiom of choice (ZFC). This framework is, however, laden with philosophical difficulties. One important alternative foundational programme that is actively pursued today is predicativistic constructivism based on Martin-Löf type theory. Associated philosophical foundations are meaning theories in the tradition of Wittgenstein, Dummett, Prawitz and Martin-Löf. What is the relation between proof-theoretical semantics in the tradition of Gentzen, Prawitz, and Martin-Löf and Wittgensteinian or other accounts of meaning-as-use? What can proof-theoretical analyses tell us about the scope and limits of constructive and predicative mathematics?

The Social Life of Numbers

The Social Life of Numbers

Author: Gary Urton

Publisher: University of Texas Press

Published: 2010-07-05

Total Pages: 294

ISBN-13: 0292786840

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Unraveling all the mysteries of the khipu--the knotted string device used by the Inka to record both statistical data and narrative accounts of myths, histories, and genealogies--will require an understanding of how number values and relations may have been used to encode information on social, familial, and political relationships and structures. This is the problem Gary Urton tackles in his pathfinding study of the origin, meaning, and significance of numbers and the philosophical principles underlying the practice of arithmetic among Quechua-speaking peoples of the Andes. Based on fieldwork in communities around Sucre, in south-central Bolivia, Urton argues that the origin and meaning of numbers were and are conceived of by Quechua-speaking peoples in ways similar to their ideas about, and formulations of, gender, age, and social relations. He also demonstrates that their practice of arithmetic is based on a well-articulated body of philosophical principles and values that reflects a continuous attempt to maintain balance, harmony, and equilibrium in the material, social, and moral spheres of community life.


Book Synopsis The Social Life of Numbers by : Gary Urton

Download or read book The Social Life of Numbers written by Gary Urton and published by University of Texas Press. This book was released on 2010-07-05 with total page 294 pages. Available in PDF, EPUB and Kindle. Book excerpt: Unraveling all the mysteries of the khipu--the knotted string device used by the Inka to record both statistical data and narrative accounts of myths, histories, and genealogies--will require an understanding of how number values and relations may have been used to encode information on social, familial, and political relationships and structures. This is the problem Gary Urton tackles in his pathfinding study of the origin, meaning, and significance of numbers and the philosophical principles underlying the practice of arithmetic among Quechua-speaking peoples of the Andes. Based on fieldwork in communities around Sucre, in south-central Bolivia, Urton argues that the origin and meaning of numbers were and are conceived of by Quechua-speaking peoples in ways similar to their ideas about, and formulations of, gender, age, and social relations. He also demonstrates that their practice of arithmetic is based on a well-articulated body of philosophical principles and values that reflects a continuous attempt to maintain balance, harmony, and equilibrium in the material, social, and moral spheres of community life.

Ontology and the Ambitions of Metaphysics

Ontology and the Ambitions of Metaphysics

Author: Thomas Hofweber

Publisher: Oxford University Press

Published: 2016

Total Pages: 382

ISBN-13: 0198769830

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Many significant problems in metaphysics are tied to ontological questions, but ontology and its relation to larger questions in metaphysics give rise to a series of puzzles that suggest that we don't fully understand what ontology is supposed to do, nor what ambitions metaphysics can have for finding out about what the world is like. Thomas Hofweber aims to solve these puzzles about ontology and consequently to make progress on four central metaphysical problems: the philosophy of arithmetic, the metaphysics of ordinary objects, the problem of universals, and the question whether reality is independent of us. Crucial parts of the proposed solution include considerations about quantification and its relationship to ontology, the place of reference in natural languages, the possibility of ineffable facts, the extent of empirical evidence in metaphysics, and whether metaphysics can properly be esoteric. Overall, Hofweber defends a rationalist account of arithmetic, an empiricist picture in the philosophy of ordinary objects, a restricted from of nominalism, and realism about reality, understood as all there is, but idealism about reality, understood as all that is the case. He defends metaphysics as having some questions of fact that are distinctly its own, with a limited form of autonomy from other parts of inquiry, but rejects several metaphysical projects and approaches as being based on a mistake.


Book Synopsis Ontology and the Ambitions of Metaphysics by : Thomas Hofweber

Download or read book Ontology and the Ambitions of Metaphysics written by Thomas Hofweber and published by Oxford University Press. This book was released on 2016 with total page 382 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many significant problems in metaphysics are tied to ontological questions, but ontology and its relation to larger questions in metaphysics give rise to a series of puzzles that suggest that we don't fully understand what ontology is supposed to do, nor what ambitions metaphysics can have for finding out about what the world is like. Thomas Hofweber aims to solve these puzzles about ontology and consequently to make progress on four central metaphysical problems: the philosophy of arithmetic, the metaphysics of ordinary objects, the problem of universals, and the question whether reality is independent of us. Crucial parts of the proposed solution include considerations about quantification and its relationship to ontology, the place of reference in natural languages, the possibility of ineffable facts, the extent of empirical evidence in metaphysics, and whether metaphysics can properly be esoteric. Overall, Hofweber defends a rationalist account of arithmetic, an empiricist picture in the philosophy of ordinary objects, a restricted from of nominalism, and realism about reality, understood as all there is, but idealism about reality, understood as all that is the case. He defends metaphysics as having some questions of fact that are distinctly its own, with a limited form of autonomy from other parts of inquiry, but rejects several metaphysical projects and approaches as being based on a mistake.

Toward an Ontology of Number, Mind and Sign

Toward an Ontology of Number, Mind and Sign

Author: Charles B. Daniels

Publisher: MacMillan Publishing Company

Published: 1986

Total Pages: 176

ISBN-13:

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Book Synopsis Toward an Ontology of Number, Mind and Sign by : Charles B. Daniels

Download or read book Toward an Ontology of Number, Mind and Sign written by Charles B. Daniels and published by MacMillan Publishing Company. This book was released on 1986 with total page 176 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Philosophy of Mathematics

Philosophy of Mathematics

Author:

Publisher:

Published: 1997

Total Pages: 0

ISBN-13:

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Book Synopsis Philosophy of Mathematics by :

Download or read book Philosophy of Mathematics written by and published by . This book was released on 1997 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Universality in Set Theories

Universality in Set Theories

Author: Manuel Bremer

Publisher: Walter de Gruyter

Published: 2013-05-02

Total Pages: 125

ISBN-13: 3110326108

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The book discusses the fate of universality and a universal set in several set theories. The book aims at a philosophical study of ontological and conceptual questions around set theory. Set theories are ontologies. They posit sets and claim that these exhibit the essential properties laid down in the set theoretical axioms. Collecting these postulated entities quantified over poses the problem of universality. Is the collection of the set theoretical entities itself a set theoretical entity? What does it mean if it is, and what does it mean if it is not? To answer these questions involves developing a theory of the universal set. We have to ask: Are there different aspects to universality in set theory, which stand in conflict to each other? May inconsistency be the price to pay to circumvent ineffability? And most importantly: How far can axiomatic ontology take us out of the problems around universality?


Book Synopsis Universality in Set Theories by : Manuel Bremer

Download or read book Universality in Set Theories written by Manuel Bremer and published by Walter de Gruyter. This book was released on 2013-05-02 with total page 125 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book discusses the fate of universality and a universal set in several set theories. The book aims at a philosophical study of ontological and conceptual questions around set theory. Set theories are ontologies. They posit sets and claim that these exhibit the essential properties laid down in the set theoretical axioms. Collecting these postulated entities quantified over poses the problem of universality. Is the collection of the set theoretical entities itself a set theoretical entity? What does it mean if it is, and what does it mean if it is not? To answer these questions involves developing a theory of the universal set. We have to ask: Are there different aspects to universality in set theory, which stand in conflict to each other? May inconsistency be the price to pay to circumvent ineffability? And most importantly: How far can axiomatic ontology take us out of the problems around universality?

Ontological Economy

Ontological Economy

Author: Dale Gottlieb

Publisher: Oxford University Press, USA

Published: 1980

Total Pages: 184

ISBN-13:

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Shows that when Qyuine's criterion of ontological commitment is modified to allow for the legitimacy of substitutional quantification, two consequences follow: (i) fundamental questions of ontology cease to be settled by mere appeal to logical form and truth, and (ii) a powerful method for reducing ontological commitments becomes available.


Book Synopsis Ontological Economy by : Dale Gottlieb

Download or read book Ontological Economy written by Dale Gottlieb and published by Oxford University Press, USA. This book was released on 1980 with total page 184 pages. Available in PDF, EPUB and Kindle. Book excerpt: Shows that when Qyuine's criterion of ontological commitment is modified to allow for the legitimacy of substitutional quantification, two consequences follow: (i) fundamental questions of ontology cease to be settled by mere appeal to logical form and truth, and (ii) a powerful method for reducing ontological commitments becomes available.