Arithmetic And Ontology

Author : Philip Hugly,Charles Sayward
Publisher : BRILL
Page : 397 pages
File Size : 54,9 Mb
Release : 2016-08-09
Category : Philosophy
ISBN : 9789004333680

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Arithmetic and Ontology by Philip Hugly,Charles Sayward Pdf

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.

Author : Stewart Shapiro
Publisher : Oxford University Press
Page : 290 pages
File Size : 48,6 Mb
Release : 1997-08-07
Category : Philosophy
ISBN : 9780190282523

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Philosophy of Mathematics by Stewart Shapiro Pdf

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.

Author : P. Dybjer,Sten Lindström,Erik Palmgren,B.G. Sundholm
Publisher : Springer Science & Business Media
Page : 399 pages
File Size : 43,9 Mb
Release : 2012-07-10
Category : Philosophy
ISBN : 9789400744356

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Epistemology versus Ontology by P. Dybjer,Sten Lindström,Erik Palmgren,B.G. Sundholm Pdf

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?

Author : Eric Snyder
Publisher : Cambridge University Press
Page : 144 pages
File Size : 54,8 Mb
Release : 2021-05-20
Category : Science
ISBN : 9781108653053

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Semantics and the Ontology of Number by Eric Snyder Pdf

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.

Author : Dale Gottlieb
Publisher : Oxford University Press, USA
Page : 184 pages
File Size : 42,8 Mb
Release : 1980
Category : Language Arts & Disciplines
ISBN : UCAL:B4410951

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Ontological Economy by Dale Gottlieb Pdf

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.

Author : Gary Urton,Primitivo Nina Llanos
Publisher : University of Texas Press
Page : 294 pages
File Size : 46,6 Mb
Release : 2010-07-05
Category : Social Science
ISBN : 9780292786844

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The Social Life of Numbers by Gary Urton,Primitivo Nina Llanos Pdf

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.

Author : Penelope Rush
Publisher : Cambridge University Press
Page : 93 pages
File Size : 47,6 Mb
Release : 2022-02-10
Category : Science
ISBN : 9781108626569

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Ontology and the Foundations of Mathematics by Penelope Rush Pdf

This Element looks at the problem of inter-translation between mathematical realism and anti-realism and argues that so far as realism is inter-translatable with anti-realism, there is a burden on the realist to show how her posited reality differs from that of the anti-realist. It also argues that an effective defence of just such a difference needs a commitment to the independence of mathematical reality, which in turn involves a commitment to the ontological access problem – the problem of how knowable mathematical truths are identifiable with a reality independent of us as knowers. Specifically, if the only access problem acknowledged is the epistemological problem – i.e. the problem of how we come to know mathematical truths – then nothing is gained by the realist notion of an independent reality and in effect, nothing distinguishes realism from anti-realism in mathematics.

Author : Ernest Davis,Philip J. Davis
Publisher : Springer
Page : 379 pages
File Size : 46,8 Mb
Release : 2015-11-17
Category : Mathematics
ISBN : 9783319214733

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Mathematics, Substance and Surmise by Ernest Davis,Philip J. Davis Pdf

The seventeen thought-provoking and engaging essays in this collection present readers with a wide range of diverse perspectives on the ontology of mathematics. The essays address such questions as: What kind of things are mathematical objects? What kinds of assertions do mathematical statements make? How do people think and speak about mathematics? How does society use mathematics? How have our answers to these questions changed over the last two millennia, and how might they change again in the future? The authors include mathematicians, philosophers, computer scientists, cognitive psychologists, sociologists, educators and mathematical historians; each brings their own expertise and insights to the discussion. Contributors to this volume: Jeremy Avigad Jody Azzouni David H. Bailey David Berlinski Jonathan M. Borwein Ernest Davis Philip J. Davis Donald Gillies Jeremy Gray Jesper Lützen Ursula Martin Kay O’Halloran Alison Pease Steven Piantadosi Lance Rips Micah T. Ross Nathalie Sinclair John Stillwell Hellen Verran

Author : Fabio Gironi
Publisher : Springer
Page : 236 pages
File Size : 41,9 Mb
Release : 2014-11-23
Category : Philosophy
ISBN : 9781137463470

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Naturalizing Badiou by Fabio Gironi Pdf

Crossing the boundaries between 'continental' and 'analytic' philosophical approaches, this book proposes a naturalistic revision of the mathematical ontology of Alain Badiou, establishing links with structuralist projects in the philosophy of science and mathematics.

Author : Manuel Bremer
Publisher : Walter de Gruyter
Page : 125 pages
File Size : 55,9 Mb
Release : 2013-05-02
Category : Philosophy
ISBN : 9783110326109

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Universality in Set Theories by Manuel Bremer Pdf

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?

Author : Ivette Fred-Rivera,Jessica Leech
Publisher : Oxford University Press
Page : 320 pages
File Size : 52,5 Mb
Release : 2018-09-26
Category : Philosophy
ISBN : 9780192510600

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Being Necessary by Ivette Fred-Rivera,Jessica Leech Pdf

What is the relationship between ontology and modality - between what there is, and what there could be, must be, or might have been? Bob Hale interwove these two strands of metaphysics throughout his long and distinguished career, putting forward his theses in his book, Necessary Beings: An Essay on Ontology, Modality, and the Relations Between Them (OUP 2013). Hale addressed questions of ontology and modality on a number of fronts: through the development of a Fregean approach to ontology, an essentialist theory of modality, and in his work on neo-logicism in the philosophy of mathematics. The essays in this volume engage with these themes in Hale's work in order to progress our understanding of ontology, modality, and the relations between them. Some directly address questions in modal metaphysics, drawing on ontological concerns, while others raise questions in modal epistemology and of its links to matters of ontology, such as the challenge to give an epistemology of essence. Several essays also engage with questions of what might be called 'modal ontology': the study of whether and what things exist necessarily or contingently. Such issues have an important bearing on the kinds of semantic commitments engendered in logic and mathematics (to the existence of sets, or numbers, or properties, and so on) and the extent to which one's ontology of necessary beings interacts with other plausible assumptions and commitments.

Author : Jody Azzouni
Publisher : Cambridge University Press
Page : 284 pages
File Size : 52,5 Mb
Release : 1994-02-25
Category : Mathematics
ISBN : 9780521442237

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Metaphysical Myths, Mathematical Practice by Jody Azzouni Pdf

This original and exciting study offers a completely new perspective on the philosophy of mathematics. Most philosophers of mathematics try to show either that the sort of knowledge mathematicians have is similar to the sort of knowledge specialists in the empirical sciences have or that the kind of knowledge mathematicians have, although apparently about objects such as numbers, sets, and so on, isn't really about those sorts of things at all. Jody Azzouni argues that mathematical knowledge is a special kind of knowledge that must be gathered in its own unique way. He analyzes the linguistic pitfalls and misperceptions philosophers in this field are often prone to, and explores the misapplications of epistemic principles from the empirical sciences to the exact sciences. What emerges is a picture of mathematics sensitive both to mathematical practice and to the ontological and epistemological issues that concern philosophers. The book will be of special interest to philosophers of science, mathematics, logic, and language. It should also interest mathematicians themselves.

Author : Thomas Hofweber
Publisher : Oxford University Press
Page : 382 pages
File Size : 53,8 Mb
Release : 2016
Category : Mathematics
ISBN : 9780198769835

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Ontology and the Ambitions of Metaphysics by Thomas Hofweber Pdf

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.

Author : Charles B. Daniels,James B. Freeman,Gerald W. Charlwood
Publisher : MacMillan Publishing Company
Page : 176 pages
File Size : 48,5 Mb
Release : 1986
Category : Knowledge, Theory of
ISBN : UCAL:B3931965

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Toward an Ontology of Number, Mind and Sign by Charles B. Daniels,James B. Freeman,Gerald W. Charlwood Pdf

Author : Anonim
Publisher : Unknown
Page : 0 pages
File Size : 45,8 Mb
Release : 1997
Category : Electronic
ISBN : OCLC:874809893

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Philosophy of Mathematics by Anonim Pdf