A Structural Account Of Mathematics

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A Structural Account of Mathematics

Charles S. Chihara
A Structural Account of Mathematics Book PDF

Author : Charles S. Chihara
Publisher : Clarendon Press
Page : 395 pages
File Size : 52,8 Mb
Release : 2003-11-20
Category : Philosophy
ISBN : 9780191533105

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A Structural Account of Mathematics by Charles S. Chihara Pdf

Charles Chihara's new book develops and defends a structural view of the nature of mathematics, and uses it to explain a number of striking features of mathematics that have puzzled philosophers for centuries. The view is used to show that, in order to understand how mathematical systems are applied in science and everyday life, it is not necessary to assume that its theorems either presuppose mathematical objects or are even true. Chihara builds upon his previous work, in which he presented a new system of mathematics, the constructibility theory, which did not make reference to, or presuppose, mathematical objects. Now he develops the project further by analysing mathematical systems currently used by scientists to show how such systems are compatible with this nominalistic outlook. He advances several new ways of undermining the heavily discussed indispensability argument for the existence of mathematical objects made famous by Willard Quine and Hilary Putnam. And Chihara presents a rationale for the nominalistic outlook that is quite different from those generally put forward, which he maintains have led to serious misunderstandings. A Structural Account of Mathematics will be required reading for anyone working in this field.

A Structural Account of Mathematics

Charles S. Chihara
A Structural Account of Mathematics Book PDF

Author : Charles S. Chihara
Publisher : Clarendon Press
Page : 395 pages
File Size : 42,6 Mb
Release : 2004
Category : Language Arts & Disciplines
ISBN : 9780199267538

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A Structural Account of Mathematics by Charles S. Chihara Pdf

Charles Chihara's new book develops and defends a structural view of the nature of mathematics, and uses it to explain a number of striking features of mathematics that have puzzled philosophers for centuries. The view is used to show that, in order to understand how mathematical systems areapplied in science and everyday life, it is not necessary to assume that its theorems either presuppose mathematical objects or are even true.Chihara builds upon his previous work, in which he presented a new system of mathematics, the constructibility theory, which did not make reference to, or presuppose, mathematical objects. Now he develops the project further by analysing mathematical systems currently used by scientists to show howsuch systems are compatible with this nominalistic outlook. He advances several new ways of undermining the heavily discussed indispensability argument for the existence of mathematical objects made famous by Willard Quine and Hilary Putnam. And Chihara presents a rationale for the nominalisticoutlook that is quite different from those generally put forward, which he maintains have led to serious misunderstandings.A Structural Account of Mathematics will be required reading for anyone working in this field.

Mathematical Structuralism

Geoffrey Hellman,Stewart Shapiro
Mathematical Structuralism Book PDF

Author : Geoffrey Hellman,Stewart Shapiro
Publisher : Cambridge University Press
Page : 167 pages
File Size : 47,6 Mb
Release : 2018-11-29
Category : Science
ISBN : 9781108630740

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Mathematical Structuralism by Geoffrey Hellman,Stewart Shapiro Pdf

The present work is a systematic study of five frameworks or perspectives articulating mathematical structuralism, whose core idea is that mathematics is concerned primarily with interrelations in abstraction from the nature of objects. The first two, set-theoretic and category-theoretic, arose within mathematics itself. After exposing a number of problems, the Element considers three further perspectives formulated by logicians and philosophers of mathematics: sui generis, treating structures as abstract universals, modal, eliminating structures as objects in favor of freely entertained logical possibilities, and finally, modal-set-theoretic, a sort of synthesis of the set-theoretic and modal perspectives.

Proof and Other Dilemmas

Bonnie Gold,Roger A. Simons
Proof and Other Dilemmas Book PDF

Author : Bonnie Gold,Roger A. Simons
Publisher : MAA
Page : 392 pages
File Size : 47,5 Mb
Release : 2008
Category : Mathematics
ISBN : 0883855674

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Proof and Other Dilemmas by Bonnie Gold,Roger A. Simons Pdf

Sixteen original essays exploring recent developments in the philosophy of mathematics, written in a way mathematicians will understand.

Structuralism and Structures

Charles Earl Rickart
Structuralism and Structures Book PDF

Author : Charles Earl Rickart
Publisher : World Scientific
Page : 240 pages
File Size : 55,8 Mb
Release : 1995
Category : Mathematics
ISBN : 9810218605

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Structuralism and Structures by Charles Earl Rickart Pdf

This book is devoted to an analysis of the way that structures must enter into a serious study of any subject, and the term ?structuralism? refers to the general method of approaching a subject from the viewpoint of structure. A proper appreciation of this approach requires a deeper understanding of the concept of structure than is provided by the simple intuitive notion of structures that everyone posseses to some degree. Therefore, a large part of the discussion is devoted directly or indirectly to a study of the nature of structures themselves. A formal definition of a structure, plus some basic general properties and examples, is given early in the discussion. Also, in order to clarify the general notions and to see how they are used, the later chapters are devoted to an examination of how structures enter into some special fields, including linguistics, mental phenomena, mathematics (and its applications), and biology (especially in the theory of evolution). Because the author is a mathematician, certain mathematical ideas have influenced greatly the choice and approach to the material covered. In general, however, the mathematical influence is not on a technical level and is often only implicit. Even the chapter on mathematical structures is nontechnical and is about rather than on mathematics. Only in the last chapter and earlier in three short sections does one find any of the expected ?formal? mathematics. In other words, the great bulk of the material is accessible to someone without a mathematical background.

Structures Mères: Semantics, Mathematics, and Cognitive Science

Alberto Peruzzi,Silvano Zipoli Caiani
Structures M res Semantics Mathematics and Cognitive Science Book PDF

Author : Alberto Peruzzi,Silvano Zipoli Caiani
Publisher : Springer Nature
Page : 191 pages
File Size : 40,8 Mb
Release : 2020-09-14
Category : Philosophy
ISBN : 9783030518219

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Structures Mères: Semantics, Mathematics, and Cognitive Science by Alberto Peruzzi,Silvano Zipoli Caiani Pdf

This book reports on cutting-edge concepts related to Bourbaki’s notion of structures mères. It merges perspectives from logic, philosophy, linguistics and cognitive science, suggesting how they can be combined with Bourbaki’s mathematical structuralism in order to solve foundational, ontological and epistemological problems using a novel category-theoretic approach. By offering a comprehensive account of Bourbaki’s structuralism and answers to several important questions that have arisen in connection with it, the book provides readers with a unique source of information and inspiration for future research on this topic.

Reasoning in Science and Mathematics

Richard L Epstein
Reasoning in Science and Mathematics Book PDF

Author : Richard L Epstein
Publisher : Advanced Reasoning Forum
Page : 133 pages
File Size : 50,9 Mb
Release : 2018-11-05
Category : Philosophy
ISBN : 9780983452133

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Reasoning in Science and Mathematics by Richard L Epstein Pdf

This series of books is meant to present the fundamentals of reasoning well in a clear manner accessible to both scholars and students. The body of each essay gives the main development of the subject, while the footnotes and appendices place the research within a larger scholarly context. The topic of this volume is the nature and evaluation of reasoning in science and mathematics. Science and mathematics can both be understood as proceeding by a method of abstraction from experience. Mathematics is distinguished from other sciences only in its greater abstraction and its demand for necessity in its inferences. That methodology of abstraction is the main focus here. The study of these subjects is not just of academic interest but can lead to better research in science and mathematics. First comes clear thinking, then comes clear research and clear writing. The essays: • Background • Models and Theories • Experiments • Mathematics as the Art of Abstraction.

Philosophy of Mathematics

Stewart Shapiro
Philosophy of Mathematics Book PDF

Author : Stewart Shapiro
Publisher : Oxford University Press
Page : 290 pages
File Size : 46,5 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.

Rigor and Structure

John P. Burgess
Rigor and Structure Book PDF

Author : John P. Burgess
Publisher : Oxford University Press, USA
Page : 241 pages
File Size : 44,9 Mb
Release : 2015
Category : Mathematics
ISBN : 9780198722229

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Rigor and Structure by John P. Burgess Pdf

John P. Burgess presents an illuminating study of the nature of mathematical rigour and of mathematical structure, and above all of the relation between them. He considers recent developments in the field including experimental mathematics and computerised formal proofs, and surveys many historical developments in mathematics, philosophy, and logic.

The Prehistory of Mathematical Structuralism

Erich H. Reck,Georg Schiemer
The Prehistory of Mathematical Structuralism Book PDF

Author : Erich H. Reck,Georg Schiemer
Publisher : Oxford University Press
Page : 469 pages
File Size : 55,7 Mb
Release : 2020
Category : Mathematics
ISBN : 9780190641221

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The Prehistory of Mathematical Structuralism by Erich H. Reck,Georg Schiemer Pdf

This edited volume explores the previously underacknowledged 'pre-history' of mathematical structuralism, showing that structuralism has deep roots in the history of modern mathematics. The contributors explore this history along two distinct but interconnected dimensions. First, they reconsider the methodological contributions of major figures in the history of mathematics. Second, they re-examine a range of philosophical reflections from mathematically-inclinded philosophers like Russell, Carnap, and Quine, whose work led to profound conclusions about logical, epistemological, and metaphysic.

Mathematics and Scientific Representation

Christopher Pincock
Mathematics and Scientific Representation Book PDF

Author : Christopher Pincock
Publisher : OUP USA
Page : 347 pages
File Size : 40,7 Mb
Release : 2012-01-13
Category : Mathematics
ISBN : 9780199757107

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Mathematics and Scientific Representation by Christopher Pincock Pdf

Mathematics plays a central role in much of contemporary science, but philosophers have struggled to understand what this role is or how significant it might be for mathematics and science. Pincock tackles this perennial question by asking how mathematics contributes to the success of our best scientific representations.

Oxford Studies in Metaphysics

Dean Zimmerman
Oxford Studies in Metaphysics Book PDF

Author : Dean Zimmerman
Publisher : Oxford University Press
Page : 334 pages
File Size : 53,6 Mb
Release : 2008-05-22
Category : Philosophy
ISBN : 9780199542987

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Oxford Studies in Metaphysics by Dean Zimmerman Pdf

The fourth volume of a series that acts as a forum for new works in the field of metaphysics. The collection offers a broad overview of the subject & features traditional topics as well as questions from neighbouring fields such as the philosophies of mind & science.

Modern Algebra and the Rise of Mathematical Structures

Leo Corry
Modern Algebra and the Rise of Mathematical Structures Book PDF

Author : Leo Corry
Publisher : Birkhäuser
Page : 463 pages
File Size : 50,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034879170

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Modern Algebra and the Rise of Mathematical Structures by Leo Corry Pdf

This book describes two stages in the historical development of the notion of mathematical structures: first, it traces its rise in the context of algebra from the mid-1800s to 1930, and then considers attempts to formulate elaborate theories after 1930 aimed at elucidating, from a purely mathematical perspective, the precise meaning of this idea.

Social Constructivism as a Philosophy of Mathematics

Paul Ernest
Social Constructivism as a Philosophy of Mathematics Book PDF

Author : Paul Ernest
Publisher : SUNY Press
Page : 338 pages
File Size : 55,6 Mb
Release : 1998-01-01
Category : Philosophy
ISBN : 0791435881

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Social Constructivism as a Philosophy of Mathematics by Paul Ernest Pdf

Extends the ideas of social constructivism to the philosophy of mathematics, developing a powerful critique of traditional absolutist conceptions of mathematics, and proposing a reconceptualization of the philosophy of mathematics.

Mathematics and Its Logics

Geoffrey Hellman
Mathematics and Its Logics Book PDF

Author : Geoffrey Hellman
Publisher : Cambridge University Press
Page : 0 pages
File Size : 54,7 Mb
Release : 2022-11-10
Category : Science
ISBN : 1108714005

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Mathematics and Its Logics by Geoffrey Hellman Pdf

In these essays Geoffrey Hellman presents a strong case for a healthy pluralism in mathematics and its logics, supporting peaceful coexistence despite what appear to be contradictions between different systems, and positing different frameworks serving different legitimate purposes. The essays refine and extend Hellman's modal-structuralist account of mathematics, developing a height-potentialist view of higher set theory which recognizes indefinite extendability of models and stages at which sets occur. In the first of three new essays written for this volume, Hellman shows how extendability can be deployed to derive the axiom of Infinity and that of Replacement, improving on earlier accounts; he also shows how extendability leads to attractive, novel resolutions of the set-theoretic paradoxes. Other essays explore advantages and limitations of restrictive systems - nominalist, predicativist, and constructivist. Also included are two essays, with Solomon Feferman, on predicative foundations of arithmetic.