A Structural Account of Mathematics

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Author: Charles S. Chihara

Publisher: Clarendon Press

ISBN: 0199267537

Category: Philosophy

Page: 380

View: 8430

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 Chiharapresents 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.

Proof and Other Dilemmas

Mathematics and Philosophy

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Author: Roger Simons

Publisher: MAA

ISBN: 9780883855676

Category: Mathematics

Page: 346

View: 8883

For the majority of the twentieth century, philosophers of mathematics focused their attention on foundational questions. However, in the last quarter of the century they began to return to basics, and two new schools of thought were created: social constructivism and structuralism. The advent of the computer also led to proofs and development of mathematics assisted by computer, and to questions concerning the role of the computer in mathematics. This book of sixteen original essays is the first to explore this range of new developments in the philosophy of mathematics, in a language accessible to mathematicians. Approximately half the essays were written by mathematicians, and consider questions that philosophers have not yet discussed. The other half, written by philosophers of mathematics, summarise the discussion in that community during the last 35 years. A connection is made in each case to issues relevant to the teaching of mathematics.

Modern Algebra and the Rise of Mathematical Structures

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Author: Leo Corry

Publisher: Birkhäuser

ISBN: 3034879172

Category: Mathematics

Page: 452

View: 4876

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.

Oxford Studies in Metaphysics

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Author: Dean Zimmerman

Publisher: OUP Oxford

ISBN: 0191562319

Category: Philosophy

Page: 336

View: 6684

Oxford Studies in Metaphysics is the forum for the best new work in this flourishing field. Much of the most interesting work in philosophy today is metaphysical in character: this new series is a much-needed focus for it. OSM offers a broad view of the subject, featuring not only the traditionally central topics such as existence, identity, modality, time, and causation, but also the rich clusters of metaphysical questions in neighbouring fields, such as philosophy of mind and philosophy of science. Besides independent essays, volumes will often contain a critical essay on a recent book, or a symposium that allows participants to respond to one another's criticisms and questions. A special feature of this volume is an unpublished paper on nominalism by W. V. Quine, arguably the most influential figure in philosophy in the second half of the twentieth century. It is accompanied by five specially commissioned commentaries. Topics discussed by other papers in this volume include ontology, location, truthmaking, and physicalism. Anyone who wants to know what's happening in metaphysics can start here.

Why Is There Philosophy of Mathematics At All?

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Author: Ian Hacking

Publisher: Cambridge University Press

ISBN: 1107050170

Category: Mathematics

Page: 308

View: 8690

Hacking explores how mathematics became possible for the human race, and how it ensured our status as the dominant species.

Rigor and Structure

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Author: John P. Burgess

Publisher: OUP Oxford

ISBN: 019103360X

Category: Philosophy

Page: 224

View: 2488

While we are commonly told that the distinctive method of mathematics is rigorous proof, and that the special topic of mathematics is abstract structure, there has been no agreement among mathematicians, logicians, or philosophers as to just what either of these assertions means. John P. Burgess clarifies the nature of mathematical rigor and of mathematical structure, and above all of the relation between the two, taking into account some of the latest developments in mathematics, including the rise of experimental mathematics on the one hand and computerized formal proofs on the other hand. The main theses of Rigor and Structure are that the features of mathematical practice that a large group of philosophers of mathematics, the structuralists, have attributed to the peculiar nature of mathematical objects are better explained in a different way, as artefacts of the manner in which the ancient ideal of rigor is realized in modern mathematics. Notably, the mathematician must be very careful in deriving new results from the previous literature, but may remain largely indifferent to just how the results in the previous literature were obtained from first principles. Indeed, the working mathematician may remain largely indifferent to just what the first principles are supposed to be, and whether they are set-theoretic or category-theoretic or something else. Along the way to these conclusions, a great many historical developments in mathematics, philosophy, and logic are surveyed. Yet very little in the way of background knowledge on the part of the reader is presupposed.

Mathematical Principles of Human Conceptual Behavior

The Structural Nature of Conceptual Representation and Processing

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Author: Ronaldo Vigo

Publisher: Psychology Press

ISBN: 1134690436

Category: Psychology

Page: 246

View: 6499

The ability to learn concepts lies at the very core of human cognition, enabling us to efficiently classify, organize, identify, and store complex information. In view of the basic role that concepts play in our everyday physical and mental lives, the fields of cognitive science and psychology face three long standing challenges: discovering the laws that govern concept learning and categorization behavior in organisms, showing how they inform other areas of cognitive research, and describing them with the mathematical systematicity and precision found in the physical sciences. In light of these theoretical and methodological shortcomings, this volume will introduce a set of general mathematical principles for predicting and explaining conceptual behavior. The author’s theory is based on seven fundamental constructs of universal science: invariance, complexity, information, similarity, dissimilarity, pattern, and representation. These constructs are joined by a novel mathematical framework that does not depend on probability theory, and derives key results from conceptual behavior research with other key areas of cognitive research such as pattern perception, similarity assessment, and contextual choice. The result is a unique and systematic unifying foundation for cognitive science in the tradition of classical physics.

EPSA Philosophical Issues in the Sciences

Launch of the European Philosophy of Science Association

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Author: Mauricio Suárez,Mauro Dorato,Miklós Rédei

Publisher: Springer Science & Business Media

ISBN: 9789048132522

Category: Science

Page: 332

View: 8573

This volume collects papers presented at the Founding Conference of the European Philosophy of Science Association meeting, held November 2007. It provides an excellent overview of the state of the art in philosophy of science in different European countries.

Contemporary Kantian Metaphysics

New Essays on Space and Time

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Author: R. Baiasu,G. Bird,A. Moore

Publisher: Springer

ISBN: 0230358918

Category: Philosophy

Page: 300

View: 7269

Responding to growing interest in the Kantian tradition and in issues concerning space and time, this volume offers an insightful and original contribution to the literature by bringing together analytical and phenomenological approaches in a productive exchange on topical issues such as action, perception, the body, and cognition and its limits.

Representation and Productive Ambiguity in Mathematics and the Sciences

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Author: Emily R. Grosholz

Publisher: Clarendon Press

ISBN: 0191538515

Category: Mathematics

Page: 330

View: 6673

Emily Grosholz offers an original investigation of demonstration in mathematics and science, examining how it works and why it is persuasive. Focusing on geometrical demonstration, she shows the roles that representation and ambiguity play in mathematical discovery. She presents a wide range of case studies in mechanics, topology, algebra, logic, and chemistry, from ancient Greece to the present day, but focusing particularly on the seventeenth and twentieth centuries. She argues that reductive methods are effective not because they diminish but because they multiply and juxtapose modes of representation. Such problem-solving is, she argues, best understood in terms of Leibnizian 'analysis' - the search for conditions of intelligibility. Discovery and justification are then two aspects of one rational way of proceeding, which produces the mathematician's formal experience. Grosholz defends the importance of iconic, as well as symbolic and indexical, signs in mathematical representation, and argues that pragmatic, as well as syntactic and semantic, considerations are indispensable for mathematical reasoning. By taking a close look at the way results are presented on the page in mathematical (and biological, chemical, and mechanical) texts, she shows that when two or more traditions combine in the service of problem solving, notations and diagrams are sublty altered, multiplied, and juxtaposed, and surrounded by prose in natural language which explains the novel combination. Viewed this way, the texts yield striking examples of language and notation that are irreducibly ambiguous and productive because they are ambiguous. Grosholtz's arguments, which invoke Descartes, Locke, Hume, and Kant, will be of considerable interest to philosophers and historians of mathematics and science, and also have far-reaching consequences for epistemology and philosophy of language.

The Blackwell Companion to Natural Theology

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Author: William Lane Craig,J. P. Moreland

Publisher: John Wiley & Sons

ISBN: 1444350854

Category: Philosophy

Page: 697

View: 5922

With the help of in–depth essays from some of the world′s leading philosophers, The Blackwell Companion to Natural Theology explores the nature and existence of God through human reason and evidence from the natural world. Provides in–depth and cutting–edge treatment of natural theology′s main arguments Includes contributions from first–rate philosophers well known for their work on the relevant topics Updates relevant arguments in light of the most current, state–of–the–art philosophical and scientific discussions Stands in useful contrast and opposition to the arguments of the ′new atheists′

New Directions for Situated Cognition in Mathematics Education

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Author: Anne Watson,Peter Winbourne

Publisher: Springer Science & Business Media

ISBN: 0387715797

Category: Education

Page: 360

View: 8372

This book draws together a range of papers by experienced writers in mathematics education who have used the concept of situated cognition in their research within recent years. No other books are available which take this view specifically in mathematics education. Thus it provides an up-to-date overview of developments and applications to which other researchers can refer and which will inspire future research.

Reasonable Faith (3rd edition)

Christian Truth and Apologetics

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Author: William Lane Craig

Publisher: Crossway

ISBN: 1433521180

Category: Religion

Page: 416

View: 7630

Perfect as a textbook yet excellent for lay readers, this updated edition builds a positive case for Christianity by applying the latest thought to core theological themes. J. Gresham Machen once said, "False ideas are the greatest obstacles to the reception of the gospel"-which makes apologetics that much more important. Wanting to engage not just academics and pastors but Christian laypeople and seekers, William Lane Craig has revised and updated key sections in this third edition of his classic text to reflect the latest work in astrophysics, philosophy, probability calculus, the arguments for the existence of God, and Reformed epistemology. His approach-that of positive apologetics-gives careful attention to crucial questions and concerns, including: the relationship of faith and reason, the existence of God, the problems of historical knowledge and miracles, the personal claims of Christ, and the historicity of the resurrection of Jesus. He shows that there is good reason to think Christianity is true. As Craig says, "If you have a sound and persuasive case for Christianity, you don't have to become an expert in comparative religions and Christian cults. A positive justification of the Christian faith automatically overwhelms all competing world views lacking an equally strong case."

Constructibility and Mathematical Existence

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Author: Charles S. Chihara

Publisher: Oxford University Press

ISBN: 0198248172

Category: Law

Page: 282

View: 2386

This book is concerned with `the problem of existence in mathematics'. It develops a mathematical system in which there are no existence assertions but only assertions of the constructibility of certain sorts of things. It explores the philosophical implications of such an approach in an examination of the writings of Field, Burgess, Maddy, Kitcher, and others.

Neo-Aristotelian Perspectives in Metaphysics

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Author: Daniel D. Novotný,Lukáš Novák

Publisher: Routledge

ISBN: 1134630166

Category: Philosophy

Page: 342

View: 4931

This volume re-examines some of the major themes at the intersection of traditional and contemporary metaphysics. The book uses as a point of departure Francisco Suárez’s Metaphysical Disputations published in 1597. Minimalist metaphysics in empiricist/pragmatist clothing have today become mainstream in analytic philosophy. Independently of this development, the progress of scholarship in ancient and medieval philosophy makes clear that traditional forms of metaphysics have affinities with some of the streams in contemporary analytic metaphysics. The book brings together leading contemporary metaphysicians to investigate the viability of a neo-Aristotelian metaphysics.

Philosophy of Mathematics

Structure and Ontology

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Author: Stewart Shapiro

Publisher: Oxford University Press

ISBN: 9780198025450

Category: Philosophy

Page: 296

View: 9281

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.

The Mathematics of Syntactic Structure

Trees and their Logics

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Author: Hans-Peter Kolb,Uwe Mönnich

Publisher: Walter de Gruyter

ISBN: 3110806789

Category: Language Arts & Disciplines

Page: 347

View: 5845

The architecture of the human language faculty has been one of the main foci of the linguistic research of the last half century. This branch of linguistics, broadly known as Generative Grammar, is concerned with the formulation of explanatory formal accounts of linguistic phenomena with the ulterior goal of gaining insight into the properties of the 'language organ'. The series comprises high quality monographs and collected volumes that address such issues. The topics in this series range from phonology to semantics, from syntax to information structure, from mathematical linguistics to studies of the lexicon.

Structuralism and Structures

A Mathematical Perspective

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Author: Charles Earl Rickart

Publisher: World Scientific

ISBN: 9789810218607

Category: Mathematics

Page: 219

View: 1615

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.