A Structural Account of Mathematics

Author: Charles S. Chihara

Publisher: Clarendon Press

ISBN: 0199267537

Category: Philosophy

Page: 380

View: 7589

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.

Oxford Studies in Metaphysics

Author: Dean Zimmerman

Publisher: OUP Oxford

ISBN: 0191562319

Category: Philosophy

Page: 336

View: 9000

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 A special feature of this volume is an unpublished paper on nominalism by W. V. Quine, arguably the most influential figure in philosophy in the second half of the twentieth century.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?

Author: Ian Hacking

Publisher: Cambridge University Press

ISBN: 1107050170

Category: Mathematics

Page: 308

View: 1163

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

Rigor and Structure

Author: John P. Burgess

Publisher: OUP Oxford

ISBN: 019103360X

Category: Philosophy

Page: 224

View: 5656

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.

Proof and Other Dilemmas

Mathematics and Philosophy

Author: Roger Simons

Publisher: MAA

ISBN: 9780883855676

Category: Mathematics

Page: 346

View: 7859

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.

EPSA Philosophical Issues in the Sciences

Launch of the European Philosophy of Science Association

Author: Mauricio Suárez,Mauro Dorato,Miklós Rédei

Publisher: Springer Science & Business Media

ISBN: 9789048132522

Category: Science

Page: 332

View: 2216

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

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

Publisher: Springer

ISBN: 0230358918

Category: Philosophy

Page: 300

View: 8664

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

Author: Emily R. Grosholz

Publisher: Clarendon Press

ISBN: 0191538515

Category: Mathematics

Page: 330

View: 5939

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.

New Directions for Situated Cognition in Mathematics Education

Author: Anne Watson,Peter Winbourne

Publisher: Springer Science & Business Media

ISBN: 0387715797

Category: Education

Page: 360

View: 9524

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.

Modern Algebra and the Rise of Mathematical Structures

Author: Leo Corry

Publisher: Springer Science & Business Media

ISBN: 9783764370022

Category: Mathematics

Page: 452

View: 7786

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.

Neo-Aristotelian Perspectives in Metaphysics

Author: Daniel D. Novotný,Lukáš Novák

Publisher: Routledge

ISBN: 1134630166

Category: Philosophy

Page: 342

View: 8348

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

Author: Stewart Shapiro

Publisher: Oxford University Press

ISBN: 9780198025450

Category: Philosophy

Page: 296

View: 7151

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.

Laruelle and Non-Philosophy

Author: John Mullarkey

Publisher: Edinburgh University Press

ISBN: 0748664769

Category: Philosophy

Page: 272

View: 5733

The first collection of critical essays on the work of Francois Laruelle.

Structural/Process Models of Complex Human Behavior

Author: M. N. Scandura,Nato Advanced Study Institute on Structural/Process Theories of Complex Human Behavior (1977 : Banff, Alta.)

Publisher: Springer Science & Business Media

ISBN: 9789028605787

Category: Psychology

Page: 612

View: 6922

An Architectonic for Science

The Structuralist Program

Author: W. Balzer,C.U. Moulines,J.D. Sneed

Publisher: Springer Science & Business Media

ISBN: 9400937652

Category: Science

Page: 440

View: 3616

This book has grown out of eight years of close collaboration among its authors. From the very beginning we decided that its content should come out as the result of a truly common effort. That is, we did not "distribute" parts of the text planned to each one of us. On the contrary, we made a point that each single paragraph be the product of a common reflection. Genuine team-work is not as usual in philosophy as it is in other academic disciplines. We think, however, that this is more due to the idiosyncrasy of philosophers than to the nature of their subject. Close collaboration with positive results is as rewarding as anything can be, but it may also prove to be quite difficult to implement. In our case, part of the difficulties came from purely geographic separation. This caused unsuspected delays in coordinating the work. But more than this, as time passed, the accumulation of particular results and ideas outran our ability to fit them into an organic unity. Different styles of exposition, different ways of formalization, different levels of complexity were simultaneously present in a voluminous manuscript that had become completely unmanageable. In particular, a portion of the text had been conceived in the language of category theory and employed ideas of a rather abstract nature, while another part was expounded in the more conventional set-theoretic style, stressing intui tivity and concreteness.

Mathematics Education and Language

Interpreting Hermeneutics and Post-Structuralism

Author: Tony Brown

Publisher: Springer Science & Business Media

ISBN: 9401007268

Category: Education

Page: 306

View: 6240

Contemporary thinking on philosophy and the social sciences has primarily focused on the centrality of language in understanding societies and individuals; important developments which have been under-utilised by researchers in mathematics education. In this revised and extended edition this book reaches out to contemporary work in these broader fields, adding new material on how progression in mathematical learning might be variously understood. A new concluding chapter considers how teachers experience the new demands they face.

The Many Sides of Logic

Author: Walter Carnielli,Marcelo E. Coniglio,Itala M. L. D'Ottaviano

Publisher: N.A

ISBN: N.A

Category: Computers

Page: 586

View: 661

The Many Sides of Logic'' is a volume containing a selection of the papers delivered at three simultaneous events held between 11-17 May 2008 in Paraty, RJ, Brazil, continuing a tradition of three decades of Brazilian and Latin-American meetings and celebrating the 30th anniversary of an institution congenital with the mature interest for logic, epistemology and history of sciences in Brazil: CLE 30 - 30th Anniversary of the Centre for Logic, Epistemology and the History of Science at the State University of Campinas (UNICAMP) XV EBL -15th Brazilian Logic Conference XIV SLALM - 14th Latin-American Symposium on Mathematical Logic Several renowned logicians, philosophers and mathematicians gathered in colonial Paraty, a historic village on the Brazilian coast founded in the 17th Century and surrounded by the luscious Atlantic rain forest to deliver lectures and talks celebrating the many sides of logic: the philosophical, the mathematical, the computational, the historical, and the multiple facets therein. The topics of the joint conferences, well represented here, included philosophical and mathematical Logic and applications with emphasis on model theory and proof theory, set theory, non-classical logics and applications, history and philosophy of logic, philosophy of the formal sciences and issues on the foundations of mathematics. The events have been preceded by a Logic School planned for students and young researchers held at the UNICAMP campus in Campinas, SP.

Contemporary Theories of Knowledge

Author: John L. Pollock,Joseph Cruz

Publisher: Rowman & Littlefield Publishers

ISBN: 0742575144

Category: Science

Page: 288

View: 6799

This new edition of the classic Contemporary Theories of Knowledge has been significantly updated to include analyses of the recent literature in epistemology. Rather than merely making slight amendments to the first edition, Pollock and Cruz have undertaken a groundbreaking assessment of twentieth-century epistemology. This book is both an advanced textbook offering traditional discussions of foundationalism, coherentism, and reliabilism and a detailed treatise on the authors' own distinctive view, direct realism.