A Structural Account of Mathematics

Author: Charles S. Chihara

Publisher: Clarendon Press

ISBN: 0199267537

Category: Philosophy

Page: 380

View: 3193

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

Author: Roger Simons

Publisher: MAA

ISBN: 9780883855676

Category: Mathematics

Page: 346

View: 2689

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.

From Logic to Practice

Italian Studies in the Philosophy of Mathematics

Author: Gabriele Lolli,Marco Panza,Giorgio Venturi

Publisher: Springer

ISBN: 3319104349

Category: Philosophy

Page: 336

View: 4871

This book brings together young researchers from a variety of fields within mathematics, philosophy and logic. It discusses questions that arise in their work, as well as themes and reactions that appear to be similar in different contexts. The book shows that a fairly intensive activity in the philosophy of mathematics is underway, due on the one hand to the disillusionment with respect to traditional answers, on the other to exciting new features of present day mathematics. The book explains how the problem of applicability once again plays a central role in the development of mathematics. It examines how new languages different from the logical ones (mostly figural), are recognized as valid and experimented with and how unifying concepts (structure, category, set) are in competition for those who look at this form of unification. It further shows that traditional philosophies, such as constructivism, while still lively, are no longer only philosophies, but guidelines for research. Finally, the book demonstrates that the search for and validation of new axioms is analyzed with a blend of mathematical historical, philosophical, psychological considerations.

Oxford Studies in Metaphysics

Author: Dean Zimmerman

Publisher: OUP Oxford

ISBN: 0191562319

Category: Philosophy

Page: 336

View: 4166

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.

Rigor and Structure

Author: John P. Burgess

Publisher: OUP Oxford

ISBN: 019103360X

Category: Philosophy

Page: 224

View: 8468

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.

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: 8774

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.

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: 7031

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: 4233

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.

The Age of Alternative Logics

Assessing Philosophy of Logic and Mathematics Today

Author: Johan van Benthem,Gerhard Heinzmann,Manuel Rebuschi,Henk Visser

Publisher: Springer Science & Business Media

ISBN: 1402050127

Category: Philosophy

Page: 348

View: 6063

In the last century, developments in mathematics, philosophy, physics, computer science, economics and linguistics have proven important for the development of logic. There has been an influx of new ideas, concerns, and logical systems reflecting a great variety of reasoning tasks in the sciences. This book embodies the multi-dimensional interplay between logic and science, presenting contributions from the world's leading scholars on new trends and possible developments for research.

Neo-Aristotelian Perspectives in Metaphysics

Author: Daniel D. Novotný,Lukáš Novák

Publisher: Routledge

ISBN: 1134630166

Category: Philosophy

Page: 342

View: 420

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: 340

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: 6356

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

The Many Sides of Logic

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

Publisher: N.A


Category: Computers

Page: 586

View: 5449

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.

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: 1337

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: 4832

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: 6032

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.

Contemporary Theories of Knowledge

Author: John L. Pollock,Joseph Cruz

Publisher: Rowman & Littlefield Publishers

ISBN: 0742575144

Category: Science

Page: 288

View: 766

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.

Formen der Anschauung

eine Philosophie der Mathematik

Author: Pirmin Stekeler-Weithofer

Publisher: Walter de Gruyter


Category: Philosophy

Page: 400

View: 7163

What are pure geometric forms? In what sense are there an infinite number of points on a line? What is the relationship between empirically correct statements about real bodily figures (or movements)and the ideal truths of a pure mathematical geometry (also in space-time)? Starting from Kant and Wittgenstein, the book demonstrates how our dealings with figures and symbols is to be understood beyond the technical mastery of forms of calculation and proof.