The book poses basic questions, including: What are our aims of teaching and learning mathematics? What is mathematics anyway? How is mathematics related to society in the 21st century? How do students learn mathematics?

Author: Paul Ernest

Publisher: Springer

ISBN: 9783319777603

Category: Education

Page: 377

View: 647

This book offers an up-to-date overview of the research on philosophy of mathematics education, one of the most important and relevant areas of theory. The contributions analyse, question, challenge, and critique the claims of mathematics education practice, policy, theory and research, offering ways forward for new and better solutions. The book poses basic questions, including: What are our aims of teaching and learning mathematics? What is mathematics anyway? How is mathematics related to society in the 21st century? How do students learn mathematics? What have we learnt about mathematics teaching? Applied philosophy can help to answer these and other fundamental questions, and only through an in-depth analysis can the practice of the teaching and learning of mathematics be improved. The book addresses important themes, such as critical mathematics education, the traditional role of mathematics in schools during the current unprecedented political, social, and environmental crises, and the way in which the teaching and learning of mathematics can better serve social justice and make the world a better place for the future.

Representing the state of the art in the field of the philosophy of mathematics, this collection of 20 essays deals with fundamental issues, ranging from the nature of mathematical knowledge to sets and natural 'number'.

Author: Matthias Schirn

Publisher: Oxford University Press

ISBN: 0199262624

Category: Philosophy

Page: 638

View: 434

Representing the state of the art in the field of the philosophy of mathematics, this collection of 20 essays deals with fundamental issues, ranging from the nature of mathematical knowledge to sets and natural 'number'.

This is the Hermeneutic strand of research in the philosophy of mathematics education in Brazil. This illustrates one orientation towards research inquiry in the philosophy of mathematics education.

Author: Paul Ernest

Publisher: Springer

ISBN: 3319405683

Category: Education

Page: 26

View: 695

This survey provides a brief and selective overview of research in the philosophy of mathematics education. It asks what makes up the philosophy of mathematics education, what it means, what questions it asks and answers, and what is its overall importance and use? It provides overviews of critical mathematics education, and the most relevant modern movements in the philosophy of mathematics. A case study is provided of an emerging research tradition in one country. This is the Hermeneutic strand of research in the philosophy of mathematics education in Brazil. This illustrates one orientation towards research inquiry in the philosophy of mathematics education. It is part of a broader practice of ‘philosophical archaeology’: the uncovering of hidden assumptions and buried ideologies within the concepts and methods of research and practice in mathematics education. An extensive bibliography is also included.

What is needed now is a new beginning , not a continuation of the various " schools ” of logicism , formalism or intuitionism . [ . . . ] Many of the difficulties and stumbling blocks in the philosophy of mathematics are created by ...

Author: E. Agazzi

Publisher: Springer Science & Business Media

ISBN: 9789401156905

Category: Mathematics

Page: 361

View: 550

Mathematics is often considered as a body of knowledge that is essen tially independent of linguistic formulations, in the sense that, once the content of this knowledge has been grasped, there remains only the problem of professional ability, that of clearly formulating and correctly proving it. However, the question is not so simple, and P. Weingartner's paper (Language and Coding-Dependency of Results in Logic and Mathe matics) deals with some results in logic and mathematics which reveal that certain notions are in general not invariant with respect to different choices of language and of coding processes. Five example are given: 1) The validity of axioms and rules of classical propositional logic depend on the interpretation of sentential variables; 2) The language dependency of verisimilitude; 3) The proof of the weak and strong anti inductivist theorems in Popper's theory of inductive support is not invariant with respect to limitative criteria put on classical logic; 4) The language-dependency of the concept of provability; 5) The language dependency of the existence of ungrounded and paradoxical sentences (in the sense of Kripke). The requirements of logical rigour and consistency are not the only criteria for the acceptance and appreciation of mathematical proposi tions and theories.

The text describes the questions about mathematics that motivated philosophers throughout history and covers historical figures such as Plato, Aristotle, Kant, and Mill.

Author: Stewart Shapiro

Publisher: OUP Oxford

ISBN: 9780192893062

Category: Philosophy

Page: 328

View: 205

Thinking about Mathematics covers the range of philosophical issues and positions concerning mathematics. The text describes the questions about mathematics that motivated philosophers throughout history and covers historical figures such as Plato, Aristotle, Kant, and Mill. It also presents the major positions and arguments concerning mathematics throughout the twentieth century, bringing the reader up to the present positions and battle lines.

The present book is an introduction to the philosophy of mathematics. It asks philosophical questions concerning fundamental concepts, constructions and methods - this is done from the standpoint of mathematical research and teaching.

Author: Thomas Bedürftig

Publisher: Walter de Gruyter GmbH & Co KG

ISBN: 9783110470772

Category: Mathematics

Page: 474

View: 357

The present book is an introduction to the philosophy of mathematics. It asks philosophical questions concerning fundamental concepts, constructions and methods - this is done from the standpoint of mathematical research and teaching. It looks for answers both in mathematics and in the philosophy of mathematics from their beginnings till today. The reference point of the considerations is the introducing of the reals in the 19th century that marked an epochal turn in the foundations of mathematics. In the book problems connected with the concept of a number, with the infinity, the continuum and the infinitely small, with the applicability of mathematics as well as with sets, logic, provability and truth and with the axiomatic approach to mathematics are considered. In Chapter 6 the meaning of infinitesimals to mathematics and to the elements of analysis is presented. The authors of the present book are mathematicians. Their aim is to introduce mathematicians and teachers of mathematics as well as students into the philosophy of mathematics. The book is suitable also for professional philosophers as well as for students of philosophy, just because it approaches philosophy from the side of mathematics. The knowledge of mathematics needed to understand the text is elementary. Reports on historical conceptions. Thinking about today‘s mathematical doing and thinking. Recent developments. Based on the third, revised German edition. For mathematicians - students, teachers, researchers and lecturers - and readersinterested in mathematics and philosophy. Contents On the way to the reals On the history of the philosophy of mathematics On fundamental questions of the philosophy of mathematics Sets and set theories Axiomatic approach and logic Thinking and calculating infinitesimally – First nonstandard steps Retrospection

Many practical problems and impasses confronting mathematics today have philosophical aspects . The dearth of well - founded philosophical discourse on mathematics has observable harmful consequences , in teaching , in research , and in ...

Author: Thomas Tymoczko

Publisher: Princeton University Press

ISBN: 0691034982

Category: Mathematics

Page: 436

View: 395

The traditional debate among philosophers of mathematics is whether there is an external mathematical reality, something out there to be discovered, or whether mathematics is the product of the human mind. This provocative book, now available in a revised and expanded paperback edition, goes beyond foundationalist questions to offer what has been called a "postmodern" assessment of the philosophy of mathematics--one that addresses issues of theoretical importance in terms of mathematical experience. By bringing together essays of leading philosophers, mathematicians, logicians, and computer scientists, Thomas Tymoczko reveals an evolving effort to account for the nature of mathematics in relation to other human activities. These accounts include such topics as the history of mathematics as a field of study, predictions about how computers will influence the future organization of mathematics, and what processes a proof undergoes before it reaches publishable form. This expanded edition now contains essays by Penelope Maddy, Michael D. Resnik, and William P. Thurston that address the nature of mathematical proofs. The editor has provided a new afterword and a supplemental bibliography of recent work.

In this chapter, I won't try to defend a particular philosophical view about mathematics, but, in the spirit of New Waves, ... during the rise of logic over 100 years ago, and they are ubiquitous in the philosophy of mathematics today ...

Author: O. Bueno

Publisher: Springer

ISBN: 9780230245198

Category: Philosophy

Page: 327

View: 104

Thirteen promising young researchers write on what they take to be the right philosophical account of mathematics and discuss where the philosophy of mathematics ought to be going. New trends are revealed, such as an increasing attention to mathematical practice, a reassessment of the canon, and inspiration from philosophical logic.

Direct proofs often carry information about how to construct the mathematical objects whose existence is being ... known as structuralism , in one form perhaps the most widely held philosophical position amongst mathematicians today .

Author: Joel David Hamkins

Publisher: MIT Press

ISBN: 9780262542234

Category: Mathematics

Page: 352

View: 345

An introduction to the philosophy of mathematics grounded in mathematics and motivated by mathematical inquiry and practice. In this book, Joel David Hamkins offers an introduction to the philosophy of mathematics that is grounded in mathematics and motivated by mathematical inquiry and practice. He treats philosophical issues as they arise organically in mathematics, discussing such topics as platonism, realism, logicism, structuralism, formalism, infinity, and intuitionism in mathematical contexts. He organizes the book by mathematical themes--numbers, rigor, geometry, proof, computability, incompleteness, and set theory--that give rise again and again to philosophical considerations.

Now apply Exercise 6.74. References 1 Chapter 7 Intuitionist Logic 7.1 Inference We are going to. Boolos, G. S., & Heck, R. G. (1998). Die Grundlagen der Arithmetik SS82-83. In M. Schirn (Ed.), Philosophy of mathematics today (pp.

Author: Stephen Pollard

Publisher: Springer

ISBN: 9783319058160

Category: Science

Page: 202

View: 930

This book is based on two premises: one cannot understand philosophy of mathematics without understanding mathematics and one cannot understand mathematics without doing mathematics. It draws readers into philosophy of mathematics by having them do mathematics. It offers 298 exercises, covering philosophically important material, presented in a philosophically informed way. The exercises give readers opportunities to recreate some mathematics that will illuminate important readings in philosophy of mathematics. Topics include primitive recursive arithmetic, Peano arithmetic, Gödel's theorems, interpretability, the hierarchy of sets, Frege arithmetic and intuitionist sentential logic. The book is intended for readers who understand basic properties of the natural and real numbers and have some background in formal logic.

History and Philosophy of Logic, 17, 209–220. Heck, R. (1998). The finite and the infinite in Frege's Grundgesetze der Arithmetik. In M. Schirn (Ed.), The Philosophy of Mathematics Today (pp. 429–466). Oxford: Clarendon Press.

Author: Hassan Tahiri

Publisher: Springer

ISBN: 9783319937335

Category: Mathematics

Page: 320

View: 679

This book explores the unique relationship between two different approaches to understand the nature of knowledge, reality, and existence. It collects essays that examine the distinctive historical relationship between mathematics and philosophy. Readers learn what key philosophers throughout the ages thought about mathematics. This includes both thinkers who recognized the relevance of mathematics to their own work as well as those who chose to completely ignore its many achievements. The essays offer insight into the role that mathematics played in the formation of each included philosopher’s doctrine as well as the impact its remarkable expansion had on the philosophical systems each erected. Conversely, the authors also highlight the ways that philosophy contributed to the growth and transformation of mathematics. Throughout, significant historical examples help to illustrate these points in a vivid way. Mathematics has often been a favored interlocutor of philosophers and a major source of inspiration. This book is the outcome of an international conference held in honor of Roshdi Rashed, a renowned historian of mathematics. It provides researchers, students, and interested readers with remarkable insights into the history of an important relationship throughout the ages.

“The Turning Point and the Revolution: Philosophy of Mathematics in Logical Empiricism from Tractatus to Logical ... “What Mathematical Truth Could Not Be—I.” In The Philosophy of Mathematics Today, edited by Matthias Schirn, 33–76.

Author: Russell Marcus

Publisher: Bloomsbury Publishing

ISBN: 9781472532916

Category: Philosophy

Page: 896

View: 770

A comprehensive collection of historical readings in the philosophy of mathematics and a selection of influential contemporary work, this much-needed introduction reveals the rich history of the subject. An Historical Introduction to the Philosophy of Mathematics: A Reader brings together an impressive collection of primary sources from ancient and modern philosophy. Arranged chronologically and featuring introductory overviews explaining technical terms, this accessible reader is easy-to-follow and unrivaled in its historical scope. With selections from key thinkers such as Plato, Aristotle, Descartes, Hume and Kant, it connects the major ideas of the ancients with contemporary thinkers. A selection of recent texts from philosophers including Quine, Putnam, Field and Maddy offering insights into the current state of the discipline clearly illustrates the development of the subject. Presenting historical background essential to understanding contemporary trends and a survey of recent work, An Historical Introduction to the Philosophy of Mathematics: A Reader is required reading for undergraduates and graduate students studying the philosophy of mathematics and an invaluable source book for working researchers.

'Wittgenstein on Some Questions in Foundations of Mathematics', Journal of Philosophy, vol. ... Philosophy and Language, A. Ambrose and M. Iazerowitz (eds) (London, George Allen & Unwin, 1972). ... Mathematics Today.

Author: S.G. Shanker

Publisher: Routledge

ISBN: 9781317832041

Category: Philosophy

Page: 376

View: 188

First published in 2005. Routledge is an imprint of Taylor & Francis, an informa company.

This volume covers these disciplines in a comprehensive and accessible manner, giving the reader an overview of the major problems, positions, and battle lines.

Author: Stewart Shapiro

Publisher: OUP USA

ISBN: 0195325923

Category: Mathematics

Page: 850

View: 990

Mathematics and logic have been central topics of concern since the dawn of philosophy. Since logic is the study of correct reasoning, it is a fundamental branch of epistemology and a priority in any philosophical system. Philosophers have focused on mathematics as a case study for general philosophical issues and for its role in overall knowledge- gathering. Today, philosophy of mathematics and logic remain central disciplines in contemporary philosophy, as evidenced by the regular appearance of articles on these topics in the best mainstream philosophical journals; in fact, the last decade has seen an explosion of scholarly work in these areas. This volume covers these disciplines in a comprehensive and accessible manner, giving the reader an overview of the major problems, positions, and battle lines. The 26 contributed chapters are by established experts in the field, and their articles contain both exposition and criticism as well as substantial development of their own positions. The essays, which are substantially self-contained, serve both to introduce the reader to the subject and to engage in it at its frontiers. Certain major positions are represented by two chapters--one supportive and one critical. The Oxford Handbook of Philosophy of Math and Logic is a ground-breaking reference like no other in its field. It is a central resource to those wishing to learn about the philosophy of mathematics and the philosophy of logic, or some aspect thereof, and to those who actively engage in the discipline, from advanced undergraduates to professional philosophers, mathematicians, and historians.

''On the Philosophical Relations Between Ramsey and Wittgenstein,'' in Hintikka and Puhl, eds., pp. 150–162. Schirn, M., ed. 1998. The Philosophy of Mathematics Today, Oxford, Clarendon Press. Shanker, S.G. 1987.

Author: Stewart Shapiro

Publisher: OUP USA

ISBN: 9780195148770

Category: Mathematics

Page: 833

View: 192

Covers the state of the art in the philosophy of maths and logic, giving the reader an overview of the major problems, positions, and battle lines. The chapters in this book contain both exposition and criticism as well as substantial development of their own positions. It also includes a bibliography.

Axiomathes 1–3, 125–142 (1997) G.E. Rosado Haddock, Husserl's Philosophy of Mathematics: Its Origin and Relevance, Husserl Stud. 22, (2006). Reprinted as Chapter 5 of Against the Current, pp. 145–181 G.E. Rosado Haddock, ...

Author: Stefania Centrone

Publisher: Springer

ISBN: 9789402411324

Category: Philosophy

Page: 526

View: 233

Essays on Husserl’s Logic and Philosophy of Mathematics sets out to fill up a lacuna in the present research on Husserl by presenting a precise account of Husserl’s work in the field of logic, of the philosophy of logic and of the philosophy of mathematics. The aim is to provide an in-depth reconstruction and analysis of the discussion between Husserl and his most important interlocutors, and to clarify pivotal ideas of Husserl’s by considering their reception and elaboration by some of his disciples and followers, such as Oskar Becker and Jacob Klein, as well as their influence on some of the most significant logicians and mathematicians of the past century, such as Luitzen E. J. Brouwer, Rudolf Carnap, Kurt Gödel and Hermann Weyl. Most of the papers consider Husserl and another scholar – e.g. Leibniz, Kant, Bolzano, Brentano, Cantor, Frege – and trace out and contextualize lines of influence, points of contact, and points of disagreement. Each essay is written by an expert of the field, and the volume includes contributions both from the analytical tradition and from the phenomenological one.

Author: Oxford University PressPublish On: 2010-06-01

An up-to-date survey of the main forms of mathematical Platonism as well as the central arguments for this conception and the main objections that have been raised against it. ▻Schirn, Matthias, ed. The Philosophy of Mathematics Today.

Author: Oxford University Press

Publisher: Oxford University Press, USA

ISBN: 0199808937

Category: Philosophy

Page: 26

View: 339

This ebook is a selective guide designed to help scholars and students of social work find reliable sources of information by directing them to the best available scholarly materials in whatever form or format they appear from books, chapters, and journal articles to online archives, electronic data sets, and blogs. Written by a leading international authority on the subject, the ebook provides bibliographic information supported by direct recommendations about which sources to consult and editorial commentary to make it clear how the cited sources are interrelated related. This ebook is a static version of an article from Oxford Bibliographies Online: Philosophy, a dynamic, continuously updated, online resource designed to provide authoritative guidance through scholarship and other materials relevant to the study Philosophy. Oxford Bibliographies Online covers most subject disciplines within the social science and humanities, for more information visit www.oxfordbibligraphies.com.

Janiszewski's programme 'generated' the set-theoretic foundations of mathematics as a non-philosophical but ... Today even a lecture on the foundations of higher mathematics cannot omit certain information from set theory (1915 ...

Author: Roman Murawski

Publisher: Springer

ISBN: 9783034808316

Category: Mathematics

Page: 228

View: 477

The aim of this book is to present and analyze philosophical conceptions concerning mathematics and logic as formulated by Polish logicians, mathematicians and philosophers in the 1920s and 1930s. It was a remarkable period in the history of Polish science, in particular in the history of Polish logic and mathematics. Therefore, it is justified to ask whether and to what extent the development of logic and mathematics was accompanied by a philosophical reflection. We try to answer those questions by analyzing both works of Polish logicians and mathematicians who have a philosophical temperament as well as their research practice. Works and philosophical views of the following Polish scientists will be analyzed: Wacław Sierpiński, Zygmunt Janiszewski, Stefan Mazurkiewicz, Stefan Banach Hugo Steinhaus, Eustachy Żylińsk and Leon Chwistek, Jan Łukasiewicz, Zygmunt Zawirski, Stanisław Leśniewski, Tadeusz Kotarbiński, Kazimierz Ajdukiewicz, Alfred Tarski, Andrzej Mostowski and Henryk Mehlberg, Jan Sleszyński, Stanisław Zaremba and Witold Wilkosz. To indicate the background of scientists being active in the 1920s and 1930s we consider in Chapter 1 some predecessors, in particular: Jan Śniadecki, Józef Maria Hoene-Wroński, Samuel Dickstein and Edward Stamm.

STANFIELD-POTWOROWSKI, J. (1988) 'Socialising Mathematics', Mathematics Teaching, 125, pp. 3–8. STAPLEDON, O. (1937) Star Maker, Harmondsworth, Penguin Books. STEEN, L.A. (ed.) (1978) Mathematics Today, New York, Springer Verlag.

Author: Paul Ernest

Publisher: Routledge

ISBN: 9781135387532

Category: Education

Page: 344

View: 956

Although many agree that all teaching rests on a theory of knowledge, there has been no in-depth exploration of the implications of the philosophy of mathematics for education. This is Paul Ernest's aim. Building on the work of Lakatos and Wittgenstein it challenges the prevalent notion that mathematical knowledge is certain, absolute and neutral, and offers instead an account of mathematics as a social construction. This has profound educational implications for social issues, including gender, race and multiculturalism; for pedagogy, including investigations and problem solving; and challenges hierarchical views of mathematics, learning and ability. Beyond this, the book offers a well-grounded model of five educational ideologies, each with its own epistemology, values, aims and social group of adherents. An analysis of the impact of these groups on the National Curriculum results in a powerful critique, revealing the questionable assumptions, values and interests upon which it rests. The book finishes on an optimistic note, arguing that pedagogy, left unspecified by the National Curriculum, is the way to achieve the radical aims of educating confident problem posers and solvers who are able to critically evaluate the social uses of mathematics.

There has been an influx of new ideas, concerns, and logical systems reflecting a great variety of reasoning tasks in the sciences. This volume reflects the multi-dimensional nature of the interplay between logic and science.

Author: Johan van Benthem

Publisher: Springer Science & Business Media

ISBN: 9781402050121

Category: Philosophy

Page: 348

View: 616

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.