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 is one of the most successful human endeavors—a paradigm of precision and objectivity. It is also one of our most puzzling endeavors, as it seems to deliver non-experiential knowledge of a non-physical reality consisting of numbers, sets, and functions. How can the success and objectivity of mathematics be reconciled with its puzzling features, which seem to set it apart from all the usual empirical sciences? This book offers a short but systematic introduction to the philosophy of (...) mathematics. Readers are introduced to all of the classical approaches to the field, including logicism, formalism, intuitionism, empiricism, and structuralism. The book also contains accessible introductions to some more specialized issues, such as mathematical intuition, potential infinity, the iterative conception of sets, and the search for new mathematical axioms. The exposition is always closely informed by ongoing research in the field and sometimes draws on the author’s own contributions to this research. This means that Gottlob Frege—a German mathematician and philosopher widely recognized as one of the founders of analytic philosophy—figures prominently in the book, both through his own views and his criticism of other thinkers. (shrink)

Why study notations, diagrams, or more broadly the variety of nonverbal “representations” or “signs” that are used in mathematical practice? This chapter maps out recent work on the topic by distinguishing three main philosophical motivations for doing so. First, some work (like that on diagrammatic reasoning) studies signs to recover norms of informal or historical mathematical practices that would get lost if the particular signs that these practices rely on were translated away; work in this vein has the potential to (...) broaden our view of the specific normativity of mathematics beyond logical proof. Second, some work focuses on signs to highlight the open-ended and “nonrigorous” aspects of mathematical practice, and the role of these aspects in the historical development of mathematics. The third motivation is based on the following observation: the reason differences in signs matter in mathematics is that humans are finite agents, with cognitive and computational limitations. Accordingly, studying signs can be a way to study how human computational constraints shape the mathematics that we do, and to tackle the topic of mathematical understanding. (shrink)

The present collection brings together in a convenient form the seminal articles in the philosophy of mathematics by these and other major thinkers.

In recent years, philosophical work directly concerned with the practice of mathematics has intensified, giving rise to a movement known as the philosophy of mathematical practice . In this paper we offer a survey of this movement aimed at mathematics educators. We first describe the core questions philosophers of mathematical practice investigate as well as the philosophical methods they use to tackle them. We then provide a selective overview of work in the philosophy of mathematical practice (...) covering topics including the distinction between formal and informal proofs, visualization and artefacts, mathematical explanation and understanding, value judgments, and mathematical design. We conclude with some remarks on the potential connections between the philosophy of mathematical practice and mathematics education. (shrink)

_Philosophy of Mathematics_ is an excellent introductory text. This student friendly book discusses the great philosophers and the importance of mathematics to their thought. It includes the following topics: * the mathematical image * platonism * picture-proofs * applied mathematics * Hilbert and Godel * knots and nations * definitions * picture-proofs and Wittgenstein * computation, proof and conjecture. The book is ideal for courses on philosophy of mathematics and logic.

A survey of Euclid's Elements, this text provides an understanding of the classical Greek conception of mathematics and its similarities to modern views as well as its differences. It focuses on philosophical, foundational, and logical questions — rather than strictly historical and mathematical issues — and features several helpful appendixes.

There is an urgent need in philosophy of mathematics for new approaches which pay closer attention to mathematical practice. This book will blaze the trail: it offers philosophical analyses of important characteristics of contemporary mathematics and of many aspects of mathematical activity which escape purely formal logical treatment.

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. (shrink)

A number of examples of studies from the field ‘The Philosophy of Mathematical Practice’ (PMP) are given. To characterise this new field, three different strands are identified: an agent-based, a historical, and an epistemological PMP. These differ in how they understand ‘practice’ and which assumptions lie at the core of their investigations. In the last part a general framework, capturing some overall structure of the field, is proposed.

In his long-awaited new edition of Philosophy of Mathematics, James Robert Brown tackles important new as well as enduring questions in the mathematical sciences. Can pictures go beyond being merely suggestive and actually prove anything? Are mathematical results certain? Are experiments of any real value?" "This clear and engaging book takes a unique approach, encompassing nonstandard topics such as the role of visual reasoning, the importance of notation, and the place of computers in mathematics, as well as (...) traditional topics such as formalism, Platonism, and constructivism. The combination of topics and clarity of presentation make it suitable for beginners and experts alike. The revised and updated second edition of Philosophy of Mathematics contains more examples, suggestions for further reading, and expanded material on several topics including a novel approach to the continuum hypothesis."--BOOK JACKET. (shrink)

Without attempting to cover all the philosophical questions posed by modern mathematics, provides a glimpse of a broad vision of the subject. Covering general philosophical perspectives, foundational approaches, the applicability of mathematics, and history, treats selected topics from a variety of perspectives to demonstrate the range of practices in the discipline. Among them are moderate mathematical fictionism, categorical foundations of the protean character of mathematics, the mathematical overdetermination of physics, and Hungarian traditions and the philosophy of (...) non-classical logic. The 21 papers were presented at a meeting of the International Academy of Philosophy of Science, held in Budapest in May 1993. No index. Annotation copyrighted by Book News, Inc., Portland, OR. (shrink)

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. (shrink)

Philosophy of mathematics is moving in a new direction: away from a foundationalism in terms of formal logic and traditional ontology, and towards a broader range of approaches that are united by a focus on mathematical practice. The scientific research network PhiMSAMP (Philosophy of Mathematics: Sociological Aspects and Mathematical Practice) consisted of researchers from a variety of backgrounds and fields, brought together by their common interest in the shift of philosophy of mathematics towards mathematical (...) practice. Hosted by the Rheinische Friedrich- Wilhelms-Universitat Bonn and funded by the Deutsche Forschungsgemeinschaft (DFG) from 2006-2010, the network organized and contributed to a number of workshops and conferences on the topic of mathematical practice. The refereed contributions in this volume represent the research results of the network and consists of contributions of the network members as well as selected paper versions of presentations at the network's mid-term conference, "Is mathematics special?" (PhiMSAMP-3) held in Vienna 2008. (shrink)

An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, (...) are parts of the physical world and are objects of mathematics. Though some mathematical structures such as infinities may be too big to be realized in fact, all of them are capable of being realized. Informed by the author's background in both philosophy and mathematics, but keeping to simple examples, the book shows how infant perception of patterns is extended by visualization and proof to the vast edifice of modern pure and applied mathematical knowledge. (shrink)

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. (shrink)

This volume offers a selection of the most interesting and important work from recent years in the philosophy of mathematics, which has always been closely linked to, and has exerted a significant influence upon, the main stream of analytical philosophy. The issues discussed are of interest throughout philosophy, and no mathematical expertise is required of the reader. Contributors include W.V. Quine, W.D. Hart, Michael Dummett, Charles Parsons, Paul Benacerraf, Penelope Maddy, W.W. Tait, Hilary Putnam, George Boolos, (...) Daniel Isaacson, Stewart Shapiro, and Hartry Field. (shrink)

In this book, which is both a philosophical and historiographical study, the author investigates the fallibility and the rationality of mathematics by means of rational reconstructions of developments in mathematics. The initial chapters are devoted to a critical discussion of Lakatos' philosophy of mathematics. In the remaining chapters several episodes in the history of mathematics are discussed, such as the appearance of deduction in Greek mathematics and the transition from Eighteenth-Century to Nineteenth-Century analysis. The (...) author aims at developing a notion of mathematical rationality that agrees with the historical facts. A modified version of Lakatos' methodology is proposed. The resulting constructions show that mathematical knowledge is fallible, but that its fallibility is remarkably weak. (shrink)

Moving beyond both realist and anti-realist accounts of mathematics, Shapiro articulates a "structuralist" approach, arguing that the subject matter of a mathematical theory is not a fixed domain of numbers that exist independent of each other, but rather is the natural structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle.

The twentieth century has witnessed an unprecedented 'crisis in the foundations of mathematics', featuring a world-famous paradox (Russell's Paradox), a challenge to 'classical' mathematics from a world-famous mathematician (the 'mathematical intuitionism' of Brouwer), a new foundational school (Hilbert's Formalism), and the profound incompleteness results of Kurt Gödel. In the same period, the cross-fertilization of mathematics and philosophy resulted in a new sort of 'mathematical philosophy', associated most notably (but in different ways) with Bertrand Russell, W. (...) V. Quine, and Gödel himself, and which remains at the focus of Anglo-Saxon philosophical discussion. The present collection brings together in a convenient form the seminal articles in the philosophy of mathematics by these and other major thinkers. It is a substantially revised version of the edition first published in 1964 and includes a revised bibliography. The volume will be welcomed as a major work of reference at this level in the field. (shrink)

Both Carnap and Quine made significant contributions to the philosophy of mathematics despite their diversedviews. Carnap endorsed the dichotomy between analytic and synthetic knowledge and classified certainmathematical questions as internal questions appealing to logic and convention. On the contrary, Quine wasopposed to the analytic-synthetic distinction and promoted a holistic view of scientific inquiry. The purpose of thispaper is to argue that in light of the recent advancement of experimental mathematics such as Monte Carlosimulations, limiting mathematical inquiry to (...) the domain of logic is unjustified. Robustness studies implemented inMonte Carlo Studies demonstrate that mathematics is on par with other experimental-based sciences. (shrink)

Both Carnap and Quine made significant contributions to the philosophy of mathematics despite their diversed views. Carnap endorsed the dichotomy between analytic and synthetic knowledge and classified certain mathematical questions as internal questions appealing to logic and convention. On the contrary, Quine was opposed to the analytic-synthetic distinction and promoted a holistic view of scientific inquiry. The purpose of this paper is to argue that in light of the recent advancement of experimental mathematics such as Monte Carlo (...) simulations, limiting mathematical inquiry to the domain of logic is unjustified. Robustness studies implemented in Monte Carlo Studies demonstrate that mathematics is on par with other experimental-based sciences. (shrink)

The aim of this paper is to present and discuss the philosophical views concerning mathematics of the founders of the so called Warsaw Mathematical School, i.e., Wacław Sierpiński, Zygmunt Janiszewski and Stefan Mazurkiewicz. Their interest in the philosophy of mathematics and their philosophical papers will be considered. We shall try to answer the question whether their philosophical views influenced their proper mathematical investigations. Their views towards set theory and its rôle in mathematics will be emphasized.

This is a book that no one but Weyl could have written--and, indeed, no one has written anything quite like it since.

This lucid and comprehensive essay by a distinguished philosopher surveys the views of Plato, Aristotle, Leibniz, and Kant on the nature of mathematics. It examines the propositions and theories of the schools these philosophers inspired, and it concludes by discussing the relationship between mathematical theories, empirical data, and philosophical presuppositions. 1968 edition.

Finally the book concludes with a discussion of the most recent debates between realists and nominalists.

If mathematics is regarded as a science, then the philosophy of mathematics can be regarded as a branch of the philosophy of science, next to disciplines such as the philosophy of physics and the philosophy of biology. However, because of its subject matter, the philosophy of mathematics occupies a special place in the philosophy of science. Whereas the natural sciences investigate entities that are located in space and time, it is not (...) at all obvious that this is also the case with respect to the objects that are studied in mathematics. In addition to that, the methods of investigation of mathematics differ markedly from the methods of investigation in the natural sciences. Whereas the latter acquire general knowledge using inductive methods, mathematical knowledge appears to be acquired in a different way, namely, by deduction from basic principles. The status of mathematical knowledge also appears to differ from the status of knowledge in the natural sciences. The theories of the natural sciences appear to be less certain and more open to revision than mathematical theories. For these reasons mathematics poses problems of a quite distinctive kind for philosophy. Therefore philosophers have accorded special attention to ontological and epistemological questions concerning mathematics. (shrink)

Moving beyond both realist and anti-realist accounts of mathematics, Shapiro articulates a "structuralist" approach, arguing that the subject matter of a mathematical theory is not a fixed domain of numbers that exist independent of each other, but rather is the natural structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle.

This text is organized around the distinction between finite and infinite. It includes a brief overview of what different philosophers have said about infinity, and looks at some of the arguments to the effect that one should adopt a pro-infinity attitude. Other chapters contain an exposition of the ontological schools; interactions among these schools and various theories of truth; the relationship between mathematics and values; a history of mathematics; an analysis of mathematical knowledge; the role of mathematics (...) in eduction; the implications of religion for the philosophy of mathematics; and referenc eto mathematical objects. (shrink)

This introduction to the philosophy of mathematics focuses on contemporary debates in an important and central area of philosophy. The reader is taken on a fascinating and entertaining journey through some intriguing mathematical and philosophical territory, including such topics as the realism/anti-realism debate in mathematics, mathematical explanation, the limits of mathematics, the significance of mathematical notation, inconsistent mathematics and the applications of mathematics. Each chapter has a number of discussion questions and recommended further (...) reading from both the contemporary literature and older sources. Very little mathematical background is assumed and all of the mathematics encountered is clearly introduced and explained using a wide variety of examples. The book is suitable for an undergraduate course in philosophy of mathematics and, more widely, for anyone interested in philosophy and mathematics. (shrink)

This chapter contains sections titled: Basic Tasks for a Philosophy of Mathematics Basic Responses New Procedures, New Problems: Analytic Geometry and the Infinite Foundations Beyond Foundationalism and Analytic Philosophy.

The twentieth century has witnessed an unprecedented 'crisis in the foundations of mathematics', featuring a world-famous paradox, a challenge to 'classical' mathematics from a world-famous mathematician, a new foundational school, and the profound incompleteness results of Kurt Gödel. In the same period, the cross-fertilization of mathematics and philosophy resulted in a new sort of 'mathematical philosophy', associated most notably with Bertrand Russell, W. V. Quine, and Gödel himself, and which remains at the focus of Anglo-Saxon (...) philosophical discussion. The present collection brings together in a convenient form the seminal articles in the philosophy of mathematics by these and other major thinkers. It is a substantially revised version of the edition first published in 1964 and includes a revised bibliography. The volume will be welcomed as a major work of reference at this level in the field. (shrink)

One of the important discussions in the philosophy of mathematics, is that centered on Benacerraf’s Dilemma. Benacerraf’s dilemma challenges theorists to provide an epistemology and semantics for mathematics, based on their favourite ontology. This challenge is the point on which all philosophies of mathematics are judged, and clarifying how we might acquire mathematical knowledge is one of the main occupations of philosophers of mathematics. In this thesis I argue that this discussion has overlooked an important (...) part of mathematics, namely mathematics as it is exercised by ordinary people. I do so by looking at the different theories that have been put forward in the recent debate, and showing for each of these that they are unable to account for the mathematical practices of ordinary people. In order to show that these practices do need to be accounted for, I also argue that ordinary people are doing mathematics, i.e. that they engage in properly mathematical practices. Because these practices are properly mathematical, they should be accounted for by any philosophy of mathematics. The conclusion of my thesis, then, is that current theories fail to do something that they should do, while remaining neutral on how well they perform when it comes to accounting for the practices of professional mathematicians. (shrink)

Although many agree that all teaching rests on a theory of knowledge, this is an in-depth exploration of the philosophy of mathematics for education, building on the work of Lakatos and Wittgenstein.

The Philosophy of Mathematics Today gives a panorama of the best current work in this lively field, through twenty essays specially written for this collection by leading figures. The topics include indeterminacy, logical consequence, mathematical methodology, abstraction, and both Hilbert's and Frege's foundational programmes. The collection will be an important source for research in the philosophy of mathematics for years to come. Contributors Paul Benacerraf, George Boolos, John P. Burgess, Charles S. Chihara, Michael Detlefsen, Michael Dummett, (...) Hartry Field, Kit Fine, Bob Hale, Richard G. Heck, Jnr., Geoffrey Hellman, Penelope Maddy, Karl-Georg Niebergall, Charles D. Parsons, Michael D. Resnik, Matthias Schirn, Stewart Shapiro, Peter Simons, W.W. Tait, Crispin Wright. (shrink)

The philosophy of mathematics plays an important role in analytic philosophy, both as a subject of inquiry in its own right, and as an important landmark in the broader philosophical landscape. Mathematical knowledge has long been regarded as a paradigm of human knowledge with truths that are both necessary and certain, so giving an account of mathematical knowledge is an important part of epistemology. Mathematical objects like numbers and sets are archetypical examples of abstracta, since we treat (...) such objects in our discourse as though they are independent of time and space; finding a place for such objects in a broader framework of thought is a central task of ontology, or metaphysics. The rigor and precision of mathematical language depends on the fact that it is based on a limited vocabulary and very structured grammar, and semantic accounts of mathematical discourse often serve as a starting point for the philosophy of language. Although mathematical thought has exhibited a strong degree of stability through history, the practice has also evolved over time, and some developments have evoked controversy and debate; clarifying the basic goals of the practice and the methods that are appropriate to it is therefore an important foundational and methodological task, locating the philosophy of mathematics within the broader philosophy of science. (shrink)

For many philosophers of science, mathematics lies closer to logic than it does to the ordinary sciences like physics, biology and economics. While this view may account for the relative neglect of the philosophy of mathematics by philosophers of science, it ignores at least two pressing questions about mathematics that philosophers of science need to be able to answer. First, do the similarities between mathematics and science support the view that mathematics is, after all, (...) another science? Second, does the central role of mathematics in science shed any light on traditional philosophical debates about science like scientific realism, the nature of explanation or reduction? When faced with these kinds of questions many philosophers of science have little to say. Unfortunately, most philosophers of mathematics also fail to engage with questions about the relationship between mathematics and science and so a peculiar isolation has emerged between philosophy of science and philosophy of mathematics. In this introductory survey I aim to equip the interested philosopher of science with a roadmap that can guide her through the often intimidating terrain of contemporary philosophy of mathematics. I hope that such a survey will make clear how fruitful a more sustained interaction between philosophy of science and philosophy of mathematics could be. (shrink)

What is mathematics about? Does the subject-matter of mathematics exist independently of the mind or are they mental constructions? How do we know mathematics? Is mathematical knowledge logical knowledge? And how is mathematics applied to the material world? In this introduction to the philosophy of mathematics, Michele Friend examines these and other ontological and epistemological problems raised by the content and practice of mathematics. Aimed at a readership with limited proficiency in mathematics (...) but with some experience of formal logic it seeks to strike a balance between conceptual accessibility and correct representation of the issues. Friend examines the standard theories of mathematics - Platonism, realism, logicism, formalism, constructivism and structuralism - as well as some less standard theories such as psychologism, fictionalism and Meinongian philosophy of mathematics. In each case Friend explains what characterises the position and where the divisions between them lie, including some of the arguments in favour and against each. This book also explores particular questions that occupy present-day philosophers and mathematicians such as the problem of infinity, mathematical intuition and the relationship, if any, between the philosophy of mathematics and the practice of mathematics. Taking in the canonical ideas of Aristotle, Kant, Frege and Whitehead and Russell as well as the challenging and innovative work of recent philosophers like Benacerraf, Hellman, Maddy and Shapiro, Friend provides a balanced and accessible introduction suitable for upper-level undergraduate courses and the non-specialist. (shrink)