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.

This book discusses both the philosophy of mathematics and of mathematics education. The first part is a critique of existing approaches and a new philosophy of mathematics. Chapters include: (1) "A Critique of Absolutist Philosophies of Mathematics," (2) "The Philosophy of Mathematics Reconceptualized," (3) "Social Constructivism as a Philosophy of Mathematics," (4) "Social Constructivism and Subjective Knowledge," and (5) "The Parallels of Social Constructivism." The second part of the book (...) explores the philosophy of mathematics education and shows that many aspects of mathematics education rest on underlying, usually implicit, philosophical assumptions. Making these assumptions explicit puts a critical tool into the hands of teachers and researchers. Chapters in this section are: (6) "Aims and Ideologies of Mathematics Education," (7) "Groups with Utilitarian Ideologies," (8) "Groups with Purist Ideologies," (9) "The Social Change Ideology of the Public Educators," (10) "Critical Review of Cockcroft and the National Curriculum," (11) "Hierarchy in Mathematics, Learning, Ability and Society," (12) "Mathematics, Values and Equal Opportunities," and (13) "Investigation, Problem Solving and Pedagogy." Each chapter contains notes. (Contains 654 references.) (MKR). (shrink)

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. (shrink)

Andrew Davis; Philosophy of Mathematics Education, Journal of Philosophy of Education, Volume 26, Issue 1, 30 May 2006, Pages 121–126, https://doi.org/10.1111/j.

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)

I articulate Charles S. Peirce’s philosophy of mathematical education as related to his conception of mathematics, the nature of its method of inquiry, and especially, the reasoning abilities required for mathematical inquiry. The main thesis is that Peirce’s philosophy of mathematical education primarily aims at fostering the development of the students’ semeiotic abilities of imagination, concentration, and generalization required for conducting mathematical inquiry by way of experimentation upon diagrams. This involves an emphasis on the relation (...) between theory and practice and between mathematics and other fields including the arts and sciences. For achieving its goals, the article is divided in three sections. First, I expound Peirce’s philosophical account of mathematical reasoning. Second, I illustrate this account by way of a geometrical example, placing special emphasis on its relation to mathematical education. Finally, I put forth some Peircean philosophical principles for mathematical education. (shrink)

A Volume in Cognition, Equity & Society: International Perspectives formally known as International Perspectives on Mathematics Education - Cognition, Equity & Society Series Editor Bharath Sriraman, The University of Montana and Lyn English, Queensland University of Technology The diversity of research in mathematics education has been addressed as both, a problem and a strength. When manifested through adherence to different intellectual roots and theoretical orientations, diversions constitute 'refractions' of mathematics education. The collection and analysis (...) of empirical data in a study are by necessity refracted through the specific analytical lens employed, as well as the aim of the study itself. Refractions can also refer to looking at old phenomena through new lenses. The chapters in this book are refracted through philosophical, political, mathematical and personal lenses by distinguished authors in the field, addressing issues about the elusive experience of doing mathematics, purification of texts, refractions, mathematics and ethnomathematics, political messages in textbook tasks, mathematics education policy debate, the political in mathematics education research, philosophy and mathematics, meanings and representations, identity of mathematical modeling, and dilemmas in the teaching of calculus. An ancient Sanskrit adage states that Knowledge is something that grows when shared, but shrinks when hoarded. Academics engaged in the generation of new Knowledge are blessed with both the time and the freedom to engage in pursuits that allow for intellectual pleasure. As a phenomenon of the Zeitgeist many have succumbed to the increased corporatization of academic work, engaging in activities for monetary and self advancement purposes. Are there any real intellectuals left in academia, a l Adorno, Bourdieu, Chomsky, Foucault, among others? This Festschrift is dedicated to academics that don't bother with self promotion or aggrandizement of themselves or their ideas in simplistic terms. (shrink)

The article examines the current state of the mathematical education of the students-philosophers that depends on language of the humanitarian mathematics, evidence of its statements and methodological problem of the cognition of the mathematical facts. One of important tasks of philosophy of mathematical education consists in motivation of the need for training mathematics of students-philosophers. The main criterion of the usefulness of mathematics for philosophers is revealed in the ways of justification of its truth (...) and completeness of reasoning of mathematical statements. This reflects the attractiveness of mathematical knowledge for philosophers that is characterized by the fact that the truth is revealed along with a proof in mathematics. One of the main so-called ‘acupuncture points‘ of mathematical education of philosophers, as the author believes, is the language of modern mathematics. In General, the linguistic aspect is very important in modern science. The second ‘acupuncture point‘ of mathematical education of philosophers is the level of evidence claims. Third, according to the author, is responsible for knowledge of mathematical truths, is involved in not only our existence, but also transcendental knowledge. In the context of philosophy of the future of modern civilization, mathematics is indispensable as a necessary element of scientific and philosophical picture of the world. To support the conclusions the author draws a broad philosophical and scientific context. (shrink)

This volume contains ten papers that have been collected by the Canadian Society for History and Philosophy of Mathematics/Société canadienne d’histoire et de philosophie des mathématiques. It showcases rigorously-reviewed contemporary scholarship on an interesting variety of topics in the history and philosophy of mathematics from the seventeenth century to the modern era. -/- The volume begins with an exposition of the life and work of Professor Bolesław Sobociński. It then moves on to cover a collection of (...) topics about twentieth-century philosophy of mathematics, including Fred Sommers’s creation of Traditional Formal Logic and Alexander Grothendieck’s work as a starting point for discussing analogies between commutative algebra and algebraic geometry. Continuing the focus on the philosophy of mathematics, the next selections discuss the mathematization of biology and address the study of numerical cognition. The volume then moves to discussing various aspects of mathematics education, including Charles Davies’s early book on the teaching of mathematics and the use of Gaussian Lemniscates in the classroom. A collection of papers on the history of mathematics in the nineteenth century closes out the volume, presenting a discussion of Gauss’s “Allgemeine Theorie des Erdmagnetismus” and a comparison of the geometric works of Desargues and La Hire. Written by leading scholars in the field, these papers are accessible not only to mathematicians and students of the history and philosophy of mathematics, but also to anyone with a general interest in mathematics. (shrink)

In the mathematics education research literature, there is a growing body of scholarship on how mathematicians practice their craft. The purpose of this chapter is to survey some of this literature and explain how it can contribute to the philosophy of mathematical practice. We first describe how mathematics educators use empirical methodologies to investigate the behaviors of mathematicians and argue that findings from these studies can inform the philosophy of mathematical practice. We then illustrate this (...) by summarizing research on mathematicians’ understandings and mathematicians’ proof reading. (shrink)

The Mathematics Education and Neurosciences project is an interdisciplinary research program that bridges mathematics education research with neuroscientific research. The bidirectional collaboration will provide greater insight into young children's (aged four to six years) mathematical abilities. Specifically, by combining qualitative ‘design research’ with quantitative ‘experimental research’, we aim to come to a more thorough understanding of prerequisites that are involved in the development of early spatial and number sense. The mathematics education researchers are concerned (...) with kindergartner's spatial structuring ability, while the neuroscientists are studying kindergartner's automatic quantity processing and its effect on mathematical development. The outcomes of these investigations should contribute to practical ways of fostering and supporting young children's mathematical thinking and learning. (shrink)

This paper surveys central justifications and approaches adopted by educators interested in incorporating history of mathematics into mathematics teaching and learning. This interest itself has historical roots and different historical manifestations; these roots are examined as well in the paper. The paper also asks what it means for history of mathematics to be treated as genuine historical knowledge rather than a tool for teaching other kinds of mathematical knowledge. If, however, history of mathematics is not subordinated (...) to the ideas and methods at the heart of the usual mathematical curriculum – algebraic equations, functions, derivatives, analytic geometry – if it becomes an essential subject for mathematics, then the attempt to find a place for history of mathematics requires refining our understanding of the nature of mathematics education itself. Thus, the paper asks not only how can history of mathematics be incorporated into mathematics education but also how the idea of mathematics education may need to be adjusted to accommodate history. (shrink)

本书主要内容包括:初恋风云——数学教育主心骨的的百年寻觅之旅、践约时代——“现代数学教育方式”渐行渐的近、热题冷谈——数学教育焦点话题的哲学断想.

It is argued that the philosophical and epistemological beliefs about the nature of mathematics have a significant influence on the way mathematics is taught at school. In this paper, the philosophy of mathematics of the NCTM's Standards is investigated by examining is explicit assumptions regarding the teaching and learning of school mathematics. The main conceptual tool used for this purpose is the model of two dichotomous philosophies of mathematics-absolutist versus- fallibilist and their relation to (...) mathematics pedagogy. The main conclusion is that a fallibilist view of mathematics is assumed in the Standards and that most of its pedagogical assumptions and approaches are based on this philosophy. (shrink)

This paper describes an attempt to develop a program for teaching history and philosophy of mathematics to inservice mathematics teachers. I argue briefly for the view that philosophical positions and epistemological accounts related to mathematics have a significant influence and a powerful impact on the way mathematics is taught. But since philosophy of mathematics without history of mathematics does not exist, both philosophy and history of mathematics are necessary components of (...) programs for the training of preservice as well as inservice mathematics teachers. (shrink)

The circumstance that the text of Imre Lakatos' doctoral thesis from the University of Debrecen did not survive makes the evaluation of his career in Hungary and the research of aspects of continuity of his lifework difficult. My paper tries to reconstruct these newer aspects of continuity, introducing the influence of László Kalmár the mathematician and his fellow student, and Sándor Karácsony the philosopher and his mentor on Lakatos' work. The connection between the understanding of the empirical basis of exact (...) ideas—which is a common feature in the papers of the members of the Karácsony-circle—and Lakatos' way of thinking regarding mathematics is more direct and can be documented through his connection to Kalmár. The central element of Lakatos' philosophy of mathematics is criticism of formalism and his tendency is to use the empirical view. Discussions at the 1965 International Colloquium in the Philosophy of Science in London were very helpful in clarifying the quasi-empirical conception. Kalmár's lecture in London, based on one of his papers published by Karácsony in 1942, emphasized the empirical character of mathematics. After this colloquium some elements of the heritage of the Karácsony-circle were integrated again in the development of Lakatos' way of thinking. First I will analyze the Kalmár lecture of 1965 at the Colloquium of Philosophy of Science and Lakatos's reflections on the problem of the foundation of mathematics. Then I will present their common Hungarian background, their education and the beginning of their career, which have many important common features; third I draw attention to the network of contacts of Karácsony-disciples. (shrink)

. The circumstance, that the text of Imre Lakatos' doctoral thesis from the University of Debrecen did not survive, makes the evaluation of his career in Hungary and the research of aspects of continuity of his lifework difficult. My paper tries to reconstruct these newer aspects of continuity, introducing the influence of László Kalmár the mathematician and his fellow student, and Sándor Karácsony the philosopher and his mentor on Lakatos' work. The connection between the understanding of the empirical basis of (...) exact ideas—which is a common feature in the papers of the members of the Karácsonycircle—and Lakatos' way of thinking regarding mathematics is more direct and can be documented through his connection to Kalmár. The central element of Lakatos' philosophy of mathematics is criticism of formalism and his tendency is to use the empirical view. Discussions at the 1965 International Colloquium in the Philosophy of Science in London were very helpful in clarifying the quasi-empirical conception. Kalmár's lecture in London, based on one of his papers published by Karácsony in 1942, emphasized the empirical character of mathematics. After this colloquium some elements of the heritage of the Karácsony-circle were integrated again in the development of Lakatos' way of thinking. First I will analyze the Kalmár lecture of 1965 at the Colloquium of Philosophy of Science and Lakatos's reflections on the problem of the foundation of mathematics. Then I will present their common Hungarian background, their education and the beginning of their career, which have many important common features; third I draw attention to the network of contacts of Karácsony-disciples. (shrink)

Philosophy of Education and the Curriculum is a six-chapter book that first elucidates the forms of knowledge argument and religious education. Subsequent chapters detail the mathematics, natural science, and history forms of knowledge. Moral philosophy and moral education are then explained. The last chapter provides an explanation of learning.

In this article we contend that attempts to foster democratic education in the United States' public schools rarely include mathematics class in meaningful ways. We begin with Dewey's conception of democracy and then argue that current ways of thinking about mathematics do not provide adequate foundations for democratic mathematics education. Our reconceptualization of mathematics draws on Dewey's uniquely humanistic philosophy of mathematics. We conclude with some implications of democratic mathematics education (...) for school and society. Thus, this project seeks to blur the theory-practice dualism, developing a theoretical argument which draws sustenance from and seeks to contribute back to educational practice. (shrink)

This very short introduction will first outline how ontological investigations and questions of practice go together. The second section will bring in the next pole of this entire book, history of mathematics. How do ontology, practice, and history go together? Is this a forced marriage or one born in true love? That is, do these three belong together in some very basic way? One chapter in the section argues that the philosophy of mathematical practice intersects with the history (...) of mathematics essentially. If the first section of this introduction is persuasive that will also show that the history of mathematics intersects with ontological questions as well, specifically, that historical questions are very much about the ontological commitments of past mathematicians. The last section will take up what might seem a very different theme, namely, how these things matter to mathematics education. This section is connected closely to the second section since mathematics education has taken an interest in the role of history of mathematics in mathematics education, and much insight can be gained by seeing the issues in relation to the ontology-practice-history triad. The question of education is also important since teachers must consider what it is exactly that they are teaching, or, to adapt the title of Reuben Hersh’s well-known book, “What is mathematics really about?” And the “aboutness” of mathematics is really the ontological question. (shrink)

The history of the continuous inclusion of mathematics in liberal education in the West, from ancient times through the modern period, is sketched in the first two sections of this chapter. Next, the heart of this essay (Sects. 3, 4, 5, 6, and 7) delineates the central role mathematics has played throughout the history of Western civilization: not just a tool for science and technology, mathematics continually illuminates, interacts with, and sometimes challenges fields like art, music, (...) literature, and philosophy – subjects now universally considered to be liberal arts. Section 8 adds an international perspective to the contemporary liberal arts story by describing some instructive mathematical achievements from many cultures and societies. Finally, Sect. 9 addresses how contemporary mathematics teaching can use the history of mathematics viewed as a liberal art to enhance the appreciation and understanding of mathematics for all students. (shrink)

Current formal mathematics, being divorced from the empirical, is entirely a social construct, so that mathematical theorems are no more secure than the cultural belief in two-valued logic, incorrectly regarded as universal. Computer technology, by enhancing the ability to calculate, has put pressure on this social construct, since proof-oriented formal mathematics is awkward for computation, while computational mathematics is regarded as epistemo-logically insecure. Historically, a similar epistemological fissure between computational/practical Indian mathematics and formal/spiritual Western mathematics (...) persisted for centuries, during a dialogue of civilizations, when texts on "algorismus" and "infinitesimal" calculus were imported into Europe, enhancing the ability to calculate. It is argued here that this epistemological tension should be resolved by accepting mathematics as empirically based and fallible, and by revising accordingly the mathematics syllabus outlined by Plato. (shrink)

It is a rather safe statement to claim that the social dimensions of the scientific process are accepted in a fair share of studies in the philosophy of science. It is a somewhat safe statement to claim that the social dimensions are now seen as an essential element in the understanding of what human cognition is and how it functions. But it would be a rather unsafe statement to claim that the social is fully accepted in the philosophy (...) of mathematics. And we are not quite sure what kind of statement it is to claim that the social dimensions in theories of mathematics education are becoming more prominent, compared to the psychological dimensions. In our contribution we will focus, after a brief presentation of the above claims, on this particular domain to understand the successes and failures of the development of theories of mathematics education that focus on the social and not primarily on the psychological. (shrink)

This book explores the results of applying empirical methods to the philosophy of logic and mathematics. Much of the work that has earned experimental philosophy a prominent place in twenty-first century philosophy is concerned with ethics or epistemology. But, as this book shows, empirical methods are just as much at home in logic and the philosophy of mathematics. -/- Chapters demonstrate and discuss the applicability of a wide range of empirical methods including experiments, surveys, (...) interviews, and data-mining. Distinct themes emerge that reflect recent developments in the field, such as issues concerning the logic of conditionals and the role played by visual elements in some mathematical proofs. -/- Featuring leading figures from experimental philosophy and the fields of philosophy of logic and mathematics, this collection reveals that empirical work in these disciplines has been quietly thriving for some time and stresses the importance of collaboration between philosophers and researchers in mathematics education and mathematical cognition. (shrink)

Current formal mathematics, being divorced from the empirical, is entirely a social construct, so that mathematical theorems are no more secure than the cultural belief in two-valued logic, incorrectly regarded as universal. Computer technology, by enhancing the ability to calculate, has put pressure on this social construct, since proof-oriented formal mathematics is awkward for computation, while computational mathematics is regarded as epistemo-logically insecure. Historically, a similar epistemological fissure between computational/practical Indian mathematics and formal/spiritual Western mathematics (...) persisted for centuries, during a dialogue of civilizations, when texts on "algorismus" and "infinitesimal" calculus were imported into Europe, enhancing the ability to calculate. It is argued here that this epistemological tension should be resolved by accepting mathematics as empirically based and fallible, and by revising accordingly the mathematics syllabus outlined by Plato. (shrink)

This book offers an innovative introduction to the psychological basis of mathematics and the nature of mathematical thinking and learning, using an approach that empowers students by fostering their own construction of mathematical structures. Through accessible and engaging writing, award-winning mathematician and educator Anderson Norton reframes mathematics as something that exists first in the minds of students, rather than something that exists first in a textbook. By exploring the psychological basis for mathematics at every level - including (...) geometry, algebra, calculus, complex analysis and more - Norton unlocks students' personal power to construct mathematical objects based on their own mental activity, and illustrates the power of mathematics in organizing the world as we know it. Including reflections and activities designed to inspire awareness of the mental actions and processes coordinated in practicing mathematics, the book is geared towards current and future secondary and elementary mathematics teachers who will empower the next generation of mathematicians and STEM majors. Those interested in the history and philosophy that underpins mathematics will also benefit from this book, as well as those informed and curious minds attentive to the human experience more generally. Anderson Norton is Professor in the Department of Mathematics at Virginia Tech, USA, where he has been teaching mathematics courses for future teachers for 15 years. He is the editor of Constructing Number, and the co-author of Developing Fractions Knowledge and Numeracy for All Learners. (shrink)

An international group of distinguished scholars brings a variety of resources to bear on the major issues in the study and teaching of mathematics, and on the problem of understanding mathematics as a cultural and social phenomenon. All are guided by the notion that our understanding of mathematical knowledge must be grounded in and reflect the realities of mathematical practice. Chapters on the philosophy of mathematics illustrate the growing influence of a pragmatic view in a field (...) traditionally dominated by platonic perspectives. In a section on mathematics, politics, and pedagogy, the emphasis is on politics and values in mathematics education. Issues addressed include gender and mathematics, applied mathematics and social concerns, and the reflective and dialogical nature of mathematical knowledge. The concluding section deals with the history and sociology of mathematics, and with mathematics and social change. Contributors include Philip J. Davis, Helga Jungwirth, Nel Noddings, Yehuda Rav, Michael D. Resnik, Ole Skovsmose, and Thomas Tymoczko. (shrink)

The article is devoted to identifying the value of the phenomenon of aesthetic value and beauty of mathematical knowledge and the beauty of mathematical theory of teaching mathematics. The aesthetic potential of mathematical knowledge allows the use of theater technology in the educational process with the active dialogic interaction between teacher and students. The criteria of beauty in mathematical theories are distinguished: the realization of beauty as the unity of the whole, and in the disclosure of the complex through (...) the elementary; methodological interpretation of the beauty in the community of mathematical structures and optimal information content of the meta-language of mathematics; the practical embodiment of beauty in the formalization of the infinite through the finite. The beauty of mathematics is the force that permeates all the ‘layers of knowledge‘ not along, and across, although the effectiveness of mathematical activity due to aesthetic laws, which do not always lend themselves to unambiguous interpretation. In the article it is stated that, depending on the educational goals of communicative impact on the audience, in fact, ‘mathematical lectures theatricality‘ can have different characteristics, the most important of which are teachers artistry and artistic director’s work of a teacher. This cultural phenomenon that includes the theatrical talent, helps create an atmosphere of cooperation needed in varying degrees of activity for pedagogical interaction. The author believes that such approach, developed on the basis of the Stanislavsky system, allows university professors of mathematics significantly improve mathematical lectures. (shrink)

In this article, modern standards of early years mathematics education are criticized and a proposal for change is presented. Today's early years mathematics education standards rest on a view of mathematics that became obsolete already at the end of the 19th century while the spirit of children's mathematics is precisely the spirit of modern mathematics. The proposal for change is not a return to the “new mathematics” movement, but something different.

This paper addresses the context of emergence, development, and current status of the use of history and philosophy of science in science education in Brazil. After a short overview of the three areas (history of science, philosophy of science, and science education) in Brazil, the paper focuses on the application of this approach to teaching physics, chemistry, and biology at the secondary school level. The first Brazilian researches along this line appeared more consistently in the decade (...) of 1970. From 1980 onwards, the importance of this approach became widely accepted, and the subject became a common theme of dissertations and theses, appearing in conferences and educational journals. Since 1998, the use of history and philosophy of science was included among the government recommendations for secondary school science teaching in Brazil. Nowadays, this is an important line of research in graduate programs on science and mathematics education. However, the actual use of this approach in secondary education is still a desideratum. (shrink)

The contributors to this collection of essays offer a stimulating and varied range of approaches to this developing area. The volume includes discussions on the concept of education and such related topics as indoctrination and the nature and scope of the theory of education. Aspects of education including the field of moral education, and issues which are reflected prominently in the curricula of such subjects as Mathematics and Science in schools and colleges are considered.

The nature of this book is fourfold: First, it provides comprehensive education in ontology, epistemology, logic, and ethics. From this perspective, it can be treated as a philosophical textbook. Second, it provides comprehensive education in mathematical analysis and analytic geometry, including significant aspects of set theory, topology, mathematical logic, number systems, abstract algebra, linear algebra, and the theory of differential equations. From this perspective, it can be treated as a mathematical textbook. Third, it makes a student and a (...) researcher in philosophy and/or mathematics capable of developing a holistic approach to reality, of undertaking interdisciplinary endeavors, of understanding (and possibly contributing to) advances and research projects in different academic disciplines, and of having more sources of inspiration and pleasure. From this perspective, it can be treated as a contribution to pedagogy and as an attempt to refresh and, indeed, revitalize modern philosophy. Fourth, it seeks to defend, refresh, and enrich philosophical and scientific structuralism and dynamical philosophy (known also as dynamism). From this perspective, this book can be treated as a research monograph on structuralism and dynamism, tackling the fundamental problems of reality, truth, and consciousness. In this context, Nicolas Laos expounds and proposes: (i) the concepts of dynamized time and dynamized space; (ii) a theory and method that he calls the "dialectic of rational dynamicity"; and (iii) his attempt to consider the fundamental problems of philosophy and science from the perspective of the dialectic of rational dynamicity. Thus, this book pertains to every field that is controlled by the function of consciousness, namely, being, knowing, and acting. The philosophy of rational dynamicity, as the author explains in this book, is a way of contemplating the laws of motion of nature, history, and spirit. (shrink)

The Companion Encyclopedia is the first comprehensive work to cover all the principal lines and themes of the history and philosophy of mathematics from ancient times up to the twentieth century. In 176 articles contributed by 160 authors of 18 nationalities, the work describes and analyzes the variety of theories, proofs, techniques, and cultural and practical applications of mathematics. The work's aim is to recover our mathematical heritage and show the importance of mathematics today by treating (...) its interactions with the related disciplines of physics, astronomy, engineering and philosophy. It also covers the history of higher education in mathematics and the growth of institutions and organizations connected with the development of the subject. Part 1 deals with mathematics in various ancient and non-Western cultures from antiquity up to medieval and Renaissance times. Part 2 treats developments in all the main areas of mathematics during the medieval and Renaissance periods up to and including the early 17th century. Parts 3-10 are divided into the main branches into which mathematics developed from the early 17th century onwards: calculus and mathematical analysis, logic and foundations, algebras, geometries, mechanics, mathematical physics and engineering, and probability and statistics. Parts 11-13 review the history of mathematics from an international perspective. The teaching of mathematics in higher education is examined in various countries, and mathematics in culture, art and society is covered. The Companion Encyclopedia features annotated bibliographies of both classic and contemporary sources; black and white illustrations, line figures and equations; biographies of major mathematicians and historians and philosophers of mathematics; a chronological table of main events in the developments of mathematics; and a fully integrated index of people, events and topics. (shrink)