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)

The main objective of this paper is to present a mathematics education of helplessness in a problematization movement of political affects. So, we produce a discussion about Circuit of Affects, by the Brazilian philosopher Vladimir Safatle, in which fear and hope are constituted as affects that paralyze us and prevent us from creating new forms of life. Helplessness could be constituted as an affect for us to move with the contingency outside the temporality of expectation. The Covid-19 pandemic is (...) used as example for our considerations in the direction of presenting, in a first draft, three concepts to think/operate a mathematics education of helplessness: scale, belonging and border. A mathematics education with Gaia between humans and non-humans in becoming movements. (shrink)

It is often said to-day that mathematics is nothing but an extension or development of logic; indeed, the identity of logic and pure mathematics is alleged so confidently by persons whose mathematical attainments entitle them to consideration when they talk about the subject-matter of mathematics, as to be in danger of being ranked with the truths that an educated man should accept on the authority of the specialist. Yet a little reflection might at least make one hesitate. For whatever (...) else may be said about logic, it is generally allowed to study thinking. Some would say merely that it studies inference; but inference is thinking. Now the mathematician thinks and infers, but he does not study the activity of thinking and inferring; and a study of that activity would never make the discoveries credited to the mathematicians. (shrink)

The field of mathematics education has been fashioned by a diversity of theoretical and philosophical perspectives. The purpose of this study is to add to this field an analysis of the philosophical position of critical realism. To achieve this objective, the study addresses the following questions: what does critical realism have to offer mathematics education? How may critical realism underlabour for this discipline? In addressing these questions, the study provides an overview of the basic theories and the possible (...) weak points of and arguments against critical realism and realism in general. It then draws upon the notion of a dialectical phenomenology to provide a sequence of Achilles' heel critiques of some of the perspectives that constitute broadly the theoretical landscape of mathematics education research. This critique proceeds with an analysis of didactically oriented empiricism, hermeneutics, pragmatism, postmodernism, traditionally recognized forms of constructivism, traditionally recognized theories of activity, and ethnomathematics. The last section summarizes the implications of philosophical underlabouring for a more beneficial science of mathematics education. (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)

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)

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)

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)

This article traces the development of the theory and practice of what is known as ‘community of inquiry’ as an ideal of classroom praxis. The concept has ancient and uncertain origins, but was seized upon as a form of pedagogy by the originators of the Philosophy for Children program in the 1970s. Its location at the intersection of the discourses of argumentation theory, communications theory, semiotics, systems theory, dialogue theory, learning theory and group psychodynamics makes of it a rich site (...) for the dialogue between theory and practice in education. This article is an exploration of those intersections, and a prospectus of its possible role in the formation and reformulation of school curriculum. It will be argued here that, when formulated as community of philosophical inquiry in particular, it offers the possibility of ‘philosophising’ the school curriculum in general, by extending the concept-work that doing philosophy entails to all of the disciplines. The article begins with an attempt at an operational definition of the term as, move to an analysis of its dynamics, offers an example of its use in a mathematics classroom, and finishes with a schematic view of its whole-curriculum and whole-school possibilities. (shrink)

This editorial, introducing the Journal for Theoretical & Marginal Mathematics Education, is historically situated in a moment when the field of mathematics education research is on the precipice of acknowledging that the old world is dying. That is to say, the way research has been done before is no longer adequate for operating within the White, Colonial, cis-hetero Patriarchal, Abled Capitalist dystopia we find ourselves. This inadequacy is, however, not a reason for despair but instead for celebration. Instead (...) of directing our resources towards reifying existing norms and structures, towards adopting exhausted methods and theories—as is the way of the old world—we can instead foster a ‘new aesthetics’ of mathematics education research, wherein we remake the very material that constitutes mathematics education research itself. In doing so, we engage with the politics of the margins of mathematics education research. We discuss the critical, philosophical, and psychoanalytic as three potential, but non-exhaustive, departures from mathematics education research as it has been. Each of these marginal approaches, in their own way, points to the ways that presuming a fixed mathematics education research enacts exclusionary politics and precludes change. Let this be our manifesto, written from the margins, for a more root-grasping, more self-reflective, and freer mathematics education research. (shrink)

Mathematics has always been a core part of western education, from the medieval quadrivium to the large amount of arithmetic and algebra still compulsory in high schools. It is an essential part. Its commitment to exactitude and to rigid demonstration balances humanist subjects devoted to appreciation and rhetoric as well as giving the lie to postmodernist insinuations that all “truths” are subject to political negotiation. In recent decades, the character of mathematics has changed – or rather broadened: it has (...) become the enabling science behind the complexity of contemporary knowledge, from gene interpretation to bank risk. Mathematical understanding is all the more necessary for future jobs, as well as remaining, as ever, a prophylactic against the more corrosive philosophical views emanating from the humanities. (shrink)

Epistemology is a conceptual template for how we think about the world, and the study of how we come to know the world around us. The world does not dictate unequivocally how to interpret it. This article will explore this position on the fluidity of epistemic constructs through two prominent philosophical perspectives, those being derived from the works of Ludwig Wittgenstein and Michael Foucault, respectively. These insights will be used to more deeply unfold the current situation for Aboriginal students within (...) dominant approaches to mathematics curriculum in Australia, and the subsequent approaches to the inclusion of Aboriginal knowledge and epistemologies. It is suggested that the epistemic constructs most valued and thus credited as conveyors of ‘truth’, and therefore positioned as powerful forms of knowledge within dominant curriculum and education policy, are those derived from Western, Eurocentric origins. This privileging of particular epistemological constructs over others is reinforced unconsciously through the articulation of educational goals deemed most appropriate, or ‘socially just’, for the Aboriginal student population. The place of Aboriginal knowledge within such constructs is therefore reflective of broader ideation around the role of education within society and its failure to challenge existing structures of power and injustice. (shrink)

The theoretical solid-state physicist Walter Kohn was awarded one-half of the 1998 Nobel Prize in Chemistry for his mid-1960s creation of an approach to the many-particle problem in quantum mechanics called density functional theory (DFT). In its exact form, DFT establishes that the total charge density of any system of electrons and nuclei provides all the information needed for a complete description of that system. This was a breakthrough for the study of atoms, molecules, gases, liquids, and solids. Before DFT, (...) it was thought that knowledge of the vastly more complicated many-electron wave function was essential for a complete description of such systems. Today, 50 years after its introduction, DFT (in one of its approximate forms) is the method of choice used by most scientists to calculate the physical properties of materials of all kinds. In this paper, I present a biographical essay of Kohn’s educational experiences and professional career up to and including the creation of DFT. My account begins with Kohn’s student years in Austria, England, and Canada during World War II and continues with his graduate and postgraduate training at Harvard University and Niels Bohr’s Institute for Theoretical Physics in Copenhagen. I then study the research choices he made during the first 10 years of his career (when he was a faculty member at the Carnegie Institute of Technology and a frequent visitor to the Bell Telephone Laboratories) in the context of the theoretical solid-state physics agenda of the late 1950s and early 1960s. Subsequent sections discuss his move to the University of California, San Diego, identify the research issue which led directly to DFT, and analyze the two foundational papers of the theory. The paper concludes with an explanation of how the chemists came to award “their” Nobel Prize to the physicist Kohn and a discussion of why he was unusually well suited to create the theory in the first place. (shrink)

This volume presents the proceedings from the Eleventh Brazilian Logic Conference on Mathematical Logic held by the Brazilian Logic Society in Salvador, Bahia, Brazil. The conference and the volume are dedicated to the memory of professor Mario Tourasse Teixeira, an educator and researcher who contributed to the formation of several generations of Brazilian logicians. Contributions were made from leading Brazilian logicians and their Latin-American and European colleagues. All papers were selected by a careful refereeing processs and were revised and (...) updated by their authors for publication in this volume. There are three sections: Advances in Logic, Advances in Theoretical Computer Science, and Advances in Philosophical Logic. Well-known specialists present original research on several aspects of model theory, proof theory, algebraic logic, category theory, connections between logic and computer science, and topics of philosophical logic of current interest. Topics interweave proof-theoretical, semantical, foundational, and philosophical aspects with algorithmic and algebraic views, offering lively high-level research results. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Abraham, Planter of Mathematics":Histories of Mathematics and Astrology in Early Modern EuropeNicholas PopperFrancis Bacon's 1605 Advancement of Learning proposed to dedicatee James I a massive reorganization of the institutions, goals, and methods of generating and transmitting knowledge. The numerous defects crippling the contemporary educational regime, Bacon claimed, should be addressed by strengthening emphasis on philosophy and natural knowledge. To that end, university positions were to be created devoted to (...) "Artes and Sciences at large," rather than to the professions. High salaries would render lecturers "able and sufficient," undistracted from their task. Most famously, he argued that teaching of the "operatiue studie of many Scyences" should involve sophisticated technical education. The study of natural philosophy demanded not only books, but globes, astrolabes, and other "instrumentals." Most significantly, yielding reliable and meaningful knowledge from experiential gleanings required a rigorous system of deductive reasoning.The legacy of this colossal proposal has earned Bacon honored status as devisor of the scientific method.1 But Bacon's educational reform extended [End Page 87] beyond the methods of producing and transmitting knowledge. To facilitate more efficient "vse and administration" of the knowledge produced by his system, he also demanded a searching examination of the history of learning. This history would provide a mirror enabling his contemporaries to deploy the fruits of his method, by considering how learning in the past had been used successfully or ill-advisedly. Producing this history was the ambition of the Advancement, for he explained: "no man hath propounded to himselfe the generall state of learning to bee described and represented from age to age, as many have done the works of Nature, & the State civile and Ecclesiastical." He continued:And yet I am not ignorant that in divers particular sciences, as of the Iurisconsults, the Mathematicians, the Rhetoricians, the Philosophers, there are set down some smal memorials of the Schooles, Authors, and Bookes: and so likewise some barren relations touching the Invention of Arts, or usages. But a iust story of learning, containing the Antiquities and Originalls of Knowledges, & their Sects; their Inventions, their Traditions; their diverse Administrations and Managings; their Flourishings, their Oppositions, Decayes, Depressions, Oblivions, Removes, with the causes, and occasions of them, and all other events concerning learning, throughout the ages of the world; I may truly affirme to be wanting.2Bacon thus positioned himself not only as a Father of Modern Science, but also a Father of the History of Science.Following Bacon's suggestion, I will examine the "small memorials" of the history of mathematics—and particularly the mathematical art of astrology—in the sixteenth and seventeenth centuries. My conclusion, however, will not bear out Bacon's claims. Despite his frustration, Bacon was only one of many early modern scholars appraising the role of mathematics within history. And he devoted less energy than others to mapping its origins and tracing its transmissions between communities. In fact, Bacon's 1605 proposal for a history of mathematics was already out-of-date. Discussions [End Page 88] of the history of mathematics had been rife on the continent and in England throughout the previous century.3The evidence that Renaissance scholars inherited was sprawling and inconclusive. Several genealogies for mathematics could be found within classical Greek, Latin, and patristic references. One lineage claimed mathematics began in ancient Assyria, where the priestly caste, the Chaldeans, practiced a form of mathematics that seemed a corrupt admixture of philosophy, medicine, and religion, and relied heavily on observation of the heavens. Other scholars traced the origins to Egypt, claiming that the field developed to survey lands frequently flooded by the Nile. The real problem for ancient, late antique, and medieval scholars had not been the origins of mathematics, but what exactly mathematics were. For some, the term strictly referred to the quadrivium: geometry, arithmetic, astronomy, and music. But for others, mathematics' origins amongst the Egyptians or Chaldeans inextricably linked it to forms of astrological divination, augury, and necromancy that were unsavory to both Latin and Christian traditions. Mathematici were included alongside ghastly lists of Magi, Brahmins, Aruspices, Genethliaci, and other diabolic practitioners of idolatrous magic. These might or might not be distinguished from mathematici such as Pythagoras... (shrink)

Since the methods employed during teacher-learner interchange are constrained by the internal structure of a discipline, a study of the interaction amongst verbal language, technical language and structure of disciplines is at the heart of the classic problem of transfer in teaching-learning situations. This paper utilizes the analytic method of philosophy to explore aspects of the role of language in mathematics education, and attempts to harmonize mathematical meanings exposed by verbal language and the precise meanings expressed by the (...) mathematics register (MR) formulated in verbal language. While focusing on the integration of language use and meaning construction in mathematics education, the paper explores the relationship between the conceptual understanding revealed by the mathematics register and the procedural knowledge that refers to the mathematical content through ordinary discourse. (shrink)

Discover how mathematical sequences abound in our natural world in this definitive exploration of the geography of the cosmos You need not be a philosopher or a botanist, and certainly not a mathematician, to enjoy the bounty of the world around us. But is there some sort of order, a pattern, to the things that we see in the sky, on the ground, at the beach? In A Beginner's Guide to Constructing the Universe, Michael Schneider, an education writer (...) and computer consultant, combines science, philosophy, art, and common sense to reaffirm what the ancients observed: that a consistent language of geometric design underpins every level of the universe, from atoms to galaxies, cucumbers to cathedrals. Schneider also discusses numerical and geometric symbolism through the ages, and concepts such as periodic renewal and resonance. This book is an education in the world and everything we can't see within it. Contains numerous b&w photos and illustrations. (shrink)

In this handbook, I put into practice my philosophical views on children's mathematics. The handbook contains brief instructions and examples of mathematical activities. In the INSTRUCTIONS section, instructions are given on how, and in part why that way, to help preschool children in their mathematical development. In the ACTIVITIES section, there are examples of activities through which the child develops her mathematical abilities.

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)

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)

Upshot: This is a book for thoughtful science and mathematics teachers and curriculum developers and educational philosophers. Quale helps us to challenge pernicious received “truths‘ and offers us intriguing perspectives, valuable discourse ventures and practical paedagogic strategies to engage the youth of today who are turning away from science in droves, to their and our cost.

In mathematics education, students' difficulties with negative numbers are well known. To explain these difficulties, researchers traditionally refer to obstacles raised by the concept of NEGATIVE NUMBERS itself throughout its historical evolution. In order to improve our understanding, I propose to take into consideration another point of view, based on Vygotsky's principles, which define a strong relationship between signs such as language or symbols and cognitive development. I show how it is of great interest to consider students' difficulties with (...) negatives in the light of the use of symbols, in particular the minus sign. I describe the results of an empirical study about students' strategies in solving arithmetical equations with negatives. My analysis shows that the problems raised by the presence of negatives in the equations are attributable to students' very restricted and often inadequate use of the minus sign, which prevents them from finding a negative solution or making sense of it. (shrink)

Imre Lakatos' philosophical and scientific papers are published here in two volumes. Volume I brings together his very influential but scattered papers on the philosophy of the physical sciences, and includes one important unpublished essay on the effect of Newton's scientific achievement. Volume 2 presents his work on the philosophy of mathematics (much of it unpublished), together with some critical essays on contemporary philosophers of science and some famous polemical writings on political and educational issues.

Using the work of philosopher Henri Bergson to examine the nature of movement and memory, this article contributes to recent research on the role of the body in learning mathematics. Our aim in this paper is to introduce the ideas of Bergson and to show how these ideas shed light on mathematics classroom activity. Bergson’s monist philosophy provides a framework for understanding the materiality of both bodies and mathematical concepts. We discuss two case studies of classrooms to show how (...) the mathematical concepts of number and function are themselves mobile and full of potentiality, open to deformation and the remapping of the possible. Bergson helps us look differently at mathematical activity in the classroom, not as a closed set of distinct interacting bodies groping after abstract concepts, but as a dynamic relational assemblage. (shrink)

Imre Lakatos' philosophical and scientific papers are published here in two volumes. Volume I brings together his very influential but scattered papers on the philosophy of the physical sciences, and includes one important unpublished essay on the effect of Newton's scientific achievement. Volume 2 presents his work on the philosophy of mathematics, together with some critical essays on contemporary philosophers of science and some famous polemical writings on political and educational issues.

The cognitive foundations of geometry have puzzled academics for a long time, and even today are mostly unknown to many scholars, including mathematical cognition researchers. -/- Foundations of Geometric Cognition shows that basic geometric skills are deeply hardwired in the visuospatial cognitive capacities of our brains, namely spatial navigation and object recognition. These capacities, shared with non-human animals and appearing in early stages of the human ontogeny, cannot, however, fully explain a uniquely human form of geometric cognition. In the (...) book, Hohol argues that Euclidean geometry would not be possible without the human capacity to create and use abstract concepts, demonstrating how language and diagrams provide cognitive scaffolding for abstract geometric thinking, within a context of a Euclidean system of thought. -/- Taking an interdisciplinary approach and drawing on research from diverse fields including psychology, cognitive science, and mathematics, this book is a must-read for cognitive psychologists and cognitive scientists of mathematics, alongside anyone interested in mathematical education or the philosophical and historical aspects of geometry. (shrink)

This book showcases Elisabeth of Bohemia, Princess Palatine, one of the foremost female minds of the 17th century. Best known today for her important correspondence with the philosopher René Descartes, Elisabeth was famous in her own time for her learning, philosophical acumen, and mathematical brilliance. She was also well-connected in the seventeenth-century intellectual circles. Elisabeth’s status as a woman philosopher is emblematic of both the possibilities and limitations of women's participation in the republic of letters and of their subsequent (...) fate in history. Few sources containing her own views survive, and until recently there has been no work on Elisabeth as a thinker in her own right. This volume brings together an international team of scholars to discuss her work from a cross-disciplinary perspective on the occasion of her fourth centenary. It is the first collection of essays to examine a range of her interests and to discuss them in relation to her historical context. The studies presented here discuss her educational background, her friendships and contacts, her interest in politics, religion, and astronomy, as well as her views on politics, her moral philosophy and her engagement with Cartesianism. The volume will appeal to historians of philosophy, historians of political thought, philosophers, feminists and seventeenth-century historians. (shrink)

This collection presents significant contributions from an international network project on mathematical cultures, including essays from leading scholars in the history and philosophy of mathematics and mathematics education.​ Mathematics has universal standards of validity. Nevertheless, there are local styles in mathematical research and teaching, and great variation in the place of mathematics in the larger cultures that mathematical practitioners belong to. The reflections on mathematical cultures collected in this book are of interest to mathematicians, (...) class='Hi'>philosophers, historians, sociologists, cognitive scientists and mathematics educators. (shrink)

Imre Lakatos' philosophical and scientific papers are published here in two volumes. Volume I brings together his very influential but scattered papers on the philosophy of the physical sciences, and includes one important unpublished essay on the effect of Newton's scientific achievement. Volume 2 presents his work on the philosophy of mathematics (much of it unpublished), together with some critical essays on contemporary philosophers of science and some famous polemical writings on political and educational issues.

Using the work of philosopher Henri Bergson to examine the nature of movement and memory, this article contributes to recent research on the role of the body in learning mathematics. Our aim in this paper is to introduce the ideas of Bergson and to show how these ideas shed light on mathematics classroom activity. Bergson’s monist philosophy provides a framework for understanding the materiality of both bodies and mathematical concepts. We discuss two case studies of classrooms to show how (...) the mathematical concepts of number and function are themselves mobile and full of potentiality, open to deformation and the remapping of the possible. Bergson helps us look differently at mathematical activity in the classroom, not as a closed set of distinct interacting bodies groping after abstract concepts, but as a dynamic relational assemblage. (shrink)

Educational assessments, specifically standardized and normalized exams, owe most of their foundations to psychological test theory in psychometrics. While the theoretical assumptions of these practices are widespread and relatively uncontroversial in the testing community, there are at least two that are philosophically and mathematically suspect and have troubling implications in education. Assumption 1 is that repeated assessment measures that are calculated into an arithmetic mean are thought to represent some real stable, quantitative psychological trait or ability plus some error. (...) Assumption 2 is that aggregated, group-level educational data collected from assessments can then be interpreted to make inferences about a given individual person over time without explicit justification. It is argued that the former assumption cannot be taken for granted; it is also argued that, while it is typically attributed to 20th century thought, the assumption in a rigorous form can be traced back at least to the 1830s via an unattractive Platonistic statistical thesis offered by one of the founders of the social sciences—Belgian mathematician Adolphe Quetelet (1796–1874). While contemporary research has moved away from using his work directly, it is demonstrated that cognitive psychology is still facing the preservation of assumption 1, which is becoming increasingly challenged by current paradigms that pitch human cognition as a dynamical, complex system. However, how to deal with assumption 1 and whether it is broadly justified is left as an open question. It is then argued that assumption 2 is only justified by assessments having ergodic properties, which is a criterion rarely met in education; specifically, some forms of normalized standardized exams are intrinsically non-ergodic and should be thought of as invalid assessments for saying much about individual students and their capability. The article closes with a call for the introduction of dynamical mathematics into educational assessment at a conceptual level (e.g., through Bayesian networks), the critical analysis of several key psychological testing assumptions, and the introduction of dynamical language into philosophical discourse. Each of these prima facie distinct areas ought to inform each other more closely in educational studies. (shrink)

Greek mathematics is usually seen as having reached its height in a “golden age” around 300 b.c., after which it declined, reaching a rather sad stage in late antiquity. In this latter period Pappus of Alexandria stands out as one of the last competent mathematicians, although even his Mathematical Collection has been valued by historians mainly for its wealth of information on earlier mathematical achievements. In her readable book, Serafina Cuomo sets out to correct the conventional view of (...) mathematics in late antiquity: her general goal is “to show that the mathematics of late antiquity deserves a place both in the history of science and in the history of antiquity” . To that end she focuses on Pappus's Collection, for which she attempts “to produce a historical analysis … and at the same time to explore its wider cultural contexts” . Thus her text is not intended primarily as a discussion of the mathematics contained in the Collection; instead Cuomo looks into the historical circumstances in which Pappus produced this work and the image he wanted to convey both of mathematics and of himself.The first chapter presents some general background on the public's perception of mathematics and its practitioners in late antiquity. Emphasizing evidence “apart from what we find in the books of the famous mathematicians” , Cuomo looks at astrological treatises such as the Mathesis by Julius Firmicus Maternus and considers the social and fiscal status of various “technical” professions, such as those of land surveyor, architect, and public administrator. She uses Diocletian's edict on wages and salaries and examines the role of mathematics in education to show that, “far from being invisible or confined to ivory towers” , mathematics was perceived as an integral part of life and regarded positively. All this material is very interesting, but how relevant is it? Cuomo's broadening of the perspective to include more than just the works of famous mathematicians is certainly commendable. But should one equate the mere skill of calculating with numbers, or even that of applying some “higher” mathematics, with theoretical, philosophical, mathematical thought? Of course, Cuomo is aware of this problem. Taking it to be irrelevant for her purpose, she makes a good point: claiming affiliation with mathematics to enhance one's status, as those professional practitioners would do, presupposes a positive image of mathematics among the public . Nevertheless, such evidence would probably not have swayed previous historians' opinion of the decline of mathematics in late antiquity.Chapters 2–4 are devoted to Book 5 of the Collection, which covers isoperimetric problems and Platonic solids; Book 8, on mechanics; Book 3, on cube duplication; and Book 4, on special curves and their classification. In Chapter 2 Cuomo argues that Pappus used the widely known problems discussed in Book 5 to address a general audience so as to promote mathematics and show that “mathematicians are better qualified than philosophers … to talk about isoperimetry and the five Platonic bodies” . At the same time, pointing out the complementarity of mathematics and mechanics , Pappus stresses the usefulness of mathematics in educational and material matters. In Chapter 4 Cuomo examines how Pappus, addressing experts, “marks out his territory within the mathematical field itself” .In the final chapter Cuomo tries “to get a clearer grasp of Pappus' mathematical agenda” . She concludes that Pappus is not just a compiler admiring the past; rather, he selected his subjects and manipulated the mathematical tradition to serve his own immediate goals in the present.This book contains an abundance of interesting historical material, although, regrettably, the mathematical examples are only roughly sketched. Cuomo shows that we can learn much from looking at Pappus's Collection in its own right. Her book deserves careful study. (shrink)

This volume is the first extensive study of the historical and philosophical connections between technology and mathematics. Coverage includes the use of mathematics in ancient as well as modern technology, devices and machines for computation, cryptology, mathematics in technological education, the epistemology of computer-mediated proofs, and the relationship between technological and mathematical computability. The book also examines the work of such historical figures as Gottfried Wilhelm Leibniz, Charles Babbage, Ada Lovelace, and Alan Turing.

This inaugural handbook documents the distinctive research field that utilizes history and philosophy in investigation of theoretical, curricular and pedagogical issues in the teaching of science and mathematics. It is contributed to by 130 researchers from 30 countries; it provides a logically structured, fully referenced guide to the ways in which science and mathematics education is, informed by the history and philosophy of these disciplines, as well as by the philosophy of education more generally. The first handbook to (...) cover the field, it lays down a much-needed marker of progress to date and provides a platform for informed and coherent future analysis and research of the subject. -/- The publication comes at a time of heightened worldwide concern over the standard of science and mathematics education, attended by fierce debate over how best to reform curricula and enliven student engagement in the subjects There is a growing recognition among educators and policy makers that the learning of science must dovetail with learning about science; this handbook is uniquely positioned as a locus for the discussion. -/- The handbook features sections on pedagogical, theoretical, national, and biographical research, setting the literature of each tradition in its historical context. Each chapter engages in an assessment of the strengths and weakness of the research addressed, and suggests potentially fruitful avenues of future research. A key element of the handbook’s broader analytical framework is its identification and examination of unnoticed philosophical assumptions in science and mathematics research. It reminds readers at a crucial juncture that there has been a long and rich tradition of historical and philosophical engagements with science and mathematics teaching, and that lessons can be learnt from these engagements for the resolution of current theoretical, curricular and pedagogical questions that face teachers and administrators. (shrink)

Many students experience mathematics as ahistorical and acultural. We review the philosophical roots of this experience and pose alternatives. We argue that there is evidence that the inclusion of a historical dimension into the teaching of mathematics courses at all levels, combined with an ‘active’ approach to learning, will improve motivation and achievement.

An inclusive vision of mathematics-- its beauty, its humanity, and its power to build virtues that help us all flourish. For mathematician Francis Su, a society without mathematical affection is like a city without museums. To miss out on mathematics is to live without experiencing some of humanity's most beautiful ideas. In this profound book, written for a diverse audience but especially for those disenchanted by their past experiences, an award-winning mathematician and educator weaves personal reflections, puzzles, and stories (...) to show how mathematics meets basic human desires and cultivates virtues essential for human flourishing. Readers will explore mathematical concepts-- and see how mathematical thinking can even fulfill such longings as for love, play, freedom, justice, and community. Some lessons come from those who have struggled, including philosopher Simone Weil, whose own mathematical contributions were overshadowed by her brother's, and Christopher Jackson, who discovered mathematics as an inmate in a federal prison. Christopher Jackson's letters to the author appear throughout the book and show how this intellectual pursuit can-- and must-- be open to all. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Method and Mathematics:Peter Ramus's Histories of the SciencesRobert GouldingPeter Ramus (1515–72) was, at first sight, the least likely person to write an influential history of mathematics. For one thing, he was clearly no great mathematician himself. His sympathetic biographer Nicholas Nancel related that Ramus would spend the mornings being coached in mathematics by a team of experts he had assembled, and in the afternoon would lecture on the very (...) same subjects.1 But there can be no doubt that Ramus held mathematics in particular esteem and even played a crucial role in linking philosophical discourse to mathematics and in promoting the use of quadrivial reasoning in the study of the natural world.2The origins for his enthusiasm for mathematics are to be found in his account of the nature of the arts and critique of the curriculum of the universities. From his earliest career, Ramus was a logician and remained one in all his works, whether writing on mathematics or Virgil, and he conceived [End Page 63] of all the arts, especially mathematics, as unchanging structures of necessarily true propositions—not prima facie a position which lends itself to notions of historical change and development.3 His career as a historian of mathematics, I will argue, was directed by problems that arose in his theoretical understanding of mathematics as he began to immerse himself in the sciences of the ancient world.Ramus and the Reform of DialecticRamus set out his fundamental logical positions in his very first printed work, the Dialecticae institutiones (Education in Dialectic) of 1543, a contribution to the ongoing humanist attack on scholastic logic, in which Ramus rehearsed many of the commonplace criticisms of the university dialectic. Humanists complained that logic no longer concerned itself with real human reasoning; instead, it had become a discipline studied for its own sake, using its own incomprehensible jargon, and was of no practical interest. The new humanist dialectics of Lorenzo Valla and Rudolph Agricola attempted to teach the kind of practical reasoning useful for composing a speech or letter, and they borrowed extensively from rhetorical works of Cicero and Quintilian to develop a highly rhetoricized logic. Questions about the formal validity of arguments were of little interest; what mattered was whether the arguments were persuasive.4In his Dialecticae institutiones, Ramus took this humanist reformulation of dialectic a step further. Dialectic—or, in fact, any art or science—consisted of nature, doctrine, and exercise. The natural workings of the mind formed the most significant element. Doctrine was nothing more than a record of natural reasoning; in importance it paled next to nature and practice.5 The logic of the universities bore no resemblance to the true, natural dialectic, as he argued at length in the companion volume to the Institutiones, the innocuously titled but exuberantly offensive Aristotelicae animadversiones [End Page 64] (Observations on Aristotle).6 The question, then, remained: how can we gain access to this "natural reasoning"?Ramus's answer was surprising. He asks us to find a group of men—not scholars, but completely uneducated vineyard workers. Question them about the coming year: the fertility of the soil, the quality and quantity of the crop. "And then," he writes "from their minds as from a mirror an image of nature will be reflected."7 In the reasoned replies of these uneducated men, says Ramus, we discover every part of logic needed for any purpose, whether everyday discourse or composition of poetry: the invention of arguments, the assessment of their truth and their proper and orderly presentation. Other humanists had praised man's natural logical faculties, elevating them over artificial technique, but Ramus was the first to look beyond the walls of the university and the writings of the ancients to find natural dialectic at work.If even the uneducated have some possession of the arts, then the arts taught at the university should do no more than clear away the misleading junk in the mind, and allow its natural clarity to shine through.8 Logic should be easy to learn—not because this is a pedagogic ideal, but from the very nature of the art. And if logic as it is... (shrink)

This volume is the first extensive study of the historical and philosophical connections between technology and mathematics. Coverage includes the use of mathematics in ancient as well as modern technology, devices and machines for computation, cryptology, mathematics in technological education, the epistemology of computer-mediated proofs, and the relationship between technological and mathematical computability. The book also examines the work of such historical figures as Gottfried Wilhelm Leibniz, Charles Babbage, Ada Lovelace, and Alan Turing.

Research on the nature of science and science education enjoys a longhistory, with its origins in Ernst Mach's work in the late nineteenthcentury and John Dewey's at the beginning of the twentieth century.As early as 1909 the Central Association for Science and MathematicsTeachers published an article – ‘A Consideration of the Principles thatShould Determine the Courses in Biology in Secondary Schools’ – inSchool Science and Mathematics that reflected foundational concernsabout science and how school curricula should be informed by them. (...) Sincethen a large body of literature has developed related to the teaching andlearning about nature of science – see, for example, the Lederman and Meichtry reviews cited below. As well there has been intensephilosophical, historical and philosophical debate about the nature of scienceitself, culminating in the much-publicised ‘Science Wars’ of recent time. Thereferences listed here primarily focus on the empirical research related to thenature of science as an educational goal; along with a few influential philosophicalworks by such authors as Kuhn, Popper, Laudan, Lakatos, and others. Whilenot exhaustive, the list should prove useful to educators, and scholars in otherfields, interested in the nature of science and how its understanding can berealised as a goal of science instruction. The authors welcome correspondenceregarding omissions from the list, and on-going additions that can be made to it. (shrink)

John Locke is widely regarded as one of the most influential of the Enlightenment philosophers. This volume, edited by J. W. Adamson and published as a second edition in 1922, contains two of John Locke's essays concerning education; Some Thoughts Concerning Education and Of the Conduct of the Understanding. Some Thoughts Concerning Education expands on Locke's pioneering theory of mind by explaining how to educate a child using three complementary methods: the development of a healthy body; (...) the formation of a virtuous mind; and the pursuit of an academic curriculum including the emerging sciences, mathematics and languages. Of the Conduct of the Understanding continues the theme of the earlier essay by describing how to develop rational thought. For over a century after the publication of these essays, John Locke's views on education were considered authoritative, and his work was translated into almost all major European languages. (shrink)

At least since the seventeenth century, the strange combination of epistemological certainty and ontological power that characterizes mathematics has made it a major focus of philosophical, social, and cultural negotiation. In the eighteenth century, all of these factors were at play as mathematical thinkers struggled to assimilate and extend the analysis they had inherited from the seventeenth century. A combination of educational convictions and historical assumptions supported a humanistic mathematics essentially defined by its flexibility and breadth. This mathematics was (...) an expression of l’esprit humain, which was unfolding in a progressive historical narrative. The French Revolution dramatically altered the historical and educational landscapes that had supported this eighteenth‐century approach, and within thirty years Augustin Louis Cauchy had radically reconceptualized and restructured mathematics to be rigorous rather than narrative. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:RUSSELL AND WHITEHEAD ON THE PROCESS OF GROWTH IN EDUCATION1 HOWARD WOODHOUSE Educational Foundations / University of Saskatchewan Saskatoon, Sask., Canada S7N owo 1. RUSSELL, WHITEHEAD, AND PROCESS PHILOSOPHY W ere there no similarities between the philosophies of education of Bertrand Russell and Alfred North Whitehead, one would want to know why. Russell, after all, was Whitehead 's student as an undergraduate at Cambridge, his colleague and collaborator (...) for a dozen years working on the manuscript of Pri1!cipia Mathematica published in three volumes from 1910 to 1913, as well as his friend. Moreover, it was the sight of Whitehead's wife, Evelyn, in paroxysms of pain that prompted Russell's mystical experience in 1901, during which he tells us that he became a humanist, pacifist, and advocate of free schooling.2 In this paper, I consider the question of whether or not Russell's account of the process of growth in education is compatible with that of Whitehead. The question is important because it enables one to 1 Presented ar the annual meeting of rhe Association of Process Philosophy of Education, American Philosophical Association Central Division Meeting, Chicago, 25-7 April 1991. Brian Hendley was the respondent. I would like to thank the audience for its penetrating questions, as well as Don Cochrane, Brian Hendley, John McMurtry and Viola Safr for their comments on an earlier draft. 2 Auto., I: 149. For a psychoanalytic interpretation of Russell's "mystical illumination ", see Bennett and Nancy Simon, "The Pacifist Turn: an Episode of Mystic Illumination in Russell's Life", Russell, no. 13 (Spring 1974): Il-12, 17-24. Jo Vellacott believes that the term "pragmatic pacifist" more accurately desctibes Russell's ". Russell and Whitehead on the Process ofGrowth. 139 Whitehead the precise investigation of the natural world. Both the logical method and the precision of ideas that are learned in mathematics form the foundations of science and philosophy. Indeed, the structures of the physical world mirror those of mathematics, since they, too, are based upon relations among entities.1O The kind of understanding afforded by mathematics, therefore, is also the basis of philosophy, since it puts students in contact with long-lasting ideas having fundamental value, and enables them to gain a kind of liberation from the concerns of the everyday world. (2) Both Russell and Whitehead uphold the idea· that science is based on those kinds of sense-experience to which human beings have ready access (colours, sounds, smells, and observable objects, etc.).II The problem facing science at this point is to show how its generalizations are based upon these experiences. For Russell, induction fails because we can never prove the principle on the basis of experience without thereby begging the question.I:'- This is because the prin~iple of induction appeals to the future or to those unexperienced parts of the past or present with which we are not acquainted. As a result, Russell prefers to adopt a hypothetico-deductive method by which it is possible· to move from the sense-experience of everyday life to a systematic understanding of the structures of the universe. In this way, science is able to progress by means of "an application of mathematical probability to premisses arrived at independently of induction."I3 Whitehead also argues for the need for "careful scrutiny" in the manner in which we infer the existence of "the physical world [which] is, in some general sense of the term, a deduced concept."I4 In other words, like Russell, he prefers a more deductive kind of approach, suggesting that the theory of induction is the despair of philosophy.I5 10 Whitehead, The Aims ofEducation, pp. 82, 84, 89, 134, 155-7. Russell, PoM, pp. 448-9,471; PLA in LK, p. 207. II Russell, "The Relation of Sense-Data to Physics" in Mysticism and Logic, pp. 145-79. Whitehead, of course, also includes feelings as integrating features of sense experience that allow us to relate the diverse elements of such experience into a unitary whole; see Process and Reality (New York: Free P., 1969), p. 244 and Part III. 12 Russell, PP, Chap. 6. His argument is mirrored by Whitehead, op. cit., pp. 235-6. lJ Russell, HK, p... (shrink)

The emergence of systems biology raises many fascinating questions: What does it mean to take a systems approach to problems in biology? To what extent is the use of mathematical and computational modelling changing the life sciences? How does the availability of big data influence research practices? What are the major challenges for biomedical research in the years to come? This book addresses such questions of relevance not only to philosophers and biologists but also to readers interested in (...) the broader implications of systems biology for science and society. The book features reflections and original work by experts from across the disciplines including systems biologists, philosophers, and interdisciplinary scholars investigating the social and educational aspects of systems biology. In response to the same set of questions, the experts develop and defend their personal perspectives on the distinctive character of systems biology and the challenges that lie ahead. Readers are invited to engage with different views on the questions addressed, and may explore numerous themes relating to the philosophy of systems biology. This edited work will appeal to scholars and all levels, from undergraduates to researchers, and to those interested in a variety of scholarly approaches such as systems biology, mathematical and computational modelling, cell and molecular biology, genomics, systems theory, and of course, philosophy of biology. (shrink)

The article analyzes the philosophical course of Stephan Kalynovskyi taught at Kyiv-Mohyla Academy in 1729–1731. By now researchers considered this course to be one of the best monuments of Ukrainian philosophical thought of that time. However, the textual analysis has shown that the course was borrowed verbatim from François le Reés, a French author who wrote it in Paris in the 1630s. The article also analyzes all other currently known educational courses of Kalynovskyi (courses on rhetoric, history, mathematics). Neither they (...) were written by the professor himself, but borrowed, mostly verbatim, from other authors. The article drew conclusions about the (un)originality of Kalynovskyi's courses based on a comparison of his courses with their sources, identifying authorial changes, deviations, etc. In addition, an attempt was made to find out whether the verbatim borrowing of someone else's text without indicating the author during teaching could be considered plagiarism at that time. Biographical information about Kalynovskyi, as well as an overview of the church sermons that he delivered as a bishop, allows a better understanding of his educational courses. (shrink)

The 1990s could be called The Decade of Sociology in mathematics education. It was during those years that the sociology of mathematics became a core ingredient of discourse in mathematics education and the philosophy of mathematics and mathematics education. Unresolved questions and uncertainties have emerged out of this discourse that hinge on the key concept of social construction. More generally, what is at issue is the very idea of “the social”. Within the framework of the general problem (...) of “the social”, we want to open a discussion of boundaries and margins in mathematics and mathematics education. By theorizing the divisions of purity and danger, we will be able to better understand the intersection of logic, mathematics, and thinking with gender, race, and class, and morals, ethics, and values in the classroom. The process of transforming the sociology of mathematics and the sociology of mind into pedagogical tools for mathematics educators and philosophers of education has already begun. One of the tasks before us is the development of a more profound and at the same time more practical grasp of “the social”. Our objective in this paper is to move ourselves and our readers in the direction of just such a grasp of the social. (shrink)

That process philosophy can be the foundation of the theory and practice of educating human beings is the main argument of this book. The process philosophy of Alfred North Whitehead (1861-1947) is the particular thinking on which this book is based. Readers are shown that Whitehead's process philosophy provides a frame, a conceptual matrix, that addresses their concerns about education and offers direction for their educative acts. Whitehead theorized that all living entities are connected in some way. Relatedness, connectedness, (...) and holism are recurring themes in this exploration of Whitehead's implied philosophy of education. Whitehead never wrote a philosophy of education, but his writings over a period of nearly thirty years reveal a persistent interest and concern with education. His work, ranging from _Introduction to Mathematics_ (1911) to _Adventures of Ideas_ (1938), is drawn on here to construct, not Whitehead's philosophy of education, but, a Whiteheadian philosophy of education. _Whitehead and Philosophy of Education_ brings to scholars and students of education an understanding of Whitehead as an important figure in philosophy, particularly philosophy of education; an acquaintance with process philosophy; a brief treatment of Whitehead's life and an account of events and experiences that influenced his philosophizing; and an exploration of the educationally salient concepts found in Whitehead's formal and informal philosophy with special attention to Whitehead's ideas about creativity, process, rhythm, wisdom, and knowledge. Whitehead writes of phases of the rhythm of education - romance, precision, and generalization. The book is organized with attention to these three phases. Part One-Romance introduces readers to Whitehead the person, and the change of context for educating from a mechanistic world-view to an organismic one. Part Two-Precision examines Whitehead's writings, as they relate to process philosophy and to educating. Part Three-Generalization is an application of the explorations of Parts One and Two, yielding a construction of a Whiteheadian philosophy of education and suggestions for educational practice. (shrink)