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This volume is a collection of papers on philosophy of mathematics which deal with a series of questions quite different from those which occupied the minds of the proponents of the three classic schools: logicism, formalism, and intuitionism. The questions of the volume are not to do with justification in the traditional sense, but with a variety of other topics. Some are concerned with discovery and the growth of mathematics. How does the semantics of mathematics change as the subject develops? What heuristics are involved in mathematical discovery, and do such heuristics constitute a logic of mathematical discovery? What new problems have been introduced by the development of mathematics since the 1930s? Other questions are concerned with the applications of mathematics both to physics and to the new field of computer science. Then there is the new question of whether the axiomatic method is really so essential to mathematics as is often supposed, and the question, which goes back to Wittgenstein, of the sense in which mathematical proofs are compelling. Taking these questions together they give part of an emerging agenda which is likely to carry philosophy of mathematics forward into the twenty first century.

How can we advance knowledge? Which methods do we need in order to make new discoveries? How can we rationally evaluate, reconstruct and offer discoveries as a means of improving the ‘method’ of discovery itself? And how can we use findings about scientific discovery to boost funding policies, thus fostering a deeper impact of scientific discovery itself? The respective chapters in this book provide readers with answers to these questions. They focus on a set of issues that are essential to the development of types of reasoning for advancing knowledge, such as models for both revolutionary findings and paradigm shifts; ways of rationally addressing scientific disagreement, e.g. when a revolutionary discovery sparks considerable disagreement inside the scientific community; frameworks for both discovery and inference methods; and heuristics for economics and the social sciences.

The Mathematics Enthusiast (TME) is an eclectic internationally circulated peer reviewed journal which focuses on mathematics content, mathematics education research, innovation, interdisciplinary issues and pedagogy. The journal exists as an independent entity. It is published on a print?on?demand basis by Information Age Publishing and the electronic version is hosted by the Department of Mathematical Sciences? University of Montana. The journal is not affiliated to nor subsidized by any professional organizations but supports PMENA [Psychology of Mathematics Education? North America] through special issues on various research topics.

China, Korea, Singapore, Japan, Malaysia and India

Author: Bharath Sriraman,Jinfa Cai,Kyeonghwa Lee,Lianghuo Fan,Yoshinori Shimizu,Chap Sam Lim,K. Subramaniam

Publisher: IAP

ISBN: 1617358274

Category: Mathematics

Page: 269

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Mathematics and Science education have both grown in fertile directions in different geographic regions. Yet, the mainstream discourse in international handbooks does not lend voice to developments in cognition, curriculum, teacher development, assessment, policy and implementation of mathematics and science in many countries. Paradoxically, in spite of advances in information technology and the “flat earth” syndrome, old distinctions and biases between different groups of researcher’s persist. In addition limited accessibility to conferences and journals also contribute to this problem. The International Sourcebooks in Mathematics and Science Education focus on underrepresented regions of the world and provides a platform for researchers to showcase their research and development in areas within mathematics and science education. The First Sourcebook on Asian Research in Mathematics Education: China, Korea, Singapore, Japan, Malaysia and India provides the first synthesized treatment of mathematics education that has both developed and is now prominently emerging in the Asian and South Asian world. The book is organized in sections coordinated by leaders in mathematics education in these countries and editorial teams for each country affiliated with them. The purpose of unique sourcebook is to both consolidate and survey the established body of research in these countries with findings that have influenced ongoing research agendas and informed practices in Europe, North America (and other countries) in addition to serving as a platform to showcase existing research that has shaped teacher education, curricula and policy in these Asian countries. The book will serve as a standard reference for mathematics education researchers, policy makers, practitioners and students both in and outside Asia, and complement the Nordic and NCTM perspectives.

Unique in that it collects, presents, and synthesizes cutting edge research on different aspects of statistical reasoning and applies this research to the teaching of statistics to students at all educational levels, this volume will prove of great value to mathematics and statistics education researchers, statistics educators, statisticians, cognitive psychologists, mathematics teachers, mathematics and statistics curriculum developers, and quantitative literacy experts in education and government.

This report looks at a number of published studies on mathematics education that try to understand which education and skills are appropriate for innovative societies.

This Festschrift volume is published in memory of William W. McCune who passed away in 2011. William W. McCune was an accomplished computer scientist all around but especially a fantastic system builder and software engineer. The volume includes 13 full papers, which are presenting research in all aspects of automated reasoning and its applications to mathematics. These papers have been thoroughly reviewed and selected out of 15 submissions received in response to the call for paper issued in September 2011. The topics covered are: strategies, indexing, superposition-based theorem proving, model building, application of automated reasoning to mathematics, as well as to program verification, data mining, and computer formalized mathematics.

How do we get new knowledge? Following the maverick tradition in the philosophy of science, Carlo Cellucci gradually came to the conclusion that logic can only fulfill its role in mathematics, science and philosophy if it helps us to answer this question. He argues that mathematical logic is inadequate and that we need a new logic, framed in a naturalistic conception of knowledge and philosophy – the heuristic conception. This path from logic to a naturalistic conception of knowledge and philosophy explains the title, From a Heuristic Point of View, which recalls the celebrated collection of essays, From a Logical Point of View, by Willard Van Orman Quine, the father of modern naturalized epistemology. The word ‘heuristic’ points to Cellucci’s favorite theme and the main difference between him and Quine: the emphasis on discovery and building a ‘logic’ for generating new knowledge. This book is a collection of essays from leading figures in this field who discuss, criticize, or expand on the main topics in Cellucci’s work, dealing with some of the most challenging questions in logic, science and philosophy.

Imre Lakatos (1922-1974) was one of the protagonists in shaping the "new philosophy of science". More than 25 years after his untimely death, it is time for a critical re-evaluation of his ideas. His main theme of locating rationality within the scientific process appears even more compelling today, after many historical case studies have revealed the cultural and societal elements within scientific practices. Recently there has been, above all, an increasing interest in Lakatos' philosophy of mathematics, which emphasises heuristics and mathematical practice over logical justification. But suitable modifications of his approach are called for in order to make it applicable to modern axiomatised theories. Pioneering historical research in England and Hungary has unearthed hitherto unknown facts about Lakatos' personal life, his wartime activities and his involvement in the political developments of post-war Europe. From a communist activist committed to Györgyi Lukács' thinking, Lakatos developed into a staunch anti-Marxist who found his intellectual background in Popper's critical rationalism. The volume also publishes for the first time a part of his Debrecen Ph.D. thesis and it is concluded by a bibliography of his Hungarian writings.

Joint International Conferences, AISC 2002 and Calculemus 2002 Marseille, France, July 1-5, 2002 Proceedings

Author: France) Calculemus (2002 Marseille,International Conference on Artificial Intelligence,Spain) International Conference AISC 2000 (2000 : Madrid

Publisher: Springer Science & Business Media

ISBN: 3540438653

Category: Computers

Page: 341

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This book constitutes the refereed proceedings of the joint International Conferences on Artificial Intelligence and Symbolic Computation, AISC 2002, and Calculemus 2002 held in Marseille, France, in July 2002. The 24 revised full papers presented together with 2 system descriptions were carefully reviewed and selected from 52 submissions. Among the topics covered are automated theorem proving, logical reasoning, mathematical modeling, algebraic computations, computational mathematics, and applications in engineering and industrial practice.

Author: P. Mancosu,Klaus Frovin Jørgensen,S.A. Pedersen

Publisher: Springer Science & Business Media

ISBN: 1402033354

Category: Mathematics

Page: 300

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In the 20th century philosophy of mathematics has to a great extent been dominated by views developed during the so-called foundational crisis in the beginning of that century. These views have primarily focused on questions pertaining to the logical structure of mathematics and questions regarding the justi?cation and consistency of mathematics. Paradigmatic in this - spect is Hilbert’s program which inherits from Frege and Russell the project to formalize all areas of ordinary mathematics and then adds the requi- ment of a proof, by epistemically privileged means (?nitistic reasoning), of the consistency of such formalized theories. While interest in modi?ed v- sions of the original foundational programs is still thriving, in the second part of the twentieth century several philosophers and historians of mat- matics have questioned whether such foundational programs could exhaust the realm of important philosophical problems to be raised about the nature of mathematics. Some have done so in open confrontation (and hostility) to the logically based analysis of mathematics which characterized the cl- sical foundational programs, while others (and many of the contributors to this book belong to this tradition) have only called for an extension of the range of questions and problems that should be raised in connection with an understanding of mathematics. The focus has turned thus to a consideration of what mathematicians are actually doing when they produce mathematics. Questions concerning concept-formation, understanding, heuristics, changes instyle of reasoning, the role of analogies and diagrams etc.

One effect of information technology is the increasing need to present information visually. The trend raises intriguing questions. What is the logical status of reasoning that employs visualization? What are the cognitive advantages and pitfalls of this reasoning? What kinds of tools can be developed to aid in the use of visual representation? This newest volume on the Studies in Logic and Computation series addresses the logical aspects of the visualization of information. The authors of these specially commissioned papers explore the properties of diagrams, charts, and maps, and their use in problem solving and teaching basic reasoning skills. As computers make visual representations more commonplace, it is important for professionals, researchers and students in computer science, philosophy, and logic to develop an understanding of these tools; this book can clarify the relationship between visuals and information.

Fluid dynamics is fundamental to our understanding of the atmosphere and oceans. Although many of the same principles of fluid dynamics apply to both the atmosphere and oceans, textbooks tend to concentrate on the atmosphere, the ocean, or the theory of geophysical fluid dynamics (GFD). This textbook provides a comprehensive unified treatment of atmospheric and oceanic fluid dynamics. The book introduces the fundamentals of geophysical fluid dynamics, including rotation and stratification, vorticity and potential vorticity, and scaling and approximations. It discusses baroclinic and barotropic instabilities, wave-mean flow interactions and turbulence, and the general circulation of the atmosphere and ocean. Student problems and exercises are included at the end of each chapter. Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation will be an invaluable graduate textbook on advanced courses in GFD, meteorology, atmospheric science and oceanography, and an excellent review volume for researchers. Additional resources are available at www.cambridge.org/9780521849692.

This book introduces a new form of argumentative analysis: rhetorical heuremes. The method applies the concepts of heuristic thinking, probability, and contingency in order to develop a better understanding of complex arguments in classical oratory. A new theory is required because Greek and Roman rhetoric cannot provide detailed answers to problems of strategic argumentation in the analysis of speeches. Building on scholarship in Ciceronian oratory, this book moves beyond the extant terminology and employs a concept of heuristic reasoning derived from the psychology of decision making and mathematical problem solving. The author analyses selected passages from Cicero’s forensic speeches where arguments of probability are deployed, and shows that the Sophistic concept of probability can link ancient rhetoric and modern theories of argumentation. Six groups of heuremes are identified, each of which represents a form of probabilistic reasoning by which the orator plays upon the perception of the jurors.

Research in Contexts of Practice : [proceedings of the NATO Advanced Research Workshop on Advances in Mathematical Problem Solving Research, Held in Viana Do Castelo, Portugal, 27-30 April, 1991]

Author: Joao P. Ponte,Joao F. Matos,Domingos Fernandes,Jose M. Matos

Publisher: Springer Science & Business Media

ISBN: 9783540557357

Category: Computers

Page: 346

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This NATO volume discusses the implications of new information technologies and cognitive psychology for mathematical problem solving research and practice. It includes a discussion of problem solving and provides a view of developments in computerized learning environments.

Theory and Research in Psychology and Artificial Intelligence

Author: Morton Wagman

Publisher: Greenwood Publishing Group

ISBN: 9780275975258

Category: Philosophy

Page: 297

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This volume presents a critical analysis of current theory and research in psychological and computational sciences addressing reasoning processes. Distinguished from narrowly technical books on the one hand, and from general philosophical books on the other, this work gives a broad, structured, detailed, and critical account of advancing intellectual developments in theories on the nature of reasoning. The author concludes that artificial intelligence reasoning systems are deepening and broadening theories of human reasoning.

Yearbook 2015, Association of Mathematics Educators

Author: Khoon Yoong Wong

Publisher: World Scientific

ISBN: 9814696447

Category: Mathematics

Page: 324

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With this seventh volume, as part of the series of yearbooks by the Association of Mathematics Educators in Singapore, we aim to provide a range of learning experiences and teaching strategies that mathematics teachers can judiciously select and adapt in order to deliver effective lessons to their students at the primary to secondary level. Our ultimate goal is to develop successful problem solvers who are able to understand concepts, master fundamental skills, reason logically, apply mathematics, enjoy learning, and strategise their thinking. These qualities will prepare students for life-long learning and careers in the 21st century. The materials covered are derived from psychological theories, education praxis, research findings, and mathematics discourse, mediated by the author's professional experiences in mathematics education in four countries over the past four decades. They are organised into ten chapters aligned with the Singapore mathematics curriculum framework to help teachers and educators from Singapore and other countries deepen their understanding about the so-called "Singapore Maths". The book strikes a balance between mathematical rigour and pedagogical diversity, without rigid adherence to either. This is relevant to the current discussion about the relative roles of mathematics content knowledge and pedagogical content knowledge in effective teaching. It also encourages teachers to develop their own philosophy and teaching styles so that their lessons are effective, efficient, and enjoyable to teach. Contents:Curriculum: Map the Intended, Implemented, and Attained LandscapeConcepts: Build Meanings and ConnectionsSkills: Use Rules EfficientlyProcesses: Sharpen Mathematical Reasoning and Heuristic UseApplications: View the World Through Mathematical LensesICT: Be Its Prudent MasterAttitudes: Energise Learning with Emotional PowerMetacognition: Strategic Use of Cognitive ResourcesSchool Curriculum: Prepare Thoughtful PlansProfessional Development: Become Metacognitive Teachers Readership: Graduate students, researchers, practitioners and teachers in mathematics. Key Features:First, there is currently no mathematics methodology text that provides significant insights about learning and teaching based on the Singapore mathematics curriculum, yet supported by international perspectives and literatureThis fills a gap in the market about Singapore Maths, which has attracted much attention from overseas educatorsSecond, the teaching strategies discussed in the book are based on theories, research, and professional practices, and they satisfy the needs of both practitioners and researchers, hence widening the readership of the bookFinally, the author writes from the vintage point of having taught mathematics education and conducted research in Australia, Brunei Darussalam, Malaysia and Singapore and consulted with education institutes in Chile, Hong Kong, the Philippines and the US. This diverse experience allows the author to discuss mathematics education issues from an East-meets-West perspectiveKeywords:Mathematics;Pedagogy;Learning Experiences;Singapore;Teachers;Instruction;Curriculum

Author: P. CERRAI (Ed),P. FREGUGLIA (Ed),C. PELLEGRINI (Ed)

Publisher: Springer Science & Business Media

ISBN: 9780306466946

Category: Mathematics

Page: 294

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The Application of Mathematics to the Natural Sciences brings together scientists and historians of science to discuss how, in an increasingly interdisciplinary manner, mathematics and mathematical models are used in the natural sciences.

How to encourage investigative, discovery play with babies and children aged 0-5. Heuristic Play is a form of exploratory, investigative play that builds a whole range of skills for all children from birth to five. This guide looks at four age groups from birth to five and provides practical activities for setting up heuristic play sessions. Each session comes with expert advice on: - How to set up the session for each age group - The level of adult involvement - How to plan for these activities within the EYFS - How heuristic play relates to schemas of behaviour This easy to read and practical guide is the only one of its kind and an absolute essential for anyone working in the early years.

This is a charming and insightful contribution to an understanding of the "Science Wars" between postmodernist humanism and science, driving toward a resolution of the mutual misunderstanding that has driven the controversy. It traces the root of postmodern theory to a debate on the foundations of mathematics early in the 20th century, then compares developments in mathematics to what took place in the arts and humanities, discussing issues as diverse as literary theory, arts, and artificial intelligence. This is a straightforward, easily understood presentation of what can be difficult theoretical concepts It demonstrates that a pattern of misreading mathematics can be seen both on the part of science and on the part of postmodern thinking. This is a humorous, playful yet deeply serious look at the intellectual foundations of mathematics for those in the humanities and the perfect critical introduction to the bases of modernism and postmodernism for those in the sciences.