Mathematical Reasoning and Heuristics

Author: Carlo Cellucci,Donald Gillies

Publisher: Kings College Publications


Category: Mathematics

Page: 212

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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.

Heuristic Reasoning

Author: Emiliano Ippoliti

Publisher: Springer

ISBN: 331909159X

Category: Computers

Page: 191

View: 6197

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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 Issue

Volume 12 #13

Author: Bharath Sriraman

Publisher: IAP

ISBN: 1681232588

Category: Education

Page: 481

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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.

The Challenge of Developing Statistical Literacy, Reasoning and Thinking

Author: Dani Ben-Zvi,Joan Garfield

Publisher: Springer Science & Business Media

ISBN: 9781402022784

Category: Mathematics

Page: 423

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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.

Automated Reasoning and Mathematics

Essays in Memory of William W. McCune

Author: Maria Paola Bonacina,Mark E. Stickel

Publisher: Springer

ISBN: 3642366759

Category: Computers

Page: 259

View: 7285

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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.

From a Heuristic Point of View

Essays in Honour of Carlo Cellucci

Author: Emiliano Ippoliti,Cesare Cozzo

Publisher: Cambridge Scholars Publishing

ISBN: 1443863351

Category: Science

Page: 295

View: 6600

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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.

Artificial Intelligence, Automated Reasoning, and Symbolic Computation

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.

Mathematical Problem Solving and New Information Technologies

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

View: 7816

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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.

Visualization, Explanation and Reasoning Styles in Mathematics

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

Publisher: Springer Science & Business Media

ISBN: 1402033354

Category: Mathematics

Page: 300

View: 2388

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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.

Heuristic Strategies in the Speeches of Cicero

Author: Gábor Tahin

Publisher: Springer Science & Business Media

ISBN: 3319017993

Category: Philosophy

Page: 193

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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.

Effective Mathematics Lessons through an Eclectic Singapore Approach

Yearbook 2015, Association of Mathematics Educators

Author: Khoon Yoong Wong

Publisher: World Scientific

ISBN: 9814696447

Category: Mathematics

Page: 324

View: 8842

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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

Logical Reasoning with Diagrams

Author: Gerard Allwein,Jon Barwise

Publisher: Oxford University Press

ISBN: 9780195355864

Category: Computers

Page: 288

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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.

Reasoning Processes in Humans and Computers

Theory and Research in Psychology and Artificial Intelligence

Author: Morton Wagman

Publisher: Greenwood Publishing Group

ISBN: 9780275975258

Category: Philosophy

Page: 297

View: 9455

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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.

The Application of Mathematics to the Sciences of Nature

Critical Moments and Aspects


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.

Heuristic Play

Author: Sheila Riddall-Leech

Publisher: Andrews UK Limited

ISBN: 1907241965

Category: Education

Page: 52

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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.

Problem Solving, Reasoning, and Communicating, K-8

Helping Children Think Mathematically

Author: Arthur J. Baroody,Ronald T. Coslick

Publisher: Prentice Hall

ISBN: 9780023064883

Category: Education

Page: 160

View: 7147

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Examines in depth how teachers can help foster children's mathematical thinking. Provides practical suggestions, builds on the most recent research, uses case studies, encourages interactive learning, presents challenging problems, discusses the importanct of process-oriented math instruction, demonstrates "writing-to-learn" mathematics.

Problem Solving, Reasoning and Numeracy in the Early Years Foundation Stage

Author: Anita M Hughes

Publisher: Routledge

ISBN: 1136033661

Category: Education

Page: 160

View: 1737

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The Practical Guidance in the Early Years Foundation Stage series will assist practitioners in the smooth and successful implementation of the Early Years Foundation Stage. Each book gives clear and detailed explanations of each aspect of Learning and Development and encourages readers to consider each area within its broadest context to expand and develop their own knowledge and good practice. Practical ideas and activities for all age groups are offered along with a wealth of expertise of how elements from the practice guidance can be implemented within all early years settings. The books include suggestions for the innovative use of everyday ressources, popular books and stories. This book offers an in-depth understanding of children's thinking skills from a psychological perspective. The book introduces the Learning Tools model, a vital cognitive tool used by children to learn and solve problems, and gives practical ideas on how practitioners can use everyday materials to promote problem solving and early numeracy skills through play. Readers are encouraged to reflect on their own practice and understanding to help them provide learning opportunities to meet the unique needs of all children in their setting.

Mathematical Epistemology and Psychology

Author: E.W. Beth,J. Piaget

Publisher: Springer Science & Business Media

ISBN: 9401721939

Category: Language Arts & Disciplines

Page: 326

View: 6398

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One of the controversial philosophical issues of recent years has been the question of the nature of logical and mathematical entities. Platonist or linguistic modes of explanation have become fashionable, whilst abstrac tionist and constructionist theories have ceased to be so. Beth and Piaget approach this problem in their book from two somewhat different points of view. Beth's approach is largely historico-critical, although he discusses the nature of heuristic thinking in mathematics, whilst that of Piaget is psycho-genetic. The major purpose of this introduction is to summarise some of the main points of their respective arguments. In the first part of this book Beth makes a detailed study of the history of philosophical thinking about mathematics, and draws our attention to the important role played by the Aristotelian methodology of the demon strative sciences. This, he tells us, is characterised by three postulates: (a) deductivity, (b) self-evidence, and (c) reality. The last postulate asserts that the primitive notions of a demonstrative science must have reference to a domain of real entities in order to have significance. On the Aristote lian view discursive reasoning plays a major role in mathematics, whilst pure intuition plays a somewhat subordinate one.