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Mathematical Reasoning and Heuristics

Mathematical Reasoning and Heuristics Author Carlo Cellucci
ISBN-10 UOM:39015063202694
Release 2005
Pages 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

Heuristic Reasoning Author Emiliano Ippoliti
ISBN-10 9783319091594
Release 2014-09-05
Pages 191
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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.

Heuristics and Biases

Heuristics and Biases Author Thomas Gilovich
ISBN-10 0521796792
Release 2002-07-08
Pages 857
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Is our case strong enough to go to trial? Will interest rates go up? Can I trust this person? Such questions - and the judgments required to answer them - are woven into the fabric of everyday experience. This book, first published in 2002, examines how people make such judgments. The study of human judgment was transformed in the 1970s, when Kahneman and Tversky introduced their 'heuristics and biases' approach and challenged the dominance of strictly rational models. Their work highlighted the reflexive mental operations used to make complex problems manageable and illuminated how the same processes can lead to both accurate and dangerously flawed judgments. The heuristics and biases framework generated a torrent of influential research in psychology - research that reverberated widely and affected scholarship in economics, law, medicine, management, and political science. This book compiles the most influential research in the heuristics and biases tradition since the initial collection of 1982 (by Kahneman, Slovic, and Tversky).

Appraising Lakatos

Appraising Lakatos Author György Kampis
ISBN-10 9789401707695
Release 2013-06-29
Pages 382
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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.

Visualization Explanation and Reasoning Styles in Mathematics

Visualization  Explanation and Reasoning Styles in Mathematics Author P. Mancosu
ISBN-10 9781402033353
Release 2006-03-30
Pages 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.

From a Heuristic Point of View

From a Heuristic Point of View Author Emiliano Ippoliti
ISBN-10 9781443863353
Release 2014-07-03
Pages 295
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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.

Trust in Society

Trust in Society Author Karen Cook
ISBN-10 9781610441322
Release 2001-01-11
Pages 432
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Trust plays a pervasive role in social affairs, even sustaining acts of cooperation among strangers who have no control over each other's actions. But the full importance of trust is rarely acknowledged until it begins to break down, threatening the stability of social relationships once taken for granted. Trust in Society uses the tools of experimental psychology, sociology, political science, and economics to shed light on the many functions trust performs in social and political life. The authors discuss different ways of conceptualizing trust and investigate the empirical effects of trust in a variety of social settings, from the local and personal to the national and institutional. Drawing on experimental findings, this book examines how people decide whom to trust, and how a person proves his own trustworthiness to others. Placing trust in a person can be seen as a strategic act, a moral response, or even an expression of social solidarity. People often assume that strangers are trustworthy on the basis of crude social affinities, such as a shared race, religion, or hometown. Likewise, new immigrants are often able to draw heavily upon the trust of prior arrivals—frequently kin—to obtain work and start-up capital. Trust in Society explains how trust is fostered among members of voluntary associations—such as soccer clubs, choirs, and church groups—and asks whether this trust spills over into other civic activities of wider benefit to society. The book also scrutinizes the relationship between trust and formal regulatory institutions, such as the law, that either substitute for trust when it is absent, or protect people from the worst consequences of trust when it is misplaced. Moreover, psychological research reveals how compliance with the law depends more on public trust in the motives of the police and courts than on fear of punishment. The contributors to this volume demonstrate the growing analytical sophistication of trust research and its wide-ranging explanatory power. In the interests of analytical rigor, the social sciences all too often assume that people act as atomistic individuals without regard to the interests of others. Trust in Society demonstrates how we can think rigorously and analytically about the many aspects of social life that cannot be explained in those terms. A Volume in the Russell Sage Foundation Series on Trust/B/b/i

Mathematical Problem Solving

Mathematical Problem Solving Author ALAN H. SCHOENFELD
ISBN-10 9781483295480
Release 2014-06-28
Pages 409
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This book is addressed to people with research interests in the nature of mathematical thinking at any level, to people with an interest in "higher-order thinking skills" in any domain, and to all mathematics teachers. The focal point of the book is a framework for the analysis of complex problem-solving behavior. That framework is presented in Part One, which consists of Chapters 1 through 5. It describes four qualitatively different aspects of complex intellectual activity: cognitive resources, the body of facts and procedures at one's disposal; heuristics, "rules of thumb" for making progress in difficult situations; control, having to do with the efficiency with which individuals utilize the knowledge at their disposal; and belief systems, one's perspectives regarding the nature of a discipline and how one goes about working in it. Part Two of the book, consisting of Chapters 6 through 10, presents a series of empirical studies that flesh out the analytical framework. These studies document the ways that competent problem solvers make the most of the knowledge at their disposal. They include observations of students, indicating some typical roadblocks to success. Data taken from students before and after a series of intensive problem-solving courses document the kinds of learning that can result from carefully designed instruction. Finally, observations made in typical high school classrooms serve to indicate some of the sources of students' (often counterproductive) mathematical behavior.

The Mathematics Enthusiast Issue

The Mathematics Enthusiast Issue Author Bharath Sriraman
ISBN-10 9781681232584
Release 2015-10-01
Pages 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.

Rippling Meta Level Guidance for Mathematical Reasoning

Rippling  Meta Level Guidance for Mathematical Reasoning Author Alan Bundy
ISBN-10 052183449X
Release 2005-06-30
Pages 202
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Rippling is a radically new technique for the automation of mathematical reasoning. It is widely applicable whenever a goal is to be proved from one or more syntactically similar givens. It was originally developed for inductive proofs, where the goal was the induction conclusion and the givens were the induction hypotheses. It has proved to be applicable to a much wider class of tasks, from summing series via analysis to general equational reasoning. The application to induction has especially important practical implications in the building of dependable IT systems, and provides solutions to issues such as the problem of combinatorial explosion. Rippling is the first of many new search control techniques based on formula annotation; some additional annotated reasoning techniques are also described here. This systematic and comprehensive introduction to rippling, and to the wider subject of automated inductive theorem proving, will be welcomed by researchers and graduate students alike.

Effective Mathematics Lessons through an Eclectic Singapore Approach

Effective Mathematics Lessons through an Eclectic Singapore Approach Author Khoon Yoong Wong
ISBN-10 9789814696449
Release 2015-05-21
Pages 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

Common Sense Reasoning and Rationality

Common Sense  Reasoning  and Rationality Author Renee Elio
ISBN-10 9780198033684
Release 2002-02-07
Pages 288
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As the eleventh volume in the New Directions in Cognitive Science series (formerly the Vancouver Studies in Cognitive Science series), this work promises superb scholarship and interdisciplinary appeal. It addresses three areas of current and varied interest: common sense, reasoning, and rationality. While common sense and rationality often have been viewed as two distinct features in a unified cognitive map, this volume offers novel, even paradoxical, views of the relationship. Comprised of outstanding essays from distinguished philosophers, it considers what constitutes human rationality, behavior, and intelligence covering diverse areas of philosophy, psychology, cognitive science, and computer science. Indeed, it is at the forefront of cognitive research and promises to be of unprecedented influence across numerous disciplines.

Heuristic Strategies in the Speeches of Cicero

Heuristic Strategies in the Speeches of Cicero Author Gábor Tahin
ISBN-10 9783319017990
Release 2013-11-23
Pages 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.

Theories of Mathematics Education

Theories of Mathematics Education Author Bharath Sriraman
ISBN-10 9783642007422
Release 2009-10-13
Pages 668
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Advances in Mathematics Education is a new and innovative book series published by Springer that builds on the success and the rich history of ZDM—The Inter- tional Journal on Mathematics Education (formerly known as Zentralblatt für - daktik der Mathematik). One characteristic of ZDM since its inception in 1969 has been the publication of themed issues that aim to bring the state-of-the-art on c- tral sub-domains within mathematics education. The published issues include a rich variety of topics and contributions that continue to be of relevance today. The newly established monograph series aims to integrate, synthesize and extend papers from previously published themed issues of importance today, by orienting these issues towards the future state of the art. The main idea is to move the ?eld forward with a book series that looks to the future by building on the past by carefully choosing viable ideas that can fruitfully mutate and inspire the next generations. Taking ins- ration from Henri Poincaré (1854–1912), who said “To create consists precisely in not making useless combinations and in making those which are useful and which are only a small minority.

TUD Faculty of Technical Mathematics and Informatics

TUD Faculty of Technical Mathematics and Informatics Author N. Roos
ISBN-10 OCLC:976903542
Release 1988
Pages 17
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TUD Faculty of Technical Mathematics and Informatics has been writing in one form or another for most of life. You can find so many inspiration from TUD Faculty of Technical Mathematics and Informatics also informative, and entertaining. Click DOWNLOAD or Read Online button to get full TUD Faculty of Technical Mathematics and Informatics book for free.

Learning Search Control Knowledge for Equational Deduction

Learning Search Control Knowledge for Equational Deduction Author Stephan Schulz
ISBN-10 1586031503
Release 2000
Pages 182
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This thesis presents an approach to learning good search guiding heuristics for the supposition-based theorom prover E in equational deductions. Search decisions from successful proof searches are represented as sets annotated clause patterns. Term Space Mapping, an alternative learning method for recursive structures is used to learn heuristic evaluation functions for the evaluation of potential new consequences. Experimental results with extended system E/TSM show the success of the approach. Additional contributions of the thesis are an extended superposition calculus and a description of both the proof procedure and the implementation of a state-of-the-art equational theorem prover.

Automated Reasoning and Mathematics

Automated Reasoning and Mathematics Author Maria Paola Bonacina
ISBN-10 9783642366758
Release 2013-02-28
Pages 259
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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.