Formal learning theory is the mathematical embodiment of a normative epistemology. It deals with the question of how an agent should use observations about her environment to arrive at correct and informative conclusions. Philosophers such as Putnam, Glymour and Kelly have developed learning theory as a normative framework for scientific reasoning and inductive inference.

Explicit concepts and sufficiently precise definitions are the basis for further advance of a science beyond a given level. To move toward a situation where the whole population has access to the authentic results of science (italics mine) requires making explicit some general philosophical principles which can help to guide the learning, development, and use of mathematics, a science which clearly plays a pivotal role regarding the learning, development and use of all the sciences. Such philosophical principles have (...) not come from speculation but from studying and concentrating the development of actual mathematical subjects such as algebraic geometry, functional analysis, continuum mechanics, combinatorics, etc. The simplifications in the conceptual relations revealed by such philosophical/mathematical advances are of great value, not only in: (1) clarifying and guiding further research in the mathematical sciences, but also in (2) reforming pedagogy and popularization in ways that will not prove deceptive. (shrink)

The present work determines the arithmetic complexity of the index sets of u.c.e. families which are learnable according to various criteria of algorithmic learning. Specifically, we prove that the index set of codes for families that are TxtFex\-learnable is \-complete and that the index set of TxtFex\-learnable and the index set of TxtFext\-learnable families are both \-complete.

The theory of objectification offers a perspective to conceptualize learning as a collective cultural-historical process and to transform classrooms into sites of communal life where students make the experience of an ethics of solidarity, plurality, and inclusivity.

This article argues that a successful answer to Hume's problem of induction can be developed from a sub-genre of philosophy of science known as formal learning theory. One of the central concepts of formal learning theory is logical reliability: roughly, a method is logically reliable when it is assured of eventually settling on the truth for every sequence of data that is possible given what we know. I show that the principle of induction (PI) is necessary (...) and sufficient for logical reliability in what I call simple enumerative induction. This answer to Hume's problem rests on interpreting PI as a normative claim justified by a non-empirical epistemic means-ends argument. In such an argument, a rule of inference is shown by mathematical or logical proof to promote a specified epistemic end. Since the proof concerning PI and logical reliability is not based on inductive reasoning, this argument avoids the circularity that Hume argued was inherent in any attempt to justify PI. (shrink)

I develop an account of how mathematized theories in physics represent physical systems, in response to the frequent claim that any such account must presuppose a non-mathematized, and usually linguistic, description of the system represented. The account I develop contains a circularity, in that representation is a mathematical relation between the models of a theory and the system as represented by some other model --- but I argue that this circularity is not vicious, in any case refers in (...) linguistic accounts of meaning and representation, and is simply a consequence of the fact that we have no unmediated, representation-independent access to the world. (shrink)

Open peer commentary on the article “Examining the Role of Re-Presentation in Mathematical Problem Solving: An Application of Ernst von Glasersfeld’s Conceptual Analysis” by Victor V. Cifarelli & Volkan Sevim. Upshot: The value of Cifarelli & Sevim’s target article lies in the analysis of how reflective abstraction contributes to the description of mathematical learning through problem solving. The additional value of the article lies in its emphasis of some aspects of the learning process that goes beyond (...) radical constructivist learning theory. I will look for common ground between the humanist philosophy of mathematics and radical constructivism. By doing so, I want to stress two converging elements: (i) the move away from traditionalist ontological positions and (ii) the central role of the students’ activity in the learning process. (shrink)

I. Prelude -- About this Book -- II. School Mathematics and Its Consequences -- The Foundations of Mathematical Thinking -- Compression, Connection and Blending of Mathematical Ideas -- Set-befores, Met-befores and Long-term Learning -- Mathematics and the Emotions -- The Three Worlds of Mathematics -- Journeys through Embodiment and Symbolism -- Problem-Solving and Proof -- III. Interlude -- The Historical Evolution of Mathematics -- IV. University Mathematics and Beyond -- The Transition to Formal Knowledge -- Blending Knowledge (...) Structures in the Calculus -- Expert Thinking and Structure Theorems -- Contemplating the Infinitely Large and the Infinitely Small -- Expanding the Frontiers through Mathematical Research -- Reflections -- Appendix: Where the Ideas Came From. (shrink)

College students struggle with the switch from thinking of mathematics as a calculation based subject to a problem solving based subject. This book describes how the introduction to proofs course can be taught in a way that gently introduces students to this new way of thinking. This introduction utilizes recent research in neuroscience regarding how the brain learns best. Rather than jumping right into proofs, students are first taught how to change their mindset about learning, how to persevere through (...) difficult problems, how to work successfully in a group, and how to reflect on their learning. With these tools in place, students then learn logic and problem solving as a further foundation. Next various proof techniques such as direct proofs, proof by contraposition, proof by contradiction, and mathematical induction are introduced. These proof techniques are introduced using the context of number theory. The last chapter uses Calculus as a way for students to apply the proof techniques they have learned. (shrink)

Topics covered in Discrete Mathematics have become essential tools in many areas of studies in recent years. This is primarily due to the revolution in technology, communications, and cyber security. The book treats major themes in a typical introductory modern Discrete Mathematics course: Propositional and predicate logic, proof techniques, set theory (including Boolean algebra, functions and relations), introduction to number theory, combinatorics and graph theory. An accessible, precise, and comprehensive approach is adopted in the treatment of each (...) topic. The ability of abstract thinking and the art of writing valid arguments are emphasized through detailed proof of (almost) every result. Developing the ability to think abstractly and roguishly is key in any areas of science, information technology and engineering. Every result presented in the book is followed by examples and applications to consolidate its comprehension. The hope is that the reader ends up developing both the abstract reasoning as well as acquiring practical skills. All efforts are made to write the book at a level accessible to first-year students and to present each topic in a way that facilitates self-directed learning. Each chapter starts with basic concepts of the subject at hand and progresses gradually to cover more ground on the subject. Chapters are divided into sections and subsections to facilitate readings. Each section ends with its own carefully chosen set of practice exercises to reenforce comprehension and to challenge and stimulate readers. As an introduction to Discrete Mathematics, the book is written with the smallest set of prerequisites possible. Familiarity with basic mathematical concepts (usually acquired in high school) is sufficient for most chapters. However, some mathematical maturity comes in handy to grasp some harder concepts presented in the book. (shrink)

Mathematical realism is the doctrine that mathematical objects really exist, that mathematical statements are either determinately true or determinately false, and that the accepted mathematical axioms are predominantly true. A realist understanding of set theory has it that when the sentences of the language of set theory are understood in their standard meaning, each sentence has a determinate truth value, so that there is a fact of the matter whether the cardinality of the continuum (...) is א2 or whether there are measurable cardinals, whether or not those facts are knowable by us. (shrink)

Mathematical realism is the doctrine that mathematical objects really exist, that mathematical statements are either determinately true or determinately false, and that the accepted mathematical axioms are predominantly true. A realist understanding of set theory has it that when the sentences of the language of set theory are understood in their standard meaning, each sentence has a determinate truth value, so that there is a fact of the matter whether the cardinality of the continuum (...) is א2 or whether there are measurable cardinals, whether or not those facts are knowable by us. (shrink)

This lively introductory text exposes the student in the humanities to the world of discrete mathematics. A problem-solving based approach grounded in the ideas of George Pólya are at the heart of this book. Students learn to handle and solve new problems on their own. A straightforward, clear writing style and well-crafted examples with diagrams invite the students to develop into precise and critical thinkers. Particular attention has been given to the material that some students find challenging, such as proofs. (...) This book illustrates how to spot invalid arguments, to enumerate possibilities, and to construct probabilities. It also presents case studies to students about the possible detrimental effects of ignoring these basic principles. The book is invaluable for a discrete and finite mathematics course at the freshman undergraduate level or for self-study since there are full solutions to the exercises in an appendix. (shrink)

About 25 years ago, it came to light that a single combinatorial property determines both an important dividing line in model theory and machine learning. The following years saw a fruitful exchange of ideas between PAC-learning and the model theory of NIP structures. In this article, we point out a new and similar connection between model theory and machine learning, this time developing a correspondence between stability and learnability in various settings of online (...) class='Hi'>learning. In particular, this gives many new examples of mathematically interesting classes which are learnable in the online setting. (shrink)

The number of studies related to natural and artificial mechanisms of learning rapidly increases. However, there is no general theory of learning that could provide a unifying basis for exploring different directions in this growing field. For a long time the development of such a theory has been hindered by nativists' belief that the development of a biological organism during ontogeny should be viewed as parameterization of an innate, encoded in the genome structure by an innate (...) algorithm, and nothing essentially new is created during this process. Noam Chomsky has claimed, therefore, that the creation of a non-trivial general mathematical theory of learning is not feasible, since any algorithm cannot produce a more complex algorithm. This study refutes the above argumentation by developing a counter-example based on the mathematical theory of algorithms and computable functions. It introduces a novel concept of a Universal Learning System (ULS) capable of learning to control in an optimal way any given constructive system from a certain class. The necessary conditions for the existence of a ULS and its main functional properties are investigated. The impossibility of building an algorithmic ULS for a sufficiently complex class of controlled objects is shown, and a proof of the existence of a non-algorithmic ULS based on the axioms of classical mathematics is presented. It is argued that a non-algorithmic ULS is a legitimate object of not only mathematics, but also the world of nature. These results indicate that an algorithmic description of the organization and adaptive development of biological systems in general is not sufficient. At the same time, it is possible to create a rigorous non-algorithmic general theory of learning as a theory of ULS. The utilization of this framework for integrating learning-related studies is discussed. (shrink)

Theories are part and parcel of just about every human activity that involves knowing about the world and our place in it. In all areas of inquiry from the most mundane to the most esoteric and sophisticated, theorizing plays a fundamental role. What is true of our everyday existence is even more pervasive in more scholarly fields. How is thinking about the subject organized? What methods are used in moving a neophyte in a given subject matter into the position of (...) a competent student of that subject? What, in short, is the prime repository and vehicle for the knowledge that has been acquired in any discipline over the course of its development? The answer is the theories that are used, produced, and promulgated by the practitioners of those disciplines. The SAGE Encyclopedia of Theory is a landmark work serving as something of a capstone to 10 prior SAGE encyclopedias that examined theory in specific social science disciplines (psychology, sociology, education, political science, etc.). Within the pages of this more general work, readers will learn about the complex decisions that are made when framing theory, what goes into constructing a powerful theory, why some theories change or fail, how theories reflect socio-historical moments in time and how – at their best – they form the foundations for exploring and unlocking the mysteries of the world around us. Key features include: · Over 300 signed articles written by key figures across the social sciences spread across four comprehensive volumes · Further Readings and Cross References conclude each article aiding readers further in their research · A Reader’s Guide organizes entries by broad themes. · A Resource Guide takes readers to the next step in their research journeys by listing classic books in the field, key journals, associations, and websites. (shrink)

Few philosophers that have been studied as much as Ibn Sīnā have been as much misunderstood. His extraordinary ability to reflect upon and write in a variety of styles about seemingly every topic in every domain has steered his thought from philosophy and theology to mysticism and esoterism. Instead of helping us to learn and understand better Ibn Sīnā than he has previously been understood, the recent surge of Avicennan studies only adds more confusion to the already complex social context (...) which he was living in. (shrink)

For parents wanting their children to get a head start in reading, it can be a challenge to find something that will maintain their attention. Now, learning to read can become a fun and enter- taining thing to do with the help of an extraordinary cat. Join Cleo-cat-tra as she brings reading to life in the charming picture book Rhymes and Times with Cleo-cat-tra by Lucy T. Geringer and illustrated by Bernardita Cox Kollock. Rhymes and Times of Cleo-cat-tra is (...) a picture book about a special cat named Cleo-cat-tra. She is not an ordinary cat, as she cleverly rhymes about animals, people, and things she encounters inside and outside her house. The rhymes are silly which make learning a fun and creative pro- cess for kids and for parents to share. This delightful story makes use of colorful illustrations to heighten a child's imagination. Its simple text and brief sentences make reading easy for the early reader. It is also a great book for children who are learning English and can be enjoyed both at home and in school through lively interactive discussions between students and teachers. It is a book for all ages. (shrink)

This textbook for postgraduate students learning mathematical methods in economics provides a comprehensive account of mathematics required to analyse and solve problems of choice encountered by economists. It looks at a wide variety of decision-making problems, both static and dynamic, in various contexts and provides mathematical foundations for the relevant economic theory.

In this report on an interview-based school case study undertaken with seven school leaders using component theory analysis and the hermeneutic method, we reveal the relational essence of learning design at the Australian Science and Mathematics School. The phenomenon of learning togetherness presents, forged by deliberately practised notions of contributive leadership within open learning spaces and ongoing attention to new interdisciplinary curriculum forms. This case study highlights the phenomenological nature of a school that has been deliberately (...) purposed for deep collaborative learning forms, respecting student and teacher ideas in the process, and marginalising habitual industrial school design forms that constrain effective student and teacher learning. The study has relevance for school leaders and teachers wishing to pursue new school design forms within enabling learning cultures that attend more closely to the learning needs of young people poised to enter the Third and Fourth Industrial Revolutions. (shrink)

As is well known, the late Husserl warned against the dangers of reifying and objectifying the mathematical models that operate at the heart of our physical theories. Although Husserl’s worries were mainly directed at Galilean physics, the first aim of our paper is to show that many of his critical arguments are no less relevant today. By addressing the formalism and current interpretations of quantum theory, we illustrate how topics surrounding the mathematization of nature come to the fore (...) naturally. Our second aim is to consider the program of reconstructing quantum theory, a program that currently enjoys popularity in the field of quantum foundations. We will conclude by arguing that, seen from this vantage point, certain insights delivered by phenomenology and quantum theory regarding perspectivity are remarkably concordant. Our overall hope with this paper is to show that there is much room for mutual learning between phenomenology and modern physics. (shrink)

In recent years, several remarkable results have shown that certain theorems of finite combinatorics are unprovable in certain logical systems. These developments have been instrumental in stimulating research in both areas, with the interface between logic and combinatorics being especially important because of its relation to crucial issues in the foundations of mathematics which were raised by the work of Kurt Godel. Because of the diversity of the lines of research that have begun to shed light on these issues, there (...) was a need for a comprehensive overview which would tie the lines together. This volume fills that need by presenting a balanced mixture of high quality expository and research articles that were presented at the August 1985 AMS-IMS-SIAM Joint Summer Research Conference, held at Humboldt State University in Arcata, California.With an introductory survey to put the works into an appropriate context, the collection consists of papers dealing with various aspects of 'unprovable theorems and fast-growing functions'. Among the topics addressed are: ordinal notations, the dynamical systems approach to Ramsey theory, Hindman's finite sums theorem and related ultrafilters, well quasiordering theory, uncountable combinatorics, nonstandard models of set theory, and a length-of-proof analysis of Godel's incompleteness theorem. Many of the articles bring the reader to the frontiers of research in this area, and most assume familiarity with combinatorics and/or mathematical logic only at the senior undergraduate or first-year graduate level. (shrink)

This paper develops a methodology for using Activity Theory (AT) to investigate pedagogical practices in primary school mathematics classrooms by selecting object-oriented pedagogical activity as the unit of analysis. While an understanding of object-oriented activity is central to Activity Theory (AT), the notion of object is a frequently debated and often misunderstood one. The conceptual confusion surrounding the object arises both from difficulties related to translating the original Russian conceptualisation of object-oriented activity into English as well as from (...) the different interpretations of the object currently in use within two contemporary approaches in activity theory. This paper seeks to clarify understandings of the object by exploring notions of object oriented activity. To this end, the paper traces the historical development of the object through Leontiev (1978; 1981) and Engeström’s (1987; 1999) expansion of Vygotsky’s original triadic understanding of object oriented activity. Drawing on Basil Bernstein’s (1996) notion of evaluative criteria as those rules that transmit the criteria for the production of a legitimate text, the paper goes on to elaborate a methodology for using AT to analyse observational data by developing the notion of “evaluative episodes” as pedagogical moments in which the pedagogical object is made visible. Findings indicate that an evaluative episode can serve as an analytical space in which the dynamism of an activity system is momentarily frozen, enabling one to model human activity in the system under investigation and, hence, in this study, to understand pedagogy in context. (shrink)

Machine learning (ML) refers to a class of computer-facilitated methods of statistical modelling. ML modelling techniques are now being widely adopted across the sciences. A number of outspoken representatives from the general public, computer science, various scientific fields, and philosophy of science alike seem to share in the belief that ML will radically disrupt scientific practice or the variety of epistemic outputs science is capable of producing. Such a belief is held, at least in part, because its adherents take (...) ML to exist on novel epistemic footing relative to classical mathematical or statistical modelling approaches utilised in science. Namely, they take modelling with ML to be a “theory-free” enterprise, in the sense of not resting essentially on input from human conceptual grasp on the target phenomenon and domain expertise. I take this view to arise from the further, and more deeply entrenched belief that data is worldly and objective; i.e., data is viewed as recapitulating or representing with perfect fidelity the structure or properties of the systems in nature it is sampled from. Yet most contemporary philosophers of science take on board, in one version or other, the thesis of theory-ladenness or theory-mediation of observation or measurement. From this, it follows that most philosophers of science (and, I will venture, many scientists) hold irreconcilable views on the nature of data, an internal tension which threatens the integrity of their appraisals of ML in science. Taking the thesis of theory-ladenness on board---and its implications for the nature of data seriously---it follows that there is no reason to believe that ML differs fundamentally in its epistemic footing from established mathematical modelling approaches in the sciences. I show that usage and interpretation can differ from “standard practice” in scientific projects which wield the tools of ML without accepting a difference in the epistemic or representational status of such tools. (shrink)

Undergraduate students with no prior classroom instruction in mathematical logic will benefit from this evenhanded multipart text by one of the centuries greatest authorities on the subject. Part I offers an elementary but thorough overview of mathematical logic of first order. The treatment does not stop with a single method of formulating logic; students receive instruction in a variety of techniques, first learning model theory (truth tables), then Hilbert-type proof theory, and proof theory handled (...) through derived rules. Part II supplements the material covered in Part I and introduces some of the newer ideas and the more profound results of logical research in the twentieth century. Subsequent chapters introduce the study of formal number theory, with surveys of the famous incompleteness and undecidability results of Godel, Church, Turing, and others. The emphasis in the final chapter reverts to logic, with examinations of Godel's completeness theorem, Gentzen's theorem, Skolem's paradox and nonstandard models of arithmetic, and other theorems. Unabridged republication of the edition published by John Wiley & Sons, Inc. New York, 1967. Preface. Bibliography. Theorem and Lemma Numbers: Pages. List of Postulates. Symbols and Notations. Index. (shrink)

A U-shaped curve in a cognitive-developmental trajectory refers to a three-step process: good performance followed by bad performance followed by good performance once again. U-shaped curves have been observed in a wide variety of cognitive-developmental and learning contexts. U-shaped learning seems to contradict the idea that learning is a monotonic, cumulative process and thus constitutes a challenge for competing theories of cognitive development and learning. U-shaped behavior in language learning (in particular in learning English (...) past tense) has become a central topic in the Cognitive Science debate about learning models. Antagonist models (e.g., connectionism versus nativism) are often judged on their ability of modeling or accounting for U-shaped behavior. The prior literature is mostly occupied with explaining how U-shaped behavior occurs. Instead, we are interested in the necessity of this kind of apparently inefficient strategy. We present and discuss a body of results in the abstract mathematical setting of (extensions of) Gold-style computational learning theory addressing a mathematically precise version of the following question: Are there learning tasks that require U-shaped behavior? All notions considered are learning in the limit from positive data. We present results about the necessity of U-shaped learning in classical models of learning as well as in models with bounds on the memory of the learner. The pattern emerges that, for parameterized, cognitively relevant learning criteria, beyond very few initial parameter values, U-shapes are necessary for full learning power! We discuss the possible relevance of the above results for the Cognitive Science debate about learning models as well as directions for future research. (shrink)

1971. Discourse Grammars and the Structure of Mathematical Reasoning II: The Nature of a Correct Theory of Proof and Its Value, Journal of Structural Learning 3, #2, 1–16. REPRINTED 1976. Structural Learning II Issues and Approaches, ed. J. Scandura, Gordon & Breach Science Publishers, New York, MR56#15263. -/- This is the second of a series of three articles dealing with application of linguistics and logic to the study of mathematical reasoning, especially in the setting of (...) a concern for improvement of mathematical education. The present article presupposes the previous one. Herein we develop our ideas of the purposes of a theory of proof and the criterion of success to be applied to such theories. In addition we speculate at length concerning the specific kinds of uses to which a successful theory of proof may be put vis-a-vis improvement of various aspects of mathematical education. The final article will deal with the construction of such a theory. The 1st is the 1971. Discourse Grammars and the Structure of Mathematical Reasoning I: Mathematical Reasoning and Stratification of Language, Journal of Structural Learning 3, #1, 55–74. https://www.academia.edu/s/fb081b1886?source=link . (shrink)

In formal semantics, structure is treated as the essential ingredient in the creation of sentence meaning from individual word meaning. This book introduces some of the foundational concepts, principles and techniques in the formal semantics of natural language and outlines the mathematical principles that underlie linguistics meaning. Using English examples, Yoad Winter presents the most useful tools and concepts of formal semantics in an accessible style and includes a variety of practical exercises so that readers can learn to utilize (...) these tools effectively. For readers with an elementary background in set theory and linguistics or with an interest in mathematical modelling, this fascinating study is an ideal introduction to natural language semantics. Designed as a quick yet thorough introduction to one of the most vibrant areas of research in modern linguistics today this volume reveals the beauty and elegance of the mathematical study of meaning. (shrink)

This paper summarizes a theory of cognitive development and elaborates on its educational implications. The theory postulates that development occurs in cycles along multiple fronts. Cognitive competence in each cycle comprises a different profile of executive, inferential, and awareness processes, reflecting changes in developmental priorities in each cycle. Changes reflect varying needs in representing, understanding, and interacting with the world. Interaction control dominates episodic representation in infancy; attention control and perceptual awareness dominate in realistic representations in preschool; inferential (...) control and awareness dominate rule-based representation in primary school; truth and validity control and precise self-evaluation dominate in principle-based thought in adolescence. We demonstrate that the best predictors of school learning in each cycle are the cycle’s cognitive priorities. Also learning in different domains, e.g., language and mathematics, depends on an interaction between the general cognitive processes dominating in each cycle and the state of the representational systems associated with each domain. When a representational system is deficient, specific learning difficulties may emerge, e.g., dyslexia and dyscalculia. We also discuss the educational implications for evaluation and learning at school. (shrink)

The goal of this unique text is to provide an "experience" that would facilitate a better transition for mathematics majors to the advanced proof-based courses required for their major. If you "love mathematics, but I hate proofs" this book is for you. Example-based courses such as introductory Calculus transition somewhat abruptly, and without a warning label, to proof-based courses, and may leave students with the unpleasant feeling that a subject they loved has turned into material they find hard to understand. (...) The book exposes students and readers to the fundamental nature and principles of constructing mathematical proofs and in the context of main courses required for the major, e.g., probability, linear algebra, real analysis, and abstract algebra. Four short chapters, each chapter focusing on a particular course, provide a short but rigorous introduction. Students then get a preview of the discipline, its focus, language, mathematical objects of interests, and common methods of proof presented in those courses. Because which ideas apply to which future courses may not be obvious in many transition courses, this structure addresses this need. The book may also be used as a review tool at the end of course and for readers who want to learn the language and scope of the broad disciplines of linear algebra, abstract algebra, real analysis, and probability, before transitioning to these courses. (shrink)

This is a short, modern, and motivated introduction to mathematical logic for upper undergraduate and beginning graduate students in mathematics and computer science. Any mathematician who is interested in getting acquainted with logic and would like to learn Gödel’s incompleteness theorems should find this book particularly useful. The treatment is thoroughly mathematical and prepares students to branch out in several areas of mathematics related to foundations and computability, such as logic, axiomatic set theory, model theory, recursion (...) theory, and computability. In this new edition, many small and large changes have been made throughout the text. The main purpose of this new edition is to provide a healthy first introduction to model theory, which is a very important branch of logic. Topics in the new chapter include ultraproduct of models, elimination of quantifiers, types, applications of types to model theory, and applications to algebra, number theory and geometry. Some proofs, such as the proof of the very important completeness theorem, have been completely rewritten in a more clear and concise manner. The new edition also introduces new topics, such as the notion of elementary class of structures, elementary diagrams, partial elementary maps, homogeneous structures, definability, and many more. (shrink)

The present book of proceedings includes chapters related to the areas of pure, applied and inter-disciplinary mathematics reflecting the potential applications in the domains of sciences and engineering. The main areas include algebra and its applications, analysis and approximation theory, cryptography, computational fluid dynamics, continuum mechanics and vibrations, differential equations and applications, graph theory, fuzzy mathematics and logic, numerical analysis, optimization and its applications, wave propagation, etc. The scientists, engineers, academicians and researchers working in the proposed areas of (...) coding and information theory, computational fluid dynamics, differential equations, fuzzy sets and systems, numerical analysis, optimization, vibrations, etc. looking for new insight and ideas in these areas will be benefitted by the contents of this book. In fact, it will be useful to a large class of readers interested in recent findings related to mathematical sciences and their applications to the diverse domains of knowledge. The book will be a fruitful contribution to all knowledge seekers and researchers in the concerned areas who are looking for new insight and ideas. As it consists of updated research articles on emerging areas of mathematical sciences and their applications, it will be a valued addition to the learning resource centres of various universities, industries, and research and development organizations across the globe. (shrink)

An Introduction to Proof via Inquiry-Based Learning is a textbook for the transition to proof course for mathematics majors. Designed to promote active learning through inquiry, the book features a highly structured set of leading questions and explorations. The reader is expected to construct their own understanding by engaging with the material. The content ranges over topics traditionally included in transitions courses: logic, set theory including cardinality, the topology of the real line, a bit of number (...) class='Hi'>theory, and more. The exposition guides and mentors the reader through an adventure in mathematical discovery, requiring them to solve problems, conjecture, experiment, explore, create, and communicate. Ultimately, this is really a book about productive struggle and learning how to learn. This is a print version of the popular open-access online text by Dana C. Ernst. (shrink)

Product acceptance determination using similarity measure index by neutrosophic statistics / Muhammad Aslam and Rehan Ahmed Khan Sherwani -- New concepts of strongly edge irregular interval-valued neutrosophic graphs / A.A.Talebi, Hossein Rashmanlou and Masoomeh Ghasemi -- The link between neutrosophy and learning : through the related concepts of representation and compression / Philippe Schweizer -- Neutrosophic soft cubic M-subalgebras of B-algebras / Mohsin Khalid, Neha Andaleeb Khalid and Hasan Khalid -- Alpha, beta and gamma product of neutrosophic graphs / (...) Nasir Shah, Said Broumi, Abdul Raheem and Tahir Imran -- Neutrosophic cubic normal ideal and neutrosophic cubic closed normal ideal of PS-algebra / Mohsin Khalid, Hasan khalid and Neha Andaleeb Khalid -- Neutrosophic multisets : an overview / Vakkas Uluçay -- Neutrosophy as a model for knowledge : the influence of representative models on thinking / Philippe Schweizer -- When neutrosophic theory meets three-way decisions / Rui Ren, Chao Zhang and Deyu Li -- Neutrosophic economic order quantity model with more than one price breaks / R. Surya, M. Mullai and G. Madhan Kumar -- General non-square systems of linear equations in single-valued triangular neutrosophic number environment / S.A. Edalatpanah -- Hesitant neutrosophic soft set relations / Abhijit Saha -- Computer application of neutrosophic set operations / S. Saranya, M. Vigneshwaran, Said Broumi -- Triangular bipolar neutrosophic based transportation problem / Avishek Chakraborty -- Studying neutrosophic variables / Rafif Alhabib and Ahmad Salama -- Decision making for logistics center location selection in trapezoidal neutrosophic environment / Kalyan Mondal and Surapati Pramanik -- Weighted similarity measure and decision making model for clinical application of single valued neutrosophic set / R.Binu and P. Isaac -- Rough neutrosophic set : an overview / Surapati Pramanik -- Some new concepts on normalized SVTN-number and multiple criteria decision making / İrfan Deli and Emel Kırmızı Öztürk. (shrink)

Learn how to develop your reasoning skills and how to write well-reasoned proofs Learning to Reason shows you how to use the basic elements of mathematical language to develop highly sophisticated, logical reasoning skills. You'll get clear, concise, easy-to-follow instructions on the process of writing proofs, including the necessary reasoning techniques and syntax for constructing well-written arguments. Through in-depth coverage of logic, sets, and relations, Learning to Reason offers a meaningful, integrated view of modern mathematics, cuts through (...) confusing terms and ideas, and provides a much-needed bridge to advanced work in mathematics as well as computer science. Original, inspiring, and designed for maximum comprehension, this remarkable book: Clearly explains how to write compound sentences in equivalent forms and use them in valid arguments Presents simple techniques on how to structure your thinking and writing to form well-reasoned proofs Reinforces these techniques through a survey of sets-the building blocks of mathematics Examines the fundamental types of relations, which is "where the action is" in mathematics Provides relevant examples and class-tested exercises designed to maximize the learning experience Includes a mind-building game/exercise space at www.wiley.com/products/subject/mathematics/. (shrink)

"In Learning from Our Mistakes: Epistemology for the Real World, Talbott provides a new framework for understanding the history of Western epistemology and uses that framework to propose a new way of understanding rational belief. His proposal makes epistemology relevant to the real world, which he illustrates with a new theory of racial, gender and other kinds of prejudice, a new diagnosis of the sources of the inequity in the U.S. criminal justice system, and insight into the proliferation (...) of tribal and fascist epistemologies based on alt-facts and alt-truth. Talbott's new model of rational belief is not a model of a theorem prover in mathematics. It is a model of a good learner. Being a good learner requires sensitivity to clues, the maginative ability to generate alternative explanatory narratives that fit the clues, and the ability to select the most coherent explanatory narrative. Sensitivity to clues requires sensitivity not only to evidence that supports one's own beliefs, but also to evidence that casts doubt on them. One of the most important characteristics of a good learner is the ability to correct mistakes. Talbott articulates nine principles that help to explain the difference between rational and irrational belief. Talbott contrasts his approach with the approach of historically important philosophers, including Socrates, Plato, Aristotle, Hume, Kant, Wittgenstein, and Kuhn. He also contrasts his approach with a variety of contemporary approaches, including pragmatism, Bayesianism, and naturalism. He responds to the main criticisms of his method from the experimental philosophy literature"--. (shrink)

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

Is rhetoric just a new and trendy way toépater les bourgeois?Unfortunately, I think that the newfound interest of some economists in rhetoric, and particularly Donald McCloskey in his new book and subsequent responses to critics, gives that impression. After economists have worked so hard for the past five decades to learn their sums, differential calculus, real analysis, and topology, it is a fair bet that one could easily hector them about their woeful ignorance of the conjugation of Latin verbs or (...) Aristotle's Six Elements of Tragedy. Moreover, it has certainly become an academic cliché that economists write as gracefully and felicitously as a hundred monkeys chained to broken typewriters. The fact that economists still trot out Keynes's prose in their defense is itself an index of the inarticulate desperation of an inarticulate profession. (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)

We owe most scientific knowledge to methods of inquiry that are never formally analyzed. The analysis of behavior does not call for hypothetico-deductive methods. Statistics, taught in lieu of scientific method, is incompatible with major features of much laboratory research. Squeezing significance out of ambiguous data discourages the more promising step of scrapping the experiment and starting again. As a consequence, psychologists have taken flight from the laboratory. They have fled to Real People and the human interest of “real life,” (...) to Mathematical Models and the elegance of symbolic treatments, to the Inner Man and the explanatory preoccupation with inferred internal mechanisms, and to Laymanship and its appeal to “common sense.” An experimental analysis provides an alternative to these divertissements.The “theories” to which objection is raised here are not the basic assumptions essential to any scientific activity or statements that are not yet facts, but rather explanations which appeal to events taking place somewhere else, at some other level of observation, described in different terms, and measured, if at all, in different dimensions. Three types of learning theories satisfy this definition: physiological theories attempting to reduce behavior to events in the nervous system; mentalistic theories appealing to inferred inner events; and theories of the Conceptual Nervous System offered as explanatory models of behavior. It would be foolhardy to deny the achievements of such theories in the history of science. The question of whether they are necessary, however, has other implications.Experimental material in three areas illustrates the function of theory more concretely. Alternatives to behavior ratios, excitatory potentials, and so on demonstrate the utility of rate or probability of response as the basic datum in learning. Functional relations between behavior and environmental variables provide an account of why learning occurs. Activities such as preferring, choosing, discriminating, and matching can be dealt with solely in terms of behavior, without referring to processes in another dimensional system. The experiments are not offered as demonstrating that theories are not necessary but to suggest an alternative. Theory is possible in another sense. Beyond the collection of uniform relationships lies the need for a formal representation of the data reduced to a minimal number of terms. A theoretical construction may yield greater generality than any assemblage of facts; such a construction will not refer to another dimensional system. (shrink)

ABSTRACT This part of the series has a dual purpose. In the first place we will discuss two kinds of theories of proof. The first kind will be called a theory of linear proof. The second has been called a theory of suppositional proof. The term "natural deduction" has often and correctly been used to refer to the second kind of theory, but I shall not do so here because many of the theories so-called are not of (...) the second kind--they must be thought of either as disguised linear theories or theories of a third kind (see postscript below). The second purpose of this part is 25 to develop some of the main ideas needed in constructing a comprehensive theory of proof. The reason for choosing the linear and suppositional theories for this purpose is because the linear theory includes only rules of a very simple nature, and the suppositional theory can be seen as the result of making the linear theory more comprehensive. CORRECTION: At the time these articles were written the word ‘proof’ especially in the phrase ‘proof from hypotheses’ was widely used to refer to what were earlier and are now called deductions. I ask your forgiveness. I have forgiven Church and Henkin who misled me. (shrink)

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

An Elementary Transition to Abstract Mathematics will help students move from introductory courses to those where rigor and proof play a much greater role. The text is organized into five basic parts: the first looks back on selected topics from pre-calculus and calculus, treating them more rigorously, and it covers various proof techniques; the second part covers induction, sets, functions, cardinality, complex numbers, permutations, and matrices; the third part introduces basic number theory including applications to cryptography; the fourth part (...) introduces key objects from abstract algebra; and the final part focuses on polynomials. Features: The material is presented in many short chapters, so that one concept at a time can be absorbed by the student. Two "looking back" chapters at the outset (pre-calculus and calculus) are designed to start the student's transition by working with familiar concepts. Many examples of every concept are given to make the material as concrete as possible and to emphasize the importance of searching for patterns. A conversational writing style is employed throughout in an effort to encourage active learning on the part of the student. (shrink)

Assuming no previous study in logic, this informal yet rigorous text covers the material of a standard undergraduate first course in mathematical logic, using natural deduction and leading up to the completeness theorem for first-order logic. At each stage of the text, the reader is given an intuition based on standard mathematical practice, which is subsequently developed with clean formal mathematics. Alongside the practical examples, readers learn what can and can't be calculated; for example the correctness of a (...) derivation proving a given sequent can be tested mechanically, but there is no general mechanical test for the existence of a derivation proving the given sequent. The undecidability results are proved rigorously in an optional final chapter, assuming Matiyasevich's theorem characterising the computably enumerable relations. Rigorous proofs of the adequacy and completeness proofs of the relevant logics are provided, with careful attention to the languages involved. Optinal sections discuss the classification of mathematical structures by first-order theories; the required theory of cardinality is developed from scratch. Throughout the book there are notes on historical aspects of the material, and connections with linguistics and computer science, and the discussion of syntax and semantics is influenced by modern linguistic approaches. Two basic themes in recent cognitive science studies of actual human reasoning are also introduced. Including extensive exercises and selected solutions, this text is ideal for students in logic, mathematics, philosophy, and computer science. (shrink)

In every introductory course on logic, students learn that expressions like ‘somebody’, ‘nothing’ or ‘every woman’ are not names or referring expressions, but quantifiers, and that, owing to this,...

We want to highlight the strong link between learning systems such as deep learning neural networks and neutrosophy. The latter is above all a representation considering a neutral state which is at the heart of many phenomena of reality as well as mathematical and information theories. Here, we start from the recent understanding of neural networks, which considers their internal functioning and the learning that characterizes them as based on adapted representations (both of the information to (...) be processed and of the task to be performed) that are in fact optimal compressions. Compression means discriminating the similar and the different between cases by identifying representative characteristics through focusing on the most significant and rejecting of the rest. These three operations of neural processing, learning and compression owe their existence to non-linear treatments which essentially amount to distinguishing a neutral zone between two filtered zones, corresponding to the neutrosophical approach. (shrink)