The operation of developing a concept is a common procedure in mathematics and in natural science, but has traditionally seemed much less possible to philosophers and, especially, logicians. Meir Buzaglo's innovative study proposes a way of expanding logic to include the stretching of concepts, while modifying the principles which block this possibility. He offers stimulating discussions of the idea of conceptual expansion as a normative process, and of the relation of conceptual expansion to truth, meaning, reference, ontology and paradox, and analyzes the views of Kant, Wittgenstein, Godel, and others, paying especially close attention to Frege. His book will be of interest to a wide range of readers, from philosophers (of logic, mathematics, language, and science) to logicians, mathematicians, linguists, and cognitive scientists.

This book offers an up-to-date overview of the research on philosophy of mathematics education, one of the most important and relevant areas of theory. The contributions analyse, question, challenge, and critique the claims of mathematics education practice, policy, theory and research, offering ways forward for new and better solutions. The book poses basic questions, including: What are our aims of teaching and learning mathematics? What is mathematics anyway? How is mathematics related to society in the 21st century? How do students learn mathematics? What have we learnt about mathematics teaching? Applied philosophy can help to answer these and other fundamental questions, and only through an in-depth analysis can the practice of the teaching and learning of mathematics be improved. The book addresses important themes, such as critical mathematics education, the traditional role of mathematics in schools during the current unprecedented political, social, and environmental crises, and the way in which the teaching and learning of mathematics can better serve social justice and make the world a better place for the future.

Basic Mathematics is aimed primarily at management students preparing to write the GMAT test which requires a strong foundation in fundamentals of mathematics. It is also of value to anyone wanting a general revision of basic mathematics. The text focuses only on those areas of mathematics required for the GMAT test, consisting of four main topics: basic arithmetic; fundamental algebra; geometry and introductory statistics. After a brief review of each topic's basic rules and methods, there is at least one worked example followed by an extensive set of self-practice exercises. The student should attempt as many exercises as is necessary to master the topic.

From an infant's first grasp of quantity to Einstein's theory of relativity, the human experience of number has intrigued researchers for centuries. Numeracy and mathematics have played fundamental roles in the development of societies and civilisations, and yet there is an essential mystery to these concepts, evidenced by the fear many people still feel when confronted by apparently simple sums. Including perspectives from anthropology, education and psychology, The Nature and Development of Mathematics addresses three core questions: Is maths natural? What is the impact of our culture and environment on mathematical thinking? And how can we improve our mathematical ability? Examining the cognitive processes that we use, the origins of these skills and their cultural context, and how learning and teaching can be supported in the classroom, the book contextualises each issue within the wider field, arguing that only by taking a cross-disciplinary perspective can we fully understand what it means to be numerate, as well as how we become numerate in our modern world. This is a unique collection including contributions from a range of renowned international researchers. It will be of interest to students and researchers across cognitive psychology, cultural anthropology and educational research.

Mathematical Puzzle Tales from Mount Olympus uses fascinating tales from Greek Mythology as the background for introducing mathematics puzzles to the general public. A background in high school mathematics will be ample preparation for using this book, and it should appeal to anyone who enjoys puzzles and recreational mathematics. Features: Combines the arts and science, and emphasizes the fact that mathematics straddles both domains. Great resource for students preparing for mathematics competitions, and the trainers of such students.

The present book is an introduction to the philosophy of mathematics. It asks philosophical questions concerning fundamental concepts, constructions and methods - this is done from the standpoint of mathematical research and teaching. It looks for answers both in mathematics and in the philosophy of mathematics from their beginnings till today. The reference point of the considerations is the introducing of the reals in the 19th century that marked an epochal turn in the foundations of mathematics. In the book problems connected with the concept of a number, with the infinity, the continuum and the infinitely small, with the applicability of mathematics as well as with sets, logic, provability and truth and with the axiomatic approach to mathematics are considered. In Chapter 6 the meaning of infinitesimals to mathematics and to the elements of analysis is presented. The authors of the present book are mathematicians. Their aim is to introduce mathematicians and teachers of mathematics as well as students into the philosophy of mathematics. The book is suitable also for professional philosophers as well as for students of philosophy, just because it approaches philosophy from the side of mathematics. The knowledge of mathematics needed to understand the text is elementary. Reports on historical conceptions. Thinking about today's mathematical doing and thinking. Recent developments. Based on the third, revised German edition. For mathematicians - students, teachers, researchers and lecturers - and readersinterested in mathematics and philosophy. Contents On the way to the reals On the history of the philosophy of mathematics On fundamental questions of the philosophy of mathematics Sets and set theories Axiomatic approach and logic Thinking and calculating infinitesimally - First nonstandard steps Retrospection

One of the greatest mathematicians in the world, Michael Atiyah has earned numerous honors, including a Fields Medal, the mathematical equivalent of the Nobel Prize. While the focus of his work has been in the areas of algebraic geometry and topology, he has also participated in research with theoretical physicists. For the first time, these volumes bring together Atiyah's collected papers--both monographs and collaborative works-- including those dealing with mathematical education and current topics of research such as K-theory and gauge theory. The volumes are organized thematically. They will be of great interest to research mathematicians, theoretical physicists, and graduate students in these areas.

This book gathers the proceedings of the conference "Cultures of Mathematics and Logic," held in Guangzhou, China. The event was the third in a series of interdisciplinary, international conferences emphasizing the cultural components of philosophy of mathematics and logic. It brought together researchers from many disciplines whose work sheds new light on the diversity of mathematical and logical cultures and practices. In this context, the cultural diversity can be diachronical (different cultures in different historical periods), geographical (different cultures in different regions), or sociological in nature.

Offers the latest research on the topic.

Offers an English translation and in-depth commentary on the ten extant books of Diophantus of Alexandria's Arithmetica.

Mathematical Recreations from the Tournament of the Towns contains the complete list of problems and solutions to the International Mathematics Tournament of the Towns from Fall 2007 to Spring 2021. The primary audience for this book is the army of recreational mathematicians united under the banner of Martin Gardner. It should also have great value to students preparing for mathematics competitions and trainers of such students. This book also provides an entry point for students in upper elementary schools. Features Huge recreational value to mathematics enthusiasts Accessible to upper-level high school students Problems classified by topics such as two-player games, weighing problems, mathematical tasks etc.

One of the only volumes that brings the humanities, social sciences and even the natural sciences under one remit to look at how they can be researched in an integrated and useful way, with policy and real world implications in terms of how we relate in and to the world. Interdisciplinarity and Transdisciplinarity have been around for a long time, but as as we move through a digital age they are becoming more and more important and interesting to the scholarly community and beyond. There is nothing on the market that pulls all of these subjects across disciplines together and works out a framework to construct the analysis in a way that asks and answers useful questions.

This volume brings together a collection of essays on the history and philosophy of probability and statistics by one of the eminent scholars in these subjects. Written over the last fifteen years, they fall into three broad categories. The first deals with the use of symmetry arguments in inductive probability, in particular, their use in deriving rules of succession (Carnap's 'continuum of inductive methods'). The second group deals with four outstanding individuals who made lasting contributions to probability and statistics in very different ways: Frank Ramsey, R. A. Fisher, Alan Turing, and Abraham de Moivre. The last group of essays deals with the problem of 'predicting the unpredictable' - making predictions when the range of possible outcomes is unknown in advance. The essays weave together the history and philosophy of these subjects and document the fascination that they have exercised for more than three centuries.

This book presents a new nominalistic philosophy of mathematics: semantic conventionalism. Its central thesis is that mathematics should be founded on the human ability to create language - and specifically, the ability to institute conventions for the truth conditions of sentences. This philosophical stance leads to an alternative way of practicing mathematics: instead of "building" objects out of sets, a mathematician should introduce new syntactical sentence types, together with their truth conditions, as he or she develops a theory. Semantic conventionalism is justified first through criticism of Cantorian set theory, intuitionism, logicism, and predicativism; then on its own terms; and finally, exemplified by a detailed reconstruction of arithmetic and real analysis. Also included is a simple solution to the liar paradox and the other paradoxes that have traditionally been recognized as semantic. And since it is argued that mathematics is semantics, this solution also applies to Russell's paradox and the other mathematical paradoxes of self-reference. In addition to philosophers who care about the metaphysics and epistemology of mathematics or the paradoxes of self-reference, this book should appeal to mathematicians interested in alternative approaches.

This book offers an alternative to current philosophy of mathematics: heuristic philosophy of mathematics. In accordance with the heuristic approach, the philosophy of mathematics must concern itself with the making of mathematics and in particular with mathematical discovery. In the past century, mainstream philosophy of mathematics has claimed that the philosophy of mathematics cannot concern itself with the making of mathematics but only with finished mathematics, namely mathematics as presented in published works. On this basis, mainstream philosophy of mathematics has maintained that mathematics is theorem proving by the axiomatic method. This view has turned out to be untenable because of Goedel's incompleteness theorems, which have shown that the view that mathematics is theorem proving by the axiomatic method does not account for a large number of basic features of mathematics. By using the heuristic approach, this book argues that mathematics is not theorem proving by the axiomatic method, but is rather problem solving by the analytic method. The author argues that this view can account for the main items of the mathematical process, those being: mathematical objects, demonstrations, definitions, diagrams, notations, explanations, applicability, beauty, and the role of mathematical knowledge.

This book provides, for the first time, a critical edition and an English translation of the chapter on spheres (goladhyaya) from Nityananda's Sarvasiddhantaraja, a Sanskrit astronomical text written in seventeenth-century Mughal India.

Recursive Functions and Metamathematics deals with problems of the completeness and decidability of theories, using as its main tool the theory of recursive functions. This theory is first introduced and discussed. Then G del's incompleteness theorems are presented, together with generalizations, strengthenings, and the decidability theory. The book also considers the historical and philosophical context of these issues and their philosophical and methodological consequences. Recent results and trends have been included, such as undecidable sentences of mathematical content, reverse mathematics. All the main results are presented in detail. The book is self-contained and presupposes only some knowledge of elementary mathematical logic. There is an extensive bibliography. Readership: Scholars and advanced students of logic, mathematics, philosophy of science.

New to the Fourth Edition Reorganised and revised chapter seven and thirteen New exercises and examples Expanded, updated references Further historical material on figures besides Galois: Omar Khayyam, Vandermonde, Ruffini, and Abel A new final chapter discussing other directions in which Galois Theory has developed: the inverse Galois problem, differential Galois theory, and a (very) brief introduction to p-adic Galois representations.

Brilliant introduction to the philosophy of mathematics, from the question 'what is a number?' up to the concept of infinity, descriptions, classes and axioms Russell deploys all his skills and brilliant prose to write an introductory book - a real gem by one of the 20th century's most celebrated philosophers New foreword by Michael Potter to the Routledge Classics edition places the book in helpful context and explains why it's a classic

Russell's first book on philosophy and a fascinating insight into his early thinking A classic in the history and philosophy of mathematics and logic by one of the greatest philosophers of the 20th century This Routledge Classics edition includes a new foreword by Michael Potter, a renowned expert on analytic philosophy

This monograph considers several well-known mathematical theorems and asks the question, "Why prove it again?" while examining alternative proofs. It explores the different rationales mathematicians may have for pursuing and presenting new proofs of previously established results, as well as how they judge whether two proofs of a given result are different. While a number of books have examined alternative proofs of individual theorems, this is the first that presents comparative case studies of other methods for a variety of different theorems. The author begins by laying out the criteria for distinguishing among proofs and enumerates reasons why new proofs have, for so long, played a prominent role in mathematical practice. He then outlines various purposes that alternative proofs may serve. Each chapter that follows provides a detailed case study of alternative proofs for particular theorems, including the Pythagorean Theorem, the Fundamental Theorem of Arithmetic, Desargues' Theorem, the Prime Number Theorem, and the proof of the irreducibility of cyclotomic polynomials. Why Prove It Again? will appeal to a broad range of readers, including historians and philosophers of mathematics, students, and practicing mathematicians. Additionally, teachers will find it to be a useful source of alternative methods of presenting material to their students.

"The old logic put thought in fetters, while the new logic gives it wings." For the past century, philosophers working in the tradition of Bertrand Russell - who promised to revolutionise philosophy by introducing the 'new logic' of Frege and Peano - have employed predicate logic as their formal language of choice. In this book, Dr David Corfield presents a comparable revolution with a newly emerging logic - modal homotopy type theory. Homotopy type theory has recently been developed as a new foundational language for mathematics, with a strong philosophical pedigree. Modal Homotopy Type Theory: The Prospect of a New Logic for Philosophy offers an introduction to this new language and its modal extension, illustrated through innovative applications of the calculus to language, metaphysics, and mathematics. The chapters build up to the full language in stages, right up to the application of modal homotopy type theory to current geometry. From a discussion of the distinction between objects and events, the intrinsic treatment of structure, the conception of modality as a form of general variation to the representation of constructions in modern geometry, we see how varied the applications of this powerful new language can be.

What is the source of logical and mathematical truth? This volume revitalizes conventionalism as an answer to this question. Conventionalism takes logical and mathematical truth to have their source in linguistic conventions. This was an extremely popular view in the early 20th century, but it was never worked out in detail and is now almost universally rejected in mainstream philosophical circles. In Shadows of Syntax, Jared Warren offers the first book-length treatment and defense of a combined conventionalist theory of logic and mathematics. He argues that our conventions, in the form of syntactic rules of language use, are perfectly suited to explain the truth, necessity, and a priority of logical and mathematical claims. In Part I, Warren explains exactly what conventionalism amounts to and what linguistic conventions are. Part II develops an unrestricted inferentialist theory of the meanings of logical constants that leads to logical conventionalism. This conventionalist theory is elaborated in discussions of logical pluralism, the epistemology of logic, and of the influential objections that led to the historical demise of conventionalism. Part III aims to extend conventionalism from logic to mathematics. Unlike logic, mathematics involves both ontological commitments and a rich notion of truth that cannot be generated by any algorithmic process. To address these issues Warren develops conventionalist-friendly but independently plausible theories of both metaontology and mathematical truth. Finally, Part IV steps back to address big picture worries and meta-worries about conventionalism. This book develops and defends a unified theory of logic and mathematics according to which logical and mathematical truths are reflections of our linguistic rules, mere shadows of syntax.

With a never-before published paper by Lord Henry Cavendish, as well as a biography on him, this book offers a fascinating discourse on the rise of scientific attitudes and ways of knowing. A pioneering British physicist in the late 18th and early 19th centuries, Cavendish was widely considered to be the first full-time scientist in the modern sense. Through the lens of this unique thinker and writer, this book is about the birth of modern science.

The Ga?itatilaka and its Commentary: Two Medieval Sanskrit Mathematical Texts presents the first English annotated translation and analysis of the Ga?itatilaka by Sripati and its Sanskrit commentary by the Jaina monk Si?hatilakasuri (13th century CE). Si?hatilakasuri's commentary upon the Ga?itatilaka is a key text for the study of Sanskrit mathematical jargon and a precious source of information on mathematical practices of medieval India; this is, in fact, the first known Sanskrit mathematical commentary written by a Jaina monk, about whom we have substantial information, to survive to the present day. In presenting the first annotated translation of these two Sanskrit mathematical texts, this volume focusses on language in mathematics and puts forward a novel, fresh approach to Sanskrit mathematical literature which favours linguistic, literary features and textual data. This key resource makes these important texts available in English for the first time for students of Sanskrit, ancient and medieval mathematics, South Asian history, and philology.