This monograph offers a fresh perspective on the applicability of mathematics in science. It explores what mathematics must be so that its applications to the empirical world do not constitute a mystery. In the process, readers are presented with a new version of mathematical structuralism. The author details a philosophy of mathematics in which the problem of its applicability, particularly in physics, in all its forms can be explained and justified. Chapters cover: mathematics as a formal science, mathematical ontology: what does it mean to exist, mathematical structures: what are they and how do we know them, how different layers of mathematical structuring relate to each other and to perceptual structures, and how to use mathematics to find out how the world is. The book simultaneously develops along two lines, both inspired and enlightened by Edmund Husserl's phenomenological philosophy. One line leads to the establishment of a particular version of mathematical structuralism, free of "naturalist" and empiricist bias. The other leads to a logical-epistemological explanation and justification of the applicability of mathematics carried out within a unique structuralist perspective. This second line points to the "unreasonable" effectiveness of mathematics in physics as a means of representation, a tool, and a source of not always logically justified but useful and effective heuristic strategies.

This book gathers the best presentations from the Topic Study Group 30: Mathematics Competitions at ICME-13 in Hamburg, and some from related groups, focusing on the field of working with gifted students. Each of the chapters includes not only original ideas, but also original mathematical problems and their solutions. The book is a valuable resource for researchers in mathematics education, secondary and college mathematics teachers around the globe as well as their gifted students.

This book traces the history of the concept of work from its earliest stages and shows that its further formalization leads to equilibrium principle and to the principle of virtual works, and so pointing the way ahead for future research and applications. The idea that something remains constant in a machine operation is very old and has been expressed by many mathematicians and philosophers such as, for instance, Aristotle. Thus, a concept of energy developed. Another important idea in machine operation is Archimedes' lever principle. In modern times the concept of work is analyzed in the context of applied mechanics mainly in Lazare Carnot mechanics and the mechanics of the new generation of polytechnical engineers like Navier, Coriolis and Poncelet. In this context the word "work" is finally adopted. These engineers are also responsible for the incorporation of the concept of work into the discipline of economics when they endeavoured to combine the study of the work of machines and men together.

This book is the “Study Book” of ICMI-Study no. 20, which was run in cooperation with the International Congress on Industry and Applied Mathematics (ICIAM). The editors were the co-chairs of the study (Damlamian, Straesser) and the organiser of the Study Conference (Rodrigues). The text contains a comprehensive report on the findings of the Study Conference, original plenary presentations of the Study Conference, reports on the Working Groups and selected papers from all over world. This content was selected by the editors as especially pertinent to the study each individual chapter represents a significant contribution to current research.

In this new text, Steven Givant—the author of several acclaimed books, including works co-authored with Paul Halmos and Alfred Tarski—develops three theories of duality for Boolean algebras with operators. Givant addresses the two most recognized dualities (one algebraic and the other topological) and introduces a third duality, best understood as a hybrid of the first two. This text will be of interest to graduate students and researchers in the fields of mathematics, computer science, logic, and philosophy who are interested in exploring special or general classes of Boolean algebras with operators. Readers should be familiar with the basic arithmetic and theory of Boolean algebras, as well as the fundamentals of point-set topology.

This work examines the main directions of research conducted on the history of mathematics education. It devotes substantial attention to research methodologies and the connections between this field and other scholarly fields. The results of a survey about academic literature on this subject are accompanied by a discussion of what has yet to be done and problems that remain unsolved. The main topics you will find in "ICME-13 Topical Survey" include: * Discussions of methodological issues in the history of mathematics education and of the relation between this field and other scholarly fields. * The history of the formation and transformation of curricula and textbooks as a reflection of trends in social-economic, cultural and scientific-technological development. * The influence of politics, ideology and economics on the development of mathematics education, from a historical perspective. * The history of the preeminent mathematics education organizations and the work of leading figures in mathematics education. * Mathematics education practices and tools and the preparation of mathematics teachers, from a historical perspective.

This book collects research papers on the philosophical foundations of probability, causality, spacetime and quantum theory. The papers are related to talks presented in six subsequent workshops organized by The Budapest-Krakow Research Group on Probability, Causality and Determinism. Coverage consists of three parts. Part I focuses on the notion of probability from a general philosophical and formal epistemological perspective. Part II applies probabilistic considerations to address causal questions in the foundations of quantum mechanics. Part III investigates the question of indeterminism in spacetime theories. It also explores some related questions, such as decidability and observation. The contributing authors are all philosophers of science with a strong background in mathematics or physics. They believe that paying attention to the finer formal details often helps avoiding pitfalls that exacerbate the philosophical problems that are in the center of focus of contemporary research. The papers presented here help make explicit the mathematical-structural assumptions that underlie key philosophical argumentations. This formally rigorous and conceptually precise approach will appeal to researchers and philosophers as well as mathematicians and statisticians.

This volume contains the texts and translations of two Arabic treatises on magic squares, which are undoubtedly the most important testimonies on the early history of that science. It is divided into the three parts: the first and most extensive is on tenth-century construction methods, the second is the translations of the texts, and the third contains the original Arabic texts, which date back to the tenth century.

First published in 1903, Principles of Mathematics was Bertrand Russell's first major work in print. It was this title which saw him begin his ascent towards eminence. In this groundbreaking and important work, Bertrand Russell argues that mathematics and logic are, in fact, identical and what is commonly called mathematics is simply later deductions from logical premises. Highly influential and engaging, this important work led to Russell's dominance of analytical logic on western philosophy in the twentieth century.

This Element shows that Plato keeps a clear distinction between mathematical and metaphysical realism and the knife he uses to slice the difference is method. The philosopher's dialectical method requires that we tether the truth of hypotheses to existing metaphysical objects. The mathematician's hypothetical method, by contrast, takes hypotheses as if they were first principles, so no metaphysical account of their truth is needed. Thus, we come to Plato's methodological as-if realism: in mathematics, we treat our hypotheses as if they were first principles, and, consequently, our objects as if they existed, and we do this for the purpose of solving problems. Taking the road suggested by Plato's Republic, this Element shows that methodological commitments to mathematical objects are made in light of mathematical practice; foundational considerations; and, mathematical applicability. This title is also available as Open Access on Cambridge Core.

The papers in this volume address fundamental, and interrelated, philosophical issues concerning modality and identity, issues that have not only been pivotal to the development of analytic philosophy in the twentieth century, but remain a key focus of metaphysical debate in the twenty-first. How are we to understand the concepts of necessity and possibility? Is chance a basic ingredient of reality? How are we to make sense of claims about personal identity? Do numbers require distinctive identity criteria? Does the capacity to identify an object presuppose an ability to bring it under a sortal concept? Rather than presenting a single, partisan perspective, Identity and Modality enriches our understanding of identity and modality by bringing together papers written by leading researchers working in metaphysics, the philosophy of mind, the philosophy of science, and the philosophy of mathematics. The resulting variety of perspectives correspondingly reflects both the breadth and depth of contemporary theorizing about identity and modality, each paper addressing a particular issue and advancing our knowledge of the area. This volume will provide essential reading for graduate students in the subject and professional philosophers.

Think of a number, any number, or properties like fragility and humanity. These and other abstract entities are radically different from concrete entities like electrons and elbows. While concrete entities are located in space and time, have causes and effects, and are known through empirical means, abstract entities like meanings and possibilities are remarkably different. They seem to be immutable and imperceptible and to exist "outside" of space and time. This book provides a comprehensive critical assessment of the problems raised by abstract entities and the debates about existence, truth, and knowledge that surround them. It sets out the key issues that inform the metaphysical disagreement between platonists who accept abstract entities and nominalists who deny abstract entities exist. Beginning with the essentials of the platonist-nominalist debate, it explores the key arguments and issues informing the contemporary debate over abstract reality: arguments for platonism and their connections to semantics, science, and metaphysical explanation the abstract-concrete distinction and views about the nature of abstract reality epistemological puzzles surrounding our knowledge of mathematical entities and other abstract entities. arguments for nominalism premised upon concerns about paradox, parsimony, infinite regresses, underdetermination, and causal isolation nominalist options that seek to dispense with abstract entities. Including chapter summaries, annotated further reading, and a glossary, Abstract Entities is essential reading for anyone seeking a clear and authoritative introduction to the problems raised by abstract entities.

The Symbolic Universe considers the ways in which many leading mathematicians between 1890 and 1930 attempted to apply geometry to physics. It concentrates on responses to Einstein's theories of special and general relativity, but also considers the philosophical implications of these ideas.

Sofia Kovalevskaya was a brilliant and determined young Russian woman of the 19th century who wanted to become a mathematician and who succeeded, in often difficult circumstances, in becoming arguably the first woman to have a professional university career in the way we understand it today. This memoir, written by a mathematician who specialises in symplectic geometry and integrable systems, is a personal exploration of the life, the writings and the mathematical achievements of a remarkable woman. It emphasises the originality of Kovalevskaya's work and assesses her legacy and reputation as a mathematician and scientist. Her ideas are explained in a way that is accessible to a general audience, with diagrams, marginal notes and commentary to help explain the mathematical concepts and provide context. This fascinating book, which also examines Kovalevskaya's love of literature, will be of interest to historians looking for a treatment of the mathematics, and those doing feminist or gender studies.

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.

When a doctor tells you there's a one percent chance that an operation will result in your death, or a scientist claims that his theory is probably true, what exactly does that mean? Understanding probability is clearly very important, if we are to make good theoretical and practical choices. In this engaging and highly accessible introduction to the philosophy of probability, Darrell Rowbottom takes the reader on a journey through all the major interpretations of probability, with reference to real-world situations. In lucid prose, he explores the many fallacies of probabilistic reasoning, such as the 'gambler's fallacy' and the 'inverse fallacy', and shows how we can avoid falling into these traps by using the interpretations presented. He also illustrates the relevance of the interpretation of probability across disciplinary boundaries, by examining which interpretations of probability are appropriate in diverse areas such as quantum mechanics, game theory, and genetics. Using entertaining dialogues to draw out the key issues at stake, this unique book will appeal to students and scholars across philosophy, the social sciences, and the natural sciences.

This book presents a philosophical approach to probability and probabilistic thinking, considering the underpinnings of probabilistic reasoning and modeling, which effectively underlie everything in data science. The ultimate goal is to call into question many standard tenets and lay the philosophical and probabilistic groundwork and infrastructure for statistical modeling. It is the first book devoted to the philosophy of data aimed at working scientists and calls for a new consideration in the practice of probability and statistics to eliminate what has been referred to as the "Cult of Statistical Significance." The book explains the philosophy of these ideas and not the mathematics, though there are a handful of mathematical examples. The topics are logically laid out, starting with basic philosophy as related to probability, statistics, and science, and stepping through the key probabilistic ideas and concepts, and ending with statistical models. Its jargon-free approach asserts that standard methods, such as out-of-the-box regression, cannot help in discovering cause. This new way of looking at uncertainty ties together disparate fields - probability, physics, biology, the "soft" sciences, computer science - because each aims at discovering cause (of effects). It broadens the understanding beyond frequentist and Bayesian methods to propose a Third Way of modeling.

 From the Preface: “There are three volumes. The first one contains a curriculum vitae, a «Brève Analyse des Travaux» and a Iist of publications, including books and seminars. In addition the volume contains all papers of H. Cartan on analytic functions published before 1939. The other papers on analytic functions, e.g. those on Stein manifolds and coherent sheaves, make up the second volume. The third volume contains, with a few exceptions, all further papers of H. Cartan; among them is a reproduction of exposés 2 to 11 of his 1954/55 Seminar on Eilenberg-MacLane algebras. Each volume is arranged in chronological order. The reader should be aware that these volumes do not fully reflect H. Cartan's work, a large part of which is also contained in his fifteen ENS-Seminars (1948-1964) and in his book "Homological Algebra" with S. Eilenberg... Still, we trust that mathematicians throughout the world will welcome the availability of the "Oeuvres" of a mathematician whose writing and teaching has had such an influence on our generation.”

This collection presents significant contributions from an international network project on mathematical cultures, including essays from leading scholars in the history and philosophy of mathematics and mathematics education. Mathematics has universal standards of validity. Nevertheless, there are local styles in mathematical research and teaching, and great variation in the place of mathematics in the larger cultures that mathematical practitioners belong to. The reflections on mathematical cultures collected in this book are of interest to mathematicians, philosophers, historians, sociologists, cognitive scientists and mathematics educators.

This book presents 25 selected papers from the International Conference on "Developing Synergies between Islam & Science and Technology for Mankind's Benefit" held at the International Institute for Advanced Islamic Studies Malaysia, Kuala Lumpur, in October 2014. The papers cover a broad range of issues reflecting the main conference themes: Cosmology and the Universe, Philosophy of Science and the Emergence of Biological Systems, Principles and Applications of Tawhidic Science, Medical Applications of Tawhidic Science and Bioethics, and the History and Teaching of Science from an Islamic Perspective. Highlighting the relationships between the Islamic religious worldview and the physical sciences, the book challenges secularist paradigms on the study of Science and Technology. Integrating metaphysical perspectives of Science, topics include Islamic approaches to S&T such as an Islamic epistemology of the philosophy of science, a new quantum theory, environmental care, avoiding wasteful consumption using Islamic teachings, and emotional-blasting psychological therapy. Eminent contributing scholars include Osman Bakar, Mohammad Hashim Kamali, Mehdi Golshani, Mohd. Kamal Hassan, Adi Setia and Malik Badri. The book is essential reading for a broad group of academics and practitioners, from Islamic scholars and social scientists to (physical) scientists and engineers.

From the Preface: "There are three volumes. The first one contains a curriculum vitae, a "Breve Analyse des Travaux" and a Iist of publications, including books and seminars. In addition the volume contains all papers of H. Cartan on analytic functions published before 1939. The other papers on analytic functions, e.g. those on Stein manifolds and coherent sheaves, make up the second volume. The third volume contains, with a few exceptions, all further papers of H. Cartan; among them is a reproduction of exposes 2 to 11 of his 1954/55 Seminar on Eilenberg-MacLane algebras. Each volume is arranged in chronological order. The reader should be aware that these volumes do not fully reflect H. Cartan's work, a large part of which is also contained in his fifteen ENS-Seminars (1948-1964) and in his book "Homological Algebra" with S. Eilenberg...Still, we trust that mathematicians throughout the world will welcome the availability of the "Oeuvres" of a mathematician whose writing and teaching has had such an influence on our generation."

From the Preface: "There are three volumes. The first one contains a curriculum vitae, a "Breve Analyse des Travaux" and a Iist of publications, including books and seminars. In addition the volume contains all papers of H. Cartan on analytic functions published before 1939. The other papers on analytic functions, e.g. those on Stein manifolds and coherent sheaves, make up the second volume. The third volume contains, with a few exceptions, all further papers of H. Cartan; among them is a reproduction of exposes 2 to 11 of his 1954/55 Seminar on Eilenberg-MacLane algebras. Each volume is arranged in chronological order. The reader should be aware that these volumes do not fully reflect H. Cartan's work, a large part of which is also contained in his fifteen ENS-Seminars (1948-1964) and in his book "Homological Algebra" with S. Eilenberg...Still, we trust that mathematicians throughout the world will welcome the availability of the "Oeuvres" of a mathematician whose writing and teaching has had such an influence on our generation."

This work offers a re-edition of twelve mathematical tablets from the site of Tell Harmal, in the borders of present-day Baghdad. In ancient times, Tell Harmal was Saduppum, a city representative of the region of the Diyala river and of the kingdom of Esnunna, to which it belonged for a time. These twelve tablets were originally published in separate articles in the beginning of the 1950s and mostly contain solved problem texts. Some of the problems deal with abstract matters such as triangles and rectangles with no reference to daily life, while others are stated in explicitly empirical contexts, such as the transportation of a load of bricks, the size of a vessel, the number of men needed to build a wall and the acquisition of oil and lard. This new edition of the texts is the first to group them, and takes into account all the recent developments of the research in the history of Mesopotamian mathematics. Its introductory chapters are directed to readers interested in an overview of the mathematical contents of these tablets and the language issues involved in their interpretation, while a chapter of synthesis discusses the ways history of mathematics has typically dealt with the mathematical evidence and inquires how and to what degree mathematical tablets can be made part of a picture of the larger social context. Furthermore, the volume contributes to a geography of the Old Babylonian mathematical practices, by evidencing that scribes at Saduppum made use of cultural material that was locally available. The edited texts are accompanied by translations, philological, and mathematical commentaries.

This volume contains thirteen papers that were presented at the 2014 Annual Meeting of the Canadian Society for History and Philosophy of Mathematics/La Societe Canadienne d'Histoire et de Philosophie des Mathematiques, held on the campus of Brock University in St. Catharines, Ontario, Canada. It contains rigorously reviewed modern scholarship on general topics in the history and philosophy of mathematics, as well as on the meeting's special topic, Early Scientific Computation. These papers cover subjects such as *Physical tools used by mathematicians in the seventeenth century *The first historical appearance of the game-theoretical concept of mixed-strategy equilibrium *George Washington's mathematical cyphering books *The development of the Venn diagram *The role of Euler and other mathematicians in the development of algebraic analysis *Arthur Cayley and Alfred Kempe's influence on Charles Peirce's diagrammatic logic *The influence publishers had on the development of mathematical pedagogy in the nineteenth century *A description of the 1924 International Mathematical Congress held in Toronto, told in the form of a "narrated slide show" Written by leading scholars in the field, these papers will be accessible to not only mathematicians and students of the history and philosophy of mathematics, but also anyone with a general interest in mathematics.

Mathematics as a Science of Patterns expounds a system of ideas about the nature of mathematics which Michael Resnik has been elaborating for a number of years. In calling mathematics a science he implies that it has a factual subject-matter and that mathematical knowledge is on a par with other scientific knowledge; in calling it a science of patterns he expresses his commitment to a structuralist philosophy of mathematics. He links this to a defence of realism about the metaphysics of mathematics--the view that mathematics is about things that really exist.