Chinese Remainder Theorem, CRT, is one of the jewels of mathematics. It is a perfect combination of beauty and utility or, in the words of Horace, omne tulit punctum qui miscuit utile dulci. Known already for ages, CRT continues to present itself in new contexts and open vistas for new types of applications. So far, its usefulness has been obvious within the realm of "three C's". Computing was its original field of application, and continues to be important as regards various aspects of algorithmics and modular computations. Theory of codes and cryptography are two more recent fields of application.This book tells about CRT, its background and philosophy, history, generalizations and, most importantly, its applications. The book is self-contained. This means that no factual knowledge is assumed on the part of the reader. We even provide brief tutorials on relevant subjects, algebra and information theory. However, some mathematical maturity is surely a prerequisite, as our presentation is at an advanced undergraduate or beginning graduate level. We have tried to make the exposition innovative, many of the individual results being new. We will return to this matter, as well as to the interdependence of the various parts of the book, at the end of the Introduction.A special course about CRT can be based on the book. The individual chapters are largely independent and, consequently, the book can be used as supplementary material for courses in algorithmics, coding theory, cryptography or theory of computing. Of course, the book is also a reference for matters dealing with CRT.

There are things we routinely say that may strike us as literally false but that we are nonetheless reluctant to give up. This might be something mundane, like the way we talk about the sun setting in the west (it is the earth that moves), or it could be something much deeper, like engaging in talk that is ostensibly about numbers despite believing that numbers do not literally exist. Rather than regard such behaviour as self-defeating, a "fictionalist" is someone who thinks that this kind of discourse is entirely appropriate, even helpful, so long as we treat what is said as a useful fiction, rather than as the sober truth. "Fictionalism" can be broadly understood as a view that uses a notion of pretense or fiction in order to resolve certain puzzles or problems that otherwise do not necessarily have anything to do with literature or fictional creations. Within contemporary analytic philosophy, fictionalism has been on the scene for well over a decade and has matured during that time, growing in popularity. There are now myriad competing views about fictionalism and consequently the discussion has branched out into many more subdisciplines of philosophy. Yet there is widespread disagreement on what philosophical fictionalism actually amounts to and about how precisely it ought to be pursued. This volume aims to guide these discussions, collecting some of the most up-to-date work on fictionalism and tracing the view's development over the past decade. After a detailed discussion in the book's introductory chapter of how philosophers should think of fictionalism and its connection to metaontology more generally, the remaining chapters provide readers with arguments for and against this view from leading scholars in the fields of epistemology, ethics, metaphysics, philosophy of science, philosophy of language, and others.

This monograph presents a groundbreaking scholarly treatment of the German mathematician Jost Burgi's original work on logarithms, Arithmetische und Geometrische Progress Tabulen. It provides the first-ever English translation of Burgi's text and illuminates his role in the development of the conception of logarithms, for which John Napier is traditionally given priority. High-resolution scans of each page of the his handwritten text are reproduced for the reader and as a means of preserving an important work for which there are very few surviving copies. The book begins with a brief biography of Burgi to familiarize readers with his life and work, as well as to offer an historical context in which to explore his contributions. The second chapter then describes the extant copies of the Arithmetische und Geometrische Progress Tabulen, with a detailed description of the copy that is the focus of this book, the 1620 "Graz manuscript". A complete facsimile of the text is included in the next chapter, along with a corresponding transcription and an English translation; a transcription of a second version of the manuscript (the "Gdansk manuscript") is included alongside that of the Graz edition so that readers can easily and closely examine the differences between the two. The final chapter considers two important questions about Burgi's work, such as who was the copyist of the Graz manuscript and what the relationship is between the Graz and Gdansk versions. Appendices are also included that contain a timeline of Burgi's life, the underlying concept of Napier's construction of logarithms, and scans of all 58 sheets of the tables from Burgi's text. Anyone with an appreciation for the history of mathematics will find this book to be an insightful and interesting look at an important and often overlooked work. It will also be a valuable resource for undergraduates taking courses in the history of mathematics, researchers of the history of mathematics, and professors of mathematics education who wish to incorporate historical context into their teaching.

Mathematics depends on proofs, and proofs must begin somewhere, from some fundamental assumptions. For nearly a century, the axioms of set theory have played this role, so the question of how these axioms are properly judged takes on a central importance. Approaching the question from a broadly naturalistic or second-philosophical point of view, Defending the Axioms isolates the appropriate methods for such evaluations and investigates the ontological and epistemological backdrop that makes them appropriate. In the end, a new account of the objectivity of mathematics emerges, one refreshingly free of metaphysical commitments.

We are all captivated and puzzled by the infinite, in its many varied guises; by the endlessness of space and time; by the thought that between any two points in space, however close, there is always another; by the fact that numbers go on forever; and by the idea of an all-knowing, all-powerful God. In this acclaimed introduction to the infinite, A. W. Moore takes us on a journey back to early Greek thought about the infinite, from its inception to Aristotle. He then examines medieval and early modern conceptions of the infinite, including a brief history of the calculus, before turning to Kant and post-Kantian ideas. He also gives an account of Cantor's remarkable discovery that some infinities are bigger than others. In the second part of the book, Moore develops his own views, drawing on technical advances in the mathematics of the infinite, including the celebrated theorems of Skolem and Goedel, and deriving inspiration from Wittgenstein. He concludes this part with a discussion of death and human finitude. For this third edition Moore has added a new part, 'Infinity superseded', which contains two new chapters refining his own ideas through a re-examination of the ideas of Spinoza, Hegel, and Nietzsche. This new part is heavily influenced by the work of Deleuze. Also new for the third edition are: a technical appendix on still unresolved questions about different infinite sizes; an expanded glossary; and updated references and further reading. The Infinite, Third Edition is ideal reading for anyone interested in an engaging and historically informed account of this fascinating topic, whether from a philosophical point of view, a mathematical point of view, or a religious point of view.

This text presents an intuitive and robust mathematical image of fundamental particle physics based on a novel approach to quantum field theory, which is guided by four carefully motivated metaphysical postulates. In particular, the book explores a dissipative approach to quantum field theory, which is illustrated for scalar field theory and quantum electrodynamics, and proposes an attractive explanation of the Planck scale in quantum gravity. Offering a radically new perspective on this topic, the book focuses on the conceptual foundations of quantum field theory and ontological questions. It also suggests a new stochastic simulation technique in quantum field theory which is complementary to existing ones. Encouraging rigor in a field containing many mathematical subtleties and pitfalls this text is a helpful companion for students of physics and philosophers interested in quantum field theory, and it allows readers to gain an intuitive rather than a formal understanding.

What is mathematics about? Does the subject-matter of mathematics exist independently of the mind or are they mental constructions? How do we know mathematics? Is mathematical knowledge logical knowledge? And how is mathematics applied to the material world? In this introduction to the philosophy of mathematics, Michele Friend examines these and other ontological and epistemological problems raised by the content and practice of mathematics. Aimed at a readership with limited proficiency in mathematics but with some experience of formal logic it seeks to strike a balance between conceptual accessibility and correct representation of the issues. Friend examines the standard theories of mathematics - Platonism, realism, logicism, formalism, constructivism and structuralism - as well as some less standard theories such as psychologism, fictionalism and Meinongian philosophy of mathematics. In each case Friend explains what characterises the position and where the divisions between them lie, including some of the arguments in favour and against each. This book also explores particular questions that occupy present-day philosophers and mathematicians such as the problem of infinity, mathematical intuition and the relationship, if any, between the philosophy of mathematics and the practice of mathematics. Taking in the canonical ideas of Aristotle, Kant, Frege and Whitehead and Russell as well as the challenging and innovative work of recent philosophers like Benacerraf, Hellman, Maddy and Shapiro, Friend provides a balanced and accessible introduction suitable for upper-level undergraduate courses and the non-specialist.

What is mathematics about? Does the subject-matter of mathematics exist independently of the mind or are they mental constructions? How do we know mathematics? Is mathematical knowledge logical knowledge? And how is mathematics applied to the material world? In this introduction to the philosophy of mathematics, Michele Friend examines these and other ontological and epistemological problems raised by the content and practice of mathematics. Aimed at a readership with limited proficiency in mathematics but with some experience of formal logic it seeks to strike a balance between conceptual accessibility and correct representation of the issues. Friend examines the standard theories of mathematics - Platonism, realism, logicism, formalism, constructivism and structuralism - as well as some less standard theories such as psychologism, fictionalism and Meinongian philosophy of mathematics. In each case Friend explains what characterises the position and where the divisions between them lie, including some of the arguments in favour and against each. This book also explores particular questions that occupy present-day philosophers and mathematicians such as the problem of infinity, mathematical intuition and the relationship, if any, between the philosophy of mathematics and the practice of mathematics. Taking in the canonical ideas of Aristotle, Kant, Frege and Whitehead and Russell as well as the challenging and innovative work of recent philosophers like Benacerraf, Hellman, Maddy and Shapiro, Friend provides a balanced and accessible introduction suitable for upper-level undergraduate courses and the non-specialist.

Mathematics has long suffered in the public eye through portrayals of mathematicians as socially inept geniuses devoted to an arcane discipline. In this book, Philip J. Davis addresses this image through a question-and-answer dialogue that lays to rest many of the misnomers and misunderstandings of mathematical study. He answers these questions and more: What is Mathematics? Why is mathematics difficult, and why do I spontaneously react negatively when I hear the word? Davis demonstrates how mathematics surrounds, imbues, and maintains our everyday lives: the digitization and automation of processes like pumping gas, withdrawing cash, and buying groceries are all fueled by mathematics. He takes the reader through a point-by-point explanation of many frequently asked questions about mathematics, gently introducing this Handmaiden of Science and telling you everything you've ever wanted to know about her.

An exploration of mathematical style through 99 different proofs of the same theorem This book offers a multifaceted perspective on mathematics by demonstrating 99 different proofs of the same theorem. Each chapter solves an otherwise unremarkable equation in distinct historical, formal, and imaginative styles that range from Medieval, Topological, and Doggerel to Chromatic, Electrostatic, and Psychedelic. With a rare blend of humor and scholarly aplomb, Philip Ording weaves these variations into an accessible and wide-ranging narrative on the nature and practice of mathematics. Inspired by the experiments of the Paris-based writing group known as the Oulipo-whose members included Raymond Queneau, Italo Calvino, and Marcel Duchamp-Ording explores new ways to examine the aesthetic possibilities of mathematical activity. 99 Variations on a Proof is a mathematical take on Queneau's Exercises in Style, a collection of 99 retellings of the same story, and it draws unexpected connections to everything from mysticism and technology to architecture and sign language. Through diagrams, found material, and other imagery, Ording illustrates the flexibility and creative potential of mathematics despite its reputation for precision and rigor. Readers will gain not only a bird's-eye view of the discipline and its major branches but also new insights into its historical, philosophical, and cultural nuances. Readers, no matter their level of expertise, will discover in these proofs and accompanying commentary surprising new aspects of the mathematical landscape.

This handbook features essays written by both literary scholars and mathematicians that examine multiple facets of the connections between literature and mathematics. These connections range from mathematics and poetic meter to mathematics and modernism to mathematics as literature. Some chapters focus on a single author, such as mathematics and Ezra Pound, Gertrude Stein, or Charles Dickens, while others consider a mathematical topic common to two or more authors, such as squaring the circle, chaos theory, Newton's calculus, or stochastic processes. With appeal for scholars and students in literature, mathematics, cultural history, and history of mathematics, this important volume aims to introduce the range, fertility, and complexity of the connections between mathematics, literature, and literary theory. Chapter 1 is available open access under a Creative Commons Attribution 4.0 International License via [link.springer.com|http://link.springer.com/].

Mathematical platonism is the view that mathematical statements are true of real mathematical objects like numbers, shapes, and sets. One central problem with platonism is that numbers, shapes, sets, and the like are not perceivable by our senses. In contemporary philosophy, the most common defense of platonism uses what is known as the indispensability argument. According to the indispensabilist, we can know about mathematics because mathematics is essential to science. Platonism is among the most persistent philosophical views. Our mathematical beliefs are among our most entrenched. They have survived the demise of millennia of failed scientific theories. Once established, mathematical theories are rarely rejected, and never for reasons of their inapplicability to empirical science. Autonomy Platonism and the Indispensability Argument is a defense of an alternative to indispensability platonism. The autonomy platonist believes that mathematics is independent of empirical science: there is purely mathematical evidence for purely mathematical theories which are even more compelling to believe than empirical science. Russell Marcus begins by contrasting autonomy platonism and indispensability platonism. He then argues against a variety of indispensability arguments in the first half of the book. In the latter half, he defends a new approach to a traditional platonistic view, one which includes appeals to a priori but fallible methods of belief acquisition, including mathematical intuition, and a natural adoption of ordinary mathematical methods. In the end, Marcus defends his intuition-based autonomy platonism against charges that the autonomy of mathematics is viciously circular. This book will be useful to researchers, graduate students, and advanced undergraduates with interests in the philosophy of mathematics or in the connection between science and mathematics.

Think of a number between one and ten. No, hang on, let's make this interesting. Between zero and infinity. Even if you stick to the whole numbers, there are a lot to choose from - an infinite number in fact. Throw in decimal fractions and infinity suddenly gets an awful lot bigger (is that even possible?) And then there are the negative numbers, the imaginary numbers, the irrational numbers like pi which never end. It literally never ends. The world of numbers is indeed strange and beautiful. Among its inhabitants are some really notable characters - pi, e, the "imaginary" number i and the famous golden ratio to name just a few. Prime numbers occupy a special status. Zero is very odd indeed: is it a number, or isn't it? How Numbers Work takes a tour of this mind-blowing but beautiful realm of numbers and the mathematical rules that connect them. Not only that, but take a crash course on the biggest unsolved problems that keep mathematicians up at night, find out about the strange and unexpected ways mathematics influences our everyday lives, and discover the incredible connection between numbers and reality itself. ABOUT THE SERIES New Scientist Instant Expert books are definitive and accessible entry points to the most important subjects in science; subjects that challenge, attract debate, invite controversy and engage the most enquiring minds. Designed for curious readers who want to know how things work and why, the Instant Expert series explores the topics that really matter and their impact on individuals, society, and the planet, translating the scientific complexities around us into language that's open to everyone, and putting new ideas and discoveries into perspective and context.

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.

Medieval Islamic World: An Intellectual History of Science and Politics surveys major scientific and philosophical discoveries in the medieval period within the broader Islamicate world, providing an alternative historical framework to that of the primarily Eurocentric history of science and philosophy of science and technology fields. Medieval Islamic World serves to address the history of rationalist inquiry within scholarly institutions in medieval Islamic societies, surveying developments in the fields of medicine and political theory, and the scientific disciplines of astronomy, chemistry, physics, and mechanics, as led by medieval Muslim scholarship.

Written by one of the preeminent researchers in the field, this book provides a comprehensive exposition of modern analysis of causation. It shows how causality has grown from a nebulous concept into a mathematical theory with significant applications in the fields of statistics, artificial intelligence, economics, philosophy, cognitive science, and the health and social sciences. Judea Pearl presents and unifies the probabilistic, manipulative, counterfactual, and structural approaches to causation and devises simple mathematical tools for studying the relationships between causal connections and statistical associations. The book will open the way for including causal analysis in the standard curricula of statistics, artificial intelligence, business, epidemiology, social sciences, and economics. Students in these fields will find natural models, simple inferential procedures, and precise mathematical definitions of causal concepts that traditional texts have evaded or made unduly complicated. The first edition of Causality has led to a paradigmatic change in the way that causality is treated in statistics, philosophy, computer science, social science, and economics. Cited in more than 5,000 scientific publications, it continues to liberate scientists from the traditional molds of statistical thinking. In this revised edition, Judea Pearl elucidates thorny issues, answers readers questions, and offers a panoramic view of recent advances in this field of research. Causality will be of interests to students and professionals in a wide variety of fields. Anyone who wishes to elucidate meaningful relationships from data, predict effects of actions and policies, assess explanations of reported events, or form theories of causal understanding and causal speech will find this book stimulating and invaluable."

This monograph examines the private annotations that Ludwig Wittgenstein made to his copy of G.H. Hardy's classic textbook, A Course of Pure Mathematics. Complete with actual images of the annotations, it gives readers a more complete picture of Wittgenstein's remarks on irrational numbers, which have only been published in an excerpted form and, as a result, have often been unjustly criticized. The authors first establish the context behind the annotations and discuss the historical role of Hardy's textbook. They then go on to outline Wittgenstein's non-extensionalist point of view on real numbers, assessing his manuscripts and published remarks and discussing attitudes in play in the philosophy of mathematics since Dedekind. Next, coverage focuses on the annotations themselves. The discussion encompasses irrational numbers, the law of excluded middle in mathematics and the notion of an "improper picture," the continuum of real numbers, and Wittgenstein's attitude toward functions and limits.

This monograph uses the concept and category of "event" in the study of mathematics as it emerges from an interaction between levels of cognition, from the bodily experiences to symbolism. It is subdivided into three parts.The first moves from a general characterization of the classical approach to mathematical cognition and mind toward laying the foundations for a view on the mathematical mind that differs from going approaches in placing primacy on events.The second articulates some common phenomena-mathematical thought, mathematical sign, mathematical form, mathematical reason and its development, and affect in mathematics-in new ways that are based on the previously developed ontology of events. The final part has more encompassing phenomena as its content, most prominently the thinking body of mathematics, the experience in and of mathematics, and the relationship between experience and mind. The volume is well-suited for anyone with a broad interest in educational theory and/or social development, or with a broad background in psychology.

Attempts to understand various aspects of the empirical world often rely on modelling processes that involve a reconstruction of systems under investigation. Typically the reconstruction uses mathematical frameworks like gauge theory and renormalization group methods, but more recently simulations also have become an indispensable tool for investigation. This book is a philosophical examination of techniques and assumptions related to modelling and simulation with the goal of showing how these abstract descriptions can contribute to our understanding of the physical world. Particular issues include the role of fictional models in science, how mathematical formalisms can yield physical information, and how we should approach the use of inconsistent models for specific types of systems. It also addresses the role of simulation, specifically the conditions under which simulation can be seen as a technique for measurement, replacing more traditional experimental approaches. Inherent worries about the legitimacy of simulation "knowledge " are also addressed, including an analysis of verification and validation and the role of simulation data in the search for the Higgs boson. In light of the significant role played by simulation in the Large Hadron Collider experiments, it is argued that the traditional distinction between simulation and experiment is no longer applicable in some contexts of modern science. Consequently, a re-evaluation of the way and extent to which simulation delivers empirical knowledge is required. "This is a, lively, stimulating, and important book by one of the main scholars contributing to current topics and debates in our field. It will be a major resource for philosophers of science, their students, scientists interested in examining scientific practice, and the general scientifically literate public. "-Bas van Fraassen, Distinguished Professor of Philosophy, San Francisco State University

Kit Fine develops a Fregean theory of abstraction, and suggests that it may yield a new philosophical foundation for mathematics, one that can account for both our reference to various mathematical objects and our knowledge of various mathematical truths. The Limits of Abstraction breaks new ground both technically and philosophically, and will be essential reading for all who work on the philosophy of mathematics.

A fascinating account of the breakthrough ideas that transformed probability and statistics In the sixteenth and seventeenth centuries, gamblers and mathematicians transformed the idea of chance from a mystery into the discipline of probability, setting the stage for a series of breakthroughs that enabled or transformed innumerable fields, from gambling, mathematics, statistics, economics, and finance to physics and computer science. This book tells the story of ten great ideas about chance and the thinkers who developed them. Complete with a brief probability refresher, Ten Great Ideas about Chance is certain to be a hit with anyone who wants to understand the secrets of probability and how they were discovered.

This volume presents interviews that have been conducted from the 1980s to the present with important scholars of social choice and welfare theory. Starting with a brief history of social choice and welfare theory written by the book editors, it features 15 conversations with four Nobel Laureates and other key scholars in the discipline. The volume is divided into two parts. The first part presents four conversations with the founding fathers of modern social choice and welfare theory: Kenneth Arrow, John Harsanyi, Paul Samuelson, and Amartya Sen. The second part includes conversations with scholars who made important contributions to the discipline from the early 1970s onwards. This book will appeal to anyone interested in the history of economics, and the history of social choice and welfare theory in particular.

This book investigates the process of care in mathematics teaching. The author proposes transformative educational spaces in which learning mathematics, rather than consisting of a repetitive grind of exercises and facts, can become a part of learner identity. This book describes examples of mathematics teachings in a wide range of contexts and pedagogies, coordinated to identify common features where care for mathematical learning and thinking is combined with care for learners. Along with detailing caring mathematics education practices in alternative spaces, the author demonstrates similar practices alive even with the current mainstream spaces of acquisition and performance. Care is integrated through listening, and developing responsive and trusting relationships. It will be of interest to scholars of mathematics education, as well as pre-service and in-service teachers and teacher educators.

This volume is a collection of essays in honour of Professor Mohammad Ardeshir. It examines topics which, in one way or another, are connected to the various aspects of his multidisciplinary research interests. Based on this criterion, the book is divided into three general categories. The first category includes papers on non-classical logics, including intuitionistic logic, constructive logic, basic logic, and substructural logic. The second category is made up of papers discussing issues in the contemporary philosophy of mathematics and logic. The third category contains papers on Avicenna's logic and philosophy. Mohammad Ardeshir is a full professor of mathematical logic at the Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran, where he has taught generations of students for around a quarter century. Mohammad Ardeshir is known in the first place for his prominent works in basic logic and constructive mathematics. His areas of interest are however much broader and include topics in intuitionistic philosophy of mathematics and Arabic philosophy of logic and mathematics. In addition to numerous research articles in leading international journals, Ardeshir is the author of a highly praised Persian textbook in mathematical logic. Partly through his writings and translations, the school of mathematical intuitionism was introduced to the Iranian academic community.

Contents:
Abbreviations Introduction Acknowledgements Chronology Part I. Early Foundational Work 1. Classes 2. Relations 3. Functions Part II. The Zig-Zag Theory 4. Outlines of Symbolic Logic 5. On Functions, Classes and Relations 6. On Functions 7. Fundamental Notions 8. On the Functionality of Denoting Complexes 9. On the Nature of Functions 10. On Classes and Relations Part III. The Theory of Denoting 11. On the Meaning and Denotation of Phrases 12. Dependent Variables and Denotation 13. Points about Denoting 14. On Meaning and Denotation 15. On Fundamentals 16. On Denoting Part IV. Philosophy of Logic and Mathematics 17. Meinong's Theory of Complexes and Assumptions 18. The Axiom of Infinity 19. Non-Euclidean Geometry 20. The Existential Import of Propositions 21. The Nature of Truth 22. Necessity and Possibility 23. On the Relation of Mathematics to Symbolic Logic Part V. Philosophical Reviews 24. Recent Work on the Philosophy of Leibniz 25. Review of Couturat 26. Review of Geissler 27. Principia Ethica 28. The Meaning of Good 29. Review of Delaporte 30. Review of Hinton 31. Review of Petronievics 32. Science and Hypothesis 33. Review of Poincare 34. Review of Meinong and Others Appendices I Frege on the Contradiction II Comments on Definitions of Philosophical Terms III Sur la relation des mathematiques a la logistique Missing and Unprinted Papers Contents Textual Notes Bibliographical Index General Index