This open access book collects the historical and medial perspectives of a systematic and epistemological analysis of the complicated, multifaceted relationship between model and mathematics, ranging from, for example, the physical mathematical models of the 19th century to the simulation and digital modelling of the 21st century. The aim of this anthology is to showcase the status of the mathematical model between abstraction and realization, presentation and representation, what is modeled and what models. This book is open access under a CC BY 4.0 license.

A comprehensive look at four of the most famous problems in mathematics Tales of Impossibility recounts the intriguing story of the renowned problems of antiquity, four of the most famous and studied questions in the history of mathematics. First posed by the ancient Greeks, these compass and straightedge problems-squaring the circle, trisecting an angle, doubling the cube, and inscribing regular polygons in a circle-have served as ever-present muses for mathematicians for more than two millennia. David Richeson follows the trail of these problems to show that ultimately their proofs-which demonstrated the impossibility of solving them using only a compass and straightedge-depended on and resulted in the growth of mathematics. Richeson investigates how celebrated luminaries, including Euclid, Archimedes, Viete, Descartes, Newton, and Gauss, labored to understand these problems and how many major mathematical discoveries were related to their explorations. Although the problems were based in geometry, their resolutions were not, and had to wait until the nineteenth century, when mathematicians had developed the theory of real and complex numbers, analytic geometry, algebra, and calculus. Pierre Wantzel, a little-known mathematician, and Ferdinand von Lindemann, through his work on pi, finally determined the problems were impossible to solve. Along the way, Richeson provides entertaining anecdotes connected to the problems, such as how the Indiana state legislature passed a bill setting an incorrect value for pi and how Leonardo da Vinci made elegant contributions in his own study of these problems. Taking readers from the classical period to the present, Tales of Impossibility chronicles how four unsolvable problems have captivated mathematical thinking for centuries.

This open access book is about the shaping of international relations in mathematics over the last two hundred years. It focusses on institutions and organizations that were created to frame the international dimension of mathematical research. Today, striking evidence of globalized mathematics is provided by countless international meetings and the worldwide repository ArXiv. The text follows the sinuous path that was taken to reach this state, from the long nineteenth century, through the two wars, to the present day. International cooperation in mathematics was well established by 1900, centered in Europe. The first International Mathematical Union, IMU, founded in 1920 and disbanded in 1932, reflected above all the trauma of WW I. Since 1950 the current IMU has played an increasing role in defining mathematical excellence, as is shown both in the historical narrative and by analyzing data about the International Congresses of Mathematicians. For each of the three periods discussed, interactions are explored between world politics, the advancement of scientific infrastructures, and the inner evolution of mathematics. Readers will thus take a new look at the place of mathematics in world culture, and how international organizations can make a difference. Aimed at mathematicians, historians of science, scientists, and the scientifically inclined general public, the book will be valuable to anyone interested in the history of science on an international level.

This text presents the ideas of a particular group of mathematicians of the late 18th century known as "the German combinatorial school" and its influence. The book tackles several questions concerning the emergence and historical development of the German combinatorial analysis, which was the unfinished scientific research project of that group of mathematicians. The historical survey covers the three main episodes in the evolution of that research project: its theoretical antecedents (which go back to the innovative ideas on mathematical analysis of the late 17th century) and first formulation, its consolidation as a foundationalist project of mathematical analysis, and its dissolution at the beginning of the 19th century. In addition, the book analyzes the influence of the ideas of the combinatorial school on German mathematics throughout the 19th century.

Cet ouvrage contient les correspondances actives et passives de Jules Houel avec Joseph-Marie De Tilly, Gaston Darboux et Victor-Amedee Le Besgue ainsi qu'une introduction qui se focalise sur la decouverte de l'impossibilite de demontrer le postulat des paralleles d'Euclide et l'apparition des premiers exemples de fonctions continues non derivables. Jules Houel (1823-1886) a occupe une place particuliere dans les mathematiques en France durant la seconde partie du 19eme siecle. Par ses travaux de traduction et ses recensions, il a vivement contribue a la reception de la geometrie non euclidienne de Bolyai et Lobatchevski ainsi qu'aux debats sur les fondements de l'analyse. Il se situe au centre d'un vaste reseau international de correspondances en lien avec son role de redacteur pour le Bulletin des sciences mathematiques et astronomiques.

Presents Book One of Euclid's Elements for students in humanities and for general readers. This treatment raises deep questions about the nature of human reason and its relation to the world. Dana Densmore's Questions for Discussion are intended as examples, to urge readers to think more carefully about what they are watching unfold, and to help them find their own questions in a genuine and exhilarating inquiry.

Emil Artin was one of the great mathematicians of the twentieth century. He had the rare distinction of having solved two of the famous problems posed by David Hilbert in 1900. He showed that every positive definite rational function of several variables was a sum of squares. He also discovered and proved the Artin reciprocity law, the culmination of over a century and a half of progress in algebraic number theory. Artin had a great influence on the development of mathematics in his time, both by means of his many contributions to research and by the high level and excellence of his teaching and expository writing. In this volume we gather together in one place a selection of his writings wherein the reader can learn some beautiful mathematics as seen through the eyes of a true master. The volume's Introduction provides a short biographical sketch of Emil Artin, followed by an introduction to the books and papers included in the volume. The reader will first find three of Artin's short books, titled The Gamma Function, Galois Theory, and Theory of Algebraic Numbers, respectively. These are followed by papers on algebra, algebraic number theory, real fields, braid groups, and complex and functional analysis. The three papers on real fields have been translated into English for the first time. The flavor of these works is best captured by the following quote of Richard Brauer. ""There are a number of books and sets of lecture notes by Emil Artin. Each of them presents a novel approach. There are always new ideas and new results. It was a compulsion for him to present each argument in its purest form, to replace computation by conceptual arguments, to strip the theory of unnecessary ballast. What was the decisive point for him was to show the beauty of the subject to the reader."" Information for our distributors: Copublished with the London Mathematical Society beginning with Volume 4. Members of the LMS may order directly from the AMS at the AMS member price. The LMS is registered with the Charity Commissioners.

Historical records show that there was no real concept of probability in Europe before the mid-seventeenth century, although the use of dice and other randomizing objects was commonplace. Ian Hacking presents a philosophical critique of early ideas about probability, induction, and statistical inference and the growth of this new family of ideas in the fifteenth, sixteenth, and seventeenth centuries. Hacking invokes a wide intellectual framework involving the growth of science, economics, and the theology of the period. He argues that the transformations that made it possible for probability concepts to emerge have constrained all subsequent development of probability theory and determine the space within which philosophical debate on the subject is still conducted. First published in 1975, this edition includes an introduction that contextualizes his book in light of developing philosophical trends. Ian Hacking is the winner of the Holberg International Memorial Prize 2009.

Writings by early mathematicians feature language and notations that are quite different from what we're familiar with today. Sourcebooks on the history of mathematics provide some guidance, but what has been lacking is a guide tailored to the needs of readers approaching these writings for the first time. "How to Read Historical Mathematics" fills this gap by introducing readers to the analytical questions historians ask when deciphering historical texts.

Sampling actual writings from the history of mathematics, Benjamin Wardhaugh reveals the questions that will unlock the meaning and significance of a given text--Who wrote it, why, and for whom? What was its author's intended meaning? How did it reach its present form? Is it original or a translation? Why is it important today? Wardhaugh teaches readers to think about what the original text might have looked like, to consider where and when it was written, and to formulate questions of their own. Readers pick up new skills with each chapter, and gain the confidence and analytical sophistication needed to tackle virtually any text in the history of mathematics.Introduces readers to the methods of textual analysis used by historians Uses actual source material as examples Features boxed summaries, discussion questions, and suggestions for further reading Supplements all major sourcebooks in mathematics history Designed for easy reference Ideal for students and teachers

The world around us is saturated with numbers. They are a fundamental pillar of our modern society, and accepted and used with hardly a second thought. But how did this state of affairs come to be? In this book, Leo Corry tells the story behind the idea of number from the early days of the Pythagoreans, up until the turn of the twentieth century. He presents an overview of how numbers were handled and conceived in classical Greek mathematics, in the mathematics of Islam, in European mathematics of the middle ages and the Renaissance, during the scientific revolution, all the way through to the mathematics of the 18th to the early 20th century. Focusing on both foundational debates and practical use numbers, and showing how the story of numbers is intimately linked to that of the idea of equation, this book provides a valuable insight to numbers for undergraduate students, teachers, engineers, professional mathematicians, and anyone with an interest in the history of mathematics.

Thomas Morel tells the story of subterranean geometry, a forgotten discipline that developed in the silver mines of early modern Europe. Mining and metallurgy were of great significance to the rulers of early modern Europe, required for the silver bullion that fuelled warfare and numerous other uses. Through seven lively case studies, he illustrates how geometry was used in metallic mines by practitioners using esoteric manuscripts. He describes how an original culture of accuracy and measurement paved the way for technical and scientific innovations, and fruitfully brought together the world of artisans, scholars and courts. Based on a variety of original manuscripts, maps and archive material, Morel recounts how knowledge was crafted and circulated among practitioners in the Holy Roman Empire and beyond. Specific chapters deal with the material culture of surveying, map-making, expertise and the political uses of quantification. By carefully reconstructing the religious, economic and cultural context of mining cities, Underground Mathematics contextualizes the rise of numbered information, practical mathematics and quantification in the early modern period.

Praise for the Second Edition "An amazing assemblage of worldwide contributions in mathematics and, in addition to use as a course book, a valuable resource ...essential." CHOICE This Third Edition of The History of Mathematics examines the elementary arithmetic, geometry, and algebra of numerous cultures, tracing their usage from Mesopotamia, Egypt, Greece, India, China, and Japan all the way to Europe during the Medieval and Renaissance periods where calculus was developed. Aimed primarily at undergraduate students studying the history of mathematics for science, engineering, and secondary education, the book focuses on three main ideas: the facts of who, what, when, and where major advances in mathematics took place; the type of mathematics involved at the time; and the integration of this information into a coherent picture of the development of mathematics. In addition, the book features carefully designed problems that guide readers to a fuller understanding of the relevant mathematics and its social and historical context. Chapter-end exercises, numerous photographs, and a listing of related websites are also included for readers who wish to pursue a specialized topic in more depth. Additional features of The History of Mathematics, Third Edition include: * Material arranged in a chronological and cultural context * Specific parts of the history of mathematics presented as individual lessons * New and revised exercises ranging between technical, factual, and integrative * Individual PowerPoint presentations for each chapter and a bank of homework and test questions (in addition to the exercises in the book) * An emphasis on geography, culture, and mathematics In addition to being an ideal coursebook for undergraduate students, the book also serves as a fascinating reference for mathematically inclined individuals who are interested in learning about the history of mathematics.

Greenwich has been a centre for scientific computing since the foundation of the Royal Observatory in 1675. Early Astronomers Royal gathered astronomical data with the purpose of enabling navigators to compute their longitude at sea. Nevil Maskelyne in the 18th century organised the work of computing tables for the Nautical Almanac, anticipating later methods used in safety-critical computing systems. The 19th century saw influential critiques of Charles Babbage's mechanical calculating engines, and in the 20th century Leslie Comrie and others pioneered the automation of computation. The arrival of the Royal Naval College in 1873 and the University of Greenwich in 1999 has brought more mathematicians and different kinds of mathematics to Greenwich. In the 21st century computational mathematics has found many new applications. This book presents an account of the mathematicians who worked at Greenwich and their achievements. Features A scholarly but accessible history of mathematics at Greenwich, from the seventeenth century to the present day, with each chapter written by an expert in the field The book will appeal to astronomical and naval historians as well as historians of mathematics and scientific computing.

A renowned puzzle master and game inventor presents 315 new and traditional puzzles. Longlisted, 2016 School Library Association Information Book Award 12+ 'A great read for anyone interested in puzzles or mathematics.' Publishers Weekly. 'Gift Guide Selection 2015. Moscovich, a celebrated puzzle inventor, makes a compelling case for puzzle solving as a means of developing creativity and even intelligence.' American Scientist. 'The Puzzle Universe is a quixotic, informative and enlightening encyclopedia of recreational mathematics. It should prove to be an inspiration to mathematical idlers, and a rich resource for learners and teachers who wish to be attuned to the playful and creative side of mathematics.' Mathrecreation Blog. The Puzzle Universe is intended for general readers and devoted puzzlers but it has also found its way into school libraries and curricula. It is about the latent beauty of mathematics, its history and the puzzles that have emerged from the science of numbers. It is full of challenging historical facts, thinking puzzles, paradoxes, illusions and problem solving. A historical and pedagogical dimension sets The Puzzle Universe apart. The 315 puzzles are described in extended captions that explain in easy terms the story of the puzzles’ origins and attempts to solve them, the value of puzzles to education, and the development of the mathematical sciences in light of recent research and unmet challenges. The puzzles appear in a dynamic layout for a visual experience that is the author’s trademark. There are ten chapters complete with answers. Icons rate the challenge and indicate the tools needed (pencil, scissors, ruler and, of course, your thinking cap). With this book, Ivan Moscovich invites readers to join him in the puzzle universe, an edifying environment of creative discovery, problem solving and fun.

This challenging book argues that a new way of speaking of mathematics and describing it emerged at the end of the sixteenth century. Leading mathematicians like Hariot, Stevin, Galileo, and Cavalieri began referring to their field in terms drawn from the exploration accounts of Columbus and Magellan. As enterprising explorers in search of treasures of knowledge, these mathematicians described themselves as sailing the treacherous seas of mathematics, facing shipwreck on the shoals of paradox, and seeking shelter and refuge on the shores of geometrical demonstrations. Mathematics, formerly praised for its logic, clarity, and inescapable truths, was for them a hazardous voyage in inhospitable geometrical lands.
Significantly, many of the same practitioners who promoted the vision of mathematics as heroic exploration also played central roles in developing the most important mathematical innovation of the period--the infinitesimal methods. This was no coincidence: the heroic tales of exploration and discovery helped shape a new form of mathematical practice, complete with new questions, new acceptable answers, and new standards of evidence. It was this new vision of mathematics as a grand adventure that allowed for the development of the new techniques that led to the Newtonian calculus.
In demonstrating this, the book moves from real voyages to imaginary ones, from the coasts of the Canadian Arctic to the tropical forests of Guyana, and from the inner structure of matter to the intricacies of the mathematical continuum. Throughout, a common rhetoric and imagery of exploration and discovery run like a thread through these diverse elements and bind them together.

"An enchanting history of Japanese geometry--of a time and place where 'geometers did not cede place to poets.' This intersection of science and culture, of the mathematical, the artistic, and the spiritual, is packed, like circles within circles, with rewarding Aha! epiphanies that drive a mathematician's curiosity."--Siobhan Roberts, author of "King of Infinite Space"

"Teachers will welcome this remarkable collection of mathematical problems, history, and art, which will enrich their curriculum and promote both logical thinking and critical evaluation. It is especially important that we maintain an interest in geometry, which needs, and for once gets, more than its share."--Richard Guy, coauthor of "The Book of Numbers"

"This remarkable book provides a novel insight into the Japanese mathematics of the past few hundred years. It is fascinating to see the difference in mathematical style from that which we are used to in the Western world, but the book also elegantly illustrates the cross-cultural Platonic nature and profound beauty of mathematics itself."--Roger Penrose, author of "The Road to Reality"

"A significant contribution to the history of mathematics. The wealth of mathematical problems--from the very simple to quite complex ones--will keep the interested reader busy for years. And the beautiful illustrations make this book a work of art as much as of science. Destined to become a classic!"--Eli Maor, author of "The Pythagorean Theorem: A 4,000-Year History"

"A pleasure to read. "Sacred Mathematics" brings to light the unique style and character of geometry in the traditional Japanese sources--in particular the "sangaku" problems. These problems range from trivialto utterly devilish. I found myself captivated by them, and regularly astounded by the ingenuity and sophistication of many of the traditional solutions."--Glen Van Brummelen, coeditor of "Mathematics and the Historian's Craft"

This book offers an original and informative view of the development of fundamental concepts of computability theory. The treatment is put into historical context, emphasizing the motivation for ideas as well as their logical and formal development. In Part I the author introduces computability theory, with chapters on the foundational crisis of mathematics in the early twentieth century, and formalism. In Part II he explains classical computability theory, with chapters on the quest for formalization, the Turing Machine, and early successes such as defining incomputable problems, c.e. (computably enumerable) sets, and developing methods for proving incomputability. In Part III he explains relative computability, with chapters on computation with external help, degrees of unsolvability, the Turing hierarchy of unsolvability, the class of degrees of unsolvability, c.e. degrees and the priority method, and the arithmetical hierarchy. Finally, in the new Part IV the author revisits the computability (Church-Turing) thesis in greater detail. He offers a systematic and detailed account of its origins, evolution, and meaning, he describes more powerful, modern versions of the thesis, and he discusses recent speculative proposals for new computing paradigms such as hypercomputing. This is a gentle introduction from the origins of computability theory up to current research, and it will be of value as a textbook and guide for advanced undergraduate and graduate students and researchers in the domains of computability theory and theoretical computer science. This new edition is completely revised, with almost one hundred pages of new material. In particular the author applied more up-to-date, more consistent terminology, and he addressed some notational redundancies and minor errors. He developed a glossary relating to computability theory, expanded the bibliographic references with new entries, and added the new part described above and other new sections.

The book offers an extensive study on the convoluted history of the research of algebraic surfaces, focusing for the first time on one of its characterizing curves: the branch curve. Starting with separate beginnings during the 19th century with descriptive geometry as well as knot theory, the book focuses on the 20th century, covering the rise of the Italian school of algebraic geometry between the 1900s till the 1930s (with Federigo Enriques, Oscar Zariski and Beniamino Segre, among others), the decline of its classical approach during the 1940s and the 1950s (with Oscar Chisini and his students), and the emergence of new approaches with Boris Moishezon's program of braid monodromy factorization. By focusing on how the research on one specific curve changed during the 20th century, the author provides insights concerning the dynamics of epistemic objects and configurations of mathematical research. It is in this sense that the book offers to take the branch curve as a cross-section through the history of algebraic geometry of the 20th century, considering this curve as an intersection of several research approaches and methods. Researchers in the history of science and of mathematics as well as mathematicians will certainly find this book interesting and appealing, contributing to the growing research on the history of algebraic geometry and its changing images.

In this book, Johnny Ball tells one of the most important stories in world history - the story of mathematics.

By introducing us to the major characters and leading us through many historical twists and turns, Johnny slowly unravels the tale of how humanity built up a knowledge and understanding of shapes, numbers and patterns from ancient times, a story that leads directly to the technological wonderland we live in today. As Galileo said, 'Everything in the universe is written in the language of mathematics', and Wonders Beyond Numbers is your guide to this language.

Mathematics is only one part of this rich and varied tale; we meet many fascinating personalities along the way, such as a mathematician who everyone has heard of but who may not have existed; a Greek philosopher who made so many mistakes that many wanted his books destroyed; a mathematical artist who built the largest masonry dome on earth, which builders had previously declared impossible; a world-renowned painter who discovered mathematics and decided he could no longer stand the sight of a brush; and a philosopher who lost his head, but only after he had died.

Enriched with tales of colourful personalities and remarkable discoveries, there is also plenty of mathematics for keen readers to get stuck into. Written in Johnny Ball's characteristically light-hearted and engaging style, this book is packed with historical insight and mathematical marvels; join Johnny and uncover the wonders found beyond the numbers.

Der Bericht uber das vielleicht grosste mathematische Genie des 20. Jahrhunderts liest sich wie ein spannender Roman."

This book offers a new and original hypothesis on the origin of modal ontology, whose roots can be traced back to the mathematical debate about incommensurable magnitudes, which forms the implicit background for Plato's later dialogues and culminates in the definition of being as dynamis in the Sophist. Incommensurable magnitudes - also called dynameis by Theaetetus - are presented as the solution to the problem of non-being and serve as the cornerstone for a philosophy of difference and becoming. This shift also marks the passage to another form of rationality - one not of the measure, but of the mediation. The book argues that the ontology and the rationality which arise out of the discovery of incommensurable constitutes a thread that runs through the entire history of philosophy, one that leads to Kantian transcendentalism and to the philosophies derived from it, such as Hegelianism and philosophical hermeneutics. Readers discover an insightful exchange with some of the most important issues in philosophy, newly reconsidered from the point of view of an ontology of the incommensurable. These issues include the infinite, the continuum, existence, and difference. This text appeals to students and researchers in the fields of ancient philosophy, German idealism, philosophical hermeneutics and the history of mathematics.

This undergraduate textbook promotes an active transition to higher mathematics. Problem solving is the heart and soul of this book: each problem is carefully chosen to demonstrate, elucidate, or extend a concept. More than 300 exercises engage the reader in extensive arguments and creative approaches, while exploring connections between fundamental mathematical topics. Divided into four parts, this book begins with a playful exploration of the building blocks of mathematics, such as definitions, axioms, and proofs. A study of the fundamental concepts of logic, sets, and functions follows, before focus turns to methods of proof. Having covered the core of a transition course, the author goes on to present a selection of advanced topics that offer opportunities for extension or further study. Throughout, appendices touch on historical perspectives, current trends, and open questions, showing mathematics as a vibrant and dynamic human enterprise. This second edition has been reorganized to better reflect the layout and curriculum of standard transition courses. It also features recent developments and improved appendices. An Invitation to Abstract Mathematics is ideal for those seeking a challenging and engaging transition to advanced mathematics, and will appeal to both undergraduates majoring in mathematics, as well as non-math majors interested in exploring higher-level concepts. From reviews of the first edition: Bajnok's new book truly invites students to enjoy the beauty, power, and challenge of abstract mathematics. ... The book can be used as a text for traditional transition or structure courses ... but since Bajnok invites all students, not just mathematics majors, to enjoy the subject, he assumes very little background knowledge. Jill Dietz, MAA ReviewsThe style of writing is careful, but joyously enthusiastic.... The author's clear attitude is that mathematics consists of problem solving, and that writing a proof falls into this category. Students of mathematics are, therefore, engaged in problem solving, and should be given problems to solve, rather than problems to imitate. The author attributes this approach to his Hungarian background ... and encourages students to embrace the challenge in the same way an athlete engages in vigorous practice. John Perry, zbMATH

In this book the authors aim to endow the reader with an operational, conceptual, and methodological understanding of the discrete mathematics that can be used to study, understand, and perform computing. They want the reader to understand the elements of computing, rather than just know them. The basic topics are presented in a way that encourages readers to develop their personal way of thinking about mathematics. Many topics are developed at several levels, in a single voice, with sample applications from within the world of computing. Extensive historical and cultural asides emphasize the human side of mathematics and mathematicians. By means of lessons and exercises on "doing" mathematics, the book prepares interested readers to develop new concepts and invent new techniques and technologies that will enhance all aspects of computing. The book will be of value to students, scientists, and engineers engaged in the design and use of computing systems, and to scholars and practitioners beyond these technical fields who want to learn and apply novel computational ideas.

This volume contains eleven papers that have been collected by the Canadian Society for History and Philosophy of Mathematics/Societe canadienne d'histoire et de philosophie des mathematiques. It showcases rigorously-reviewed contemporary scholarship on an interesting variety of topics in the history and philosophy of mathematics, as well as the teaching of the history of mathematics. Topics considered include The mathematics and astronomy in Nathaniel Torperly's only published work, Diclides Coelometricae, seu valvae astronomicae universal Connections between the work of Urbain Le Verrier, Carl Gustav Jacob Jacobi, and Augustin-Louis Cauchy on the algebraic eigenvalue problem An evaluation of Ken Manders' argument against conceiving of the diagrams in Euclid's Elements in semantic terms The development of undergraduate modern algebra courses in the United States Ways of using the history of mathematics to teach the foundations of mathematical analysis Written by leading scholars in the field, these papers are accessible not only to mathematicians and students of the history and philosophy of mathematics, but also to anyone with a general interest in mathematics.