The first World Meeting for Women in Mathematics - (WM)(2) - was a satellite event of the International Congress of Mathematicians (ICM) 2018 in Rio de Janeiro. With a focus on Latin America, the first (WM)(2) brought together mathematicians from all over the world to celebrate women mathematicians, and also to reflect on gender issues in mathematics, challenges, initiatives, and perspectives for the future. Its activities were complemented by a panel discussion organized by the Committee for Women in Mathematics (CWM) of the International Mathematical Union (IMU) inside the ICM 2018 entitled "The gender gap in mathematical and natural sciences from a historical perspective". This historical proceedings book, organized by CWM in coordination with the Association for Women in Mathematics, records the first (WM)(2) and the CWM panel discussion at ICM 2018. The first part of the volume includes a report of activities with pictures of the first (WM)(2) and a tribute to Maryam Mirzakhani, the first woman to be awarded the Fields medal. It also comprises survey research papers from invited lecturers, which provide panoramic views of different fields in pure and applied mathematics. The second part of the book contains articles from the panelists of the CWM panel discussion, which consider the historical context of the gender gap in mathematics. It includes an analysis of women lecturers in the ICM since its inception. This book is dedicated to the memory of Maryam Mirzakhani.

This is the first comprehensive International Handbook on the History of Mathematics Education, covering a wide spectrum of epochs and civilizations, countries and cultures. Until now, much of the research into the rich and varied history of mathematics education has remained inaccessible to the vast majority of scholars, not least because it has been written in the language, and for readers, of an individual country. And yet a historical overview, however brief, has become an indispensable element of nearly every dissertation and scholarly article. This handbook provides, for the first time, a comprehensive and systematic aid for researchers around the world in finding the information they need about historical developments in mathematics education, not only in their own countries, but globally as well. Although written primarily for mathematics educators, this handbook will also be of interest to researchers of the history of education in general, as well as specialists in cultural and even social history.

Emily Grosholz offers an original investigation of demonstration in mathematics and science, examining how it works and why it is persuasive. Focusing on geometrical demonstration, she shows the roles that representation and ambiguity play in mathematical discovery. She presents a wide range of case studies in mechanics, topology, algebra, logic, and chemistry, from ancient Greece to the present day, but focusing particularly on the seventeenth and twentieth centuries. She argues that reductive methods are effective not because they diminish but because they multiply and juxtapose modes of representation. Such problem-solving is, she argues, best understood in terms of Leibnizian "analysis"--the search for conditions of intelligibility. Discovery and justification are then two aspects of one rational way of proceeding, which produces the mathematician's formal experience.
Grosholz defends the importance of iconic, as well as symbolic and indexical, signs in mathematical representation, and argues that pragmatic, as well as syntactic and semantic, considerations are indispensable fore mathematical reasoning. By taking a close look at the way results are presented on the page in mathematical (and biological, chemical, and mechanical) texts, she shows that when two or more traditions combine in the service of problem solving, notations and diagrams are subtly altered, multiplied, and juxtaposed, and surrounded by prose in natural language which explains the novel combination. Viewed this way, the texts yield striking examples of language and notation that are irreducibly ambiguous and productive because they are ambiguous. Grosholtz's arguments, which invoke Descartes, Locke, Hume, and Kant, will be of considerable interest to philosophers and historians of mathematics and science, and also have far-reaching consequences for epistemology and philosophy of language.

Ada's Legacy illustrates the depth and diversity of writers, thinkers, and makers who have been inspired by Ada Lovelace, the English mathematician and writer. The volume, which commemorates the bicentennial of Ada's birth in December 1815, celebrates Lovelace's many achievements as well as the impact of her life and work, which reverberated widely since the late nineteenth century. In the 21st century we have seen a resurgence in Lovelace scholarship, thanks to the growth of interdisciplinary thinking and the expanding influence of women in science, technology, engineering and mathematics. Ada's Legacy is a unique contribution to this scholarship, thanks to its combination of papers on Ada's collaboration with Charles Babbage, Ada's position in the Victorian and Steampunk literary genres, Ada's representation in and inspiration of contemporary art and comics, and Ada's continued relevance in discussions around gender and technology in the digital age. With the 200th anniversary of Ada Lovelace's birth on December 10, 2015, we believe that the timing is perfect to publish this collection of papers. Because of its broad focus on subjects that reach far beyond the life and work of Ada herself, Ada's Legacy will appeal to readers who are curious about Ada's enduring importance in computing and the wider world.

This is the second part of a two volume anthology comprising a selection of 49 articles that illustrate the depth, breadth and scope of Nigel Kalton's research. Each article is accompanied by comments from an expert on the respective topic, which serves to situate the article in its proper context, to successfully link past, present and hopefully future developments of the theory and to help readers grasp the extent of Kalton's accomplishments. Kalton's work represents a bridge to the mathematics of tomorrow, and this book will help readers to cross it. Nigel Kalton (1946-2010) was an extraordinary mathematician who made major contributions to an amazingly diverse range of fields over the course of his career.

Einfuhrung.- Qualitat und Quantitat in der antiken Philosophie: Zur Genese einer Fragestellung.- Quantity and Quality in Scholastic Aristotelian Natural Philosophy: John Dumbleton's Summa Logicae et Philosophiae Naturalis.- Der Arzt Paracelsus zwischen Physica und Virtus.- (Bearbeitung: Hans Bieri; Nachwort: Gunter Scharf) Qualitat und Quantitat in Keplers Weltharmonik.- Farbenlehre bei Newton und Goethe.- Laboratoriumspraxis, Quantitaten und die Produktion von Erkenntnis: Transformationen der Chemie zwischen Lavoisier und Liebig.- Die Einheit von Mikro- und Makrokosmos: Quantitat und Qualitat in der Physik.- Qualitas und Quantitas und die beiden Hirnhalften.- Rechnen und Unterscheiden: Qualitat und Quantitat zwischen Philosophie und Wissenschaft.- Gibt es eine Wiedergeburt der Qualitat in der Mathematik?.- Literaturauswahl zur Einfuhrung.- UEber die Autoren.

This book provides the first critical edition of Ibn al-Haytham's On the Shape of the Eclipse with English translation and commentary, which records the first scientific analysis of the camera obscura. On the Shape of the Eclipse includes pioneering research on the conditions of formation of the image, in a time deemed to be committed to aniconism. It also provides an early attempt to merge the two branches of Ancient optics-the theory of light and theory of vision. What perhaps most strongly characterizes this treatise is the close interaction of a geometric analysis of light and experimental reasoning. Ibn al-Haytham conducted his experiments in a systematic way by varying all that could be changed: the shape and size of the aperture, the focal length of the camera obscura, the distance and shape of the celestial bodies. This way, he achieved a thorough understanding. This work represents a decisive step in both the history of optics and the application of the experimental method that was just as efficient in medieval Islam as today.

This book is the first part of a two volume anthology comprising a selection of 49 articles that illustrate the depth, breadth and scope of Nigel Kalton's research. Each article is accompanied by comments from an expert on the respective topic, which serves to situate the article in its proper context, to successfully link past, present and hopefully future developments of the theory, and to help readers grasp the extent of Kalton's accomplishments. Kalton's work represents a bridge to the mathematics of tomorrow, and this book will help readers to cross it. Nigel Kalton (1946-2010) was an extraordinary mathematician who made major contributions to an amazingly diverse range of fields over the course of his career.

This book provides the only critical edition and English translation of Mahmud al-Jaghmini's al-Mulakhkhas fi al-hay'a al-basita, the most widely circulated Arabic treatise on Ptolemaic astronomy ever written. Composed in the early 13th century, this introductory textbook played a crucial role in the teaching, dissemination, and institutional instruction of Islamic astronomy well into the 19th century (and beyond). Establishing the base text is a fundamental prerequisite for gaining insights into what was considered an elementary astronomical textbook in Islam and also for understanding the extensive commentary tradition that built upon it. Within this volume, the Mulakhkhas is situated within the broader context of the genre of literature termed 'ilm al-hay'a, which has become the subject of intensive research over the past 25 years. In so doing, it provides a survey of summary accounts of theoretical astronomy of Jaghmini's predecessors, both Ancient and Islamic, which could have served as potential sources for the Mulakhkhas. Jaghmini's dates (which until now remained unsettled) are established, and it is definitively shown that he composed not only the Mulakhkhas but also other scientific treatises, including the popular medical treatise al-Qanunca, during a period that has been deemed one of scientific decline and stagnation in Islamic lands. The book will be of particular interest to scholars engaged in the study of Islamic theoretical astronomy, but is accessible to a general readership interested in learning what constituted an introduction to Ptolemaic astronomy in Islamic lands.

This book explains the development of theoretical computer science in its early stages, specifically from 1965 to 1990. The author is among the pioneers of theoretical computer science, and he guides the reader through the early stages of development of this new discipline. He explains the origins of the field, arising from disciplines such as logic, mathematics, and electronics, and he describes the evolution of the key principles of computing in strands such as computability, algorithms, and programming. But mainly it's a story about people - pioneers with diverse backgrounds and characters came together to overcome philosophical and institutional challenges and build a community. They collaborated on research efforts, they established schools and conferences, they developed the first related university courses, they taught generations of future researchers and practitioners, and they set up the key publications to communicate and archive their knowledge. The book is a fascinating insight into the field as it existed and evolved, it will be valuable reading for anyone interested in the history of computing.

This book deals with a topic that has been largely neglected by philosophers of science to date: the ability to refer and analyze in tandem. On the basis of a set of philosophical case studies involving both problems in number theory and issues concerning time and cosmology from the era of Galileo, Newton and Leibniz up through the present day, the author argues that scientific knowledge is a combination of accurate reference and analytical interpretation. In order to think well, we must be able to refer successfully, so that we can show publicly and clearly what we are talking about. And we must be able to analyze well, that is, to discover productive and explanatory conditions of intelligibility for the things we are thinking about. The book's central claim is that the kinds of representations that make successful reference possible and those that make successful analysis possible are not the same, so that significant scientific and mathematical work typically proceeds by means of a heterogeneous discourse that juxtaposes and often superimposes a variety of kinds of representation, including formal and natural languages as well as more iconic modes. It demonstrates the virtues and necessity of heterogeneity in historically central reasoning, thus filling an important gap in the literature and fostering a new, timely discussion on the epistemology of science and mathematics.

Chapters "Turing and Free Will: A New Take on an Old Debate" and "Turing and the History of Computer Music" are available open access under a Creative Commons Attribution 4.0 International License via link.springer.com.

This is the first extensive biography of the influential German mathematician, Peter Gustav Lejeune Dirichlet (1805 - 1859). Dirichlet made major contributions to number theory in addition to clarifying concepts such as the representation of functions as series, the theory of convergence, and potential theory. His mathematical methodology was explicitly based on a thorough knowledge of the work of his predecessors and his belief in the underlying unity of the branches of mathematics. This unified approach is exemplified in a paper that effectively launched the field of analytic number theory. The same orientation pervaded his teaching, which had a profound influence on the work of many mathematicians of subsequent generations. Chapters dealing with his mathematical work alternate with biographical chapters that place Dirichlet's life and those of some of his notable associates in the context of the political, social, and artistic culture of the period. This book will appeal not only to mathematicians but also to historians of mathematics and sciences, and readers interested in the cultural and intellectual history of the nineteenth century.

This book studies the brilliant Danish 19th Century astronomer, T.N. Thiele who made important contributions to statistics, actuarial science, astronomy and mathematics. The most important of these contributions in statistics are translated into English for the first time, and the text includes comments that set his achievements in a modern and historical perspective.

This is a volume of chapters on the historical study of information, computing, and society written by seven of the most senior, distinguished members of the History of Computing field. These are edited, expanded versions of papers presented in a distinguished lecture series in 2018 at the University of Colorado Boulder - in the shadow of the Flatirons, the front range of the Rocky Mountains. Topics range widely across the history of computing. They include the digitalization of computer and communication technologies, gender history of computing, the history of data science, incentives for innovation in the computing field, labor history of computing, and the process of standardization. Authors were given wide latitude to write on a topic of their own choice, so long as the result is an exemplary article that represents the highest level of scholarship in the field, producing articles that scholars in the field will still look to read twenty years from now. The intention is to publish articles of general interest, well situated in the research literature, well grounded in source material, and well-polished pieces of writing. The volume is primarily of interest to historians of computing, but individual articles will be of interest to scholars in media studies, communication, computer science, cognitive science, general and technology history, and business.

In 1842 the Belgian mathematician Eugene Charles Catalan asked whether 8 and 9 are the only consecutive pure powers of non-zero integers. 160 years after, the question was answered affirmatively by the Swiss mathematician of Romanian origin Preda Mihailescu. In other words, 32 - 23 = 1 is the only solution of the equation xp - yq = 1 in integers x, y, p, q with xy 0 and p, q 2. In this book we give a complete and (almost) self-contained exposition of Mihailescu's work, which must be understandable by a curious university student, not necessarily specializing in Number Theory. We assume a very modest background:a standard university course of algebra, including basic Galois theory, and working knowledge of basic algebraic number theory.

This book fills an important gap in studies on D. D. Kosambi. For the first time, the mathematical work of Kosambi is described, collected and presented in a manner that is accessible to non-mathematicians as well. A number of his papers that are difficult to obtain in these areas are made available here. In addition, there are essays by Kosambi that have not been published earlier as well as some of his lesser known works. Each of the twenty four papers is prefaced by a commentary on the significance of the work, and where possible, extracts from technical reviews by other mathematicians.

The book deals with expounding the nature of Reality as it is understood in contemporary times in Quantum Physics. It also explains the classical Indian theory of Sunya in its diverse facets. Thereafter it undertakes comparison between the two which is an area of great topical interest. It is a cross-disciplinary study by erudite Indian and western scholars between traditional Indian knowledge system and contemporary researches in Physical sciences. It points out how the theory of 'Sunyata has many seminal ideas and theories in common with contemporary Quantum Physics. The learned authors have tried to dissolve the "mysteries" of Quantum Physics and resolved its "weird paradoxes" with the help of theory of Sunyata. The issue of non-separability or entanglement has been approached with the help of the Buddhist theory of Pratityasamutpada. The paradoxical situation of "wave-particle duality" has been explained with the help of Upanisadic theory of complementarity of the two opposites. The measurement problem represented by "Schrodinger's cat" has been dealt with by resorting to two forms of the calculation of probabilities. Some writers have argued for Sunyata-like non-essentialist position to understand quantum reality. To make sense of quantum theory some papers provide a happy symbiosis of technical understanding and personal meditative experience by drawing multifarious parallels. This book will be of interest to philosophically inclined physicists and philosophers with interest in quantum mechanics.

Science is based not only on observation and experiment, but on theory as well. As Einstein said, "Theory tells us what to measure." And theories are often crystallized into succinct calculations, like those made using Einstein's famous E = mc2. This book looks at fifty such great calculations, exploring how and why they were developed and assessing their impact on the history of science. As the author shows, many significant scientific calculations are quite simple and fairly easy to understand, even for readers will little math background. But their implications can be surprising and profound. For example, what links a famous comet and the cost of an annuity? Why do scientists claim there is "dark matter" in the universe if it can't be observed? How does carbon-based life on Earth depend on a quirk of nuclear physics? The answer to each question is an illuminating calculation. This accessible, engaging book will help you understand these breakthroughs and how they changed our view of life and the world.

Shortlisted for the Royal Society Science Book Prize 2019 A magisterial history of calculus (and the people behind it) from one of the world's foremost mathematicians. This is the captivating story of mathematics' greatest ever idea: calculus. Without it, there would be no computers, no microwave ovens, no GPS, and no space travel. But before it gave modern man almost infinite powers, calculus was behind centuries of controversy, competition, and even death. Taking us on a thrilling journey through three millennia, professor Steven Strogatz charts the development of this seminal achievement from the days of Archimedes to today's breakthroughs in chaos theory and artificial intelligence. Filled with idiosyncratic characters from Pythagoras to Fourier, Infinite Powers is a compelling human drama that reveals the legacy of calculus on nearly every aspect of modern civilisation, including science, politics, medicine, philosophy, and much besides.

The contributions in this proceedings volume offer a new perspective on the mathematical ties between France and Italy, and reveal how mathematical developments in these two countries affected one another. The focus is above all on the Peninsula's influence on French mathematicians, counterbalancing the historically predominant perception that French mathematics was a model for Italian mathematicians. In the process, the book details a subtle network of relations between the two countries, where mathematical exchanges fit into the changing and evolving framework of Italian political and academic structures. It reconsiders the issue of nationalities in all of its complexity, an aspect often neglected in research on the history of mathematics. The works in this volume are selected contributions from a conference held in Lille and Lens (France) in November 2013 on Images of Italian Mathematics in France from Risorgimento to Fascism. The authors include respected historians of mathematics, philosophers of science, historians, and specialists for Italy and intellectual relations, ensuring the book will be of great interest to their peers.

Charles Dupin was a multifaceted figure in the history of France, where his life spanned several regimes. He produced an enormous number of publications in mathematics, engineering, economics, and education. Long neglected by historians, he is at last beginning to receive attention. In his youth, he championed many causes, including the education of women, perhaps because of the influence of his dynamic and learned mother. He was already very ambitious as a youth and left behind the usual youthful desires in pursuit of his goals. Dupin began as a brilliant mathematician as a student at the Ecole polytechnique in Paris and proceeded to become a fine naval engineer, that is until visits to Britain inspired him to change his course of direction. As the French industry was undergoing expansion, Dupin saw in Britain that workers were more efficient and healthier if they were educated. He greatly admired the freedom he witnessed in Britain, and this did not endear him to the French government of the Restoration. Indeed, the high honours to which he so much aspired eluded him for a considerable time. He saw the British savings banks and regular saving by workers within industry as a system to be introduced in France and one that should be propagated. As an economist, he considered the welfare of French workers as vital to an efficient industry. He was particularly concerned with the protection of children in work and the education of workers. In fact, he might be considered the father of workers' education in France. This was a subject very close to his heart and, from his early years, he devoted himself to making public lectures available to all, including women. However, Dupin's popularity declined as the importance of a thriving economy began to take precedence over the workers' needs, with the workers focusing mainly on having a living wage. This is the first published study of Charles Dupin and his entire life's work. It illuminates his work and contribution in so many spheres, as well as his contacts with other scientists and educators. His mathematics have long interested scholars in the field, and he would have been an outstanding naval engineer. He was a linguist and highly cultured; with his aesthetic sense he might well have rivaled San, but because of his driving ambition he was a great man manqu . Against a background of tremendous changes in France, he made important contributions in many areas, as evidenced by the bibliography in this book. This work will be of interest to mathematicians, historians of science, sociologists, economists, engineers, and educators.

This volume contains thirteen papers that were presented at the 2017 Annual Meeting of the Canadian Society for History and Philosophy of Mathematics/Societe canadienne d'histoire et de philosophie des mathematiques, which was held at Ryerson University in Toronto. It showcases rigorously reviewed modern scholarship on an interesting variety of topics in the history and philosophy of mathematics from Ancient Greece to the twentieth century. A series of chapters all set in the eighteenth century consider topics such as John Marsh's techniques for the computation of decimal fractions, Euler's efforts to compute the surface area of scalene cones, a little-known work by John Playfair on the practical aspects of mathematics, and Monge's use of descriptive geometry. After a brief stop in the nineteenth century to consider the culture of research mathematics in 1860s Prussia, the book moves into the twentieth century with an examination of the historical context within which the Axiom of Choice was developed and a paper discussing Anatoly Vlasov's adaptation of the Boltzmann equation to ionized gases. The remaining chapters deal with the philosophy of twentieth-century mathematics through topics such as an historically informed discussion of finitism and its limits; a reexamination of Mary Leng's defenses of mathematical fictionalism through an alternative, anti-realist approach to mathematics; and a look at the reasons that mathematicians select specific problems to pursue. Written by leading scholars in the field, these papers are accessible to not only mathematicians and students of the history and philosophy of mathematics, but also anyone with a general interest in mathematics.