The philosopher Immanuel Kant writes in the popular introduction to his philosophy: "There is no single book about metaphysics like we have in mathematics. If you want to know what mathematics is, just look at Euclid's Elements." (Prolegomena Paragraph 4) Even if the material covered by Euclid may be considered elementary for the most part, the way in which he presents essential features of mathematics in a much more general sense, has set the standards for more than 2000 years. He displays the axiomatic foundation of a mathematical theory and its conscious development towards the solution of a specific problem. We see how abstraction works and how it enforces the strictly deductive presentation of a theory. We learn what creative definitions are and how the conceptual grasp leads to the classification of the relevant objects. For each of Euclid's thirteen Books, the author has given a general description of the contents and structure of the Book, plus one or two sample proofs. In an appendix, the reader will find items of general interest for mathematics, such as the question of parallels, squaring the circle, problem and theory, what rigour is, the history of the platonic polyhedra, irrationals, the process of generalization, and more. This is a book for all lovers of mathematics with a solid background in high school geometry, from teachers and students to university professors. It is an attempt to understand the nature of mathematics from its most important early source.

In addition to linear perspective, complex numbers and probability were notable discoveries of the Renaissance. While the power of perspective, which transformed Renaissance art, was quickly recognized, the scientific establishment treated both complex numbers and probability with much suspicion. It was only in the twentieth century that quantum theory showed how probability might be molded from complex numbers and defined the notion of "complex probability amplitude". From a theoretical point of view, however, the space opened to painting by linear perspective and that opened to science by complex numbers share significant characteristics. The Art of Science explores this shared field with the purpose of extending Leonardo's vision of painting to issues of mathematics and encouraging the reader to see science as an art. The intention is to restore a visual dimension to mathematical sciences - an element dulled, if not obscured, by historians, philosophers, and scientists themselves.

In this two-volume compilation of articles, leading researchers reevaluate the success of Hilbert's axiomatic method, which not only laid the foundations for our understanding of modern mathematics, but also found applications in physics, computer science and elsewhere. The title takes its name from David Hilbert's seminal talk Axiomatisches Denken, given at a meeting of the Swiss Mathematical Society in Zurich in 1917. This marked the beginning of Hilbert's return to his foundational studies, which ultimately resulted in the establishment of proof theory as a new branch in the emerging field of mathematical logic. Hilbert also used the opportunity to bring Paul Bernays back to Goettingen as his main collaborator in foundational studies in the years to come. The contributions are addressed to mathematical and philosophical logicians, but also to philosophers of science as well as physicists and computer scientists with an interest in foundations. Chapter 8 is available open access under a Creative Commons Attribution 4.0 International License via link.springer.com.

Contents and treatment are fresh and very different from the standard treatments Presents a fully constructive version of what it means to do algebra The exposition is not only clear, it is friendly, philosophical, and considerate even to the most naive or inexperienced reader

First published in 1202, Fibonacci's Liber abaci was one of the most important books on mathematics in the Middle Ages, introducing Arabic numerals and methods throughout Europe. Its author, Leonardo Pisano, known today as Fibonacci, was a citizen of Pisa, an active maritime power, with trading outposts on the Barbary Coast and other points in the Muslim Empire. As a youth, Fibonacci was instructed in mathematics in one of these outposts; he continued his study of mathematics while traveling extensively on business and developed contacts with scientists throughout the Mediterranean world. A member of the academic court around the Emperor Frederick II, Leonardo saw clearly the advantages for both commerce and scholarship of the Hindu positional number system and the algebraic methods developed by al-Khwarizmi and other Muslim scientists. Though it is known as an introduction to the Hindu number system and the algorithms of arithmetic that children now learn in grade school, "Liber abaci" is much more: an encyclopaedia of thirteenth-century mathematics, both theoretical and practical. It develops the tools rigorously, establishing them with Euclidean geometric proofs, and then shows how to apply them to all kinds of situations in business and trade - conversion of measures and currency, allocations of profit, computation of interest, alloying of currencies, and so forth. It is rigorous mathematics, well applied, and vividly described. As the first translation into a modern language of the "Liber abaci," this book will be of interest not only to historians of science, but to all mathematicians and mathematics teachers interested in the origins of their methods.

Lazare Carnot was the unique example in the history of science of someone who inadvertently owed the scientific recognition he eventually achieved to earlier political prominence. He and his son Sadi producedwork that derived from their training as engineering and went largely unnoticed by physicists for a generation or more, even though their respective work introduced concepts that proved fundamental when taken up later by other hands. There was, moreover, a filial as well as substantive relation between the work of father and son. Sadi applied to the functioning of heat engines the analysis that his father had developed in his study of the operation of ordinary machines. Specifically, Sadi's idea of a reversible process originated in the use his father made of geometric motions in the analysis of machines in general.

This unique book shows how the two Carnots influenced each other in their work in the fields of mechanics and thermodynamicsand how future generations of scientists have further benefited from their work."

In this book, the author writes freely and often humorously about his life, beginning with his earliest childhood days. He describes his survival of American bombing raids when he was a teenager in Japan, his emergence as a researcher in a post-war university system that was seriously deficient, and his life as a mature mathematician in Princeton and in the international academic community. Every page of this memoir contains personal observations and striking stories. Such luminaries as Chevalley, Oppenheimer, Siegel, and Weil figure prominently in its anecdotes.

Goro Shimura is Professor Emeritus of Mathematics at Princeton University. In 1996, he received the Leroy P. Steele Prize for Lifetime Achievement from the American Mathematical Society. He is the author of Elementary Dirichlet Series and Modular Forms (Springer 2007), Arithmeticity in the Theory of Automorphic Forms (AMS 2000), and Introduction to the Arithmetic Theory of Automorphic Functions (Princeton University Press 1971)."

This volume develops the major themes of time series analysis from its formal beginnings in the early part of the 20th century to the present day through the research of six distinguished British statisticians, all of whose work is characterised by the British traits of pragmatism and the desire to solve practical problems of importance.

The smartphones in our pockets and computers like brains. The vagaries of game theory and evolutionary biology. Nuclear weapons and self-replicating spacecrafts. All bear the fingerprints of one remarkable, yet largely overlooked, man: John von Neumann. Born in Budapest at the turn of the century, von Neumann is one of the most influential scientists to have ever lived. A child prodigy, he mastered calculus by the age of eight, and in high school made lasting contributions to mathematics. In Germany, where he helped lay the foundations of quantum mechanics, and later at Princeton, von Neumann's colleagues believed he had the fastest brain on the planet-bar none. He was instrumental in the Manhattan Project and the design of the atom bomb; he helped formulate the bedrock of Cold War geopolitics and modern economic theory; he created the first ever programmable digital computer; he prophesized the potential of nanotechnology; and, from his deathbed, he expounded on the limits of brains and computers-and how they might be overcome. Taking us on an astonishing journey, Ananyo Bhattacharya explores how a combination of genius and unique historical circumstance allowed a single man to sweep through a stunningly diverse array of fields, sparking revolutions wherever he went. The Man from the Future is an insightful and thrilling intellectual biography of the visionary thinker who shaped our century.

In the 18th century, purely scientific interests as well as the practical necessities of navigation motivated the development of new theories and techniques to accurately describe celestial and lunar motion.

"Between Theory and Observations" presents a detailed and accurate account, not to be found elsewhere in the literature, of Tobias Mayer's important contributions to the study of lunar motion including the creation of his famous set of lunar tables, which were the most accurate of their time.

This book presents a novel methodology to study economic texts. The author investigates discrepancies in these writings by focusing on errors, mistakes, and rounding numbers. In particular, he looks at the acquisition, use, and development of practical mathematics in an ancient society: The Old Babylonian kingdom of Larsa (beginning of the second millennium BCE Southern Iraq). In so doing, coverage bridges a gap between the sciences and humanities. Through this work, the reader will gain insight into discrepancies encountered in economic texts in general and rounding numbers in particular. They will learn a new framework to explain error as a form of economic practice. Researchers and students will also become aware of the numerical and metrological basis for calculation in these writings and how the scribes themselves conceptualized value. This work fills a void in Assyriological studies. It provides a methodology to explore, understand, and exploit statistical data. The anlaysis also fills a void in the history of mathematics by presenting historians of mathematics a method to study practical texts. In addition, the author shows the importance mathematics has as a tool for ancient practitioners to cope with complex economic processes. This serves as a useful case study for modern policy makers into the importance of education in any economy.

Numbers: A Cultural History provides students with a compelling interdisciplinary view of the development of mathematics and its relationship to world cultures over 4,500 years of human history. Mathematics is often referred to as a "universal language," and that is a fitting description. Many cultures have contributed to mathematics in fascinating ways, but despite its "universal" character, mathematics is also a human endeavor. It has played pivotal roles in societies at particular times; and it has influenced, and been influenced by, a wide range of ideas and institutions, from commerce to philosophy. Ancient Egyptian views of mathematics, for example, are tied closely to engineering and agriculture. Some European Renaissance views, on the other hand, relate the study of number to that of the natural world. Numbers, A Cultural History seeks to place the history of mathematics into a broad cultural context. While it treats mathematical material in detail, it also relates that material to other subject matter: science, philosophy, navigation, commerce, religion, art, and architecture. It examines how mathematical thinking grows in specific cultural settings and how it has shaped those settings in turn. It also explores the movement of ideas between cultures and the evolution of modern mathematics and the quantitative, data-driven world in which we live. Presents mathematics as a human endeavor, a product of human inquiry and human society Provides readers with a cumulative history of mathematics that draws on global cultures over time Places mathematics in multiple cultural contexts and demonstrates its relationship with other areas of thought Demonstrates the link between mathematical knowledge and such practical endeavors as timekeeping, navigation, and commerce Illustrates the movement of ideas between cultures and the complexity of intellectual history Explores the complex relationship between mathematics and technology over time and cultural space

This open access book brings together for the first time all aspects of the tragic life and fascinating work of the polymath Robert Leslie Ellis (1817-1859), placing him at the heart of early-Victorian intellectual culture. Written by a diverse team of experts, the chapters in the book's first part contain in-depth examinations of, among other things, Ellis's family, education, Bacon scholarship and mathematical contributions. The second part consists of annotated transcriptions of a selection of Ellis's diaries and correspondence. Taken together, A Prodigy of Universal Genius: Robert Leslie Ellis, 1817-1859 is a rich resource for historians of science, historians of mathematics and Victorian scholars alike. Robert Leslie Ellis was one of the most intriguing and wide-ranging intellectual figures of early Victorian Britain, his contributions ranging from advanced mathematical analysis to profound commentaries on philosophy and classics and a decisive role in the orientation of mid-nineteenth century scholarship. This very welcome collection offers both new and authoritative commentaries on the work, setting it in the context of the mathematical, philosophical and cultural milieux of the period, together with fascinating passages from the wealth of unpublished papers Ellis composed during his brief and brilliant career. - Simon Schaffer, Department of History and Philosophy of Science, University of Cambridge

This book explores the origins of mathematical analysis in an accessible, clear, and precise manner. Concepts such as function, continuity, and convergence are presented with a unique historical point of view. In part, this is accomplished by investigating the impact of and connections between famous figures, like Newton, Leibniz, Johann Bernoulli, Euler, and more. Of particular note is the treatment of Karl Weierstrass, whose concept of real numbers has been frequently overlooked until now. By providing such a broad yet detailed survey, this book examines how analysis was formed, how it has changed over time, and how it continues to evolve today. A Brief History of Analysis will appeal to a wide audience of students, instructors, and researchers who are interested in discovering new historical perspectives on otherwise familiar mathematical ideas.

When Sir Cyril Burt died in 1971, he was widely recognized as Britain's most eminent educational psychologist whose studies of gifted and delinquent children, contributions to the development of factor analysis, and research on the inheritance of intelligence brought widespread acclaim. Within five years of his death, however, he was publicly denounced as a fraud who had fabricated data to conclude that intelligence is genetically determined. Examiners of the published data found serious inconsistencies that raised questions about their authenticity; the case has divided the scientific community ever since. Were the charges justified, or was he a victim of critics fearful of validating such a politically unacceptable scientific theory? This is an up-to-date and unbiased analysis of one of the most notorious scandals in science, now more timely and widely discussed than ever with the publication of The Bell Curve, the best-selling polemic that raises arguments comparable to Burt's. The distinguished contributors examine the controversial areas of Burt's work and argue that his defenders have sometimes, but by no means always, been correct, and that his critics have often jumped to hasty conclusions. In their haste, however, these critics have missed crucial evidence that is not easily reconciled with Burt's total innocence, leaving the perception that both cases are seriously flawed. An introductory chapter lays the background to the case, followed by an examination of Burt's work that relates to the controversy. The book concludes with a chapter on Burt's character, other cases of apparent scientific fraud, and the impact of Burt's alleged fabrications. These findings have profound implications not only for the study of psychology, but for the wider issues relating to integrity in scientific research, and the impact of intelligence testing on social policy.

This book is dedicated to V.A. Yankov's seminal contributions to the theory of propositional logics. His papers, published in the 1960s, are highly cited even today. The Yankov characteristic formulas have become a very useful tool in propositional, modal and algebraic logic. The papers contributed to this book provide the new results on different generalizations and applications of characteristic formulas in propositional, modal and algebraic logics. In particular, an exposition of Yankov's results and their applications in algebraic logic, the theory of admissible rules and refutation systems is included in the book. In addition, the reader can find the studies on splitting and join-splitting in intermediate propositional logics that are based on Yankov-type formulas which are closely related to canonical formulas, and the study of properties of predicate extensions of non-classical propositional logics. The book also contains an exposition of Yankov's revolutionary approach to constructive proof theory. The editors also include Yankov's contributions to history and philosophy of mathematics and foundations of mathematics, as well as an examination of his original interpretation of history of Greek philosophy and mathematics.

Hilbert's Programs & Beyond presents the foundational work of David Hilbert in a sequence of thematically organized essays. They first trace the roots of Hilbert's work to the radical transformation of mathematics in the 19th century and bring out his pivotal role in creating mathematical logic and proof theory. They then analyze techniques and results of "classical" proof theory as well as their dramatic expansion in modern proof theory. This intellectual experience finally opens horizons for reflection on the nature of mathematics in the 21st century: Sieg articulates his position of reductive structuralism and explores mathematical capacities via computational models.

This is a translated autobiography of applied mathematician N. N. Moiseev, providing an insider's view of the history of the Soviet Union from its founding in 1917 to its collapse in 1991, as well as a little of the aftermath.We see vividly the precariousness of life just after the October Revolution; his happy family life during the years 1921-28 of Lenin's New Economic Policy; the subsequent destruction of his family by Stalin's regime; his trials as a social outcast; his student days at Moscow State University; his experiences as a Soviet Air Force Engineer in World War II, including sorties as a gunner and a brush with an NKVD agent; post-war euphoria, marriage, and another round of ostracism; and then the vicissitudes of a highly varied academic career. Here we meet many famous Soviet and Western engineers and scientists. The last several chapters are devoted more to wide-ranging reflections on God, philosophy, science, communism, modelling the biosphere, and the threat of nuclear winter. His thoughts concerning the impending and then final collapse of the USSR, as well as hopes for Russia's future, conclude the journey through Moiseev's life.

Louis Couturat (1868-1914) was an outstanding intellectual of the turn of the nineteenth to the twentieth century. He is known for his work in the philosophy of mathematics, for his critical and editorial work on Leibniz, for his attempt to popularise modern logic in France, for his commitment to an international auxiliary language, as well as for his extended correspondence with scholars and mathematicians from Great Britain, the United States, Italy, and Germany. From his correspondence we know of four unpublished manuscripts on logic and its history, which were largely complete and some of which must have been of considerable size. We publish here for the ?rst time in a critical edition the only one of these manuscripts that has been rediscovered: the Traite de Logique algorithmique, presumably written in the years 1899-1901. It is a highly interesting document of the academic reception and popularisation of symbolic logic in France. It provides evidence of the discussions and controversies which accompanied the creation of logic as a new branch of science. At the same time it completes the picture of Couturat's work, which has been opened up to systematic study by the publication of important parts of his correspondence during the last decade. We append the article on Symbolic Logic of 1902 which Couturat wrote in collaboration with Christine Ladd- Franklin for Baldwin's Dictionary of Philosophy and Psychology."

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/].

Thomas Harriot's "Artis analyticae praxis" is an essential work in the history of algebra. To some extent it is a development work of Viete, who was among the first to use literal symbols to stand for known and unknown quantities. But it was Harriot who took the crucial step of creating an entirely symbolic algebra, so that reasoning could be reduced to a quasi-mechanical manipulation of symbols. Although his algebra was still limited in scope (he insisted. for example, on strict homogeneity, so only terms of the same powers could be added or equated to one another), it is recognizably modern. Although Harriot's book was highly influential in the development of analysis in England before Newton, it has recently become clear that the posthumously published Praxis contains only an incomplete account of Harriot's achievement: his editor substantially rearranged the work before publishing it, and omitted sections that were apparently beyond his comprehension, such as negative and complex roots of equations. The commentary included with the translation attempts to restore the Praxis to the state of Harriot's draft. Basing their work on manuscripts in the British Library, Pentworth House, and Lambeth Palace, the commentary contains some of Harriot's most novel and advanced mathematics, very little of which has been published in the past. It will provide the basis for a reassessment of the development of algebra.

The present work is the first ever English translation of the original text of Thomas Harriota (TM)s Artis Analyticae Praxis, first published in 1631 in Latin. Thomas Harriota (TM)s Praxis is an essential work in the history of algebra. Even though Harriota (TM)s contemporary, Viete, was among the first to use literal symbols to stand for known and unknown quantities, it was Harriott who took the crucial step of creating an entirely symbolic algebra. This allowed reasoning to be reduced to a quasi-mechanical manipulation of symbols. Although Harriota (TM)s algebra was still limited in scope (he insisted, for example, on strict homogeneity, so only terms of the same powers could be added or equated to one another), it is recognizably modern.

While Harriota (TM)s book was highly influential in the development of analysis in England before Newton, it has recently become clear that the posthumously published Praxis contains only an incomplete account of Harriota (TM)s achievement: his editor substantially rearranged the work before publishing it, and omitted sections that were apparently beyond comprehension, such as negative and complex roots of equations.

The commentary included with this translation relates the contents of the Praxis to the corresponding pages in his manuscript papers, which enables much of Harriot's most novel and advanced mathematics to be explored. This publication will become an important contribution to the history of mathematics, and it will provide the basis for a reassessment of the development of algebra.

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 proceedings volume collects the stories of mathematicians and scientists who have spent and developed parts of their careers and life in countries other than those of their origin. The reasons may have been different in different periods but were often driven by political or economic circumstances: The lack of suitable employment opportunities in their home countries, adverse political systems, and wars have led to the emigration of scientists. The volume shows that these movements have played an important role in spreading scientific knowledge and have often changed the scientific landscape, tradition and future of studies and research fields. The book analyses in particular: aspects of Euler's, Lagrange's and Boscovich's scientific biographies, migrations of scientists from France, Spain and Greece to Russia in the eighteenth and nineteenth centuries, and from Russia to France in the twentieth century, exiles from Italy before the Italian Risorgimento, migrations inside Europe and the escape of mathematicians from Nazi-fascist Europe, between the two World Wars, as well as the mobility of experts around the world. It includes selected contributions from the symposium In Foreign Lands: The Migration of Scientists for Political or Economic Reasons held at the Conference of the International Academy of the History of Science in Athens (September 2019).