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.

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.

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

This book provides a critical edition, translation, and study of the version of Euclid's treatise made by Thabit ibn Qurra, which is the earliest Arabic version that we have in its entirety. This monograph study examines the conceptual differences between the Greek and Arabic versions of the treatise, beginning with a discussion of the concept of "given" as it was developed by Greek mathematicians. This is followed by a short account of the various medieval versions of the text and a discussion of the manuscripts used in this volume. Finally, the Arabic text and an English translation are provided, followed by a critical commentary.

This book presents, in his own words, the life of Hugo Steinhaus (1887-1972), noted Polish mathematician of Jewish background, educator, and mathematical popularizer. A student of Hilbert, a pioneer of the foundations of probability and game theory, and a contributor to the development of functional analysis, he was one of those instrumental to the extraordinary flowering of Polish mathematics before and after World War I. In particular, it was he who "discovered" the great Stefan Banach. Exhibiting his great integrity and wit, Steinhaus's personal story of the turbulent times he survived - including two world wars and life postwar under the Soviet heel - cannot but be of consuming interest. His account of the years spent evading Nazi terror is especially moving. The steadfast honesty and natural dignity he maintained while pursuing a life of demanding scientific and intellectual enquiry in the face of encroaching calamity and chaos show him to be truly a mathematician for all seasons. The present work will be of great interest not only to mathematicians wanting to learn some of the details of the mathematical blossoming that occurred in Poland in the first half of the 20th century, but also to anyone wishing to read a first-hand account of the history of those unquiet times in Europe - and indeed world-wide - by someone of uncommon intelligence and forthrightness situated near an eye of the storm.

This volume contains essays that examine infinity in early modern philosophy. The essays not only consider the ways that key figures viewed the concept. They also detail how these different beliefs about infinity influenced major philosophical systems throughout the era. These domains include mathematics, metaphysics, epistemology, ethics, science, and theology. Coverage begins with an introduction that outlines the overall importance of infinity to early modern philosophy. It then moves from a general background of infinity (before early modern thought) up through Kant. Readers will learn about the place of infinity in the writings of key early modern thinkers. The contributors profile the work of Descartes, Spinoza, Leibniz, and Kant. Debates over infinity significantly influenced philosophical discussion regarding the human condition and the extent and limits of human knowledge. Questions about the infinity of space, for instance, helped lead to the introduction of a heliocentric solar system as well as the discovery of calculus. This volume offers readers an insightful look into all this and more. It provides a broad perspective that will help advance the present state of knowledge on this important but often overlooked topic.

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.

These two volumes present the collected works of James Serrin. He did seminal work on a number of the basic tools needed for the study of solutions of partial differential equations. Many of them have been and are being applied to solving problems in science and engineering. Among the areas which he studied are maximum principle methods and related phenomena such as Harnack's inequality, the compact support principle, dead cores and bursts, free boundary problems, phase transitions, the symmetry of solutions, boundary layer theory, singularities and fine regularity properties. The volumes include commentaries by leading mathematicians to indicate the significance of the articles and to discuss further developments along the lines of these articles.

This monograph considers several well-known mathematical theorems and asks the question, "Why prove it again?" while examining alternative proofs. It explores the different rationales mathematicians may have for pursuing and presenting new proofs of previously established results, as well as how they judge whether two proofs of a given result are different. While a number of books have examined alternative proofs of individual theorems, this is the first that presents comparative case studies of other methods for a variety of different theorems. The author begins by laying out the criteria for distinguishing among proofs and enumerates reasons why new proofs have, for so long, played a prominent role in mathematical practice. He then outlines various purposes that alternative proofs may serve. Each chapter that follows provides a detailed case study of alternative proofs for particular theorems, including the Pythagorean Theorem, the Fundamental Theorem of Arithmetic, Desargues' Theorem, the Prime Number Theorem, and the proof of the irreducibility of cyclotomic polynomials. Why Prove It Again? will appeal to a broad range of readers, including historians and philosophers of mathematics, students, and practicing mathematicians. Additionally, teachers will find it to be a useful source of alternative methods of presenting material to their students.

We bring into full light some excerpts on musical subjects which were until now scattered throughout the most famous scientific texts. The main scientific and musical cultures outside of Europe are also taken into consideration. The first and most important property to underline in the scientific texts examined here is the language they are written in. This means that our multicultural history of the sciences necessarily also becomes a review of the various dominant languages used in the different historical contexts. In this volume, the history of the development of the sciences is told as it happened in real contexts, not in an alienated ideal world.

This book tells one of the greatest stories in the history of school mathematics. Two of the names in the title-Samuel Pepys and Isaac Newton-need no introduction, and this book draws attention to their special contributions to the history of school mathematics. According to Ellerton and Clements, during the last quarter of the seventeenth century Pepys and Newton were key players in defining what school mathematics beyond arithmetic and elementary geometry might look like. The scene at which most of the action occurred was Christ's Hospital, which was a school, ostensibly for the poor, in central London. The Royal Mathematical School (RMS) was established at Christ's Hospital in 1673. It was the less well-known James Hodgson, a fine mathematician and RMS master between 1709 and 1755, who demonstrated that topics such as logarithms, plane and spherical trigonometry, and the application of these to navigation, might systematically and successfully be taught to 12- to 16-year-old school children. From a wider history-of-school-education perspective, this book tells how the world's first secondary-school mathematics program was created and how, slowly but surely, what was being achieved at RMS began to influence school mathematics in other parts of Great Britain, Europe, and America. The book has been written from the perspective of the history of school mathematics. Ellerton and Clements's analyses of pertinent literature and of archival data, and their interpretations of those analyses, have led them to conclude that RMS was the first major school in the world to teach mathematics-beyond-arithmetic, on a systematic basis, to students aged between 12 and 16. Throughout the book, Ellerton and Clements examine issues through the lens of a lag-time theoretical perspective. From a historiographical perspective, this book emphasizes how the history of RMS can be portrayed in very different ways, depending on the vantage point from which the history is written. The authors write from the vantage point of international developments in school mathematics education and, therefore, their history of RMS differs from all other histories of RMS, most of which were written from the perspective of the history of Christ's Hospital.

The aim of this book is to analyse historical problems related to the use of mathematics in physics as well as to the use of physics in mathematics and to investigate "Mathematical Physics" as precisely the new discipline which is concerned with this dialectical link itself. So the main question is: "When and why did the tension between mathematics and physics, explicitly practised at least since Galileo, evolve into such a new scientific theory? "

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The authors explain the various ways in which this science allowed an advanced mathematical modelling in physics on the one hand, and the invention of new mathematical ideas on the other hand. Of course this problem is related to the links between institutions, universities, schools for engineers, and industries, and so it has social implications as well.

The link by which physical ideas had influenced the world of mathematics was not new in the 19th century, but it came to a kind of maturity at that time. Recently, much historical research has been done into mathematics and physics and their relation in this period. The purpose of the Symposium and this book is to gather and re-evaluate the current thinking on this subject. It brings together contributions from leading experts in the field, and gives much-needed insight in the subject of mathematical physics from a historical point of view.

This book explores facets of Otto Neugebauer's career, his impact on the history and practice of mathematics, and the ways in which his legacy has been preserved or transformed in recent decades, looking ahead to the directions in which the study of the history of science will head in the twenty-first century. Neugebauer, more than any other scholar of recent times, shaped the way we perceive premodern science. Through his scholarship and influence on students and collaborators, he inculcated both an approach to historical research on ancient and medieval mathematics and astronomy through precise mathematical and philological study of texts, and a vision of these sciences as systems of knowledge and method that spread outward from the ancient Near Eastern civilizations, crossing cultural boundaries and circulating over a tremendous geographical expanse of the Old World from the Atlantic to India.

This volume presents the collection of mathematical articles by Martin Kneser, reprinted in the original language - mostly German -, including one yet unpublished. Moreover, also included is an article by Raman Parimala, discussing Kneser's work concerning algebraic groups and the Hasse principle, which has been written especially for this volume, as well as an article by Rudolf Scharlau about Kneser's work on quadratic forms, published elsewhere before. Another commentary article, written by Gunter M. Ziegler especially for this volume, describes the astounding influence on the field of combinatorics of what was published as "Aufgabe 360" and its subsequent solution in 1955 resp. 1957 in the "Jahresbericht der Deutschen Mathematiker-Vereinigung". However, as the titles of the articles show, Kneser's mathematical interests were much broader, which is beautifully discussed in an obituary by Ulrich Stuhler, included as well in this volume.

We bring into full light some excerpts on musical subjects which were until now scattered throughout the most famous scientific texts. The main scientific and musical cultures outside of Europe are also taken into consideration. The first and most important property to underline in the scientific texts examined here is the language they are written in. This means that our multicultural history of the sciences necessarily also becomes a review of the various dominant languages used in the different historical contexts. In this volume, the history of the development of the sciences is told as it happened in real contexts, not in an alienated ideal world.

How can we invent new certain knowledge in a methodical manner? This question stands at the heart of Salomon Maimon's theory of invention. Chikurel argues that Maimon's contribution to the ars inveniendi tradition lies in the methods of invention which he prescribes for mathematics. Influenced by Proclus' commentary on Elements, these methods are applied on examples taken from Euclid's Elements and Data. Centering around methodical invention and scientific genius, Maimon's philosophy is unique in an era glorifying the artistic genius, known as Geniezeit. Invention, primarily defined as constructing syllogisms, has implications on the notion of being given in intuition as well as in symbolic cognition. Chikurel introduces Maimon's notion of analysis in the broader sense, grounded not only on the principle of contradiction but on intuition as well. In philosophy, ampliative analysis is based on Maimon's logical term of analysis of the object, a term that has yet to be discussed in Maimonian scholarship. Following its introduction, a new version of the question quid juris? arises. In mathematics, Chikurel demonstrates how this conception of analysis originates from practices of Greek geometrical analysis.

Archimedes is held in high esteem by mathematicians, physicists and engineers as one of the most brilliant scientists of all time. These proceedings contain original, unpublished papers with the primary emphasis on the scientific work of Archimedes and his influence on the fields of mathematics, science, and engineering. There are also papers dealing with archaeological aspects and the myths and legends about Archimedes and about the Archimedes Palimpsest. Papers on the following subjects form part of the book: Hydrostatics (buoyancy, fluid pressure and density, stability of floating bodies); Mechanics (levers, pulleys, centers of gravity, laws of equilibrium); Pycnometry (measurement of volume and density); Integral Calculus (Archimedes as the father of the integral calculus, method of exhaustion, approximation of pi, determination of areas and volumes); Mathematical Physics (Archimedes as the father of mathematical physics, Law of the Lever, Law of Buoyancy, Axiomatization of Physics); History of Mathematics and Mechanics (Archimedes' influence in antiquity, the middle ages, the Renaissance, and modern times; his influence on Leonado da Vinci, Galileo, Newton, and other giants of science and mathematics); Ancient Machines and Mechanisms (catapults, water screws, iron hands, compound pulleys, planetaria, water clocks, celestial globes, the Antikythera Mechanism); Archimedean Solids (their rediscovery in the Rennaisance and their applications in materials science and chemistry); Archimedean Legends (how stories of golden crowns, eureka moments, naked runs, burning mirrors, steam cannons, etc., have influenced us through the ages, whether true or not); The Cattle Problem (how its 18th century rediscovery inspired the study of equations with integer solutions); Teaching the Ideas of Archimedes (how his life and works have influenced the teaching of science, mathematics, and engineering).

The essays in this volume concern the points of intersection between analytic philosophy and the philosophy of the exact sciences. More precisely, it concern connections between knowledge in mathematics and the exact sciences, on the one hand, and the conceptual foundations of knowledge in general. Its guiding idea is that, in contemporary philosophy of science, there are profound problems of theoretical interpretation-- problems that transcend both the methodological concerns of general philosophy of science, and the technical concerns of philosophers of particular sciences. A fruitful approach to these problems combines the study of scientific detail with the kind of conceptual analysis that is characteristic of the modern analytic tradition. Such an approach is shared by these contributors: some primarily known as analytic philosophers, some as philosophers of science, but all deeply aware that the problems of analysis and interpretation link these fields together.

Up to now there have been scarcely any publications on Leibniz dedicated to investigating the interrelations between philosophy and mathematics in his thought. In part this is due to the previously restricted textual basis of editions such as those produced by Gerhardt. Through recent volumes of the scientific letters and mathematical papers series of the Academy Edition scholars have obtained a much richer textual basis on which to conduct their studies - material which allows readers to see interconnections between his philosophical and mathematical ideas which have not previously been manifested. The present book draws extensively from this recently published material. The contributors are among the best in their fields. Their commissioned papers cover thematically salient aspects of the various ways in which philosophy and mathematics informed each other in Leibniz's thought.

Galileo and Newton s work towards the mathematisation of the physical world; Leibniz s universal logical calculus; the Enlightenment s mathematique sociale. John von Neumann inherited all these aims and philosophical intuitions, together with an idea that grew up around the Vienna Circle of an ethics in the form of an exact science capable of guiding individuals to make correct decisions. With the help of his boundless mathematical capacity, von Neumann developed a conception of the world as a mathematical game, a world globally governed by a universal logic in which individual consciousness moved following different strategies: his vision guided him from set theory to quantum mechanics, to economics and to his theory of automata (anticipating artificial intelligence and cognitive science). This book provides the first comprehensive scientific and intellectual biography of John von Neumann, a man who perhaps more than any other is representative of twentieth century science.

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This open access book examines the many contributions of Paul Lorenzen, an outstanding philosopher from the latter half of the 20th century. It features papers focused on integrating Lorenzen's original approach into the history of logic and mathematics. The papers also explore how practitioners can implement Lorenzen's systematical ideas in today's debates on proof-theoretic semantics, databank management, and stochastics. Coverage details key contributions of Lorenzen to constructive mathematics, Lorenzen's work on lattice-groups and divisibility theory, and modern set theory and Lorenzen's critique of actual infinity. The contributors also look at the main problem of Grundlagenforschung and Lorenzen's consistency proof and Hilbert's larger program. In addition, the papers offer a constructive examination of a Russell-style Ramified Type Theory and a way out of the circularity puzzle within the operative justification of logic and mathematics. Paul Lorenzen's name is associated with the Erlangen School of Methodical Constructivism, of which the approach in linguistic philosophy and philosophy of science determined philosophical discussions especially in Germany in the 1960s and 1970s. This volume features 10 papers from a meeting that took place at the University of Konstanz.

In 1644 the Qing dynasty seized power in China. Its Manchu elite were at first seen by most of their subjects as foreigners from beyond the Great Wall, and the consolidation of Qing rule presented significant cultural and political problems, as well as military challenges. It was the Kangxi emperor (r. 1662-1722) who set the dynasty on a firm footing, and one of his main stratagems to achieve this was the appropriation for imperial purposes of the scientific knowledge brought to China by the Jesuit mission (1582-1773). For almost two centuries, the Jesuits put the sciences in the service of evangelization, teaching and practising what came to be known as 'Western learning' among Chinese scholars, many of whom took an active interest in it. After coming to the throne as a teenager, Kangxi began his life-long intervention in mathematical and scientific matters when he forced a return to the use of Western methods in official astronomy. In middle life, he studied astronomy, musical theory and mathematics, with Jesuits as his teachers. In his last years he sponsored a great compilation covering these three disciplines, and set several of his sons to work on this project. All of this activity formed a vital part of his plan to establish Manchu authority over the Chinese. This book explains why Kangxi made the sciences a tool for laying the foundations of empire, and to show how, as part of this process, mathematics was reconstructed as a branch of imperial learning.

Seki was a Japanese mathematician in the seventeenth century known for his outstanding achievements, including the elimination theory of systems of algebraic equations, which preceded the works of Etienne Bezout and Leonhard Euler by 80 years. Seki was a contemporary of Isaac Newton and Gottfried Wilhelm Leibniz, although there was apparently no direct interaction between them.

The Mathematical Society of Japan andthe History of Mathematics Society of Japan hosted the International Conference on History of Mathematics in Commemoration of the 300th Posthumous Anniversary of Seki in 2008. This book is the official record of the conference and includes supplements of collated texts of Seki's original writings with notes in English on these texts.

Hikosaburo Komatsu (Professor emeritus, The University of Tokyo), one of the editors, is known for partial differential equations and hyperfunction theory, and for his study on the history of Japanese mathematics. He served as the President of the International Congress of Mathematicians Kyoto 1990."