This book focuses on the ancient Near East, early imperial China, South-East Asia, and medieval Europe, shedding light on mathematical knowledge and practices documented by sources relating to the administrative and economic activities of officials, merchants and other actors. It compares these to mathematical texts produced in related school contexts or reflecting the pursuit of mathematics for its own sake to reveal the diversity of mathematical practices in each of these geographical areas of the ancient world. Based on case studies from various periods and political, economic and social contexts, it explores how, in each part of the world discussed, it is possible to identify and describe the different cultures of quantification and computation as well as their points of contact. The thirteen chapters draw on a wide variety of texts from ancient Near East, China, South-East Asia and medieval Europe, which are analyzed by researchers from various fields, including mathematics, history, philology, archaeology and economics. The book will appeal to historians of science, economists and institutional historians of the ancient and medieval world, and also to Assyriologists, Indologists, Sinologists and experts on medieval Europe.

This volume offers an integrated understanding of how the theory of general relativity gained momentum after Einstein had formulated it in 1915. Chapters focus on the early reception of the theory in physics and philosophy and on the systematic questions that emerged shortly after Einstein's momentous discovery. They are written by physicists, historians of science, and philosophers, and were originally presented at the conference titled Thinking About Space and Time: 100 Years of Applying and Interpreting General Relativity, held at the University of Bern from September 12-14, 2017. By establishing the historical context first, and then moving into more philosophical chapters, this volume will provide readers with a more complete understanding of early applications of general relativity (e.g., to cosmology) and of related philosophical issues. Because the chapters are often cross-disciplinary, they cover a wide variety of topics related to the general theory of relativity. These include: Heuristics used in the discovery of general relativity Mach's Principle The structure of Einstein's theory Cosmology and the Einstein world Stability of cosmological models The metaphysical nature of spacetime The relationship between spacetime and dynamics The Geodesic Principle Symmetries Thinking About Space and Time will be a valuable resource for historians of science and philosophers who seek a deeper knowledge of the (early and later) uses of general relativity, as well as for physicists and mathematicians interested in exploring the wider historical and philosophical context of Einstein's theory.

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 recounting of the fearful 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 presents different conceptions of logic and mathematics and discuss their philosophical foundations and consequences. This concerns first of all topics of Wittgenstein's ideas on logic and mathematics; questions about the structural complexity of propositions; the more recent debate about Neo-Logicism and Neo-Fregeanism; the comparison and translatability of different logics; the foundations of mathematics: intuitionism, mathematical realism, and formalism. The contributing authors are Matthias Baaz, Francesco Berto, Jean-Yves Beziau, Elena Dragalina-Chernya, Gunther Eder, Susan Edwards-McKie, Oliver Feldmann, Juliet Floyd, Norbert Gratzl, Richard Heinrich, Janusz Kaczmarek, Wolfgang Kienzler, Timm Lampert, Itala Maria Loffredo D'Ottaviano, Paolo Mancosu, Matthieu Marion, Felix Muhlhoelzer, Charles Parsons, Edi Pavlovic, Christoph Pfisterer, Michael Potter, Richard Raatzsch, Esther Ramharter, Stefan Riegelnik, Gabriel Sandu, Georg Schiemer, Gerhard Schurz, Dana Scott, Stewart Shapiro, Karl Sigmund, William W. Tait, Mark van Atten, Maria van der Schaar, Vladimir Vasyukov, Jan von Plato, Jan Wolenski and Richard Zach.

During Song (960 to 1279) and Yuan (1279 to 1368) dynasties, China experienced a peak in high-level algebraic investigation through the works of famous mathematicians such as Qin Jiushao, Zhu Shijie, Yang Hui and Li Ye. Among these is Li Ye's short treatise on a curious ancient geometrical procedure: The Development of Pieces of Areas According to the Collection Augmenting the Ancient Knowledge (Yigu yanduan). The aim of this monography is to contradict traditional scholarship which has long discredited the importance of Li Ye's treatise, considering it a mere popular handbook. The author aims to show that Li Ye's work actually epitomizes a completely new aspect of ancient Chinese mathematics: a crossroad between algebra, geometry, and combinatorics containing elements reminiscent of the Book of Changes (Yi Jing). As well as Li Ye used field measurement as pretext for investigations on quadratic equations and Changes, the present study uses Li Ye's small treatise as pretext for philosophical investigations on link between mathematics and their history. The real topic of the study is the exploration of another expression of proof and generality in Chinese mathematics. This book not only completes the edition of Li Ye's works and presents new features of Chinese mathematics, but also fills a gap in the translation of Chinese mathematics treatises.It is the first book entirely dedicated to the diagrammatic practice of algebra in the history of Chinese mathematics. This practice is more important than expected. While being a monograph, the book is short and detailed enough to be used by students in class. It can also be used as an entry door to the research field of history of Chinese mathematics.

Drawing on published works, correspondence and manuscripts, this book offers the most comprehensive reconstruction of Boscovich's theory within its historical context. It explains the genesis and theoretical as well as epistemological underpinnings in light of the Jesuit tradition to which Boscovich belonged, and contrasts his ideas with those of Newton, Leibniz, and their legacy. Finally, it debates crucial issues in early-modern physical science such as the concept of force, the particle-like structure of matter, the idea of material points and the notion of continuity, and shares novel insights on Boscovich's alleged influence on later developments in physics. With its attempt to reduce all natural forces to one single law, Boscovich's Theory of Natural Philosophy, published in 1758, left a lasting impression on scientists and philosophers of every age regarding the fundamental unity of physical phenomena. The theory argues that every pair of material points is subject to one mutual force - and always the same force - which is their propensity to be mutually attracted or repelled, depending on their distance from one another. Furthermore, the action of this unique force is visualized through a famous diagram that fascinated generations of scientists. But his understanding of key terms of the theory - such as the notion of force involved and the very idea of a material point - is only ostensibly similar to our current conceptual framework. Indeed, it needs to be clarified within the plurality of contexts in which it has emerged rather than being considered in view of later developments. The book is recommended for scholars and students interested in the ideas of the early modern period, especially historians and philosophers of science, mathematicians and physicists with an interest in the history of the discipline, and experts on Jesuit science and philosophy.

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 book documents the process of transformation from natural philosophy, which was considered the most important of the sciences until the early modern era, into modern disciplines such as mathematics, physics, natural history, chemistry, medicine and engineering. It focuses on the 18th century, which has often been considered uninteresting for the history of science, representing the transition from the age of genius and the birth of modern science (the 17th century) to the age of prodigious development in the 19th century. Yet the 18th century, the century of Enlightenment, as will be demonstrated here, was in fact characterized by substantial ferment and novelty. To make the text more accessible, little emphasis has been placed on the precise genesis of the various concepts and methods developed in scientific enterprises, except when doing so was necessary to make them clear. For the sake of simplicity, in several situations reference is made to the authors who are famous today, such as Newton, the Bernoullis, Euler, d'Alembert, Lagrange, Lambert, Volta et al. - not necessarily because they were the most creative and original minds, but mainly because their writings represent a synthesis of contemporary and past studies. The above names should, therefore, be considered more labels of a period than references to real historical characters.

Felix Hausdorff war nicht nur einer der herausragenden Mathematiker des ersten Drittels des 20. Jahrhunderts, sondern unter Pseudonym auch Verfasser eines Aphorismenbandes, eines erkenntniskritischen Buches, eines Gedichtbandes, eines Theaterstucks und zahlreicher literarischer und philosophischer Essays. Der Band enthalt alle Briefe von und an Hausdorff, die bisher in Bibliotheken und Archiven in aller Welt aufgefunden werden konnten. Unter seinen Korrespondenzpartnern sind neben bedeutenden Mathematikern auch Philosophen, Schriftsteller, Kunstler und Feuilletonisten. Die gesamte Korrespondenz ist sorgfaltig kommentiert. Jeder Korrespondenzpartner wird dem Leser mit einer Kurzbiographie vorgestellt.

This book presents the classical theory of curves in the plane and three-dimensional space, and the classical theory of surfaces in three-dimensional space. It pays particular attention to the historical development of the theory and the preliminary approaches that support contemporary geometrical notions. It includes a chapter that lists a very wide scope of plane curves and their properties. The book approaches the threshold of algebraic topology, providing an integrated presentation fully accessible to undergraduate-level students.

At the end of the 17th century, Newton and Leibniz developed differential calculus, thus making available the very wide range of differentiable functions, not just those constructed from polynomials. During the 18th century, Euler applied these ideas to establish what is still today the classical theory of most general curves and surfaces, largely used in engineering. Enter this fascinating world through amazing theorems and a wide supply of surprising examples. Reach the doors of algebraic topology by discovering just how an integer (= the Euler-Poincare characteristics) associated with a surface gives you a lot of interesting information on the shape of the surface. And penetrate the intriguing world of Riemannian geometry, the geometry that underlies the theory of relativity.

The book is of interest to all those who teach classical differential geometry up to quite an advanced level. The chapter on Riemannian geometry is of great interest to those who have to intuitively introduce students to the highly technical nature of this branch of mathematics, in particular when preparing students for courses on relativity."

Winner of the Mathematics Association of America's 2021 Euler Book Prize, this is an inclusive vision of mathematics—its beauty, its humanity, and its power to build virtues that help us all flourish   “This is perhaps the most important mathematics book of our time. Francis Su shows mathematics is an experience of the mind and, most important, of the heart.�—James Tanton, Global Math Project   “A good book is an entertaining read. A great book holds up a mirror that allows us to more clearly see ourselves and the world we live in. Francis Su’s Mathematics for Human Flourishing is both a good book and a great book.�—MAA Reviews   For mathematician Francis Su, a society without mathematical affection is like a city without concerts, parks, or museums. To miss out on mathematics is to live without experiencing some of humanity’s most beautiful ideas.   In this profound book, written for a wide audience but especially for those disenchanted by their past experiences, an award‑winning mathematician and educator weaves parables, puzzles, and personal reflections to show how mathematics meets basic human desires—such as for play, beauty, freedom, justice, and love—and cultivates virtues essential for human flourishing. These desires and virtues, and the stories told here, reveal how mathematics is intimately tied to being human. Some lessons emerge from those who have struggled, including philosopher Simone Weil, whose own mathematical contributions were overshadowed by her brother’s, and Christopher Jackson, who discovered mathematics as an inmate in a federal prison. Christopher’s letters to the author appear throughout the book and show how this intellectual pursuit can—and must—be open to all.

This book presents an overview of the ways in which women have been able to conduct mathematical research since the 18th century, despite their general exclusion from the sciences. Grouped into four thematic sections, the authors concentrate on well-known figures like Sophie Germain and Grace Chisholm Young, as well as those who have remained unnoticed by historians so far. Among them are Stanislawa Nidodym, the first female students at the universities in Prague at the turn of the 20th century, and the first female professors of mathematics in Denmark. Highlighting individual biographies, couples in science, the situation at specific European universities, and sociological factors influencing specific careers from the 18th century to the present, the authors trace female mathematicians' status as it evolved from singular and anomalous to virtually commonplace. The book also offers insights into the various obstacles women faced when trying to enter perhaps the "most male" discipline of all, and how some of them continue to shape young girls' self-perceptions and career choices today. Thus, it will benefit scholars and students in STEM disciplines, gender studies and the history of science; women in science, mathematics and at institutions, and those working in mathematics education.

This book deals with the mathematical sciences in medieval Islam, and focuses on three main themes. The first is that of the translation of texts (from Greek into Arabic, then from Arabic into Latin), and close attention is paid to terminology and comparative vocabulary. The other themes are those of the technology of the sphere and of astronomical instruments, which are treated both from the mechanical and mathematical point of view. Several of the articles combine these themes, for instance the study of the self-rotating sphere of al-Khazini (12th century) or that on the transmission of spherical trigonometry to the West. Four articles also contain substantial texts, with translation and commentary.

An interdisciplinary history of trigonometry from the mid-sixteenth century to the early twentieth The Doctrine of Triangles offers an interdisciplinary history of trigonometry that spans four centuries, starting in 1550 and concluding in the 1900s. Glen Van Brummelen tells the story of trigonometry as it evolved from an instrument for understanding the heavens to a practical tool, used in fields such as surveying and navigation. In Europe, China, and America, trigonometry aided and was itself transformed by concurrent mathematical revolutions, as well as the rise of science and technology. Following its uses in mid-sixteenth-century Europe as the "foot of the ladder to the stars" and the mathematical helpmate of astronomy, trigonometry became a ubiquitous tool for modeling various phenomena, including animal populations and sound waves. In the late sixteenth century, trigonometry increasingly entered the physical world through the practical disciplines, and its societal reach expanded with the invention of logarithms. Calculus shifted mathematical reasoning from geometric to algebraic patterns of thought, and trigonometry's participation in this new mathematical analysis grew, encouraging such innovations as complex numbers and non-Euclidean geometry. Meanwhile in China, trigonometry was evolving rapidly too, sometimes merging with indigenous forms of knowledge, and with Western discoveries. In the nineteenth century, trigonometry became even more integral to science and industry as a fundamental part of the science and engineering toolbox, and a staple subject in high school classrooms. A masterful combination of scholarly rigor and compelling narrative, The Doctrine of Triangles brings trigonometry's rich historical past full circle into the modern era.

Der Band VIII der Gesammelten Werke Felix Hausdorffs enthält seine literarischen Schriften, die er unter dem Pseudonym Paul Mongré veröffentlicht hat. Dazu gehören der Gedichtband "Ekstasen", 14 Essays, die zumeist in führenden Literaturzeitschriften der damaligen Zeit erschienen sind sowie das Theaterstück "Der Arzt seiner Ehre", welches in mehr als 30 Städten über 300 mal aufgeführt wurde. In einer Einleitung des Herausgebers wird Hausdorffs literarisches Schaffen in die Literatur der Moderne eingeordnet. Ausführliche Kommentare und Erläuterungen erhellen den entstehungsgeschichtlichen Kontext, weisen alle literarischen, historischen, philosophischen und andere Anspielungen und Zitate sorgfältig nach und erklären Begriffe und Sachverhalte, die nicht allgemein geläufig sind.

Dyma gyfrol sy’n croniclo bywyd a gwaith Griffith Davies (1788–1855), o’i blentyndod tlawd yn ardal chwareli Arfon i’w waith fel sefydlydd ysgolion mathemateg yn Llundain cyn gosod sylfaen i’r proffesiwn actiwari. Tri mis yn unig o addysg ffurfiol a dderbyniodd Davies, a dyna pryd y sylweddolwyd fod ganddo allu rhyfeddol mewn mathemateg. Mentrodd i Lundain, ac ar ôl blynyddoedd wedi ymroi i hunan ddysgu cyhoeddodd lyfrau mewn mathemateg, ac yn y pen draw fe’i penodwyd yn brif actiwari i gwmni yswiriant Guardian yn y ddinas. Wrth i’w yrfa ddatblygu, daeth yn Gymrawd o’r Gymdeithas Frenhinol a derbyniodd glod ac anrhydeddau am ei waith. Bu’n weithgar ym mywyd Cymraeg Llundain gan sefydlu cyfres o ddarlithoedd gwyddonol yn ei famiaith, ac ymgyrchodd yn llwyddiannus dros hawliau tyddynwyr bro ei febyd a thros addysg i’w gyd-wladwyr. Mae hanes bywyd Griffith Davies yn stori sy’n ysbrydoli.

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.

Very Short Introductions: Brilliant, Sharp, Inspiring Kurt Goedel first published his celebrated theorem, showing that no axiomatization can determine the whole truth and nothing but the truth concerning arithmetic, nearly a century ago. The theorem challenged prevalent presuppositions about the nature of mathematics and was consequently of considerable mathematical interest, while also raising various deep philosophical questions. Goedel's Theorem has since established itself as a landmark intellectual achievement, having a profound impact on today's mathematical ideas. Goedel and his theorem have attracted something of a cult following, though his theorem is often misunderstood. This Very Short Introduction places the theorem in its intellectual and historical context, and explains the key concepts as well as common misunderstandings of what it actually states. A. W. Moore provides a clear statement of the theorem, presenting two proofs, each of which has something distinctive to teach about its content. Moore also discusses the most important philosophical implications of the theorem. In particular, Moore addresses the famous question of whether the theorem shows the human mind to have mathematical powers beyond those of any possible computer ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.

Metamathematics and the Philosophical Tradition is the first work to explore in such historical depth the relationship between fundamental philosophical quandaries regarding self-reference and meta-mathematical notions of consistency and incompleteness. Using the insights of twentieth-century logicians from Goedel through Hilbert and their successors, this volume revisits the writings of Aristotle, the ancient skeptics, Anselm, and enlightenment and seventeenth and eighteenth century philosophers Leibniz, Berkeley, Hume, Pascal, Descartes, and Kant to identify ways in which these both encode and evade problems of a priori definition and self-reference. The final chapters critique and extend more recent insights of late 20th-century logicians and quantum physicists, and offer new applications of the completeness theorem as a means of exploring "metatheoretical ascent" and the limitations of scientific certainty. Broadly syncretic in range, Metamathematics and the Philosophical Tradition addresses central and recurring problems within epistemology. The volume's elegant, condensed writing style renders accessible its wealth of citations and allusions from varied traditions and in several languages. Its arguments will be of special interest to historians and philosophers of science and mathematics, particularly scholars of classical skepticism, the Enlightenment, Kant, ethics, and mathematical logic.

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.

Searching for small gaps between consecutive primes is one way to approach the twin primes conjecture, one of the most celebrated unsolved problems in number theory. This book documents the remarkable developments of recent decades, whereby an upper bound on the known gap length between infinite numbers of consecutive primes has been reduced to a tractable finite size. The text is both introductory and complete: the detailed way in which results are proved is fully set out and plenty of background material is included. The reader journeys from selected historical theorems to the latest best result, exploring the contributions of a vast array of mathematicians, including Bombieri, Goldston, Motohashi, Pintz, Yildirim, Zhang, Maynard, Tao and Polymath8. The book is supported by a linked and freely-available package of computer programs. The material is suitable for graduate students and of interest to any mathematician curious about recent breakthroughs in the field.

This book outlines the scientific career of Arto Salomaa, a pioneer in theoretical computer science and mathematics. The author first interviewed the subject and his family and collaborators, and he then researched this fascinating biography of an intellectual who was key in the development of these fields. Early chapters progress chronologically from Academician Salomaa's origins, childhood, and education to his professional successes in science, teaching, and publishing. His most impactful direct research efforts have been in the areas of automata and formal languages. Beyond that he has influenced many more scientists and professionals through collaborations, teaching, and books on topics such as biocomputing and cryptography. The author offers insights into Finnish history, culture, and academia, while historians of computer science will appreciate the vignettes describing some of the people who have shaped the field from the 1950s to today. The author and his subject return throughout to underlying themes such as the importance of family and the value of longstanding collegial relationships, while the work and achievements are leavened with humor and references to interests such as music, sport, and the sauna.

This open access book explores commentaries on an influential text of pre-Copernican astronomy in Europe. It features essays that take a close look at key intellectuals and how they engaged with the main ideas of this qualitative introduction to geocentric cosmology. Johannes de Sacrobosco compiled his Tractatus de sphaera during the thirteenth century in the frame of his teaching activities at the then recently founded University of Paris. It soon became a mandatory text all over Europe. As a result, a tradition of commentaries to the text was soon established and flourished until the second half of the 17th century. Here, readers will find an informative overview of these commentaries complete with a rich context. The essays explore the educational and social backgrounds of the writers. They also detail how their careers developed after the publication of their commentaries, the institutions and patrons they were affiliated with, what their agenda was, and whether and how they actually accomplished it. The editor of this collection considers these scientific commentaries as genuine scientific works. The contributors investigate them here not only in reference to the work on which it comments but also, and especially, as independent scientific contributions that are socially, institutionally, and intellectually contextualized around their authors.