Das Buch zeigt, inwiefern nicht, wie man ublicherweise sagt, die Arithmetik, Logik und Mengenlehre, sondern die Geometrie die Koenigin der Mathematik ist, weil namlich die oft verpoente Anschauung allen ihren Axiomatisierungen und Anwendungen zugrunde liegt, und zwar in der Form eines diagrammtheoretischen Strukturmodells. Dessen Punkte, Geraden und Ebenen sind selbst immer schon raumlose Teilformen idealer Formen. Zu den 'reellen Zahlen' als reine Groessenproportionen gelangt man durch Ausweitung des Punktbereiches zunachst uber den Fundamentalsatz der Algebra. Aber erst Cantors Naive Mengenlehre liefert genugend Nullstellen fur beliebige stetige Funktionen. Dabei ist die euklidische Geometrie eine Theorie der Koerperformen, wahrend fur jede Theorie des Raumes, in dem sich Koerper bewegen, immer auch schon die Zeit mathematisiert werden muss, so dass der Bewegungsraum nie einfach 'dreidimensional' ist. Diese Unterscheidung zum Anschauungsraum geformter Koerper macht das vierdimensionale Minkowski-Modell der Raum-Zeit in Einsteins spezieller Relativitatstheorie allererst voll begreifbar, zumal sich im empiristischen bzw. konventionalistischenAnsatz Reichenbachs, Grunbaums und vieler anderer Autoren deutliche Mangel finden.

Originally published in 1950, this book was based on a short series of lectures given by the author at the University of Illinois in 1948. Aimed at the non-specialist, the chief aim of the text was to provide a general introduction to contemporary developments in the field of calculating instruments and machines. But there is some treatment of the historical side of the subject, with appreciation shown for the vision and foresight of key pioneers Charles Babbage and Lord Kelvin. This is a concise and informative volume that will be of value to anyone with an interest in the development and history of computation.

Im 3. und 2. Jahrhundert v. Chr. ereignete sich in der griechischen Welt ein wissenschaftlicher und technologischer Urknall. Schon im vorhergegangenen klassischen Zeitalter war die griechische Kultur in den Kunsten, der Literatur und der Philosophie zu grossen Hohen aufgestiegen; erst die Zeit eines Archimedes und eines Euklid jedoch, brachte die Entstehung von Wissenschaften in unserem Sinne. Damals entstanden hoch entwickelte Technologien, auf die man sich erst im 18. Jahrhundert wieder besinnen sollte. Gleichzeitig mit dieser wissenschaftlichen Revolution fanden auch auf vielen anderen Gebieten, wie den Kunsten und der Medizin grundlegende Veranderungen statt. Ein neues Bewusstsein entstand.

Was waren die Grundpfeiler dieses kometenhaften Aufstiegs der Wissenschaften vor 2.300 Jahren? Warum ist selbst unter Wissenschaftlern, Altphilologen und Historikern so wenig daruber bekannt? In welcher Beziehung stehen sie zu der Entwicklung der Wissenschaften seit dem 15. Jahrhundert, wie wir sie aus der Schule kennen Was fuhrte zum Ende antiker Wissenschaften? Das sind die Fragen, die in diesem Buch gestellt werden. Ihre Antworten sind von entscheidender Bedeutung auch fur Herausforderungen, vor denen wir heute stehen."

The classic Heath translation, in a completely new layout with plenty of space and generous margins. An affordable but sturdy sewn hardcover student and teacher edition in one volume, with minimal notes and a new index/glossary.

Many of the most famous results in mathematics are impossibility theorems stating that something cannot be done. Good examples include the quadrature of the circle by ruler and compass, the solution of the quintic equation by radicals, Fermat's last theorem, and the impossibility of proving the parallel postulate from the other axioms of Euclidean geometry. This book tells the history of these and many other impossibility theorems starting with the ancient Greek proof of the incommensurability of the side and the diagonal in a square. Lutzen argues that the role of impossibility results have changed over time. At first, they were considered rather unimportant meta-statements concerning mathematics but gradually they obtained the role of important proper mathematical results that can and should be proved. While mathematical impossibility proofs are more rigorous than impossibility arguments in other areas of life, mathematicians have employed great ingenuity to circumvent impossibilities by changing the rules of the game. For example, complex numbers were invented in order to make impossible equations solvable. In this way, impossibilities have been a strong creative force in the development of mathematics, mathematical physics, and social science.

Der Band 1A beginnt mit einem Vorwort zur Gesamtedition. Den Hauptteil des Bandes bilden Hausdorffs Arbeiten uber geordnete Mengen aus den Jahren 1901-1909. Diese haben der Entwicklung der Mengenlehre nachhaltige Impulse verliehen. Sie enthalten zahlreiche fur die Untersuchung geordneter Mengen grundlegende neue Begriffe sowie tiefliegendere Resultate. Alle diese Arbeiten sind sorgfaltig kommentiert. Die Kommentare zeigen, dass einige von Hausdorff's Ideen und Resultaten fur die moderne Grundlagenforschung hochaktuell sind.

Ferner enthalt der Band Hausdorff's kritische Besprechung von Russells "The Principles of Mathematics," aus dem Nachlass seine Vorlesung "Mengenlehre" von 1901 (eine der ersten Vorlesungen uber dieses Gebiet uberhaupt) sowie einen Essay "Hausdorff als akademischer Lehrer."

Essentials of Mathematical Thinking addresses the growing need to better comprehend mathematics today. Increasingly, our world is driven by mathematics in all aspects of life. The book is an excellent introduction to the world of mathematics for students not majoring in mathematical studies. The author has written this book in an enticing, rich manner that will engage students and introduce new paradigms of thought. Careful readers will develop critical thinking skills which will help them compete in today's world. The book explains: What goes behind a Google search algorithm How to calculate the odds in a lottery The value of Big Data How the nefarious Ponzi scheme operates Instructors will treasure the book for its ability to make the field of mathematics more accessible and alluring with relevant topics and helpful graphics. The author also encourages readers to see the beauty of mathematics and how it relates to their lives in meaningful ways.

Theory of Conics, Geometrical Constructions and Practical Geometry: A History of Arabic Sciences and Mathematics Volume 3, provides a unique primary source on the history and philosophy of mathematics and science from the mediaeval Arab world. The present text is complemented by two preceding volumes of A History of Arabic Sciences and Mathematics, which focused on founding figures and commentators in the ninth and tenth centuries, and the historical and epistemological development of 'infinitesimal mathematics' as it became clearly articulated in the oeuvre of Ibn al-Haytham. This volume examines the increasing tendency, after the ninth century, to explain mathematical problems inherited from Greek times using the theory of conics. Roshdi Rashed argues that Ibn al-Haytham completes the transformation of this 'area of activity,' into a part of geometry concerned with geometrical constructions, dealing not only with the metrical properties of conic sections but with ways of drawing them and properties of their position and shape. Including extensive commentary from one of world's foremost authorities on the subject, this book contributes a more informed and balanced understanding of the internal currents of the history of mathematics and the exact sciences in Islam, and of its adaptive interpretation and assimilation in the European context. This fundamental text will appeal to historians of ideas, epistemologists and mathematicians at the most advanced levels of research.

Nineteenth century Russian intellectuals perceived a Malthusian bias in Darwin's theory of evolution by means of natural selection. They identified that bias with Darwin's concept of the "struggle for existence" and his emphasis upon the evolutionary role of overpopulation and intraspecific conflict. In this book, Todes documents a historical Russian critique of Darwin's "Malthusian error", explores its relationship to such scientific work as Mechnikov's phagocytic theory, Korzhinskii's mutation theory and Kropotkin's theory of mutual aid, and finds its origins in Russia's political economy and in the very nature of its land and climate. This is the first book in English to examine in detail the scientific work of nineteenth century Russian evolutionists, and the first in any language to explore the relationship of Russian theories to the economic, political, and natural circumstances in which they were generated. It combines a broad scope (dealing with political figures and cultural movements) with a close analysis of scientific work on a range of topics.

This vividly illustrated history of the International Congress of Mathematicians - a meeting of mathematicians from around the world held roughly every four years - acts as a visual history of the 25 congresses held between 1897 and 2006, as well as a story of changes in the culture of mathematics over the past century. Because the congress is an international meeting, looking at its history allows us a glimpse into the effect of wars and strained relations between nations on the scientific community.

The first book surveying the history and ideas behind reverse mathematics Reverse mathematics is a new field that seeks to find the axioms needed to prove given theorems. In Reverse Mathematics, John Stillwell offers a historical and representative view, emphasizing basic analysis and giving a novel approach to logic. By using a minimum of mathematical logic in a well-motivated way, Reverse Mathematics will engage advanced undergraduates and all mathematicians interested in the foundations of mathematics.

An innovative, dramatic graphic novel about the treacherous pursuit of the foundations of mathematics.

This exceptional graphic novel recounts the spiritual odyssey of philosopher Bertrand Russell. In his agonized search for absolute truth, Russell crosses paths with legendary thinkers like Gottlob Frege, David Hilbert, and Kurt Godel, and finds a passionate student in the great Ludwig Wittgenstein. But his most ambitious goal--to establish unshakable logical foundations of mathematics--continues to loom before him. Through love and hate, peace and war, Russell persists in the dogged mission that threatens to claim both his career and his personal happiness, finally driving him to the brink of insanity.

This story is at the same time a historical novel and an accessible explication of some of the biggest ideas of mathematics and modern philosophy. With rich characterizations and expressive, atmospheric artwork, the book spins the pursuit of these ideas into a highly satisfying tale. Probing and ingeniously layered, the book throws light on Russell's inner struggles while setting them in the context of the timeless questions he spent his life trying to answer. At its heart, "Logicomix "is a story about the conflict between an ideal rationality and the unchanging, flawed fabric of reality.Apostolos Doxiadis studied mathematics at Columbia University. His international bestseller "Uncle Petros and Goldbach's Conjecture" spearheaded the impressive entrance of mathematics into the world of storytelling. Apart from his work in fiction, Apostolos has also worked in film and theater and is an internationally recognized expert on the relationship of mathematics to narrative. Christos H. Papadimitriou is C . Lester Hogan professor of computer science at the University of California, Berkeley. He was won numerous international awards for his pioneering work in computational complexity and algorithmic game theory. Christos is the author of the novel "Turing: A Novel about Computation." Alecos Papadatos worked for over twenty years in film animation in France and Greece. In 1997, he became a cartoonist for the major Athens daily "To Vima." He lives in Athens with his wife, Annie Di Donna, and their two children. Annie Di Donna studied graphic arts and painting in France and has worked as animator on many productions, among them "Babar" and "Tintin." Since 1991, she has been running an animation studio with her husband, Alecos Papadatos. This innovative graphic novel is based on the early life of the brilliant philosopher Bertrand Russell. Russell and his impassioned pursuit of truth. Haunted by family secrets and unable to quell his youthful curiosity, Russell became obsessed with a Promethean goal: to establish the logical foundation of all mathematics. In his agonized search for absolute truth, Russell crosses paths with legendary thinkers like Gottlob Frege, David Hilbert, and Kurt Godel, and finds a passionate student in the great Ludwig Wittgenstein. But the object of his defining quest continues to loom before him. Through love and hate, peace and war, Russell persists in the dogged mission that threatens to claim both his career and his personal happiness, finally driving him to the brink of insanity. "Logicomix" is at the same time a historical novel and an accessible explication to some of the biggest ideas of mathematics and modern philosophy. With rich characterizations and expressive, atmospheric artwork, the book spins the pursuit of these ideas into a captivating tale. Probing and ingeniously layered, the book throws light on Russell's inner struggles while setting them in the context of the timeless questions he spent his life trying to answer. At its heart, "Logicomix" is a story about the conflict between an ideal rationality and the unchanging, flawed fabric of reality. "At the heart of Logicomix stands Sir Bertrand Russell, a man determined to find a way of arriving at absolutely right answers. It's a tale within a tale, as the two authors and two graphic artists ardently pursue their own search for truth and appear as characters in the book. As one of them assures us, this won't be 'your typical, usual comic book.' Their quest takes shape and revolves around a lecture given by Russell at an unnamed American university in 1939, a lecture that is really, as he himself tells us, the story of his life and of his pursuit of real logical truth. With Proustian ambition and exhilarating artwork, "Logicomix"'s search for truth encounters head-on the horrors of the Second World War and the agonizing question of whether war can ever be the right choice. Russell himself had to confront that question personally: he endured six months in jail for his pacifism. Russell was determined to find the perfect logical method for solving all problems and attempted to remold human nature in his experimental school at Beacon Hill. Despite repeated failures, Russell never stopped being 'a sad little boy desperately seeking ways out of the deadly vortex of uncertainty.' The book is a visual banquet chronicling Russell's lifelong pursuit of 'certainty in total rationality.' As Logic and Mathematics, the last bastions of certainty, fail him, and as Reason proves not absolute, Russell is forced to face the fact that there is no Royal Road to Truth. Authors Dosiadis and Papadimitriou perfectly echo Russell's passion, with a sincere, easily grasped text amplified with breathtaking visual richness, making this the most satisfying graphic novel of 2009, a titanic artistic achievement of more than 300 pages, all of it pure reading joy."--Nick DiMartino, "Shelf Awareness" "This is an extraordinary graphic novel, wildly ambitious in daring to put into words and drawings the life and thought of one of the great philosophers of the last century, Bertrand Russell. The book is a rare intellectual and artistic achievement, which will, I am sure, lead its readers to explore realms of knowledge they thought were forbidden to them."--Howard Zinn "This magnificent book is about ideas, passions, madness, and the fierce struggle between well-defined principle and the larger good. It follows the great mathematicians--Russell, Whitehead, Frege Cantor, Hilbert--as they agonized to make the foundations of mathematics exact, consistent, and complete. And we see the band of artists and researchers--and the all-seeking dog Manga--creating, and participating in, this glorious narrative."--Barry Mazur, Gerhard Gade University Professor at Harvard University, and author of "Imagining Numbers (Particularly the Square Root of Minus Fifteen)" "The lives of ideas (and those who think them) can be as dramatic and unpredicteable as any superhero fantasy. "Logicomix" is witty, engaging, stylish, visually stunning, and full of surprising sound effects, a masterpiece in a genre for which there is as yet no name."--Michael Harris, professor of mathematics at Universite Paris 7 and member of the Institut Universitaire de France

Between inventing the concept of a universal computer in 1936 and breaking the German Enigma code during World War II, Alan Turing (1912-1954), the British founder of computer science and artificial intelligence, came to Princeton University to study mathematical logic. Some of the greatest logicians in the world--including Alonzo Church, Kurt Godel, John von Neumann, and Stephen Kleene--were at Princeton in the 1930s, and they were working on ideas that would lay the groundwork for what would become known as computer science. This book presents a facsimile of the original typescript of Turing's fascinating and influential 1938 Princeton PhD thesis, one of the key documents in the history of mathematics and computer science. The book also features essays by Andrew Appel and Solomon Feferman that explain the still-unfolding significance of the ideas Turing developed at Princeton.

A work of philosophy as well as mathematics, Turing's thesis envisions a practical goal--a logical system to formalize mathematical proofs so they can be checked mechanically. If every step of a theorem could be verified mechanically, the burden on intuition would be limited to the axioms. Turing's point, as Appel writes, is that "mathematical reasoning can be done, and should be done, in mechanizable formal logic." Turing's vision of "constructive systems of logic for practical use" has become reality: in the twenty-first century, automated "formal methods" are now routine.

Presented here in its original form, this fascinating thesis is one of the key documents in the history of mathematics and computer science."

Spherical trigonometry was at the heart of astronomy and ocean-going navigation for two millennia. The discipline was a mainstay of mathematics education for centuries, and it was a standard subject in high schools until the 1950s. Today, however, it is rarely taught. "Heavenly Mathematics" traces the rich history of this forgotten art, revealing how the cultures of classical Greece, medieval Islam, and the modern West used spherical trigonometry to chart the heavens and the Earth. Glen Van Brummelen explores this exquisite branch of mathematics and its role in ancient astronomy, geography, and cartography; Islamic religious rituals; celestial navigation; polyhedra; stereographic projection; and more. He conveys the sheer beauty of spherical trigonometry, providing readers with a new appreciation for its elegant proofs and often surprising conclusions.

"Heavenly Mathematics" is illustrated throughout with stunning historical images and informative drawings and diagrams that have been used to teach the subject in the past. This unique compendium also features easy-to-use appendixes as well as exercises at the end of each chapter that originally appeared in textbooks from the eighteenth to the early twentieth centuries.

This book presents reverse mathematics to a general mathematical audience for the first time. Reverse mathematics is a new field that answers some old questions. In the two thousand years that mathematicians have been deriving theorems from axioms, it has often been asked: which axioms are needed to prove a given theorem? Only in the last two hundred years have some of these questions been answered, and only in the last forty years has a systematic approach been developed. In Reverse Mathematics, John Stillwell gives a representative view of this field, emphasizing basic analysis--finding the "right axioms" to prove fundamental theorems--and giving a novel approach to logic. Stillwell introduces reverse mathematics historically, describing the two developments that made reverse mathematics possible, both involving the idea of arithmetization. The first was the nineteenth-century project of arithmetizing analysis, which aimed to define all concepts of analysis in terms of natural numbers and sets of natural numbers. The second was the twentieth-century arithmetization of logic and computation. Thus arithmetic in some sense underlies analysis, logic, and computation. Reverse mathematics exploits this insight by viewing analysis as arithmetic extended by axioms about the existence of infinite sets. Remarkably, only a small number of axioms are needed for reverse mathematics, and, for each basic theorem of analysis, Stillwell finds the "right axiom" to prove it. By using a minimum of mathematical logic in a well-motivated way, Reverse Mathematics will engage advanced undergraduates and all mathematicians interested in the foundations of mathematics.

On the occasion of the 200th anniversary of the birth of Hermann Grassmann (1809-1877), an interdisciplinary conference was held in Potsdam, Germany, and in Grassmann's hometown Szczecin, Poland. The idea of the conference was to present a multi-faceted picture of Grassmann, and to uncover the complexity of the factors that were responsible for his creativity. The conference demonstrated not only the very influential reception of his work at the turn of the 20th century, but also the unexpected modernity of his ideas, and their continuing development in the 21st century.

This book contains 37 papers presented at the conference. They investigate the significance of Grassmann's work for philosophical as well as for scientific and methodological questions, for comparative philology in general and for Indology in particular, for psychology, physiology, religious studies, musicology, didactics, and, last but not least, mathematics. In addition, the book contains numerous illustrations and English translations of original sources, which are published here for the first time. These include life histories of Grassmann (written by his son Justus) and of his brother Robert (written by Robert himself), as well as the paper "On the concept and extent of pure theory of number'' by Justus Grassmann (the father).<

Volume 2.3, containing the Greek translations of books II-IV, completes the edition of the Conica of Apollonius of Perga. It is arranged according to the same principle as the previous volume on book I: the critical edition of the Greek text is accompanied by numerous notes and a lexicon of all mathematical terms. An introduction and a French translation provide further insights into the text. The edition of the Greek books of the Conica will be thematically complemented by volume 3 of the SGA series, containing the critical edition of Eutocius of Ascalon's commentary on the Conica and its French translation.

This seminal collection gathers together many general writings of one of the world's leading historians of mathematics. Organized thematically, these essays ponder the intellectual underpinnings of the field, examine the major topics in the history of mathematics, and recount the bizarre history of pseudomath.

Ivor Grattan-Guinness explores how people understand mathematics--the routes of learning they take as they make important discoveries and study mathematical concepts and theories. The essays in the first part of the book discuss the history of mathematics as a field and its central philosophical issues. Those in the next part address the history of mathematics education and its importance to current modes of teaching. In the last section Grattan-Guinness investigates various understudied aspects of math, including numerology, Masonic symbols in classical music, and the links between mathematics and Christianity.

This collection includes several essays that are difficult to find anywhere else. All historians of mathematics and students of the field will want a copy of this remarkable resource on their bookshelves.

In MATH WITH BAD DRAWINGS, Ben Orlin answers math's three big questions: Why do I need to learn this? When am I ever going to use it? Why is it so hard? The answers come in various forms-cartoons, drawings, jokes, and the stories and insights of an empathetic teacher who believes that math should belong to everyone. Eschewing the tired old curriculum that begins in the wading pool of addition and subtraction and progresses to the shark infested waters of calculus (AKA the Great Weed Out Course), Orlin instead shows us how to think like a mathematician by teaching us a new game of Tic-Tac-Toe, how to understand an economic crisis by rolling a pair of dice, and the mathematical reason why you should never buy a second lottery ticket. Every example in the book is illustrated with his trademark "bad drawings," which convey both his humor and his message with perfect pitch and clarity. Organized by unconventional but compelling topics such as "Statistics: The Fine Art of Honest Lying," "Design: The Geometry of Stuff That Works," and "Probability: The Mathematics of Maybe," MATH WITH BAD DRAWINGS is a perfect read for fans of illustrated popular science.

Band 5 umfasst die Themenbereiche Astronomie, Optik und Wahrscheinlichkeitstheorie. Er enthalt Hausdorffs Dissertation uber die Refraktion des Lichtes in der Atmosphare, zwei Folgearbeiten zum gleichen Thema sowie die Habilitationsschrift uber die Extinktion des Lichtes in der Atmosphare. Es folgt eine Arbeit uber geometrische Optik, die unmittelbar an die beruhmte Publikation von H. Bruns uber das Eikonal anschliesst und in der Hausdorff die damals ganz neuen Lieschen Theorien fur die Optik nutzbar zu machen suchte.

Auf dem Gebiet der Stochastik veroffentlichte Hausdorff zwei langere Arbeiten, die in verschiedenen Bereichen der Versicherungsmathematik und der Wahrscheinlichkeitsrechnung ihre Spuren hinterlassen haben. Von besonderem historischen Interesse sind die im Band publizierten Stucke aus Hausdorffs Nachlass, etwa seine Vorlesung "Wahrscheinlichkeitsrechnung" vom Sommersemester 1923 oder seine Briefe an Richard von Mises aus dem Jahre 1919. "

The book starts with the assumption that vagueness is a fundamental property of this world. From a philosophical account of vagueness via the presentation of alternative mathematics of vagueness, the subsequent chapters explore how vagueness manifests itself in the various exact sciences: physics, chemistry, biology, medicine, computer science, and engineering.