Biographie des ungarischen Mathematikers Janos Bolyai (1802-1860), der etwa gleichzeitig mit dem russischen Mathematiker Nikolai Lobatschewski und unabhangig von ihm die nichteuklidische Revolution eingeleitet hat. Diese erbrachte den Nachweis, dass die euklidische Geometrie keine Denknotwendigkeit ist, wie Kant irrtumlicherweise annahm. Das Verstandnis fur die kuhnen Gedankengange verbreitete sich allerdings erst in der zweiten Halfte des 19. Jahrhunderts durch die Arbeiten von Riemann, Beltrami, Klein und Poincare. Die nichteuklidische Revolution war eine der Grundlagen fur die Entwicklung der Physik im 20. Jahrhundert und fur Einsteins Erkenntnis, dass der uns umgebende reale Raum gekrummt ist. Tibor Weszely schildert das wechselvolle Leben des Offiziers der K.u.K.-Armee, der krank und vereinsamt starb. Bolyai hat sich auch intensiv mit den komplexen Zahlen und mit Zahlentheorie befasst, ebenso auch mit philosophischen und sozialen Fragen ( Allheillehre ) sowie mit Logik und Grammatik.

This volume contains several invited papers as well as a selection of the other contributions. The conference was the first meeting of the Soviet logicians interested in com- puter science with their Western counterparts. The papers report new results and techniques in applications of deductive systems, deductive program synthesis and analysis, computer experiments in logic related fields, theorem proving and logic programming. It provides access to intensive work on computer logic both in the USSR and in Western countries.

In this book, David Stump traces alternative conceptions of the a priori in the philosophy of science and defends a unique position in the current debates over conceptual change and the constitutive elements in science. Stump emphasizes the unique epistemological status of the constitutive elements of scientific theories, constitutive elements being the necessary preconditions that must be assumed in order to conduct a particular scientific inquiry. These constitutive elements, such as logic, mathematics, and even some fundamental laws of nature, were once taken to be a priori knowledge but can change, thus leading to a dynamic or relative a priori. Stump critically examines developments in thinking about constitutive elements in science as a priori knowledge, from Kant's fixed and absolute a priori to Quine's holistic empiricism. By examining the relationship between conceptual change and the epistemological status of constitutive elements in science, Stump puts forward an argument that scientific revolutions can be explained and relativism can be avoided without resorting to universals or absolutes.

First published in 1982, this reissue contains a critical exposition of the views of Frege, Dedekind and Peano on the foundations of arithmetic. The last quarter of the 19th century witnessed a remarkable growth of interest in the foundations of arithmetic. This work analyses both the reasons for this growth of interest within both mathematics and philosophy and the ways in which this study of the foundations of arithmetic led to new insights in philosophy and striking advances in logic. This historical-critical study provides an excellent introduction to the problems of the philosophy of mathematics - problems which have wide implications for philosophy as a whole. This reissue will appeal to students of both mathematics and philosophy who wish to improve their knowledge of logic.

If mathematics is the purest form of knowledge, the perfect foundation of all the hard sciences, and a uniquely precise discipline, then how can the human brain, an imperfect and imprecise organ, process mathematical ideas? Is mathematics made up of eternal, universal truths? Or, as some have claimed, could mathematics simply be a human invention, a kind of tool or metaphor? These questions are among the greatest enigmas of science and epistemology, discussed at length by mathematicians, physicians, and philosophers. But, curiously enough, neuroscientists have been absent in the debate, even though it is precisely the field of neuroscience-which studies the brain's mechanisms for thinking and reasoning-that ought to be at the very center of these discussions. How our Emotions and Bodies are Vital for Abstract Thought explores the unique mechanisms of cooperation between the body, emotions, and the cortex, based on fundamental physical principles. It is these mechanisms that help us to overcome the limitations of our physiology and allow our imperfect, human brains to make transcendent mathematical discoveries. This book is written for anyone who is interested in the nature of abstract thought, including mathematicians, physicists, computer scientists, psychologists, and psychiatrists.

Hermann Grassmann, Gymnasiallehrer in Stettin und bekannt als Begrunder der n-dimensionalen Vektoralgebra, erwarb sich auch in der Physik und der Sprachforschung bleibende Verdienste. Gestutzt auf die Dialektik Schleiermachers entwickelte er in seinem Hauptwerk, der Ausdehnungslehre, mit philosophischer Methode eine vollig neue mathematische Disziplin. Zunachst von der Fachwelt abgelehnt, wurde sein Werk Jahrzehnte spater als wegweisend gefeiert. Die Biographie geht dem komplexen Geflecht innerer und ausserer Einflusse nach, innerhalb derer Grassmann sein Schopfertum entfaltete."

Uncertainty is everywhere. It lurks in every consideration of the future - the weather, the economy, the sex of an unborn child - even quantities we think that we know such as populations or the transit of the planets contain the possibility of error. It's no wonder that, throughout that history, we have attempted to produce rigidly defined areas of uncertainty - we prefer the surprise party to the surprise asteroid. We began our quest to make certain an uncertain world by reading omens in livers, tea leaves, and the stars. However, over the centuries, driven by curiosity, competition, and a desire be better gamblers, pioneering mathematicians and scientists began to reduce wild uncertainties to tame distributions of probability and statistical inferences. But, even as unknown unknowns became known unknowns, our pessimism made us believe that some problems were unsolvable and our intuition misled us. Worse, as we realized how omnipresent and varied uncertainty is, we encountered chaos, quantum mechanics, and the limitations of our predictive power. Bestselling author Professor Ian Stewart explores the history and mathematics of uncertainty. Touching on gambling, probability, statistics, financial and weather forecasts, censuses, medical studies, chaos, quantum physics, and climate, he makes one thing clear: a reasonable probability is the only certainty.

Originally published in 1976. This comprehensive study discusses in detail the philosophical, mathematical, physical, logical and theological aspects of our understanding of time and space. The text examines first the many different definitions of time that have been offered, beginning with some of the puzzles arising from our awareness of the passage of time and shows how time can be understood as the concomitant of consciousness. In considering time as the dimension of change, the author obtains a transcendental derivation of the concept of space, and shows why there has to be only one dimension of time and three of space, and why Kant was not altogether misguided in believing the space of our ordinary experience to be Euclidean. The concept of space-time is then discussed, including Lorentz transformations, and in an examination of the applications of tense logic the author discusses the traditional difficulties encountered in arguments for fatalism. In the final sections he discusses eternity and the beginning and end of the universe. The book includes sections on the continuity of space and time, on the directedness of time, on the differences between classical mechanics and the Special and General theories of relativity, on the measurement of time, on the apparent slowing down of moving clocks, and on time and probability.

Sir Walter Raleigh wollte wissen, wie Kanonenkugeln in einem Schiff am dichtesten gestapelt werden koennen. Der Astronom Johannes Kepler lieferte im Jahr 1611 die Antwort: genau so, wie Gemusehandler ihre Orangen und Tomaten aufstapeln. Noch war dies lediglich eine Vermutung - erst 1998 gelang dem amerikanischen Mathematiker Thomas Hales mit Hilfe von Computern der mathematische Beweis. Einer der besten Autoren fur popularwissenschaftliche Mathematik beschreibt auf faszinierende Art und Weise ein beruhmtes mathematisches Problem und dessen Loesung.

In this unique monograph, based on years of extensive work, Chatterjee presents the historical evolution of statistical thought from the perspective of various approaches to statistical induction. Developments in statistical concepts and theories are discussed alongside philosophical ideas on the ways we learn from experience.

On the General Science of Mathematics is the third of four surviving works out of ten by Iamblichus (c. 245 CE-early 320s) on the Pythagoreans. He thought the Pythagoreans had treated mathematics as essential for drawing the human soul upwards to higher realms described by Plato, and downwards to understand the physical cosmos, the products of arts and crafts and the order required for an ethical life. His Pythagorean treatises use edited quotation to re-tell the history of philosophy, presenting Plato and Aristotle as passing on the ideas invented by Pythagoras and his early followers. Although his quotations tend to come instead from Plato and later Pythagoreanising Platonists, this re-interpretation had a huge impact on the Neoplatonist commentators in Athens. Iamblichus' cleverness, if not to the same extent his re-interpretation, was appreciated by the commentators in Alexandria.

Analytic Philosophy: An Interpretive History explores the ways interpretation (of key figures, factions, texts, etc.) shaped the analytic tradition, from Frege to Dummet. It offers readers 17 chapters, written especially for this volume by an international cast of leading scholars. Some chapters are devoted to large, thematic issues like the relationship between analytic philosophy and other philosophical traditions such as British Idealism and phenomenology, while other chapters are tied to more fine-grained topics or to individual philosophers, like Moore and Russell on philosophical method or the history of interpretations of Wittgenstein's Tractatus. Throughout, the focus is on interpretations that are crucial to the origin, development, and persistence of the analytic tradition. The result is a more fully formed and philosophically satisfying portrait of analytic philosophy.

Analytic Philosophy: An Interpretive History explores the ways interpretation (of key figures, factions, texts, etc.) shaped the analytic tradition, from Frege to Dummet. It offers readers 17 chapters, written especially for this volume by an international cast of leading scholars. Some chapters are devoted to large, thematic issues like the relationship between analytic philosophy and other philosophical traditions such as British Idealism and phenomenology, while other chapters are tied to more fine-grained topics or to individual philosophers, like Moore and Russell on philosophical method or the history of interpretations of Wittgenstein's Tractatus. Throughout, the focus is on interpretations that are crucial to the origin, development, and persistence of the analytic tradition. The result is a more fully formed and philosophically satisfying portrait of analytic philosophy.

This collection examines the uses of quantification in climate science, higher education, and health. Numbers are both controlling and fragile. They drive public policy, figuring into everything from college rankings to vaccine efficacy rates. At the same time, they are frequent objects of obfuscation, manipulation, or outright denial. This timely collection by a diverse group of humanists and social scientists challenges undue reverence or skepticism toward quantification and offers new ideas about how to harmonize quantitative with qualitative forms of knowledge. Limits of the Numerical focuses on quantification in several contexts: climate change; university teaching and research; and health, medicine, and well-being more broadly. This volume shows the many ways that qualitative and quantitative approaches can productively interact-how the limits of the numerical can be overcome through equitable partnerships with historical, institutional, and philosophical analysis. The authors show that we can use numbers to hold the powerful to account, but only when those numbers are themselves democratically accountable.

Longlisted for the BSHS Hughes Prize 2021 A New Year's Present from a Mathematician is an exciting book dedicated to two questions: What is it that mathematicians do? And who gets to be called a 'mathematician' and why? This book seeks to answer these questions through a series of stories ranging from the beginning of modern mathematics through to the 20th century, but not in a usual, chronological manner. The author weaves her story around major questions concerning nature of mathematics, and links mathematicians by the substance of their ideas and the historical and personal context in which they were developed. Ideal as a gift for anyone with an interest in mathematics, this book gives a powerful insight into mathematical concepts in an easy-to-read-and-digest manner, without trivializing their nature. The attention given to engaging examples, framed within a poetic narrative structure, means that this book can be enjoyed by almost anyone, regardless of their level of mathematical education.

What do pure mathematicians do, and why do they do it? Looking beyond the conventional answers--for the sake of truth, beauty, and practical applications--this book offers an eclectic panorama of the lives and values and hopes and fears of mathematicians in the twenty-first century, assembling material from a startlingly diverse assortment of scholarly, journalistic, and pop culture sources.

Drawing on his personal experiences and obsessions as well as the thoughts and opinions of mathematicians from Archimedes and Omar Khayyam to such contemporary giants as Alexander Grothendieck and Robert Langlands, Michael Harris reveals the charisma and romance of mathematics as well as its darker side. In this portrait of mathematics as a community united around a set of common intellectual, ethical, and existential challenges, he touches on a wide variety of questions, such as: Are mathematicians to blame for the 2008 financial crisis? How can we talk about the ideas we were born too soon to understand? And how should you react if you are asked to explain number theory at a dinner party?

Disarmingly candid, relentlessly intelligent, and richly entertaining, "Mathematics without Apologies" takes readers on an unapologetic guided tour of the mathematical life, from the philosophy and sociology of mathematics to its reflections in film and popular music, with detours through the mathematical and mystical traditions of Russia, India, medieval Islam, the Bronx, and beyond."

Carl Friedrich Gauss, the "foremost of mathematicians," was a land surveyor. Measuring and calculating geodetic networks on the curved Earth was the inspiration for some of his greatest mathematical discoveries. This is just one example of how mathematics and geodesy, the science and art of measuring and mapping our world, have evolved together throughout history. This text is for students and professionals in geodesy, land surveying, and geospatial science who need to understand the mathematics of describing the Earth and capturing her in maps and geospatial data: the discipline known as mathematical geodesy. Map of the World: An Introduction to Mathematical Geodesy aims to provide an accessible introduction to this area, presenting and developing the mathematics relating to maps, mapping, and the production of geospatial data. Described are the theory and its fundamental concepts, its application for processing, analyzing, transforming, and projecting geospatial data, and how these are used in producing charts and atlases. Also touched upon are the multitude of cross-overs into other sciences sharing in the adventure of discovering what our world really looks like. FEATURES * Written in a fluid and accessible style, replete with exercises; adaptable for courses on different levels. * Suitable for students and professionals in the mapping sciences, but also for lovers of maps and map making.

A comprehensive look at four of the most famous problems in mathematics Tales of Impossibility recounts the intriguing story of the renowned problems of antiquity, four of the most famous and studied questions in the history of mathematics. First posed by the ancient Greeks, these compass and straightedge problems-squaring the circle, trisecting an angle, doubling the cube, and inscribing regular polygons in a circle-have served as ever-present muses for mathematicians for more than two millennia. David Richeson follows the trail of these problems to show that ultimately their proofs-which demonstrated the impossibility of solving them using only a compass and straightedge-depended on and resulted in the growth of mathematics. Richeson investigates how celebrated luminaries, including Euclid, Archimedes, Viete, Descartes, Newton, and Gauss, labored to understand these problems and how many major mathematical discoveries were related to their explorations. Although the problems were based in geometry, their resolutions were not, and had to wait until the nineteenth century, when mathematicians had developed the theory of real and complex numbers, analytic geometry, algebra, and calculus. Pierre Wantzel, a little-known mathematician, and Ferdinand von Lindemann, through his work on pi, finally determined the problems were impossible to solve. Along the way, Richeson provides entertaining anecdotes connected to the problems, such as how the Indiana state legislature passed a bill setting an incorrect value for pi and how Leonardo da Vinci made elegant contributions in his own study of these problems. Taking readers from the classical period to the present, Tales of Impossibility chronicles how four unsolvable problems have captivated mathematical thinking for centuries.

Cities house the majority of the world's population and are the dynamic centres of 21st century life, at the heart of economic, social and environmental change. They are still beset by difficult problems but often demonstrate resilience in the face of regional and national economic decline. Faced by the combined threats of globalisation and world recession, cities and their metropolitan regions have had to fight hard to maintain their global competitiveness and protect the quality of life of urban residents Transforming Urban Economies: Policy Lessons from European and Asian Cities, the first in an ongoing series of research volumes by LSE Cities, provides insights in how cities can respond positively to these challenges. The fine-grained and authoritative analysis of how Barcelona, Turin, Munich and Seoul have been transformed in the last 20 years examines comparative patterns of decline, adaptation and recovery of cities that have successfully managed to transform their economies in the face of economic hardship. This in-depth and practical analysis is aimed at urban leaders, designers, planners, policymakers and scholars who want to understand the dynamics of economic resilience while cities are still suffering from the aftershocks of the 2008 recession. The book highlights the importance of aligned and multi-level governance, the need for strategic public investments and the role of the private sector, universities and foundations in leading and guiding complex processes of urban recovery in an increasingly uncertain age.

This book offers an archeology of the undeveloped potential of mathematics for critical theory. As Max Horkheimer and Theodor W. Adorno first conceived of the critical project in the 1930s, critical theory steadfastly opposed the mathematization of thought. Mathematics flattened thought into a dangerous positivism that led reason to the barbarism of World War II. The Mathematical Imagination challenges this narrative, showing how for other German-Jewish thinkers, such as Gershom Scholem, Franz Rosenzweig, and Siegfried Kracauer, mathematics offered metaphors to negotiate the crises of modernity during the Weimar Republic. Influential theories of poetry, messianism, and cultural critique, Handelman shows, borrowed from the philosophy of mathematics, infinitesimal calculus, and geometry in order to refashion cultural and aesthetic discourse. Drawn to the austerity and muteness of mathematics, these friends and forerunners of the Frankfurt School found in mathematical approaches to negativity strategies to capture the marginalized experiences and perspectives of Jews in Germany. Their vocabulary, in which theory could be both mathematical and critical, is missing from the intellectual history of critical theory, whether in the work of second generation critical theorists such as Jurgen Habermas or in contemporary critiques of technology. The Mathematical Imagination shows how Scholem, Rosenzweig, and Kracauer's engagement with mathematics uncovers a more capacious vision of the critical project, one with tools that can help us intervene in our digital and increasingly mathematical present. The Mathematical Imagination is available from the publisher on an open-access basis.

This book, first published in 1977, discusses the Muslim contribution to mathematics during the golden age of Muslim learning from the seventh to the thirteenth century. It was during this period that Muslim culture exerted powerful economic, political and religious influence over a large part of the civilised world. The work of the Muslim scholars was by no means limited to religion, business and government. They researched and extended the theoretical and applied science of the Greeks and Romans of an earlier era in ways that preserved and strengthened man's knowledge in these important fields. Although the main object of this book is to trace the history of the Muslim contribution to mathematics during the European Dark Ages, some effort is made to explain the progress of mathematical thought and its effects upon present day culture. Certain Muslim mathematicians are mentioned because of the important nature of their ideas in the evolution of mathematical thinking during this earlier era. Muslim mathematicians invented the present arithmetical decimal system and the fundamental operations connected with it - addition, subtraction, multiplication, division, raising to a power, and extracting the square root and the cubic root. They also introduced the 'zero' symbol to Western culture which simplified considerably the entire arithmetical system and its fundamental operations; it is no exaggeration if it is said that this specific invention marks the turning point in the development of mathematics into a science.

Experience mathematics--and develop problem-solving skills that will benefit you throughout your life--with THE NATURE OF MATHEMATICS and its accompanying online learning tools. Karl Smith introduces you to proven problem-solving techniques and shows you how to use these techniques to solve unfamiliar problems. You’ll also find coverage of interesting historical topics, and practical applications to settings and situations that you encounter in your day-to-day world, such as finance (amortization, installment buying, annuities) and voting. With this book’s guidance, you’ll both understand mathematical concepts and master the techniques.

This volume brings together a selection of Solomon Feferman's most important recent writings, covering the relation between logic and mathematics, proof theory, objectivity and intensionality in mathematics, and key issues in the work of Gödel, Hilbert, and Turing.

In this book, David Stump traces alternative conceptions of the a priori in the philosophy of science and defends a unique position in the current debates over conceptual change and the constitutive elements in science. Stump emphasizes the unique epistemological status of the constitutive elements of scientific theories, constitutive elements being the necessary preconditions that must be assumed in order to conduct a particular scientific inquiry. These constitutive elements, such as logic, mathematics, and even some fundamental laws of nature, were once taken to be a priori knowledge but can change, thus leading to a dynamic or relative a priori. Stump critically examines developments in thinking about constitutive elements in science as a priori knowledge, from Kant's fixed and absolute a priori to Quine's holistic empiricism. By examining the relationship between conceptual change and the epistemological status of constitutive elements in science, Stump puts forward an argument that scientific revolutions can be explained and relativism can be avoided without resorting to universals or absolutes.