Matematikkens historie

ChatGPT og de moderne AI-chatrobotter har allerede revolutioneret mange aspekter af vores hverdag. Men vi befinder os fortsat i den spæde begyndelse af AI-epoken.Vi står i disse tider over for verdenshistoriens største teknologiske revolution. AI – eller kunstig intelligens, som teknologien kendes på dansk – er en samfundsændrende teknologi, der vil transformere alle aspekter af vores liv over de kommende år. AI vil revolutionere vores hverdag, arbejdsplads, skolevæsen og meget mere.Anders Bæk er AI-ekspert og en af landets mest benyttede foredragsholdere, når det gælder kunstig intelligens. I denne bog tager han læseren på en rundvisning ind i fremtiden. Han fortæller gennem konkrete eksempler, hvordan vores fremtid vil blive forandret på både godt og ondt takket være denne science fiction-lignende teknologi.Det kan allerede afsløres nu, at det ikke er småting vores civilisation har i vente …Anders Bæk (f. 1992) er AI-ekspert og en af landets mest benyttede foredragsholdere til at fortælle om den store AI-revolution. Er uddannet civilingeniør i software- og forretningsudvikling, og har studeret AI’s udvikling siden 2015. Han er fast paneldeltager i landets største erhvervspodcast “Millionærklubben”, udtaler sig ofte til landets medier om AI- og tech-emner, og han udgiver en af landets mest lyttede podcasts om kunstig intelligens, ”AI Revolutionen”.

A Mathematician's Apology is the famous essay by British mathematician G. H. Hardy. It concerns the aesthetics of mathematics with some personal content, and gives the layman an insight into the mind of a working mathematician. Indeed, this book is often considered one of the best insights into the mind of a working mathematician written for the layman.¿A Mathematician's Apology is the famous essay by British mathematician G. H. Hardy. It concerns the aesthetics of mathematics with some personal content, and gives the layman an insight into the mind of a working mathematician. Indeed, this book is often considered one of the best insights into the mind of a working mathematician written for the layman.

Dieses essential stellt in kondensierter Form eine Neuinterpretation der Weierstraß'schen Konstruktion der reellen Zahlen vor: Ein vergleichsweise neuer Quellenfund lässt darauf schließen, dass der Weierstraß¿sche Zahlbegriff bereits auf Mengenbegriffen basierte und somit sehr viel elementarer ist, als bislang angenommen wurde.Die beiden bislang bekanntesten Alternativdefinitionen der reellen Zahlen ¿ mittels rationaler Folgen und Konvergenz (Cantor) bzw. als Segmente (Dedekind) ¿ werden hier ebenfalls kurz erläutert und mit der Weierstraß¿schen Konstruktion verglichen.Eine ausführliche Darstellung anhand der Originalquellen findet sich in Spalt, Die Grundlegung der Analysis durch Karl Weierstraß (Springer Spektrum, 2022).

From zero to infinity, this entertaining hardback guide opens up a new world of knowledge based on the magic of numbers. Numbers have occupied our thoughts since man first realized he had not one opposable thumb but two. And from simple enumeration they have grown to be the most important and universal language there is. The Book of Numbers highlights the dominant role that numbers play in everyday life, as well as exploring how numbers and number systems evolved, and delving into the mysteries of mankind's most powerful numbers: - What are the top-ten One Hit Wonders?- What's so magnificent about 7?- Why is 13 unlucky?>From algebra to astrology, music to mythology, from religion to recreation and from science to superstition, The Book of Numbers embraces this infinitely broad subject and puts it all in order-beginning with 0.

This volume contains 8 papers that have been collected by the Canadian Society for History and Philosophy of Mathematics. It showcases rigorously reviewed contemporary scholarship on an interesting variety of topics in the history and philosophy of mathematics.Some of the topics explored include:A way to rethink how logic is taught to philosophy students by using a rejuvenated version of the Aristotelian idea of an argument schemaA quantitative approach using data from Wikipedia to study collaboration between nineteenth-century British mathematiciansThe depiction and perception of Émilie Du Châtelet¿s scientific contributions as viewed through the frontispieces designed for books written by or connected to herA study of the Cambridge Women¿s Research Club, a place where British women were able to participate in scholarly scientific discourse in the middle of the twentieth centuryAn examination of the researchand writing process of mathematicians by looking at their drafts and other preparatory notesA global history of al-Khw¿r¿zm¿¿s Kit¿b al-jabr wa-l-muq¿bala as obtained by tracing its reception through numerous translations and commentariesWritten by leading scholars in the field, these papers are accessible not only to mathematicians and students of the history and philosophy of mathematics, but also to anyone with a general interest in mathematics.

From the bestselling author of Quantum Computing for Everyone, a concise, accessible, and elegant approach to mathematics that not only illustrates concepts but also conveys the surprising nature of the digital information age.Most of us know something about the grand theories of physics that transformed our views of the universe at the start of the twentieth century: quantum mechanics and general relativity. But we are much less familiar with the brilliant theories that make up the backbone of the digital revolution. In Beautiful Math, Chris Bernhardt explores the mathematics at the very heart of the information age. He asks questions such as: What is information? What advantages does digital information have over analog? How do we convert analog signals into digital ones? What is an algorithm? What is a universal computer? And how can a machine learn?The four major themes of Beautiful Math are information, communication, computation, and learning. Bernhardt typically starts with a simple mathematical model of an important concept, then reveals a deep underlying structure connecting concepts from what, at first, appear to be unrelated areas. His goal is to present the concepts using the least amount of mathematics, but nothing is oversimplified. Along the way, Bernhardt also discusses alphabets, the telegraph, and the analog revolution; information theory; redundancy and compression; errors and noise; encryption; how analog information is converted into digital information; algorithms; and finally, neural networks. Historical anecdotes are included to give a sense of the technology at that time, its impact, and the problems that needed to be solved. Taking its readers by the hand, regardless of their math background, Beautiful Math is a fascinating journey through the mathematical ideas that undergird our everyday digital interactions.

The book offers an analysis of Joachim Jungius¿ Texturæ Contemplatio - a hitherto-unpublished manuscript written in German and Latin that deals with weaving, knitting and other textile practices, attempting to present as well various fabrics and textile techniques in a scientifical and even mathematical framework. The book aims to provide the epistemological, technical and historic framework for Jungius¿ manuscript, inspecting fabrics, weaving techniques as well as looms and other textile machines in Holy Roman Empire during the Early Modern Period. It also offers a unique investigation of the notion and metaphor of ¿texture¿ during this period, and explores, within the wider context of the ¿meeting¿ or ¿trading zones¿ thesis, the relations between artisans and natural philosophers during the 17th century. The book is of interest to historians of philosophy and mathematics, as well as historians of technology.

Dieses Standardwerk zu philosophischen Hintergründen des mathematischen Denkens und Sprechens, Lehrens und Lernens bietet einen umfangreichen Abriss zur Geschichte der Philosophie der Mathematik bis hin zu aktuellen Strömungen. Es diskutiert mathematische und philosophische Grundfragen der historischen wie der modernen Mathematik. Über Mengenlehre, Logik und Axiomatik führt es in mathematische Grundlagen ein, untersucht das Verhältnis von Wahrheit und Beweis und stellt fundamentale Ergebnisse, ungelöste und unlösbare Probleme vor.

Der Buchtitel Von Eratosthenes bis Einstein deutet einen großen Bogen an, der in einer mathematischen Zeitreise durchlaufen wird. Das Buch wendet sich an Studierende und an Personen, welche mehr über die Geschichte unseres Weltbilds von der Antike bis zur Gegenwart im Zusammenhang mit den Biografien der Protagonisten erfahren wollen. In der Antike sind dies Denker, welche nach rationalen Ursachen der Naturerscheinungen fragen und rationale Antworten versuchen, und Denker, welche Philosophie, Mathematik und Astronomie zu einer ersten Blüte bringen. In der Renaissance und Neuzeit weisen Kopernikus mit dem heliozentrischen Weltbild sowie Galilei und Kepler mit einer neuen Verknüpfung von Empirie und mathematisch geprägter Theorie den Weg zu naturwissenschaftlichem Denken, vollendet Isaac Newton mit einer tieferen Begründung und Mathematisierung der Physik die kopernikanische Wende und eröffnet gleichzeitig eine Forschungs- und Wissensvielfalt ohnegleichen. Schließlich legen Planck mit der Quantentheorie und Einstein mit den beiden Relativitätstheorien die Grundlagen unseres heutigen Weltbilds, in dem die Urknalltheorie den Beginn unserer Raumzeit vor etwa 14 Milliarden Jahren anzeigt, aber auch die Frage aufkommt, ob hinter der Entwicklung des Universums, wie wir es heute verstehen, eine zielgerichtete Strategie hin zur Existenz des Menschen steckt oder ob diese Entwicklung ein bloßer Zufall äußerst geringer Wahrscheinlichkeit ist.

The famous problems of squaring the circle, doubling the cube and trisecting an angle captured the imagination of both professional and amateur mathematicians for over two thousand years. Despite the enormous effort and ingenious attempts by these men and women, the problems would not yield to purely geometrical methods. It was only the development. of abstract algebra in the nineteenth century which enabled mathematicians to arrive at the surprising conclusion that these constructions are not possible. In this book we develop enough abstract algebra to prove that these constructions are impossible. Our approach introduces all the relevant concepts about fields in a way which is more concrete than usual and which avoids the use of quotient structures (and even of the Euclidean algorithm for finding the greatest common divisor of two polynomials). Having the geometrical questions as a specific goal provides motivation for the introduction of the algebraic concepts and we have found that students respond very favourably. We have used this text to teach second-year students at La Trobe University over a period of many years, each time refining the material in the light of student performance.

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

This volume presents the beautiful memoirs of Euler, Lagrange and Lambert on geography, translated into English and put into perspective through explanatory and historical essays as well as commentaries and mathematical notes. These works had a major impact on the development of the differential geometry of surfaces and they deserve to be studied, not only as historical documents, but most of all as a rich source of ideas.

Der Band enthält eine umfassende und problemorientierte Darstellung der antiken griechischen Mathematik von Thales bis zu Proklos Diadochos. Exemplarisch wird ein Querschnitt durch die griechische Mathematik geboten, wobei auch solche Werke von Wissenschaftlern ausführlich gewürdigt werden, von denen keine deutsche Übersetzung vorliegt. Zahlreiche Abbildungen und die Einbeziehung des kulturellen, politischen und literarischen Umfelds liefern ein großartiges Spektrum der mathematischen Wissenschaftsgeschichte und eine wahre Fundgrube für diejenigen, die biographisches und zeitgeschichtliches Hintergrundwissen suchen oder Anregungen für Unterricht bzw. Vorlesung. Die Darstellung ist aktuell und realisiert Tendenzen neuerer Geschichtsschreibung. In der Neuauflage konnten die zentralen Kapitel über Platon, Aristoteles und Alexandria aktualisiert werden. Die Ausführungen zur griechischen Rechentechnik, mathematischen Geographie und Mathematik des Frühmittelalters wurden erweitert und zeigen neue Gesichtspunkte. Völlig neu hinzugekommen ist eine einzigartige, illustrierte Darstellung der Römischen Mathematik. Neu aufgenommen sind auch mehrere Farbabbildungen, die die Thematik des Buches gelungen veranschaulichen. Mit mehr als 280 Bildern stellt der vorliegende Band ein reich bebildertes Geschichtsbuch zur antiken Mathematik dar.

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

He [Kronecker] was, in fact, attempting to describe and to initiate a new branch of mathematics, which would contain both number theory and alge- braic geometry as special cases.-Andre Weil [62] This book is about mathematics, not the history or philosophy of mathemat- ics. Still, history and philosophy were prominent among my motives for writing it, and historical and philosophical issues will be major factors in determining whether it wins acceptance. Most mathematicians prefer constructive methods. Given two proofs of the same statement, one constructive and the other not, most will prefer the constructive proof. The real philosophical disagreement over the role of con- structions in mathematics is between those-the majority-who believe that to exclude from mathematics all statements that cannot be proved construc- tively would omit far too much, and those of us who believe, on the contrary, that the most interesting parts of mathematics can be dealt with construc- tively, and that the greater rigor and precision of mathematics done in that way adds immensely to its value.

The New Mathematical Coloring Book (TNMCB) includes striking results of the past 15-year renaissance that produced new approaches, advances, and solutions to problems from the first edition. A large part of the new edition ¿Ask what your computer can do for you,¿ presents the recent breakthrough by Aubrey de Grey and works by Marijn Heule, Jaan Parts, Geoffrey Exoo, and Dan Ismailescu. TNMCB introduces new open problems and conjectures that will pave the way to the future keeping the book in the center of the field. TNMCB presents mathematics of coloring as an evolution of ideas, with biographies of their creators and historical setting of the world around them, and the world around us.A new thing in the world at the time, TMCB I is now joined by a colossal sibling containing more than twice as much of what only Alexander Soifer can deliver: an interweaving of mathematics with history and biography, well-seasoned with controversy and opinion. ¿Peter D. Johnson, Jr.Auburn UniversityLike TMCB I, TMCB II is a unique combination of Mathematics, History, and Biography written by a skilled journalist who has been intimately involved with the story for the last half-century. ¿The nature of the subject makes much of the material accessible to students, but also of interest to working Mathematicians. ¿ In addition to learning some wonderful Mathematics, students will learn to appreciate the influences of Paul Erd¿s, Ron Graham, and others.¿Geoffrey ExooIndiana State UniversityThe beautiful and unique Mathematical coloring book of Alexander Soifer is another case of ¿good mathematics¿, containing a lot of similar examples (it is not by chance that Szemerédi¿s Theorem story is included as well) and presenting mathematics as both a science and an art¿¿Peter MihókMathematical Reviews, MathSciNetA postman came to the door with a copy of the masterpiece of the century. I thank you and the mathematics community should thank you for years to come. You have set a standard for writing about mathematics and mathematicians that will be hard to match.¿ Harold W. KuhnPrinceton UniversityI have never encountered a book of this kind. The best description of it I can give is that it is a mystery novel¿ I found it hard to stop reading before I finished (in two days) the whole text. Soifer engages the reader's attention not only mathematically, but emotionally and esthetically. May you enjoy the book as much as I did!¿ Branko GrünbaumUniversity of WashingtonI am in absolute awe of your 2008 book.¿Aubrey D.N.J. de GreyLEV Foundation

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