| Description |
1 online resource (265 pages) |
| Contents |
Intro -- Preface -- For Whom Is This Book Written? -- Who Can Understand This Book? -- What Is at Stake? -- Who Has Contributed? -- Preface to the Translation -- Contents -- Introduction: The Four Big Topics of This Book -- The Configuration of Mathematics-or: Designing Mathematical Theories -- To Define Is Hard Work! -- Is a Mathematical Proof Beyond Reproach? -- From Confusion to Clarity -- Growing Insight in the Formative Power of Definitions in Mathematics -- The Change, Seen from a Philosophical Viewpoint -- The Formation of Mathematics-or: The Transformations of Analysis |
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The Foundational Years -- An Era of Pomposity: Algebraic Analysis -- The Implosion of Algebraic Analysis-and a First Attempt to Replace It -- Implementation of a Capricious Value Analysis -- Outlook: Axiomatics, Analysis Within Set-Theory and a New Kind of Formal Calculation -- The First Mathematical News in This Book: The Archetype of Today's Analysis (from Cauchy) -- The Second Mathematical News in This Book: A Third Construction of the Real Numbers (by Weierstraß) -- The Historiographical Hallmarks of This Book -- In Substance -- In Method -- All Told |
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1 The Invention of the Mathematical Formula -- Who Invented the Mathematical Formula? -- How Did Descartes Invent the Mathematical Formula? -- Transfer Arithmetic into Geometry! -- Solve Problems! -- Why Does Descartes Have Those Ideas? -- What Is x for Descartes? -- Literature -- 2 Numbers, Line Segments, Points-But No Curved Lines -- Mathematics Is in Need of Systematization -- True and False Roots -- What Are False Roots? And What Is Their Use? -- Turn False into True -- The Geometrical Advantage of Equations -- Analysis: From Problem to Equation -- Interjection: Continuity |
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Synthesis: From Points to Curved Lines? (I) -- The Admissible Curved Lines -- Synthesis: From Points to Curved Lines? (II) -- Descartes' Geometrical Successes and His Failure -- Literature -- 3 Lines and Variables -- From Two to Infinity: Leibniz' Conception of the World -- Leibniz' Mathematical Writings -- Leibniz' Theorem: Fresh from the Creator! -- The Convergence of Infinite Series -- Leibniz' Formulation of His Theorem -- Leibniz' Proof of His Theorem -- Reflection on Leibniz' Achievement -- An Idea Which Leibniz Could not Grasp and the Reason for His Inability |
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The Precise Calculation of Areas Bounded by Curves: The Integral -- The Beginning Is Easy -- The Problem -- The Solution of Leibniz-The Original Way -- Outlook -- Leibniz' Neat Construction of the Concept of a Differential -- The First Publication: A False Start -- Another False Start: The New Edition -- The Neat Construction, Part I -- Interlude: The General Rule: The Law of Continuity -- The Neat Construction, Part II -- What Is x (and What Is dx) for Leibniz? -- Literature -- 4 Indivisible: An Old Notion (Or, What Is the Continuum Made of?) -- A Modern Theory? |
| Summary |
This book explores the origins of mathematical analysis in an accessible, clear, and precise manner. Concepts such as function, continuity, and convergence are presented with a unique historical point of view. In part, this is accomplished by investigating the impact of and connections between famous figures, like Newton, Leibniz, Johann Bernoulli, Euler, and more. Of particular note is the treatment of Karl Weierstra, whose concept of real numbers has been frequently overlooked until now. By providing such a broad yet detailed survey, this book examines how analysis was formed, how it has changed over time, and how it continues to evolve today. A Brief History of Analysis will appeal to a wide audience of students, instructors, and researchers who are interested in discovering new historical perspectives on otherwise familiar mathematical ideas |
| Notes |
Leibniz Knew His Theory Was Descended from an Old Tradition |
| Bibliography |
Includes bibliographical references and index |
| Notes |
Online resource; title from PDF title page (SpringerLink, viewed August 17, 2022) |
| Subject |
Mathematical analysis -- History
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Análisis matemático -- Historia
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Mathematical analysis
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| Genre/Form |
History
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| Form |
Electronic book
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| ISBN |
9783031006500 |
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303100650X |
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