A brief history of analysis : with emphasis on philosophy, concepts, and numbers, including Weierstrass' real numbers / / Detlef D. Spalt

• Autore: Spalt Detlef D.
• Pubblicazione: Cham, Switzerland : , : Springer, , [2022]; ©2022
• Descrizione fisica: 1 online resource (265 pages)
• Disciplina: 515
• Soggetto topico: Mathematical analysis; Mathematical analysis - History; Anàlisi matemàtica; Història
• Soggetto genere / forma: Llibres electrònics
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: 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 -- 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 -- 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 -- 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 -- 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? -- Leibniz Knew His Theory Was Descended from an Old Tradition -- The Continuum and Why It Does Not Consist of Points -- What Is the Continuum? -- How Do Continuum and Point Interact? -- The Continuum Does Not Consist of Points -- The Indivisible -- Thomas Aquinas -- Nicholas of Cusa -- Buonaventura Cavalieri -- Evangelista Torricelli -- Why Are ``All the Lines'' Not the Area? -- Newton's Method of Fluxions -- Newton's Method -- An Example -- Fluxions and Fluents -- Literature -- 5 Do Infinite Numbers Exist?-An Unresolved Dispute Between Leibniz and Johann Bernoulli -- A Correspondence -- The Subject of the Controversy -- Harmony -- Johann Bernoulli's Exciting Position -- Johann Bernoulli's Prudence -- Another Shared (Mathematical) Point of View … -- … with Different, Even Contrary Consequences -- Johann Bernoulli's Position in Dispute. ; Johann Bernoulli Argues -- Leibniz Holds Against -- Johann Bernoulli Provides the Evidence for His View -- Leibniz Is Doubtful -- The End of This Debate: The Disagreement Continues to Exist -- Looking Ahead -- Considering the Real Significance of This Problem: An Inconsistency in the Actual Mathematical Thinking -- Decimal Numbers Today: Like Johann Bernoulli Then -- Natural Numbers Today: Like Leibniz Then -- Upshot: Anything Goes in Today's Mathematics! -- Literature -- 6 Johann Bernoulli's Rules for Differentials-What Does ``Equal'' Mean? -- Johann Bernoulli's Rules for Differentials-Part 1: Preparation -- Review of Leibniz' Ideas -- Johann Bernoulli Generalizes -- From Leibniz' Law of Continuity to Johann Bernoulli's First Postulate -- What Does ``Equal'' Mean? -- The Evident Facts -- What Johann Bernoulli's First Postulate Is All About -- How This Could Be Written -- What Is This Huge Step About? -- The Equalities Must Be Consistent -- Johann Bernoulli's Rules for Differentials-Part 2: Execution -- Rules 1 and 2: Addition and Subtraction -- Rules 3 and 4: Multiplication and Division -- Rule 5: Roots -- The First Book Containing the Rules for Differentials Stems from de l'Hospital -- A Precursor of de l'Hospital's Book! -- An Unsuitable Justification of the Rules for Differentials -- Literature -- 7 Euler and the Absolute Reign of Formal Calculation -- The Absolute Monarch of Eighteenth Century Mathematics -- The Invention of the Principal Notion of Analysis: ``Function'' -- The Components of Functions: Quantities -- What Is a Quantity? -- What is a quantity? -- The First Kinds of Quantities. Euler's Characterization of Quantities Is Insufficient -- The first kinds of quantities. Euler's characterization of quantities is insufficient -- The Second Kind of Quantities -- The second kind of quantities -- Euler's Algebraic Concept of Function. ; Euler's algebraic concept of function -- Simple but Important Consequences from Euler's Notion of Function -- Simple but important consequences from Euler's notion of function -- How Did Euler Denote Functions? -- A Standard Form for Functions -- Our Problems with This Theorem of Euler -- Our problems with this theorem of Euler -- A Daring Calculation with Infinite Numbers -- From the Powers of Ten to the Exponential Quantity -- From the powers of ten to the exponential quantity -- The Exponential Function -- The exponential function -- The Ingenious Trick-Or: Euler's Cheat -- The ingenious trick-or: Euler's cheat -- Euler's Concepts of Numbers -- Analysis Without Continuity and Convergence -- Continuity According to Euler -- Euler's Second Notion of Function -- Outlook -- Convergence According to Euler -- Convergence and Divergence -- Convergence and divergence -- The True Sum -- To Sum up Euler's Algebraic Analysis -- Literature -- 8 Emphases in Algebraic Analysis After Euler -- d'Alembert: Philosophical Legitimation of Algebraic Analysis as Well as His Critique of Euler's Concept of Function -- d'Alembert's Reflections on the Notion of Quantity -- d'Alembert's Critique -- d'Alembert's Notion of Quantity -- Assessment: d'Alembert's Philosophical Legitimation of Algebraic Analysis -- d'Alembert's Critique of Euler's Notion of Function -- d'Alembert's Impulse: Condorcet -- Lagrange: Making Algebra the Sole Foundation of Analysis -- Lagrange's New Foundation of Analysis: The Base -- The Idea of Lagrange -- A Contemporary Criticism on Lagrange's Plan -- How Does Lagrange Proceed? -- The Fundamental Gap in Lagrange's Proof -- Literature -- 9 Bolzano: The Republican Revolutionary of Analysis -- The Situation -- From the Academies to the University -- Bolzano: The Public Enemy -- A New Meaning of Convergence -- Euler: A Reminder -- Today. ; The Convergence of Sequences: Two Notions -- The Convergence of Series Today -- The Convergence of Series by Bolzano -- The Remaining Deficiency -- Continuity with a New Meaning -- Convergence Works with Discrete Objects -- Continuity Is Analogous to Convergence -- Continuity of Functions in Bolzano -- The Little Difference Between Then and Now -- The Differences from Euler's Continuity -- Continuity and the Continuous -- Bolzano's Revolutionary Concept of Function -- Bolzano's Definition of the Concept of Function -- Bolzano's Examples of Functions -- Judgement -- Mathematical Consequences of Bolzano's Notion of Function -- Literature -- 10 Cauchy: The Bourgeois Revolutionary as Activistof the Restoration -- Cauchy: The Atipode to Bolzano -- The Heart of Cauchy's Revolution of Analysis -- Mathematical View of Cauchy's Revolution of Analysis -- Cauchy's Concept of Variable Is Determined by ``values'' -- Cauchy Derives ``number'' from ``quantity'' -- ``Quantity'' -- ``Number'' -- The Basic Definition of ``limit'' -- The Unspoken Luxury Version of the Concept of Limit -- What Is the Difference? -- ``Function'' and ``value of a function'' in Cauchy -- The Concept of Function in Cauchy -- The New in Cauchy's Concept of Function and a New Style of Notation -- Cauchy's Concept of Function Is as Conservative as Possible for a Revolutionary -- Cauchy's Concept of the Value of a Function -- Cauchy's Concept ``value of a function'': A First Example -- A Surprise: Cauchy's ``limit'' Is Not Unambiguous! -- A Second Example Relevant to Cauchy's Concept ``value of a function'' -- Some Very Surprising Consequences from Cauchy's Concept of ``value of a function'' -- The Methodical Significance of Cauchy's Definition of This Concept -- The Historical Significance of Cauchy's Definition of This Concept. ; The Political Significance of Cauchy's Definition of This Concept.
• Titolo autorizzato: A Brief History of Analysis
• ISBN: 9783031006500; 9783031006494
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 996485660703316
• Lo trovi qui: Univ. di Salerno