A Brief History of Analysis : With Emphasis on Philosophy, Concepts, and Numbers, Including Weierstraß' Real Numbers

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 Weierstraß, 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. 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 The Technical Significance of Cauchy's Definition of This Concept Excursus: Preview of a Failed Revolution of Analysis in the Years of 1958 and 1961 History Does Not Recur, Not Even in Mathematics A Rebellion of Nonstandard-Analysis? A Digression Within the Excursus: Looking Back at a Criticism of Cauchy Back to the Upheaval of Nonstandard-Analysis Walking on Very Thin Ice Cauchy's Concept of Convergence: A Big Misunderstanding A Mystery of History: Cauchy's Concept of Convergence The Solution of the Mystery The Mathematical Significance of This Solution Cauchy's Proof of His Theorem Cauchy's Self-Defence What Are the Reasons for the Prevailing Misunderstanding of Cauchy's Notion of Convergence? Cauchy's Concept of Derivative—Again a Misunderstanding Cauchy's Concept of the Integral Cauchy's Basic Idea in His Proof of the Existence of the Definite Integral What Is Cauchy's ``x''? Literature 11 The Interregnum: Analysis on Swampy Ground On the Utility of History of Mathematics The Special Quality of Our Perspective What Additional Knowledge Have We Gained That Was Not Available to Cauchy and to His Contemporaries? The Teaching of Analysis Without a Curriculum Analysis as Freestyle Wrestling A. Missing Conceptual Precision—or—Riemann Invented the Modern Concept of Function B. Political Instead of Rational Reasoning Parallel Methods with Different Levels of Precision The Foundational Uncertainty of Limes Calculation: A Prominent Misunderstanding by Prominent Scholars What is ``x'' After Cauchy? Literature 12 Weierstraß: The Last Effort Towards a Substancial Analysis A Famous Man The Core of His Fame Dirichlet in 1829: Analysis Has Frontiers! Riemann 1854: These Limits of Analysis Can Be Shifted! Weierstraß' Shocking Function The Aftermath of Weierstraß' Construction What is Weierstraß' Understanding of a ``Function''? A Sudden Change Weierstraß' Concept of Number The Fundamental Hindrance That Obstructed the Understanding of Weierstraß' Concept of Number The Peculiarity of the Student Emil Strauß Preliminary The Construction Further Preliminaries for Tackling a Real Difficulty Solution of the (Perhaps) Real Difficulty The Benefit of Structural Thinking The Benefit and the Disadvantage of Structural Thinking Postscript Infinite Series Summability Summability for Series of Irrational Numbers Theorems on Irrational Numbers Summability for Series of Real Numbers The Concept of ``Convergence'' Upshot of Weierstraß' Concept of Number Weierstraß in Retrospect Literature 13 Analysis' Detachment from Reality—and the Introduction of the Actual Infinite into the Foundations of Mathematics A Pessimistic Mood The Initial Situation in 1817 Failure 1872: Two at Once Georg Cantor's Construction of the Real Numbers The Situation Cantor's Construction Arises Out of Weierstraß' The Philosophical Price of This Progress The Mathematical Price of This Progress Notoriously, Mathematicians Call Different Things ``Equal'' An Unexplored Mathematical Potential (Of Cantor and His Contemporaries) The Formal Character of Cantor's Analysis The Form of the Real Numbers, Invented by Traugott Müller, Joseph Bertrand, and Richard Dedekind The Basic Problem The Basic Idea Traugott Müller, the Very First Man Who Invented the ``Dedekind-Cuts'', in 1838 A Mathematical Outsider Müller's Educational Convictions Müller's Concept of Irrational Number An Incidental Idea, by Bertrand in 1849 The Dramatic Version, Given by Dedekind in 1872 The Elegant Version, Given by Russell, in 1919 The Evaluation of This Solution by Tannery in 1886 The Axiomatic Characterization of the Real Numbers by David Hilbert in the Years 1899 and 1900 The Situation in 1872: Two Definitions of One Subject Hilbert's Axiomatic System—the First Attempt, 1899 The Axiomatic System of Hilbert—the Second Attempt in 1900 Advantage and Disadvantage of Hilbert's Axiomatic System The New Social Duty of Mathematics—Concerning Hilbert's Ideas The Price of Success: The Inclusion of the Actual Infinite in Mathematics The Plain Fact The State of Affairs Until Now The Compromise Literature 14 Analysis with or Without Paradoxes? Built on Very Thin Ice: Cantor's Diagonal Argument The Presentation of Evidence The Impotence of This Reasoning The Origin of the ``Diagonal Argument'' The True Understanding of the ``Diagonal Argument'' The Significance of the ``Diagonal Argument'' Paradox I: Conditionally Convergent Series A Mathematical Monstrosity: The Riemann Theorem on Rearrangements of 1854 Mitigation Paradox II: Methods of Summation Paradox III: The Convergence of Function Series Paradox IV: The Term-by-Term Integration of Series Is an Analysis Without Those Paradoxes Possible? A Source from the Years 1948–53 Schmieden Dissolves the Paradoxes The First Formal Version of a Nonstandard-Analysis in the Year 1958 The Foundation in the Year 1958 Further Peculiarities of the New Analysis in the Year 1958 Finale Foundational Problems Axiomatics A Path to Independency Nonstandard-Analysis and the History of Analysis A Satisfying Finish Literature Author Index Subject Index