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

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