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Table of 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
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?
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?