Contents Part A Number Theory Highlights in the History of Number Theory: 1700 BC 2008

Transcription

1 Contents Part A Number Theory 1 Highlights in the History of Number Theory: 1700 BC Early Roots to Fermat Fermat Fermat s Little Theorem Sums of Two Squares Fermat s Last Theorem Bachet s Equation Pell s Equation Fermat Numbers Euler Analytic Number Theory Diophantine Equations Partitions The Quadratic Reciprocity Law Lagrange Pell s Equation Sums of Four Squares Binary Quadratic Forms Legendre Gauss Disquisitiones Arithmeticae Introduction Quadratic Reciprocity Binary Quadratic Forms Cyclotomy Algebraic Number Theory Reciprocity Laws Fermat s Last Theorem Dedekind s Ideals Summary xiii

2 xiv Contents 1.8 Analytic Number Theory The Distribution of Primes Among the Integers: Introduction The Prime Number Theorem The Riemann Zeta Function Primes in Arithmetic Progression More on the Distribution of Primes Fermat s Last Theorem Work Prior to That of Wiles Andrew Wiles References Fermat: The Founder of Modern Number Theory Introduction Fermat s Intellectual Debts Fermat s Little Theorem and Factorization A Look Ahead Sums of Squares A Look Ahead Fermat s Last Theorem A Look Ahead The Bachet and Pell Equations Bachet s Equation A Look Ahead Pell s Equation A Look Ahead Conclusion References Fermat s Last Theorem: From Fermat to Wiles Introduction The First Two Centuries Sophie Germain Lamé Pythagorean Triples Lamé s Proof Kummer Early Decades of the Twentieth Century Several Results Related to FLT, Some Major Ideas Leading to Wiles Proof of FLT Elliptic Curves Number Theory and Geometry The Shimura-Taniyama Conjecture Andrew Wiles Tributes to Wiles Is There Life After FLT? References... 63

3 Contents xv Part B Calculus/Analysis 4 History of the Infinitely Small and the Infinitely Large in Calculus, with Remarks for the Teacher Introduction Seventeenth-Century Predecessors of Newton and Leibniz Introduction Cavalieri Fermat Newton and Leibniz: The Inventors of Calculus Introduction Didactic Observation Newton Leibniz Didactic Observation The Eighteenth Century: Euler Introduction Didactic Observation The Algebraization of Calculus Didactic Observation: Discovery and Proof Foundational Issues in the Seventeenth and Eighteenth Centuries Introduction Newton and Leibniz Berkeley and d Alembert Euler Lagrange Calculus Becomes Rigorous: Cauchy, Dedekind, and Weierstrass Introduction Cauchy Dedekind and Weierstrass Didactic Observation The Twentieth Century: The Nonstandard Analysis of Robinson Introduction Hyperreal Numbers Wider Implications Robinson and Leibniz Didactic Observation References A Brief History of the Function Concept Introduction Precalculus Developments

4 xvi Contents 5.3 Euler s Introductio in Analysin Infinitorum The Vibrating-String controversy Fourier Series Dirichlet s Concept of Function Pathological Functions Baire and Analytically Representable Functions Debates About the Nature of Mathematical Objects Recent Developments References More on the History of Functions, with Remarks on Teaching Introduction Anticipations of the Function Concept Babylonian Mathematics Greek Mathematics The Latitude of Forms Precalculus Developments The Calculus of Newton and Leibniz Remark on Teaching The Emergence and Consolidation of the Function Concept Remarks on Teaching Functions Repesented by Power Series Remarks on Teaching Functions Defined by Integrals Remarks on Teaching Functions Defined as Solutions of Differential Equations Remark on Teaching Partial Differential Equations and the Representation of Functions by Fourier Series Remarks on Teaching Functions and Continuity Remarks on Teaching Conceptual Aspects of Functions Remarks on Teaching Analytically Representable Functions Remark on Teaching Conclusion References Part C Proof 7 Highlights in the Practice of Proof: 1600 BC Introduction The Babylonians Greek Axiomatics

5 Contents xvii 7.4 Symbolic Notation Leibniz Euler The Calculus of Cauchy The Calculus of Weierstrass The Reemergence of the Axiomatic Method Foundational Issues Introduction Logicism Formalism Intuitionism The Era of the Computer References Paradoxes: What Are They Good For? Introduction Numbers Incommensurables Negative Numbers Complex Numbers Logarithms Functions The Eighteenth Century Nineteenth-Century Views Continuity Euler and Cauchy Continuity and Differentiability Aspects of Calculus Other than Continuity Tangents Infinite Series Sets Curves Decomposition of Geometric Objects Doubling the Cube Squaring the Circle Conclusion References Principle of Continuity: Sixteenth Nineteenth Centuries Introduction Analysis Leibniz and Robinson Euler and Cauchy Algebra British Symbolical Algebra Cubic Equations and Complex Numbers

6 xviii Contents 9.4 Geometry Projective Geometry What Is Geometry? Number Theory The Bachet Equation Fermat s Last Theorem Conclusion References Proof: A Many-Splendored Thing Introduction Heuristics vs. Rigor Ancient Mathematics Calculus Riemann and Weierstrass Analysis vs. Synthesis Ancient Greece Leibniz and Newton Eighteenth and Nineteenth Centuries Pure vs. Applied Introduction The Vibrating-String Problem The Heat-Conduction Problem Legitimate vs. Illegitimate The Quaternions Functions Continuity Definitions in Mathematics Abstraction Idealists vs. Empiricists Ideals Pathological Functions Invariants Weyl and Von Neumann Short vs. Long Proofs Humans vs. Machines Deterministic vs. Probabilistic Proofs Theorems vs. Proofs The Recent Debate References Part D Courses Inspired by History 11 Numbers as a Source of Mathematical Ideas Introduction

7 Contents xix 11.2 Beyond the Complex Numbers A Brief History of Standard Number Systems The Quaternions Other Hypercomplex Systems What is a Number? The Algebraic-Transcendental Dichotomy Introduction Algebraic Numbers Transcendental Numbers Algebraic Numbers and Diophantine Equations Transfinite Numbers Introduction Some Implications of Cantor s Work The Personality of Numbers One, Two, Many Discovery (Invention), Use, Understanding, Justification Numbers and Geometry Numbers and Analysis The Arithmetization of Analysis Nonstandard Analysis Number Theory References History of Complex Numbers, with a Moral for Teachers Introduction Birth Growth Maturity The Moral Projects References A History-of-Mathematics Course for Teachers, Based on Great Quotations Introduction What Is Mathematics? Non-Euclidean Geometry The Infinite The Twentieth Century: Foundational Issues Conclusion References Famous Problems in Mathematics Introduction The Themes The Origin of Concepts, Results, and Theories

8 xx Contents The Roles of Intuition vs. Logic Changing Standards of Rigor The Roles of the Individual vs. the Environment Mathematics and the Physical World The Relativity of Mathematics Mathematics: Discovery or Invention? The Problems Problem 1: Diophantine Equations Problem 2: Distribution of Primes Among the Integers Problem 3: Polynomial Equations Problem 4: Are There Numbers Beyond the Complex Numbers? Problem 5: Why Is ( 1)( 1)D1? Problem 6: Euclid s Parallel Postulate Problem 7: Uniqueness of Representation of a Function in a Fourier Series Problem 8: Paradoxes in Set Theory Problem 9: Consistency, Completeness, Independence Other Problems General Remarks on the Course References Part E Brief Biographies of Selected Mathematicians 15 The Biographies Richard Dedekind ( ) Introduction Life Algebraic Numbers Real Numbers Natural Numbers Other Work Conclusion References Leonhard Euler ( ) Introduction Life Analysis Number Theory Conclusion References Carl Friedrich Gauss ( ) Life

9 Contents xxi Disquisitiones Arithmeticae Biquadratic Reciprocity Differential Geometry Probability and Statistics The Diary Personality Conclusion References David Hilbert ( ) Introduction Life Invariants Algebraic Numbers Foundations of Geometry Analysis and Physics Foundations of Mathematics Mathematical Problems Conclusion References Karl Weierstrass ( ) Life Foundations of Real Analysis Complex Analysis Other Work Conclusion References Index

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