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Invitation to Mathematics : Princeton Science Li - Konrad Jacobs

Invitation to Mathematics

Princeton Science Li

Paperback

Published: 2nd August 1992
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Based on a well-received course designed for philosophy students, this book is an informal introduction to mathematical thinking. The work will be rewarding not only for philosophers concerned with mathematical questions but also for serious amateur mathematicians with an interest in the "frontiers" as well as the foundations of mathematics. In what might be termed a sampler of the discipline, Konrad Jacobs discusses an unusually wide range of topics, including such items of contemporary interest as knot theory, optimization theory, and dynamical systems. Using Euclidean geometry and algebra to introduce the mathematical mode of thought, the author then turns to recent developments. In the process he offers what he calls a "Smithsonian of mathematical showpieces": the five Platonic Solids, the Mbius Strip, the Cantor Discontinuum, the Peano Curve, Reidemeister's Knot Table, the plane ornaments, Alexander's Horned Sphere, and Antoine's Necklace. The treatments of geometry and algebra are followed by a chapter on induction and one on optimization, game theory, and mathematical economics. The chapter on topology includes a discussion of topological spaces and continuous mappings, curves and knots, Euler's polyhedral formula for surfaces, and the fundamental group. The last chapter deals with dynamics and contains material on the Game of Life, circle rotation, Smale's "horseshoe," and stability and instability, among other topics.

"This is a wonderful book ... [which] provides an informal introduction to mathematical thinking... This book is the latest in the succession of books designed to introduce mathematical ideas to the general public."--The Times Higher Education Supplement "Should be read by all teachers of mathematics and mathematics educators."--The Mathematics Teacher

Foreword
Geometryp. 3
Some Classical Theorems of Euclidean Geometryp. 4
The Circumference Angle and the Thales Circle Theoremsp. 4
The Pythagorean Theoremp. 5
The Altitudes Intersection Theoremp. 8
Feuerbach's Nine-Point Circlep. 8
The Regular Solidsp. 9
Possible and Impossible Constructionsp. 11
Squaring the Circlep. 11
Construction of the Regular n-gonp. 13
Trisection of Anglesp. 15
Doubling the Cubep. 16
The Question of Decomposition Equivalencep. 17
Groups of Rigid Motionsp. 17
Congruence Motions of Geometric Figuresp. 18
The Classification of Symmetry Groups, Ornaments, and Crystalsp. 25
Systematizing Geometryp. 28
The Axiomatic Edifice of Euclidean Geometryp. 28
The Parallel Postulate and Non-Euclidean Geometryp. 29
Analytic Geometryp. 33
Projective Geometryp. 36
The Group-Theoretical Systematics of Geometry: Felix Klein's Erlanger Programm (1872)p. 39
Some More Views of Geometryp. 40
Elements of Algebrap. 42
The Four Basic Arithmetical Operations and the Concept of a Fieldp. 43
Square Rootsp. 52
Taking Square Roots: Questions of Principlep. 55
The Nonexistence of [actual symbol not reproducible] in [actual symbol not reproducible]p. 56
Quadratic Field Extensionp. 58
Solution of Equations and Systems of Equationsp. 61
The Principle of Mathematical Inductionp. 71
Three Summationsp. 72
Arithmetic Progressionsp. 72
Geometric Progressionsp. 74
The Divergence of the Harmonic Seriesp. 78
Proof of the Preceding Summation Results by Mathematical Inductionp. 81
Discussion of the Principle of Inductionp. 84
Fundamental Observations about the Induction Principlep. 84
Intuitive Proofs and Proofs by Inductionp. 86
Elementary Theory of Combinatoricsp. 87
Wordsp. 87
The Number of Words of a Given Lengthp. 87
The Number of 0-1-Words with a Given Number of Onesp. 89
The Number of Words with All Different Lettersp. 93
The Marriage Theoremp. 94
The Binomial Theoremp. 95
Induction Proofs of Two Fundamental Theoremsp. 97
The Well-Ordering of the System of Natural Numbersp. 97
The Pigeonhole Principle and Dedekind's Definition of a Finite Setp. 98
Ramsey's Theoremp. 99
Recursive Definitionp. 100
The Fibonacci Numbersp. 102
The Thue-Morse Sequencep. 104
Georg Cantor's Diagonal Methodp. 107
Optimization, Game Theory, and Economicsp. 109
Optimization Problemsp. 110
Sortingp. 110
A Simple Case of Linear Optimizationp. 110
An Example of Quadratic Optimizationp. 112
An Example of Cubic Optimizationp. 113
Network Optimizationp. 114
Optimal Flows in Networksp. 115
Ordered Fieldsp. 120
Orders and Partial Ordersp. 121
Ordered Fieldsp. 123
Arithmetical Means and Weighted Meansp. 126
The Behavior of Polynomial Functionsp. 130
n-Person Games with n ] 2p. 133
Stone-Paper-Scissorsp. 134
The Game of NIMp. 135
Prisoners' Dilemmap. 137
Some More Bimatrix Gamesp. 138
Game Theory and Biologyp. 140
Aggregation of Preferences and Arrow's Dictatorship Theoremp. 141
Equilibriump. 144
The Equilibrium Theorem for Tree Gamesp. 145
The Equilibrium Theorem for Noncooperative Gamesp. 150
Equilibrium Theorems of Mathematical Economicsp. 156
Topologyp. 158
Topological Spaces and Continuous Mappingsp. 158
Curves and Knotsp. 163
Surfacesp. 170
Curves on Surfacesp. 178
Euler's Polyhedral Formulap. 178
The Fundamental Groupp. 180
Compactnessp. 182
A Survey of Topologyp. 188
Dynamicsp. 189
Dynamical Systems with a Finite Number of Statesp. 191
The Injective Casep. 191
The General Casep. 193
A Glimpse of the Theory of Automatap. 193
Game of Lifep. 194
The State Space of Game of Lifep. 194
The Transition Law T of Game of Lifep. 195
The Life Stories (Orbits) of Some Configurations (States)p. 196
The Garden of Eden Theoremp. 199
Some Further Dynamical Systemsp. 202
Circle Rotation (Kronecker [1884])p. 202
The French Dough Transformation (The Baker's Transformation)p. 204
Stephen Smale's Horseshoep. 206
The Shiftp. 208
Fixed Pointsp. 212
Points of Period Twop. 212
Longer Periodsp. 213
Almost Periodicityp. 213
General Results in Dynamicsp. 216
Stability and Instabilityp. 218
Stabilityp. 220
Instabilityp. 221
Literature Citedp. 225
Indexp. 233
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780691025285
ISBN-10: 0691025282
Series: Princeton Science Li
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 264
Published: 2nd August 1992
Publisher: Princeton University Press
Country of Publication: US
Dimensions (cm): 23.5 x 15.88  x 1.91
Weight (kg): 0.41