Invitation to Mathematics

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Princeton University Press, 02/08/1992 - 247 páginas
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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.

 

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Índice

Geometry
3
2 Possible and Impossible Constructions
11
4 Systematizing Geometry
28
5 Some More Views of Geometry
40
2 Square Roots
52
Questions of Principle
55
4 Solution of Equations and Systems of Equations
61
The Principle of Mathematical Induction
71
4 nPerson Games with n 2
133
5 Equilibrium
144
Topology
158
3 Surfaces
170
4 Curves on Surfaces
178
6 A Survey of Topology
188
2 Game of Life
194
3 Some Further Dynamical Systems
202

2 Discussion of the Principle of Induction
84
4 The Marriage Theorem
94
7 Recursive Definition
100
Optimization Game Theory and Economics
109
2 Optimal Flows in Networks
115
4 The Shift
208
5 General Results in Dynamics
216
Literature Cited
225
Index
233
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Acerca do autor (1992)

Jacobs is a Professor of Mathematical Statistics at the Friedrich-Alexander-Universitat, Erlangen-Nurnberg.

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