Beyond Geometry

By: John Tabak

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Hardcover | 30 July 2011

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Hardcover
237 Pages

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15+

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10+

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22.9 x 15.2

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For millennia, Euclidean geometry, the geometry of the ancient Greeks, set the standard for rigor in mathematics--it was the only branch of mathematics that had been developed axiomatically, or based on a system of axioms. This form of geometry was the language of mathematics. But early in the 19th century, mathematicians developed geometries very different from Euclid's simply by choosing axioms different from those used by Euclid. These new geometries were internally consistent in the sense that mathematicians could find no theorems arising within these geometries that could be proven both true and false.

Beyond Geometry describes how set-theoretic topology developed and why it now occupies a central place in mathematics. Describing axiomatic method as well as providing a definition of what a geometric property is, this new resource examines how early analysts incorporated geometric thinking into their development of the calculus. It also looks at the various mathematicians who struggled to develop a new conceptual framework for mathematics and examines one of the sub-disciplines of set-theoretic topology called dimension theory.

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Preface	p. x
Acknowledgments	p. xvi
Introduction
Topology:A Prehistory	p. 1
Euclid's Axioms	p. 2
Euclidean Transformations	p. 7
The Beginnings of Calculus	p. 10
Counterexample 1: A Continuous Function That Is Not Everywhere Differentiable	p. 14
A Failure of Intuition	p. 18
An Alternative to Euclid's Axioms	p. 19
Bernhard Bolzano and Further Limitations on Geometric Reasoning	p. 23
Counterexample 2: A Continuous Nowhere Differentiable Function	p. 30
A New Mathematical Landscape	p. 33
Richard Dedekind and the Continuum	p. 34
Georg Cantor and Set Theory	p. 38
Counterexample 3: PeanoÆs Space-Filling Curve	p. 44
Different-Looking Sets, Similar Properties: Part I	p. 47
Different-Looking Sets, Similar Properties: Part n	p. 50
Tile First Topological Spaces	p. 55
Felix Hausdorff and the First Abstract Topological Spaces	p. 56
Topological Transformations	p. 66
The Role of Examples and Counterexamples in Topology	p. 74
Counterexample 4: Sierpiliskis Gasket	p. 76
The Standard Axioms and Three Topological Properties	p. 80
The Standard Axioms	p. 81
Topological Property 1: Compacmess	p. 84
Topological Property 2: Regularity	p. 91
The Role of Rigor in Mathematics	p. 92
Topological Property 3: Connectedness	p. 98
Schools of Topology	p. 102
The Polish School	p. 103
Topology at the University of Texas	p. 108
The Moore Method	p. 112
Topology in Japan	p. 114
Dimension Theory	p. 120
Continuum Theory	p. 122
Inductive Definitions of Dimension	p. 124
A Noninductive Definition of Dimension and More Consequences of Dimension Theory	p. 133
Still Another Concept of Dimension: The Hausdorff Dimension	p. 136
Topology and the Foundations of Modem Mathematics	p. 143
Topology and the Language of Mathematics	p. 143
Topology in Analysis	p. 147
Other Kinds of Topology, or What Happened to the Konigsberg Bridge Problem?	p. 151
Mathematical Structures and Topology	p. 156
Conclusion	p. 160
Afterword: An Interview with Processor Scott Williams on the Nature of Topology and the Goals of Topological Research	p. 162
Chronology	p. 173
Glossary	p. 190
Further Resources	p. 196
Index	p. 202
Table of Contents provided by Ingram. All Rights Reserved.

Beyond Geometry

At a Glance

Hardcover

Shipping

How to return your order

Defective items

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