Structuralism and Structures

By: Charles E Rickart

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Hardcover | 1 December 1994

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236 Pages

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24.13 x 16.51 x 1.91

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This text provides an analysis of the way mathematical structures must enter into a serious study of any subject. The term "structuralism" refers to the general method of approaching a subject from the view point of structure.

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Preface
Introduction
The Structuralist Approach	p. 1
The Special Role of Mathematics	p. 7
Plato's Lecture on The Good	p. 8
General Structure Concepts
The Definition Problem	p. 11
Structuralist Notions of Structure	p. 11
A Simple Example	p. 15
The Basic Definitions	p. 17
Isomorphisms of Structures	p. 21
Analogies and Isomorphisms	p. 23
An Analysis of Point-Line Structures	p. 27
Special Kinds of Relations	p. 29
Structural Stability	p. 30
Structural Information	p. 33
On Abstract Structures	p. 35
Some Examples of Structures
Atoms and Machines	p. 40
Line Drawings by Josef Albers	p. 42
Configurations	p. 44
The Pascal Configuration	p. 46
The Triangle Group	p. 48
Group Structures	p. 50
The Real Number System	p. 54
Management of Complex Structures
The Analysis of Structures	p. 57
The Approximation of Structures	p. 58
Axiomatics and Approximation	p. 58
Structural Determinism and Reductionism	p. 60
Contractions	p. 65
Contraction of Group Structures	p. 71
Language and Structure
The Role of Language	p. 73
Simple Communication	p. 75
Structural Linguistics	p. 77
Semiotics	p. 82
The Language Faculty	p. 88
Structures in Mental Phenomena
The Central Role of Structures	p. 94
The Drive for Intelligibility	p. 97
Philosophical Questions	p. 100
The Background Structure and Understanding	p. 105
Teaching and Learning	p. 108
Mathematical Structures
Mathematical Language	p. 116
How to Recognize a Mathematical Structure	p. 119
Research and Development of Mathematics	p. 120
The Role of Insight in Research	p. 122
A Structural Interpretation of Creativity	p. 128
How Mathematics is Applied	p. 131
The Effectiveness of Mathematics in Physics	p. 133
Other Applications of Mathematics	p. 138
Biological Structures
Classification of Organisms	p. 146
The Genetic Structure	p. 148
The Environment of a Structure	p. 152
The Evolutionary Process	p. 153
Complexity in Evolution	p. 157
Multiple Function	p. 163
Biological Catastrophes	p. 168
Determining Structures	p. 173
Convergent Evolution	p. 174
Anthropomorphism	p. 175
Space Structures and Stability
Euclidean Spaces	p. 180
Substructures of Euclidean Space	p. 181
The Conic Sections	p. 182
Stability in a Family of Conics	p. 186
Catastrophe Theory	p. 188
Zeeman's Catastrophe Machine	p. 190
A Mathematical Example	p. 191
Attack or Retreat	p. 196
Metric Spaces	p. 199
Stability of Point-Line Structures	p. 201
Bibliography	p. 207
Index	p. 211
Table of Contents provided by Blackwell. All Rights Reserved.

Structuralism and Structures

At a Glance

Hardcover

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