Structuralism and Structures: A Mathematical PerspectiveThis book is devoted to an analysis of the way that structures must enter into a serious study of any subject, and the term ?structuralism? refers to the general method of approaching a subject from the viewpoint of structure. A proper appreciation of this approach requires a deeper understanding of the concept of structure than is provided by the simple intuitive notion of structures that everyone posseses to some degree. Therefore, a large part of the discussion is devoted directly or indirectly to a study of the nature of structures themselves. A formal definition of a structure, plus some basic general properties and examples, is given early in the discussion. Also, in order to clarify the general notions and to see how they are used, the later chapters are devoted to an examination of how structures enter into some special fields, including linguistics, mental phenomena, mathematics (and its applications), and biology (especially in the theory of evolution). Because the author is a mathematician, certain mathematical ideas have influenced greatly the choice and approach to the material covered. In general, however, the mathematical influence is not on a technical level and is often only implicit. Even the chapter on mathematical structures is nontechnical and is about rather than on mathematics. Only in the last chapter and earlier in three short sections does one find any of the expected ?formal? mathematics. In other words, the great bulk of the material is accessible to someone without a mathematical background. |
Contents
INTRODUCTION 1 The Structuralist Approach | 1 |
The Special Role of Mathematics | 7 |
Platos Lecture on The Good | 8 |
GENERAL STRUCTURE CONCEPTS 4 The Definition Problem | 11 |
A Simple Example | 15 |
The Basic Definitions | 17 |
Isomorphisms of Structures | 21 |
Analogies and Isomorphisms | 23 |
Teaching and Learning | 108 |
MATHEMATICAL STRUCTURES 40 Introduction | 115 |
Mathematical Language | 116 |
How to Recognize a Mathematical Structure | 119 |
Research and Development of Mathematics | 120 |
The Role of Insight in Research | 122 |
A Structural Interpretation of Creativity | 128 |
How Mathematics is Applied | 131 |
An Analysis of PointLine Structures | 27 |
Special Kinds of Relations | 29 |
Structural Stability | 30 |
Structural Information | 33 |
On Abstract Structures | 35 |
SOME EXAMPLES OF STRUCTURES 15 Introduction | 39 |
Atoms and Machines | 40 |
Line Drawings by Josef Albers | 42 |
Configurations | 44 |
The Pascal Configuration | 46 |
The Triangle Group | 48 |
Group Structures | 50 |
The Real Number System | 54 |
MANAGEMENT OF COMPLEX STRUCTURES 23 The Analysis of Structures | 57 |
The Approximation of Structures | 58 |
Structural Determinism and Reductionism | 60 |
Contractions | 65 |
Contraction of Group Structures | 71 |
The Role of Language | 73 |
Simple Communication | 75 |
Structural Linguistics | 77 |
Semiotics | 82 |
The Language Faculty | 88 |
STRUCTURES IN MENTAL PHENOMENA 34 Introduction | 93 |
The Central Role of Structures | 94 |
The Drive for Intelligibility | 97 |
Philosophical Questions | 100 |
The Background Structure and Understanding | 105 |
The Effectiveness of Mathematics in Physics | 133 |
Other Applications of Mathematics | 138 |
BIOLOGICAL STRUCTURES 49 Introduction | 145 |
Classification of Organisms | 146 |
The Genetic Structure | 148 |
The Environment of a Structure | 152 |
The Evolutionary Process | 153 |
Complexity in Evolution | 157 |
Multiple Function | 163 |
Biological Catastrophes | 168 |
Determining Structures | 173 |
Convergent Evolution | 174 |
Anthropomorphism | 175 |
SPACE STRUCTURES AND STABILITY 60 Introduction | 179 |
Euclidean Spaces | 180 |
Substructures of Euclidean Space | 181 |
The Conic Sections | 182 |
Stability in a Family of Conics | 186 |
Catastrophe Theory | 188 |
Zeemans Catastrophe Machine | 190 |
A Mathematical Example | 191 |
Attack or Retreat | 196 |
Metric Spaces | 199 |
Stability of PointLine Structures | 201 |
BIBLIOGRAPHY | 207 |
211 | |
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Common terms and phrases
abstract structure actually analogy analysis applications associated atoms axioms basic binary relation biological catastrophe catastrophe theory Chapter common concept concerning concrete structure configuration conic consists contains contraction contraction mapping defined definition of structure depend Despite determined discussed elements Euclidean plane Euclidean space evolution evolutionary example exist fact fields finite formal function geometric given structure group structure hexagon human hyperbola idea identity element illustrated involve isomorphism Josef Albers kind language latter less Levi-Strauss mathematical structures mathematicians mental structures method metric space mind natural notion of structure number system objects and relations obviously organism Pascal lines perhaps phenomena physical Piaget plane point of view point-line structure possible precise problem projective plane punctuated equilibria question real numbers representation represented result role rotations Section similar simple stability struc structuralist subset substructure suggested theory tion transformation triangle triangle group ture unconscious understanding usually words