Patterns of Change: Linguistic Innovations in the Development of Classical Mathematics (Google eBook)

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Springer Science & Business Media, Oct 28, 2008 - Mathematics - 279 pages
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The book offers a reconstruction of linguistic innovations in the history of mathematics. It argues that there are at least three ways in which the language of mathematics can be changed. As illustration of changes of the first kind, called re-codings, is the development along the line: synthetic geometry, analytic geometry, fractal geometry, and set theory. In this development the mathematicians changed the very way of constructing geometric figures. As illustration of changes of the second kind, called relativization, is the development of synthetic geometry along the line: Euclid’s geometry, projective geometry, non-Euclidean geometry, Erlanger program up to Hilbert’s Grundlagen der Geometrie. Changes of the third kind, called re-formulations are for instance the changes that can be seen on the different editions of Euclid’s Elements. Perhaps the best known among them is Playfair’s change of the formulation of the fifth postulate.
  

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Contents

Introduction
1
Recoding as the First Pattern of Change in Mathematics
11
11 Historical Description of Recodings
14
111 Elementary Arithmetic
17
112 Synthetic Geometry
23
113 Algebra
29
114 Analytic Geometry
37
115 The Differential and Integral Calculus
47
223 The Coordinative Form of the Language of Algebra
173
224 The Compositive Form of the Language of Algebra
177
225 The Interpretative Form of the Language of Algebra
180
226 The Integrative Form of the Language of Algebra
184
227 The Constitutive Form of the Language of Algebra
192
228 The Conceptual Form of the Language of Algebra
197
229 An Overview of Relativizations in the Development of Algebra
198
23 Philosophical Reflections on Relativizations
201

116 Iterative Geometry
56
117 Predicate Calculus
67
118 Set Theory
76
12 Philosophical Reflections on ReCodings
85
121 Relation between Logical and Historical Reconstructions of Mathematical Theories
89
122 Perception of Shape and Motion
94
123 Epistemic Tension and the Dynamics of the Development of Mathematics
98
124 Technology and the Coordination of Activities
99
125 The PreHistory of Mathematical Theories
102
Relativizations as the Second Pattern of Change in Mathematics
107
21 Historical Description of Relativizations in Synthetic Geometry
111
211 The Perspectivist Form of Language of Synthetic Geometry
114
212 The Projective Form of Language of Synthetic Geometry
118
213 The Interpretative Form of Language of Synthetic Geometry
124
214 The Integrative Form of Language of Synthetic Geometry
133
215 The Constitutive Form of Language of Synthetic Geometry
143
216 The Conceptual Form of Language of Synthetic Geometry
154
217 An Overview of Relativizations in the Development of Synthetic Geometry
159
22 Historical Description of Relativizations in Algebra
160
221 The Perspectivist Form of the Language of Algebra
165
222 The Projective Form of the Language of Algebra
167
231 Comparison of the Development of Algebra with the Development of Geometry
202
232 Form of Language and the Development of Mathematical Theories
205
233 The Notion of the Form of Language and Philosophy of Mathematics
210
234 The Changes of the Form of Language and the Development of Subjectivity
215
235 The Problem of Understanding Mathematical Concepts
220
236 A Gap of Two Centuries in the Curricula
222
ReFormulations as a Third Pattern of Change in Mathematics
224
31 ReFormulations and ConceptFormation
228
32 ReFormulations and ProblemSolving
232
33 ReFormulations and TheoryBuilding
235
Mathematics and Change
239
41 Revolutions in Mathematics Kuhn
240
412 The Distinction between Revolutions and Epistemological Ruptures
241
413 Possible Refinements of the Kuhnian Picture
243
42 Mathematical Research Programmes Lakatos
245
421 The Lack of Connection between the Two Major Parts of Proofs and Refutations
246
422 Reduction of the Development of Mathematics to ReFormulations
247
43 Stages of Cognitive Development Piaget
249
432 The Relation of Ontogenesis and Phylogenesis
250
Bibliography
253
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