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 - 262 pages
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Kvasz’s book is a contribution to the history and philosophy of mat- matics, or, as one might say, the historical approach to the philosophy of mathematics. This approach is for mathematics what the history and philosophy of science is for science. Yet the historical approach to the philosophy of science appeared much earlier than the historical approach to the philosophy of mathematics. The ?rst signi?cant work in the history and philosophy of science is perhaps William Whewell’s Philosophy of the Inductive Sciences, founded upon their History. This was originally published in 1840, a second, enlarged edition appeared in 1847, and the third edition appeared as three separate works p- lished between 1858 and 1860. Ernst Mach’s The Science of Mech- ics: A Critical and Historical Account of Its Development is certainly a work of history and philosophy of science. It ?rst appeared in 1883, and had six further editions in Mach’s lifetime (1888, 1897, 1901, 1904, 1908, and 1912). Duhem’s Aim and Structure of Physical Theory appeared in 1906 and had a second enlarged edition in 1914. So we can say that history and philosophy of science was a well-established ?eld th th by the end of the 19 and the beginning of the 20 century. By contrast the ?rst signi?cant work in the history and philosophy of mathematics is Lakatos’s Proofs and Refutations, which was p- lished as a series of papers in the years 1963 and 1964.
  

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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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