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9783764388393

Patterns of Change

by
  • ISBN13:

    9783764388393

  • ISBN10:

    3764388390

  • Format: Hardcover
  • Copyright: 2008-09-03
  • Publisher: Birkhauser

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Summary

"This book offers a reconstruction of linguistic innovations in the history of mathematics; innovations which changed the ways in which mathematics was clone, understood and philosophically interpreted. It argues that there are at least three ways in which the language of mathematics has been changed throughout its history, thus determining the lines of development that mathematics has followed." "The book offers tools of analysis by means of which scholars and students of the history and philosophy of mathematics can attain better understanding of the various changes, which the subject of their study underwent in the course of history. The book brings also important insights for mathematics education connecting growth of language with the development of mathematical thought."--BOOK JACKET.

Table of Contents

Acknowledgementsp. V
Prefacep. VII
Introductionp. 1
Re-coding as the First Pattern of Change in Mathematicsp. 11
Historical Description of Re-codingsp. 14
Elementary Arithmeticp. 17
Synthetic Geometryp. 23
Algebrap. 29
Analytic Geometryp. 37
The Differential and Integral Calculusp. 47
Iterative Geometryp. 56
Predicate Calculusp. 67
Set Theoryp. 76
Philosophical Reflections on Re-Codingsp. 85
Relation between Logical and Historical Reconstructions of Mathematical Theoriesp. 89
Perception of Shape and Motionp. 94
Epistemic Tension and the Dynamics of the Development of Mathematicsp. 98
Technology and the Coordination of Activitiesp. 99
The Pre-History of Mathematical Theoriesp. 102
Relativizations as the Second Pattern of Change in Mathematicsp. 107
Historical Description of Relativizations in Synthetic Geometryp. 111
The Perspectivist Form of Language of Synthetic Geometryp. 114
The Projective Form of Language of Synthetic Geometryp. 118
The Interpretative Form of Language of Synthetic Geometryp. 124
The Integrative Form of Language of Synthetic Geometryp. 133
The Constitutive Form of Language of Synthetic Geometryp. 143
The Conceptual Form of Language of Synthetic Geometryp. 154
An Overview of Relativizations in the Development of Synthetic Geometryp. 159
Historical Description of Relativizations in Algebrap. 160
The Perspectivist Form of the Language of Algebrap. 165
The Projective Form of the Language of Algebrap. 167
The Coordinative Form of the Language of Algebrap. 173
The Compositive Form of the Language of Algebrap. 177
The Interpretative Form of the Language of Algebrap. 180
The Integrative Form of the Language of Algebrap. 184
The Constitutive Form of the Language of Algebrap. 192
The Conceptual Form of the Language of Algebrap. 197
An Overview of Relativizations in the Development of Algebrap. 198
Philosophical Reflections on Relativizationsp. 201
Comparison of the Development of Algebra with the Development of Geometryp. 202
Form of Language and the Development of Mathematical Theoriesp. 205
The Notion of the Form of Language and Philosophy of Mathematicsp. 210
The Changes of the Form of Language and the Development of Subjectivityp. 215
The Problem of Understanding Mathematical Conceptsp. 220
A Gap of Two Centuries in the Curriculap. 222
Re-Formulations as a Third Pattern of Change in Mathematicsp. 225
Re-Formulations and Concept-Formationp. 228
Re-Formulations and Problem-Solvingp. 232
Re-Formulations and Theory-Buildingp. 235
Mathematics and Changep. 239
Revolutions in Mathematics (Kuhn)p. 240
What Kind of Changes Should we Consider Revolutionary?p. 240
The Distinction between Revolutions and Epistemological Rupturesp. 241
Possible Refinements of the Kuhnian Picturep. 243
Mathematical Research Programmes (Lakatos)p. 245
The Lack of Connection between the Two Major Parts of Proofs and Refutationsp. 246
Reduction of the Development of Mathematics to Re-Formulationsp. 247
Stages of Cognitive Development (Piaget)p. 249
The Problem of the Omitted Stagesp. 249
The Relation of Ontogenesis and Phylogenesisp. 250
Bibliographyp. 253
Table of Contents provided by Ingram. All Rights Reserved.

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