"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.

Acknowledgements	p. V
Preface	p. VII
Introduction	p. 1
Re-coding as the First Pattern of Change in Mathematics	p. 11
Historical Description of Re-codings	p. 14
Elementary Arithmetic	p. 17
Synthetic Geometry	p. 23
Algebra	p. 29
Analytic Geometry	p. 37
The Differential and Integral Calculus	p. 47
Iterative Geometry	p. 56
Predicate Calculus	p. 67
Set Theory	p. 76
Philosophical Reflections on Re-Codings	p. 85
Relation between Logical and Historical Reconstructions of Mathematical Theories	p. 89
Perception of Shape and Motion	p. 94
Epistemic Tension and the Dynamics of the Development of Mathematics	p. 98
Technology and the Coordination of Activities	p. 99
The Pre-History of Mathematical Theories	p. 102
Relativizations as the Second Pattern of Change in Mathematics	p. 107
Historical Description of Relativizations in Synthetic Geometry	p. 111
The Perspectivist Form of Language of Synthetic Geometry	p. 114
The Projective Form of Language of Synthetic Geometry	p. 118
The Interpretative Form of Language of Synthetic Geometry	p. 124
The Integrative Form of Language of Synthetic Geometry	p. 133
The Constitutive Form of Language of Synthetic Geometry	p. 143
The Conceptual Form of Language of Synthetic Geometry	p. 154
An Overview of Relativizations in the Development of Synthetic Geometry	p. 159
Historical Description of Relativizations in Algebra	p. 160
The Perspectivist Form of the Language of Algebra	p. 165
The Projective Form of the Language of Algebra	p. 167
The Coordinative Form of the Language of Algebra	p. 173
The Compositive Form of the Language of Algebra	p. 177
The Interpretative Form of the Language of Algebra	p. 180
The Integrative Form of the Language of Algebra	p. 184
The Constitutive Form of the Language of Algebra	p. 192
The Conceptual Form of the Language of Algebra	p. 197
An Overview of Relativizations in the Development of Algebra	p. 198
Philosophical Reflections on Relativizations	p. 201
Comparison of the Development of Algebra with the Development of Geometry	p. 202
Form of Language and the Development of Mathematical Theories	p. 205
The Notion of the Form of Language and Philosophy of Mathematics	p. 210
The Changes of the Form of Language and the Development of Subjectivity	p. 215
The Problem of Understanding Mathematical Concepts	p. 220
A Gap of Two Centuries in the Curricula	p. 222
Re-Formulations as a Third Pattern of Change in Mathematics	p. 225
Re-Formulations and Concept-Formation	p. 228
Re-Formulations and Problem-Solving	p. 232
Re-Formulations and Theory-Building	p. 235
Mathematics and Change	p. 239
Revolutions in Mathematics (Kuhn)	p. 240
What Kind of Changes Should we Consider Revolutionary?	p. 240
The Distinction between Revolutions and Epistemological Ruptures	p. 241
Possible Refinements of the Kuhnian Picture	p. 243
Mathematical Research Programmes (Lakatos)	p. 245
The Lack of Connection between the Two Major Parts of Proofs and Refutations	p. 246
Reduction of the Development of Mathematics to Re-Formulations	p. 247
Stages of Cognitive Development (Piaget)	p. 249
The Problem of the Omitted Stages	p. 249
The Relation of Ontogenesis and Phylogenesis	p. 250
Bibliography	p. 253
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