## Patterns of Change: Linguistic Innovations in the Development of Classical MathematicsKvasz’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 |

253 | |

### Common terms and phrases

analysis analytic geometry axioms boundaries casus irreducibilis circle coefﬁcients complex numbers concept construction curve deﬁned deﬁnition Desargues Descartes development of geometry development of mathematics differential and integral difﬁcult discovery elementary arithmetic epistemic subject epistemological equation Euclid Euclidean geometry Euler explicit ﬁeld ﬁfth ﬁgures ﬁnd ﬁrst ﬁve form of language formula fractal framework Frege function fundamental guage history of mathematics iconic language implicit inﬁnite inﬂuence instance integral calculus integrative form intersection introduced iterative geometry Kant’s Klein Kvasz Lakatos language of algebra language of geometry language of mathematics language of synthetic line segment linguistic Lobachevski logical mathematicians means methods Nevertheless non-Euclidean geometry notion particular philosophy of mathematics picture PoincarŽe point of view polynomial possible predicate calculus problem projective form projective geometry projective plane proofs quintic equations re-codings re-formulations reconstruction reﬂection relativizations represented roots set theory solution solve space straight line structure sufﬁcient symbolic language synthetic geometry theorem tion topology transformation