Giuseppe Peano between Mathematics and Logic (eBook)

Fulvia Skof: Emeritus Professor, Department of Mathematics G. Peano, University of Torino.

Copyright Page 5
Preface 6
List of Contributors 8
Table of Contents 9
1 Giuseppe Peano and Mathematical Analysis in Italy 10
1.1 Peano and Classical Analysis 10
1.2 Symbolic Writing and New Directions of Research 13
1.3 Peano and Analysis after 1900 15
1.4 Peano and Functional Analysis 20
2 Some Contributions of Peano to Analysis in the Light of the Work of Belgian Mathematicians 22
2.1 Introduction 22
2.2 Peano, Gilbert and the Mean Value Theorem 23
2.3 Peano, Mansion and Foundations of Calculus 29
2.4 Peano, de La Vallée Poussin and Ordinary Differential Equations 30
2.5 Peano, de La Vallée Poussin and Generalized Derivatives 32
2.6 Appendix 36
3 Peano, his School and . . . Numerical Analysis 38
3.1 Introduction 38
3.2 Peano’s Works on Numerical Analysis 41
3.3 Integral Representation of Remainders 43
3.4 Applications of the Remainder Functional 47
3.5 Trapezoidal and Parabolic Formulas 52
4 Peano and the Foundations of Arithmetic 56
4.1 Prehistory 56
4.2 Dedekind 59
4.3 Peano 63
The New Method 63
The Concept of Number 67
Consistency 69
4.4 The School of Peano 72
5 Geometric Calculus and Geometry Foundations in Peano 77
5.1 Grassmann’s Legacy Background 77
5.2 Peano’s Calcolo geometrico 78
5.3 The Minimum System 84
5.4 The Axiomatic Foundations of Geometry 86
5.5 Conclusion 89
6 The Formulario between Mathematics and History 91
6.1 The Genesis and Aims of the Formulario 92
6.2 The Influence of Leibniz 96
6.3 The Stages of the Production of the Formulario 101
6.4 Outcome of the Formulario and Cultural Spin-off 109
7 Giuseppe Peano: a Revolutionary in Symbolic Logic? 142
7.1 Three Early Mathematical Interests 142
7.2 Peano on ‘Classes’ 144
7.3 Peano’s Unsung Quantification Theory 145
7.4 Two Further Figures 146
7.5 On the Aftermath 147
8 At the Origins of Metalogic 149
9 Foundations of Geometry in the School of Peano 163
9.1 Introduction 163
9.2 Projective Geometry. Two Different Views 164
9.3 Geometry of Position, according to Pieri 165
9.4 Comparing Peano 1894c to Pieri 1895–1896 Notes 167
9.5 Affinities and Segmental Transformations 170
9.6 Conclusion 173
References 176
Index of Names 192
List of Illustrations 198