Abstraction and infinity:
Paolo Mancosu provides an original investigation of historical and systematic aspects of the notions of abstraction and infinity and their interaction. A familiar way of introducing concepts in mathematics rests on so-called definitions by abstraction. An example of this is Hume's Principle, wh...
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Oxford, United Kingdom
Oxford University Press
2016
|
| Ausgabe: | First edition |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | Paolo Mancosu provides an original investigation of historical and systematic aspects of the notions of abstraction and infinity and their interaction. A familiar way of introducing concepts in mathematics rests on so-called definitions by abstraction. An example of this is Hume's Principle, which introduces the concept of number by stating that two concepts have the same number if and only if the objects falling under each one of them can be put in one-one correspondence. This principle is at the core of neo-logicism. In the first two chapters of the book, Mancosu provides a historical analysis of the mathematical uses and foundational discussion of definitions by abstraction up to Frege, Peano, and Russell. Chapter one shows that abstraction principles were quite widespread in the mathematical practice that preceded Frege's discussion of them and the second chapter provides the first contextual analysis of Frege's discussion of abstraction principles in section 64 of the Grundlagen. In the second part of the book, Mancosu discusses a novel approach to measuring the size of infinite sets known as the theory of numerosities and shows how this new development leads to deep mathematical, historical, and philosophical problems. The final chapter of the book explore how this theory of numerosities can be exploited to provide surprisingly novel perspectives on neo-logicism |
| Beschreibung: | viii, 222 Seiten |
| ISBN: | 0198746822 9780198746829 |
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| 520 | 3 | |a Paolo Mancosu provides an original investigation of historical and systematic aspects of the notions of abstraction and infinity and their interaction. A familiar way of introducing concepts in mathematics rests on so-called definitions by abstraction. An example of this is Hume's Principle, which introduces the concept of number by stating that two concepts have the same number if and only if the objects falling under each one of them can be put in one-one correspondence. This principle is at the core of neo-logicism. In the first two chapters of the book, Mancosu provides a historical analysis of the mathematical uses and foundational discussion of definitions by abstraction up to Frege, Peano, and Russell. Chapter one shows that abstraction principles were quite widespread in the mathematical practice that preceded Frege's discussion of them and the second chapter provides the first contextual analysis of Frege's discussion of abstraction principles in section 64 of the Grundlagen. In the second part of the book, Mancosu discusses a novel approach to measuring the size of infinite sets known as the theory of numerosities and shows how this new development leads to deep mathematical, historical, and philosophical problems. The final chapter of the book explore how this theory of numerosities can be exploited to provide surprisingly novel perspectives on neo-logicism | |
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Datensatz im Suchindex
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|---|---|
| adam_text | Contents
Introduction 1
1. The mathematical practice of definitions by abstraction from
Euclid to Frege (and beyond) 12
1.1 Introduction 12
1.2 Equivalence relations, invariants, and definitions by abstraction 14
1.3 Mathematical practice and definitions by abstraction in classical geometry 22
1.4 Definitions by abstraction in number theory, number systems, geometry,
and set theory during the XIXth century 28
1.4.1 Number theory 29
1.4.2 Systems of Numbers and abstraction principles 34
1.4.3 Complex numbers and geometrical calculus 41
1.4.4 Set Theory 52
1.5 Conclusion 59
2. The logical and philosophical reflection on definitions by abstraction:
From Frege to the Peano school and Russell 60
2.1 Freges Grundlagert, section 64 60
2.1.1 The Grassmannian influence on Frege: Abstraction principles in
geometry 60
2.1.2 The proper conceptual order and Freges criticism of the definition
of parallels in terms of directions 66
2.1.3 Aprioricity claims for the concept of direction: Schlomilchs
Geometrie des Maasses 72
2.1.4 The debate over Schlomilchs theory of directions 80
2.2 The logical discussion on definitions by abstraction 88
2.2.1 Peano and his school 88
2.2.2 Russell and Couturat 98
2.2.3 Padoa on definitions by abstraction and further developments 103
2.3 Conclusion 109
2.4 Appendix 113
3. Measuring the size of infinite collections of natural numbers: Was
Cantor s theory of infinite number inevitable? 116
3.1 Introduction 116
3.2 Paradoxes of the infinite up to the middle ages 117
3.3 Galileo and Leibniz 122
3.4 Emmanuel Maignan 123
3.5 Bolzano and Cantor 130
viii CONTENTS
3.6 Contemporary mathematical approaches to measuring the size of
countably infinite sets 133
3.6.1 Katzs “Sets and their Sizes” (1981) 134
3.6.2 A theory of numerosities 137
3.7 Philosophical remarks 145
3.7.1 An historiographical lesson 145
3.7.2 Godel’s claim that Cantors theory of size for infinite sets is inevitable 146
3.7.3 Generalization, explanation, fruitfulness 149
3.8 Conclusion 152
4. In good company? On Humes Principle and the assignment of
numbers to infinite concepts 154
4.1 Introduction 154
4.2 Neo-logicism and Humes Principle 155
4.3 Numerosity functions: Schroder, Peano, and Bolzano 158
4.4 A plethora of good abstractions 170
4.5 Neo-logicism and Finite Humes Principle 175
4.6 The ‘good company’ objection as a generalization of Hecks argument 183
4.7 HP’s good companions and the problem of cross-sortal identity 192
4.8 Conclusion 196
4.9 Appendix 1 197
4.10 Appendix 2 200
Bibliography 203
Name Index 219
|
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| language | English |
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| spelling | Mancosu, Paolo 1960- Verfasser (DE-588)133912035 aut Abstraction and infinity Paolo Mancosu First edition Oxford, United Kingdom Oxford University Press 2016 viii, 222 Seiten txt rdacontent n rdamedia nc rdacarrier Paolo Mancosu provides an original investigation of historical and systematic aspects of the notions of abstraction and infinity and their interaction. A familiar way of introducing concepts in mathematics rests on so-called definitions by abstraction. An example of this is Hume's Principle, which introduces the concept of number by stating that two concepts have the same number if and only if the objects falling under each one of them can be put in one-one correspondence. This principle is at the core of neo-logicism. In the first two chapters of the book, Mancosu provides a historical analysis of the mathematical uses and foundational discussion of definitions by abstraction up to Frege, Peano, and Russell. Chapter one shows that abstraction principles were quite widespread in the mathematical practice that preceded Frege's discussion of them and the second chapter provides the first contextual analysis of Frege's discussion of abstraction principles in section 64 of the Grundlagen. In the second part of the book, Mancosu discusses a novel approach to measuring the size of infinite sets known as the theory of numerosities and shows how this new development leads to deep mathematical, historical, and philosophical problems. The final chapter of the book explore how this theory of numerosities can be exploited to provide surprisingly novel perspectives on neo-logicism Geschichte gnd rswk-swf Mathematik Philosophie Philosophie (DE-588)4045791-6 gnd rswk-swf Abstraktion (DE-588)4141162-6 gnd rswk-swf Unendlichkeit (DE-588)4136067-9 gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf Mathematics / Philosophy Infinite Philosophie (DE-588)4045791-6 s Mathematik (DE-588)4037944-9 s Abstraktion (DE-588)4141162-6 s Unendlichkeit (DE-588)4136067-9 s Geschichte z DE-604 Digitalisierung BSB Muenchen - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029433400&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mancosu, Paolo 1960- Abstraction and infinity Mathematik Philosophie Philosophie (DE-588)4045791-6 gnd Abstraktion (DE-588)4141162-6 gnd Unendlichkeit (DE-588)4136067-9 gnd Mathematik (DE-588)4037944-9 gnd |
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| title | Abstraction and infinity |
| title_auth | Abstraction and infinity |
| title_exact_search | Abstraction and infinity |
| title_full | Abstraction and infinity Paolo Mancosu |
| title_fullStr | Abstraction and infinity Paolo Mancosu |
| title_full_unstemmed | Abstraction and infinity Paolo Mancosu |
| title_short | Abstraction and infinity |
| title_sort | abstraction and infinity |
| topic | Mathematik Philosophie Philosophie (DE-588)4045791-6 gnd Abstraktion (DE-588)4141162-6 gnd Unendlichkeit (DE-588)4136067-9 gnd Mathematik (DE-588)4037944-9 gnd |
| topic_facet | Mathematik Philosophie Abstraktion Unendlichkeit |
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