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
2019
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| 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: | 9780198822684 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. The mathematical practice of definitions by abstraction from Euclid to Frege (and beyond) 1.1 1.2 1.3 1.4 Introduction Equivalence relations, invariants, and definitions by abstraction Mathematical practice and definitions by abstraction in classical geometry Definitions by abstraction in number theory, number systems, geometry, and set theory during the XIXth century 1.4.1 Number theory 1.4.2 Systems of Numbers and abstraction principles 1.4.3 Complex numbers and geometrical calculus 1.4.4 Set Theory 1.5 Conclusion 1 12 12 14 22 28 29 34 41 52 59 2. The logical and philosophical reflection on definitions by abstraction: From Frege to the Peano school and Russell 60 2.1 Frege’s Grundlagen, section 64 2.1.1 The Grassmannian influence on Frege: Abstraction principles in geometry 2.1.2 The proper conceptual order and Frege’s criticism of the definition of parallels in terms of directions 2.1.3 Aprioricity claims for the concept of direction: Schlömilch’s Geometrie des Maasses 2.1.4 The debate over Schlömilch’s theory of directions 2.2 The logical discussion on definitions by abstraction 2.2.1 Peano and his school 2.2.2 Russell and Couturat 2.2.3 Padoa on definitions by abstractionand further developments 2.3 Conclusion 2.4 Appendix 60 60 66 72 80 88 88 98 103 109 113 3. Measuring the size of infinite collections of natural numbers: Was Cantor’s theory of infinite number inevitable? 116 3.1 3.2 3.3 3.4 3.5 116 117 122 123 130 Introduction Paradoxes of the infinite up to the middle ages Galileo and Leibniz Emmanuel Maignan Bolzano and Cantor
ѴП1 CONTENTS 3.6 Contemporary mathematical approaches to measuring the size of countably infinite sets 3.6.1 Katzs “Sets and their Sizes” (1981) 3.6.2 A theory of numerosities 3.7 Philosophical remarks 3.7.1 An historiographical lesson 3.7.2 Gödel’s claim that Cantor’s theory of size for infinite sets is inevitable 3.7.3 Generalization, explanation, fruitfulness 3.8 Conclusion 4. In good company? On Humes Principle and the assignment of numbers to infinite concepts 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 Introduction Neo-logicism and Humes Principle Numerosity functions: Schröder, Peano, andBolzano A plethora of good abstractions Neo-logicism and Finite Hume’s Principle The ‘good company objection as a generalization of Hecks argument HP’s good companions and the problem of cross-sortal identity Conclusion Appendix 1 Appendix 2 Bibliography Name Index 133 134 137 145 145 146 149 152 154 154 155 158 170 175 183 192 196 197 200 203 219
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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 2019 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 Abstraktion (DE-588)4141162-6 gnd rswk-swf Unendlichkeit (DE-588)4136067-9 gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf Philosophie (DE-588)4045791-6 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 UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031592670&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mancosu, Paolo 1960- Abstraction and infinity Mathematik Philosophie Abstraktion (DE-588)4141162-6 gnd Unendlichkeit (DE-588)4136067-9 gnd Mathematik (DE-588)4037944-9 gnd Philosophie (DE-588)4045791-6 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 Abstraktion (DE-588)4141162-6 gnd Unendlichkeit (DE-588)4136067-9 gnd Mathematik (DE-588)4037944-9 gnd Philosophie (DE-588)4045791-6 gnd |
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