Wittgenstein on mathematics:
"This book offers a detailed account and discussion of Ludwig Wittgenstein's philosophy of mathematics. In Part I, the stage is set with a brief presentation of Frege's logicist attempt to provide arithmetic with a foundation and Wittgenstein's criticisms of it, followed by sketc...
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York ; London
Routledge
2021
|
| Schriftenreihe: | Wittgenstein's thought and legacy
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| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | "This book offers a detailed account and discussion of Ludwig Wittgenstein's philosophy of mathematics. In Part I, the stage is set with a brief presentation of Frege's logicist attempt to provide arithmetic with a foundation and Wittgenstein's criticisms of it, followed by sketches of Wittgenstein's early views of mathematics, in the Tractatus and in the early 1930s. Then (in Part II), Wittgenstein's mature philosophy of mathematics (1937-44) is carefully presented and examined. Schroeder explains that it is based on two key ideas: the calculus view and the grammar view. On the one hand, mathematics is seen as a human activity - calculation - rather than a theory. On the other hand, the results of mathematical calculations serve as grammatical norms. The following chapters (on mathematics as grammar; rule-following; conventionalism; the empirical basis of mathematics; the role of proof) explore the tension between those two key ideas and suggest a way in which it can be resolved. Finally, there are chapters analysing and defending Wittgenstein's provocative views on Hilbert's Formalism and the quest for consistency proofs and on Gödel's incompleteness theorems"-- |
| Beschreibung: | xiii, 238 Seiten Illustrationen 24 cm |
| ISBN: | 9781844658626 |
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Datensatz im Suchindex
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| adam_text | Contents Preface List of Abbreviations PARTI Background viii xii 1 1 Foundations of Mathematics 3 2 Logicism 2.1 Frege’s Logicism 9 2.2 The Class Paradox and Russell’s Theory of Types 12 2.3 Tractatus Logico-Philosophicus: Logicism Without Classes 13 9 3 Wittgenstein’s Critique of Logicism 3.1 Can Equality of Number Be Defined in Terms of One-to-One Correlation? 15 3.2 Frege’s (and Russell’s) Definition of Numbers as Equivalence Classes Is Not Constructive: It Doesn’t Provide a Method of Identifying Numbers 21 3.3 Platonism 22 3.4 Russell’s Reconstructions of False Equations Are Not Contradictions 26 3.5 Frege’s and Russell’s Formalisation of Sums as Logical Truths Cannot Be Foundational as It Presupposes Arithmetic 27 3.6 Even If We Assumed (for Argument’s Sake) That All Arithmetic Could Be Reproduced in Russell’s Logical Calculus, That Would Not Make the Latter a Foundation of Arithmetic 31 15
vi Contents 4 The Development of Wittgenstein’s Philosophy of Mathematics: Tractatus to The Big Typescript 4.1 Tractatus Logico-Philosophicus 35 4.2 Philosophical Remarks (MSS 105-8:1929-30) to The Big Typescript (TS 213: 1933) 36 PART II Wittgenstein’s Mature Philosophy of Mathematics (1937-44) 35 55 5 The Two Strands in Wittgenstein’s Later Philosophy of Mathematics 57 6 Mathematics as Grammar 59 7 Rule-Following 7.1 Rule-Following and Community 88 78 8 Conventionalism 8.1 Quine s Circularity Objection 95 8.2 Dummett’s Objection That Conventionalism Cannot Explain Logical Inferences 101 8.3 Crispin Wright’s Infinite Regress Objection 103 8.4 The Objection to ‘Moderate Conventionalism’ From Scepticism About Rule-Following 105 8.5 The Objection From the Impossibility of a Radically Different Logic or Mathematics 109 8.6 Conclusion 124 93 9 Empirical Propositions Hardened Into Rules Synthetic A Priori 134 10 Mathematical Proof 10.1 What Is a Mathematical ProofI 142 (a) Proof That a0 = 1 150 (b) Skolem’s Inductive Proof of the Associative Law of Addition 150 (c) Cantor’s Diagonal Proof 151 (d) Euclid’s Construction of a Regular Pentagon 158 126 141
Contents vii (e) Euclid’s Proof That There Is No Greatest Prime Number 160 (f) Proof (Calculation) in Elementary Arithmetic 166 Proof and experiment 169 10.2 What Is the Relation Between a Mathematical Proposition and Its ProofÌ 171 10.3 What Is the Relation Between a Mathematical Proposition’s Proof and Its Application? 181 11 Inconsistency 12 Wittgenstein’s Remarks on Gödel’s First Incompleteness Theorem 12.1 Wittgenstein Discusses Gödel’s Informal Sketch of His Proof 206 12.2 ‘A Proposition That Says About Itself That It Is Not Provable in P’ 207 12.3 The Difference Between the Gödel Sentence and the Liar Paradox 209 12.4 Truth and Provability 210 12.5 Gödel’s Kind of Proof 213 12.6 Wittgenstein’s Pirst Objection: A Useless Paradox 216 12.7 Wittgenstein’s Second Objection: A Proof Based on Indeterminate Meaning 218 13 Concluding Remarks: Wittgenstein and Platonism Bibliography Index 189 203 220 226 234
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| adam_txt |
Contents Preface List of Abbreviations PARTI Background viii xii 1 1 Foundations of Mathematics 3 2 Logicism 2.1 Frege’s Logicism 9 2.2 The Class Paradox and Russell’s Theory of Types 12 2.3 Tractatus Logico-Philosophicus: Logicism Without Classes 13 9 3 Wittgenstein’s Critique of Logicism 3.1 Can Equality of Number Be Defined in Terms of One-to-One Correlation? 15 3.2 Frege’s (and Russell’s) Definition of Numbers as Equivalence Classes Is Not Constructive: It Doesn’t Provide a Method of Identifying Numbers 21 3.3 Platonism 22 3.4 Russell’s Reconstructions of False Equations Are Not Contradictions 26 3.5 Frege’s and Russell’s Formalisation of Sums as Logical Truths Cannot Be Foundational as It Presupposes Arithmetic 27 3.6 Even If We Assumed (for Argument’s Sake) That All Arithmetic Could Be Reproduced in Russell’s Logical Calculus, That Would Not Make the Latter a Foundation of Arithmetic 31 15
vi Contents 4 The Development of Wittgenstein’s Philosophy of Mathematics: Tractatus to The Big Typescript 4.1 Tractatus Logico-Philosophicus 35 4.2 Philosophical Remarks (MSS 105-8:1929-30) to The Big Typescript (TS 213: 1933) 36 PART II Wittgenstein’s Mature Philosophy of Mathematics (1937-44) 35 55 5 The Two Strands in Wittgenstein’s Later Philosophy of Mathematics 57 6 Mathematics as Grammar 59 7 Rule-Following 7.1 Rule-Following and Community 88 78 8 Conventionalism 8.1 Quine's Circularity Objection 95 8.2 Dummett’s Objection That Conventionalism Cannot Explain Logical Inferences 101 8.3 Crispin Wright’s Infinite Regress Objection 103 8.4 The Objection to ‘Moderate Conventionalism’ From Scepticism About Rule-Following 105 8.5 The Objection From the Impossibility of a Radically Different Logic or Mathematics 109 8.6 Conclusion 124 93 9 Empirical Propositions Hardened Into Rules Synthetic A Priori 134 10 Mathematical Proof 10.1 What Is a Mathematical ProofI 142 (a) Proof That a0 = 1 150 (b) Skolem’s Inductive Proof of the Associative Law of Addition 150 (c) Cantor’s Diagonal Proof 151 (d) Euclid’s Construction of a Regular Pentagon 158 126 141
Contents vii (e) Euclid’s Proof That There Is No Greatest Prime Number 160 (f) Proof (Calculation) in Elementary Arithmetic 166 Proof and experiment 169 10.2 What Is the Relation Between a Mathematical Proposition and Its ProofÌ 171 10.3 What Is the Relation Between a Mathematical Proposition’s Proof and Its Application? 181 11 Inconsistency 12 Wittgenstein’s Remarks on Gödel’s First Incompleteness Theorem 12.1 Wittgenstein Discusses Gödel’s Informal Sketch of His Proof 206 12.2 ‘A Proposition That Says About Itself That It Is Not Provable in P’ 207 12.3 The Difference Between the Gödel Sentence and the Liar Paradox 209 12.4 Truth and Provability 210 12.5 Gödel’s Kind of Proof 213 12.6 Wittgenstein’s Pirst Objection: A Useless Paradox 216 12.7 Wittgenstein’s Second Objection: A Proof Based on Indeterminate Meaning 218 13 Concluding Remarks: Wittgenstein and Platonism Bibliography Index 189 203 220 226 234 |
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| spelling | Schroeder, Severin 1964- Verfasser (DE-588)135897580 aut Wittgenstein on mathematics Severin Schroeder New York ; London Routledge 2021 xiii, 238 Seiten Illustrationen 24 cm txt rdacontent n rdamedia nc rdacarrier Wittgenstein's thought and legacy "This book offers a detailed account and discussion of Ludwig Wittgenstein's philosophy of mathematics. In Part I, the stage is set with a brief presentation of Frege's logicist attempt to provide arithmetic with a foundation and Wittgenstein's criticisms of it, followed by sketches of Wittgenstein's early views of mathematics, in the Tractatus and in the early 1930s. Then (in Part II), Wittgenstein's mature philosophy of mathematics (1937-44) is carefully presented and examined. Schroeder explains that it is based on two key ideas: the calculus view and the grammar view. On the one hand, mathematics is seen as a human activity - calculation - rather than a theory. On the other hand, the results of mathematical calculations serve as grammatical norms. The following chapters (on mathematics as grammar; rule-following; conventionalism; the empirical basis of mathematics; the role of proof) explore the tension between those two key ideas and suggest a way in which it can be resolved. Finally, there are chapters analysing and defending Wittgenstein's provocative views on Hilbert's Formalism and the quest for consistency proofs and on Gödel's incompleteness theorems"-- Wittgenstein, Ludwig 1889-1951 (DE-588)118634313 gnd rswk-swf Mathematische Logik (DE-588)4037951-6 gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf Mathematics / Philosophy Wittgenstein, Ludwig 1889-1951 (DE-588)118634313 p Mathematik (DE-588)4037944-9 s Mathematische Logik (DE-588)4037951-6 s DE-604 Erscheint auch als Online-Ausgabe 9781003056904 Digitalisierung BSB München - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032687702&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
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| title | Wittgenstein on mathematics |
| title_auth | Wittgenstein on mathematics |
| title_exact_search | Wittgenstein on mathematics |
| title_exact_search_txtP | Wittgenstein on mathematics |
| title_full | Wittgenstein on mathematics Severin Schroeder |
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