Reading Frege's Grundgesetze.

Format:

Book

Author/Creator:

Heck, Richard Kimberly

Language:

English

Subjects (All):

Frege, Gottlob, 1848-1925. Grundgesetze der Arithmetik.

Frege, Gottlob.

Physical Description:

xviii, 296 pages ; 25 cm

Edition:

First edition.

Place of Publication:

Oxford : Oxford University Press, 2012.

Summary:

Gottlob Frege's Grundgesetze der Arithmetik, or Basic Laws of Arithmetic, was intended to be his magnum opus, the book in which he would finally establish his logicist philosophy of arithmetic. But because of the disaster of Russell's Paradox, which undermined Frege's proofs, the more mathematical parts of the book have rarely been read. Richard G. Heck, Jr, aims to change that, and establish it as a neglected masterpiece that must be placed at the center of Frege's philosophy. Part I of Reading Frege's Grundgesetze develops an interpretation of the philosophy of logic that informs Grundgesetze, paying especially close attention to the difficult sections of Frege's book in which he discusses his notorious 'Basic Law V' and attempts to secure its status as a law of logic. Part II examines the mathematical basis of Frege's logicism, explaining and exploring Frege's formal arguments. Heck argues that Frege himself knew that his proofs could be reconstructed so as to avoid Russell's Paradox, and presents Frege's arguments in a way that makes them available to a wide audience. He shows, by example, that careful attention to the structure of Frege's arguments, to what he proved, to how he proved it, and even to what he tried to prove but could not, has much to teach us about Frege's philosophy. Book jacket.

Contents:

1 Introduction 1

1.1 The Genesis of Grundgesetze 1

1.2 The Fall and Rise of Grundgesetze 4

1.3 Frege on the Reals 13

1.4 Logic in Grundgesetze 15

1.5 The Centrality of Grundgesetze 21

1.6 Goals for the Book 22

I The Logic Behind Frege's Logicism 25

2 Frege and Semantics 27

2.1 Frege and the Justification of Logical Laws 27

2.2 Formalism and the Significance of Interpretation 38

2.3 The Regress Argument 45

3 Grundgesetze I §§29-32 (I) 51

3.1 The Character of the Induction 53

3.2 Frege on Free Variables 59

3.3 The Induction Step 64

3.4 The Basis Case: The Logical Expressions 68

3.5 The Basis Case: The Smooth Breathing (I) 72

4 Grundgesetze I §10 82

4.1 Philosophical Aspects 83

4.2 Technical Aspects 89

4.3 Is Caesar a Value-Range? 91

4.4 The Second Problem, and Frege's Response 96

4.5 Are the Arguments in §10 Syntactic or Semantic? 104

5 Grundgesetze I §§29-32 (II) 118

5.1 The Basis Case: The Smooth Breathing (II) 118

5.2 Linnebo's Alternative 121

5.3 Final Remarks on §§29-32 126

5.4 The Cost of Frege's Response to the Caesar Problem 129

II The Mathematics Behind Frege's Logicism 135

6 The Development of Arithmetic 137

6.1 Frege's Use of Basic Law V 137

6.2 Frege's Formulations of HP 140

6.3 Frege's Proofs of Axioms for Arithmetic 144

6.4 The Concept of Natural Number 151

6.5 The Basic Facts about the Ancestral 152

6.6 An Elegant Proof that Every Number has a Successor 155

6.6.1 The Strategy of the Proof 155

6.6.2 Theorem 154 157

6.6.3 An Important Lemma 159

6.6.4 Another Important Lemma 161

6.6.5 Completion of the Proof 162

6.7 Frege's Proofs of the Existence of Successor 163

6.8 HP in Grundgesetze 173

7 Definition by Recursion 179

7.1 Frege's Proof of Theorem 263 180

7.2 Frege's Use of Ordered Pairs 182

7.3 Definition by Recursion 187

7.4 The Definition of Relations by Recursion 192

7.5 Functionality and the n-Ancestral 194

7.6 Theorem 207 196

7.7 The Adequacy of Frege's Definition of the Ancestral 198

7.8 Theorem 263 in the Context of Frege's Development of Arithmetic 201

8 Frege on Finitude 207

8.1 Frege's Characterization of Finitude 208

8.2 Frege's Proof of Theorem 321 211

8.3 The Proof of Theorem 288 215

8.4 The Significance of Theorems 327 and 348 220

8.5 Finitude, the Least Number Principle, and Well-ordering 224

9 The Finite and the Infinite 228

9.1 Remarks on Volume II 228

9.2 Chapter Mu: There is No Cardinal Between the Finite Cardinals and Endlos 229

9.2.1 A Generalized Least Number Principle 229

9.2.2 Remarks on Frege's Proof of Theorem 359 232

9.2.3 The Proof of Theorem 428 234

9.2.4 The Significance of Theorem 428 245

9.3 Chapter Nu 247

10 The Definition of Addition 249

10.1 A Strengthened Version of HP 253

10.2 The Proof of Theorem 469 254

10.2.1 The Proof of Theorem 468 255

10.2.2 The Proof of Theorem 463 256

10.3 Remarks on Theorem 469 257

10.4 Cardinal Multiplication 261

11 Further unto the Infinite 263

11.1 The Theorems in Chapter Omicron 263

11.2 The Infinite and the Dedekind Infinite 265

11.3 Frege and the Axiom of Choice 268

11.4 Formalization and Frege's Conception of Logical Truth 272

12 Appendices 275

12.1 Outline of a Fregean Theory of Truth 275

12.2 Frege's Definitions 279

12.3 Theorems Concerning the Ancestral 280

12.4 Trees of Dependencies 282.

ISBN:

9780199233700

0199233705

OCLC:

829063137