The Search for Mathematical Roots, 1870-1940

CHAPTER 1 Explanations

1.1 Sallies 3

1.2 Scope and limits of the book 3

1.2.1 An outline history 3

1.2.2 Mathematical aspects 4

1.2.3 Historical presentation 6

1.2.4 Other logics, mathematics and philosophies 7

1.3 Citations, terminology and notations

1.3.1 References and the bibliography 9

1.3.2 Translations, quotations and notations 10

1.4 Permissions and acknowledgements 11

CHAPTER 2 Preludes: Algebraic Logic and Mathematical Analysis up to 1870

2.1 Plan of the chapter 14

2.2 `Logique' and algebras in French mathematics 14

2.2.1 The `logique' and clarity of `ideologie' 14

2.2.2 Lagrange's algebraic philosophy 15

2.2.3 The many senses of `analysis' 17

2.2.4 Two Lagrangian algebras: functional equations

and differential operators 17

2.2.5 Autonomy for the new algebras 19

2.3 Some English algebraists and logicians 20

2.3.1 A Cambridge revival: the `Analytical Society, Lacroix, and the professing of algebras 20

2.3.2 The advocacy of algebras by Babbage, Herschel and Peacock 20

2.3.3 An Oxford movement: Whately and the professing of logic 22

2.4 A London pioneer: De Morgan on algebras and logic 25

2.4.1 Summary of his life 25

2.4.2 De Morgan's philosophies of algebra 25

2.4.3 De Morgan's logical career 26

2.4.4 De Morgan's contributions to the foundations of logic 27

2.4.5 Beyond the syllogism 29

2.4.6 Contretemps over `the quantification of the predicate' 30

2.4.7 The logic of two place relations, 1860 32

2.4.8 Analogies between logic and mathematics 35

2.4.9 De Morgan's theory of collections 36

2.5 A Lincoln outsider: Boole on logic as applied mathematics 37

2.5.1 Summary of his career 37

2.5.2 Boole's `general method in analysis' 1844 39

2.5.3 The mathematical analysis of logic, 1847. `elective symbols' and laws 40

2.5.4 `Nothing' and the `Universe' 42

2.5.5 Propositions, expansion theorems, and solutions 43

2.5.6 The laws of thought, 1854: modified principles and extended methods 46

2.5.7 Boole's new theory of propositions 49

2.5.8 The character of Boole's system 50

2.5.9 Boole's search for mathematical roots

53

2.6 The semi-followers of Boole 54

2.6.1 Some initial reactions to Boole's theory 54

2.6.2 The reformulation by Jevons 56

2.6.3 Jevons versus Boole 59

2.6.4 Followers of Boole and/or Jevons 60

2.7 Cauchy, Weierstrass and the rise of mathematical analysis 63

2.7.1 Different traditions in the calculus 63

2.7.2 Cauchy and the Ecole Polytechnique 64

2.7.3 The gradual adoption and adaptation of Cauchy's new tradition 67

2.7.4 The refinements of Weierstrass and his followers 68

2.8 Judgement and supplement 70

2.8.1 Mathematical analysis versus algebraic logic 70

2.8.2 The places of Kant and Bolzano 71

CHAPTER 3 Cantor: Mathematics as Mengenlehre

3.1 Prefaces 75

3.1.1 Plan of the chapter 75

3.1.2 Cantor's career 75

3.2 The launching of the Mengenlehre, 1870-1883 79

3.2.1 Riemann's thesis: the realm of discontinuous functions 79

3.2.2 Heine on trigonometric series and the real line, 1870-1872 81

3.2.3 Cantor's extension of Heine's findings, 1870-1872 83

3.2.4 Dedekind on irrational numbers, 1872 85

3.2.5 Cantor on line and plane, 1874-1877 88

3.2.6 Infinite numbers and the topology of linear sets, 1878-1883 89

3.2.7 The Grundlagen, 1883: the construction of number-classes 92

3.2.8 The Grundlagen: the definition of continuity 95

3.2.9 The successor to the Grundlagen, 1884 96

3.3 Cantor's Acta mathematica phase, 1883-1885 97

3.3.1 Mittag-Lefler and the French translations, 1883 97

3.3.2 Unpublished and published 'communications' 1884-1885 98

3.3.3 Order-types and partial derivatives in the `communications' 100

3.3.4 Commentators on Cantor, 1883-1885 102

3.4 The extension of the Mengenlehre, 1886-1897 103

3.4.1 Dedekind's developing set theory, 1888 103

3.4.2 Dedekind's chains of integers 105

3.4.3 Dedekind's philosophy of arithmetic 107

3.4.4 Cantor's philosophy of the infinite, 1886-1888 109

3.4.5 Cantor's new definitions of numbers 110

3.4.6 Cardinal exponentiation: Cantor's diagonal argument, 1891 110

3.4.7 Transfinite cardinal arithmetic and simply ordered sets, 1895 112

3.4.8 Transfinite ordinal arithmetic and well-ordered sets, 1897 114

3.5 Open and hidden questions in Cantor's Mengenlehre 114

3.5.1 Well-ordering and the axioms of choice 114

3.5.2 What was Cantor's `Cantor's continuum problem'? 116

3.5.3 "Paradoxes" and the absolute infinite 117

3.6 Cantor's philosophy of mathematics 119

3.6.1 A mixed position 119

3.6.2 (No) logic and metamathematics 120

3.6.3 The supposed impossibility of infinitesimals 121

3.6.4 A contrast with Kronecker 122

3.7 Concluding comments: the character of Cantor's achievements 124

CHAPTER 4 Parallel Processes in Set Theory, Logics and Axiomatics, 1870s-1900s

4.1 Plans for the chapter 126

4.2 The splitting and selling of Cantor's Mengenlehre 126

4.2.1 National and international support 126

4.2.2 French initiatives, especially from Borel 127

4.2.3 Couturat outlining the infinite, 1896 129

4.2.4 German initiatives from Mein 130

4.2.5 German proofs of the Schroder-Bernstein theorem 132

4.2.6 Publicity from Hilbert, 1900 134

4.2.7 Integral equations and functional analysis 135

4.2.8 Kempe on `mathematical form' 137

4.2.9 Kempe-who? 139

4.3 American algebraic logic: Peirce and his followers 140

4.3.1 Peirce, published and unpublished 141

4.3.2 Influences on Peirre's logic: father's algebras 142

4.3.3 Peirce's first phase: Boolean logic and the categories, 1867-1868 144

4.3.4 Peirce's virtuoso theory of relatives, 1870 145

4.3.5 Peirce's second phase, 1880: the propositional calculus 147

4.3.6 Peirre's second phase, 1881: finite and infinite 149

4.3.7 Peirce's students, 1883: duality, and 'Quantifying' a proposition 150

4.3.8 Peirre on 'icons' and the order of `quantifiers; 1885 153

--- 4.3.9 The Peirceans in the 1890s 154

4.4 German algebraic logic: from the Grassmanns to Schr6der 156

4.4.1 The Grassmanns on duality 156

4.4.2 Schroder's Grassmannian phase 159

4.4.3 Schroder's Peirrean `lectures' on logic 161

4.4.4 Schrrider's first volume, 1890 161

4.4.5 Part of the second volume, 1891 167

4.4.6 Schroder's third volume, 1895: the `logic of relatives' 170

4.4.7 Peirce on and against Schroder in The monist, 1896-1897 172

4.4.8 Schroder on Cantorian themes, 1898 174

4.4.9 The reception and publication of Schroder in the 1900s 175

4.5 Frege: arithmetic as logic 177

4.5.1 Frege and Frege' 177

4.5.2 The `concept-script' calculus of Frege's `pure thought; 1879 179

4.5.3 Frege's arguments for logicising arithmetic, 1884 183

4.5.4 Keny's conception of Fregean concepts in the mid 1880s 187

4.5.5 Important new distinctions in the early 1890s 187

4.5.6 The `fundamental laws' of logicised arithmetic, 1893 191

4.5.7 Frege's reactions to others in the later 1890s 194

4.5.8 More `fundamental laws' of arithmetic, 1903 195

4.5.9 Frege, Korselt and Thomae on the foundations of arithmetic 197

4.6 Husserl: logic as phenomenology 199

4.6.1 A follower of Weierstrass and Cantor 199

4.6.2 The phenomenological `philosophy of arithmetic; 1891 201

4.6.3 Reviews by Frege and others 203

4.6.4 Husserl's `logical investigations; 1900-1901 204

4.6.5 Husserl's early talks in Gottingen, 1901 206

4.7 Hilbert: early proof and model theory, 1899-1905 207

4.7.1 Hilbert's growing concern with axiomatics 207

4.7.2 Hilbert's diferent axiom systems for Euclidean geometry, 1899-1902 208

4.7.3 From German completeness to American model theory 209

4.7.4 Frege, Hilbert and Korselt on the foundations of geometries 212

4.7.5 Hilbert's logic and proof theory, 1904-1905 213

4.7.6 Zermelo's logic and set theory, 1904-1909 216

CHAPTER 5 Peano: the Formulary of Mathematics

5.1 Prefaces 219

5.1.1 Plan of the chapter 219

5.1.2 Peano's career 219

5.2 Formalising mathematical analysis 221

5.2.1 Improving Genocchi, 1884 221

5.2.2 Developing Grassmann's `geometrical calculus; 1888 223

5.2.3 The logistic of arithmetic, 1889 225

5.2.4 The logistic of geometry, 1889 229

5.2.5 The logistic of analysis, 1890 230

5.2.6 Bettazzi on magnitudes, 1890 232

5.3 The Rivista: Peano and his school, 1890-1895 232

5.3.1 The `society of mathematicians' 232

5.3.2 `Mathematical logic, 1891 234

5.3.3 Developing arithmetic, 1891 235

5.3.4 Infinitesimals and limits, 1892-1895 236

5.3.5 Notations and their range, 1894 237

5.3.6 Peano on definition by equivalence classes 239

5.3.7 Burali-Forti's textbook, 1894 240

5.3.8 Burali-Forti's research, 1896-1897 241

5.4 The Formulaire and the Rivista, 1895-1900 242

5.4.1 The first edition of the Formulaire, 1895 242

5.4.2 Towards the second edition of the Formulaire, 1897 244

5.4.3 Peano on the eliminability of `the' 246

5.4.4 Frege versus Peano on logic and definitions 247

5.4.5 Schroder's steamships versus Peano's sailing boats 249

5.4.6 New presentations of arithmetic, 1898 251

5.4.7 - Padoa on classhoody 1899 253

5.4.8 Peano's new logical summary, 1900 254

5.5 Peanists in Paris, August 1900 255

5.5.1 An Italian Friday morning 255

5.5.2 Peano on definitions 256

5.5.3 Burali-Forti on definitions of numbers 257

5.5.4 Padoa on definability and independence 259

5.5.5 Pieri on the logic of geometry 261

5.6 Concluding comments: the character of Peano's achievements 262

5.6.1 Peano's little dictionary, 1901 262

5.6.2 Partly grasped opportunities 264

5.6.3 Logic without relations 266

CHAPTER 6 Russell's Way In: From Certainty to Paradoxes, 1895-1903

6.1 Prefaces 268

6.1.1 Plans for two chapters 268

6.1.2 Principal sources 269

6.1.3 Russell as a Cambridge undergraduate, 1891-1894 271

6.1.4 Cambridge philosophy in the 1890s 273

6.2 Three philosophical phases in the foundation of mathematics,

1895-1899 274

6.2.1 Russell's idealist axiomatic geometries 275

6.2.2 The importance of axioms and relations 276

6.2.3 A pair of pas de deux with Paris: Couturat and Poincare on geometries 278

6.2.4 The emergence of "itehead, 1898 280

6.2.5 The impact of G. E. Moore, 1899 282

6.2.6 Three attempted books, 1898-1899 283

6.2.7 Russell's progress with Cantor's Mengenlehre, 1896-1899 285

6.3 From neo-Hegelianism towards `Principles', 1899-1901 286

6.3.1 Changing relations 286

6.3.2 Space and time, absolutely 288

6.3.3 `Principles of Mathematics, 1899-1900 288

6.4 The first impact of Peano 290

6.4.1 The Paris Congress of Philosophy, August 1900: Schroder versus Peano on `the' 290

6.4.2 Annotating and popularising in the autumn 291

6.4.3 Dating the origins of Russell's logicism 292

6.4.4 Drafting the logic of relations, October 1900 296

6.4.5 Part 3 of The principles, November 1900: quantity and magnitude 298

6.4.6 Part 4, November 1900: order and ordinals 299

6.4.7 Part 5, November 1900: the transfinite and the continuous 300

6.4.8 Part 6, December 1900: geometries in space 301

6.4.9 Whitehead on `the algebra of symbolic logic, 1900 302

6.5 Convoluting towards logicism, 1900-1901 303

6.5.1 Logicism as generalised metageometry, January 1901 303

6.5.2 The first paper for Peano, February 1901: relations and numbers 305

6.5.3 Cardinal arithmetic with "itehead and Russell, June 1901 307

6.5.4 The second paper for Peano, March August 1901: set theory with series 308

6.6 From `fallacy' to `contradiction', 1900-1901 310

6.6.1 Russell on Cantor's `fallacy; November 1900 310

6.6.2 Russell's switch to a `contradiction' 311

6.6.3 Other paradoxes: three too large numbers 312

6.6.4 Three passions and three calamities, 1901-1902 314

6.7 Refining logicism, 1901-1902 315

6.7.1 Attempting Part 1 of The principles, May 1901 315

6.7.2 Part 2, June 1901: cardinals and classes 316

6.7.3 Part 1 again, April-May 1902: the implicational logicism 316

6.7.4 Part 1: discussing the indefinables 318

6.7.5 Part 7, June 1902: dynamics without statics; and within logic? 322

6.7.6 Sort-of finishing the book 323

6.7.7 The first impact of Frege, 1902 323

6.7.8 AppendixA on Frege 326

6.7.9 Appendix B: Russell's first attempt to solve the paradoxes 327

6.8 The roots of pure mathematics? Publishing The principles at last, 1903 328

6.8.1 Appearance and appraisal 328

6.8.2 A gradual collaboration with Whitehead 331

CHAPTER 7 Russell and Whitehead Seek the Principia Mathematica, 1903-1913

7.1 Plan of the chapter 333

7.2 Paradoxes and axioms in set theory, 1903-1906 333

7.2.1 Uniting the paradoxes of sets and numbers 333

7.2.2 New paradoxes, mostly of naming 334

7.2.3 The paradox that got away: heterology 336

7.2.4 Russell as cataloguer of the paradoxes 337

7.2.5 Controversies over axioms of choice, 1904 339

7.2.6 Uncovering Russell's 'multiplicative axiom, 1904 340

7.2.7 Keyser versus Russell over infinite classes, 1903-1905 342

7.3 The perplexities of denoting, 1903-1906 342

7.3.1 First attempts at a general system, 1903-1905 342

7.3.2 Propositional functions, reducible and identical 344

7.3.3 The mathematical importance of definite denoting functions 346

7.3.4 `On denoting' and the complex, 1905 348

7.3.5 Denoting, quantification and the mysteries of existence 350

7.3.6 Russell versus MacColl on the possible, 1904-1908 351

7.4 From mathematical induction to logical substitution, 1905-1907 354

7.4.1 Couturat's Russellian principles 354

7.4.2 A second pas de deux with Paris: Boutroux and Poincare on logicism 355

7.4.3 Poincare on the status of mathematical induction 356

7.4.4 Russell's position paper, 1905 357

7.4.5 Poincare and Russell on the vicious circle principle, 1906 358

7.4.6 The rise of the substitutional theory, 1905-1906 360

7.4.7 The fall of the substitutional theory, 1906-1907 362

7.4.8 Russell's substitutional propositional calculus 364

7.5 Reactions to mathematical logic and logicism, 1904-1907 366

7.5.1 The International Congress of Philosophy, 1904 366

7.5.2 German philosophers and mathematicians, especially Schonflies 368

7.5.3 Activities among the Peanists 370

7.5.4 American philosophers: Royce and Dewey 371

7.5.5 American mathematicians on classes 373

7.5.6 Huntington on logic and orders 375

7.5.7 Judgements fiom Shearman 376

7.6 Whitehead's role and activities, 1905-1907 377

7.6.1 Whitehead's construal of the `material world' 377

7.6.2 The axioms of geometries 379

7.6.3 Whitehead's lecture course, 1906-1907 379

7.7 The sad compromise: logic in tiers 380

7.7.1 Rehabilitating propositional functions, 1906-1907 380

7.7.2 Two reflective pieces in 1907 382

7.7.3 Russell's outline of `mathematical logic, 1908 383

7.8 The forming of Principia mathematica 384

7.8.1 Completing and funding Principia mathematica 384

7.8.2 The Organisation of Principia mathematica 386

7.8.3 The propositional calculus, and logicism 388

7.8.4 The predicate calculus, and descriptions 391

7.8.5 Classes and relations, relative to propositional functions 392

7.8.6 The multiplicative axiom: some uses and avoidance 395

7.9 Types and the treatment of mathematics in Principia mathematica 396

7.9.1 7~pes in orders 396

7.9.2 Reducing the edifice 397

7.9.3 Individuals, their nature and number 399

7.9.4 Cardinals and their finite arithmetic 401

7.9.5 The generalised ordinals 403

7.9.6 The ordinals and the alephs 404

7.9.7 The odd small ordinals 406

7.9.8 Series and continuity 406

7.9.9 Quantity with ratios 408

CHAPTER 8 The Influence and Place of Logicism, 1910-1930

8.1 Plans for two chapters 411

8.2 Whitehead's and Russell's transitions from logic to philosophy, 1910-1916 412

8.2.1 The educational concerns of "itehead, 1910-1916 412

8.2.2 Whitehead on the principles of geometry in the 1910s 413

8.2.3 British reviews of Principia mathematica 415

8.2.4 Russell and Peano on logic, 1911-1913 416

8.2.5 Russell's initial problems with epistemology, 1911-1912 417

8.2.6 Russell's first interactions with Wittgenstein, 1911-1913 418

8.2.7 Russell's confrontation with Wiener, 1913 419

8.3 Logicism and epistemology in America and with Russell, 1914-1921 421

8.3.1 Russell on logic and epistemology at Harvard, 1914 421

8.3.2 Two long American reviews 424

8.3.3 Reactions from Royce students: Sheffer and Lewis 424

8.3.4 Reactions to logicism in New York 428

8.3.5 OtherAmerican estimations 429

8.3.6 Russell's `logical atomism' and psychology, 1917-1921 430

8.3.7 Russell's `introduction'to logicism, 1918-1919 432

8.4 Revising logic and logicism at Cambridge, 1917-1925 434

8.4.1 New Cambridge authors, 1917-1921 434

8.4.2 Wittgenstein's `Abhandlung' and Tractatus, 1921-1922 436

8.4.3 The limitations of Wittgenstein's logic 437

8.4.4 Towards extensional logicism: Russell's revision of Principia mathematica, 1923-1924 440

8.4.5 Ramsey's entry into logic and philosophy, 1920-1923 443

8.4.6 Ramsey's recasting of the theory of types, 1926 444

8.4.7 Ramsey on identity and comprehensive extensionality 446

8.5 Logicism and epistemology in Britain and America, 1921-1930 448

8.5.1 Johnson on logic, 1921-1924 448

8.5.2 Other Cambridge authors, 1923-1929 450

8.5.3 American reactions to logicism in mid decade 452

8.5.4 Groping towards metalogic 454

8.5.5 Reactions in and around Columbia 456

8.6 Peripherals: Italy and France 458

8.6.1 The occasional Italian survey 458

8.6.2 New French attitudes in the Revue 459

8.6.3 Commentaries in French, 1918-1930 461

8.7 German-speaking reactions to logicism, 1910-1928 463

8.7.1 (Neo-)Kantians in the 1910s 463

8.7.2 Phenomenologists in the 1910s 467

8.7.3 Frege's positive and then negative thoughts 468

8.7.4 Hilbert's definitive `metamathematics; 1917-1930 470

8.7.5 Orders of logic and models of set theory: Lowenheim and Skolem, 1915-1923 475

8.7.6 Set theory and Mengenlehre in various forms 476

8.7.7 Intuitionistic set theory and logic: Brouwer and Weyl, 1910-1928 480

8.7.8 (Neo-)Kantians in the 1920s 484

8.7.9 Phenomenologists in the 1920s 487

8.8 The rise of Poland in the 1920s: the Lvnv-Warsaw school 489

8.8.1 From Lv6v to Warsaw: students of Twardowski 489

8.8.2 Logics with Lukasiewicz and Tarski 490

8.8.3 Russell's paradox and Lesniewski's three systems 492

8.8.4 Pole apart: Chwistek's `semantic' logicism at Cracov 495

8.9 The rise of Austria in the 1920s: the Schlick circle 497

8.9.1 Formation and influence 497

8.9.2 The impact of Russell, especially upon Camap 499

8.9.3 `Logicism ' in Camap's Abriss, 1929 500

8.9.4 Epistemology in Camap's Aufbau, 1928 502

8.9.5 Intuitionism and proof theory: Brouwer and Godel, 1928-1930 504

CHAPTER 9 Postludes: Mathematical Logic and Logicism in the 1930s

9.1 Plan of the chapter 506

9.2 Godel's incompletability theorem and its immediate reception 507

9.2.1 The consolidation of Schlick's `Vienna' Circle 507

9.2.2 News from G6del: the Konigsberg lectures, September 1930 508

9.2.3 G6del's incompletability theorem, 1931 509

9.2.4 Effects and reviews of G6del's theorem 511

9.2.5 Zermelo against Godeb the Bad Elster lectures, September 1931 512

9.3 Logic(ism) and epistemology in and around Vienna 513

9.3.1 Carnap for `metalogic' and against metaphysics 513

9.3.2 Carnap's transformed metalogic: the `logical syntax of language; 1934 515

9.3.3 Carnap on incompleteness and truth in mathematical theories, 1934-1935 517

9.3.4 Dubislav on definitions and the competing philosophies of mathematics 519

9.3.5 Behmann's new diagnosis of the paradoxes 520

9.3.6 Kaufmann and Waismann on the philosophy of mathematics 521

9.4 Logic(ism) in the U.S.A. 523

9.4.1 Mainly Eaton and Lewis 523

9.4.2 Mainly Weiss and Langer 525

9.4.3 Whitehead's new attempt to ground logicism, 1934 527

9.4.4 The debut of Quine 529

9.4.5 Two journals and an encyclopaedia, 1934-1938 531

9.4.6 Carnap's acceptance of the autonomy of semantics 533

9.5 The battle of Britain 535

9.5.1 The campaign of Stebbing for Russell and Carnap 535

9.5.2 Commentary from Black and Ayer 538

9.5.3 Mathematicians-and biologists 539

9.5.4 Retiring into philosophy: Russell's return, 1936-1937 542

9.6 European, mostly northern 543

9.6.1 Dingler and Burkamp again 543

9.6.2 German proof theory after Godel 544

9.6.3 Scholz's little circle at Munster 546

9.6.4 Historical studies, especially by Jorgensen 547

9.6.5 History philosophy, especially Cavailles 548

9.6.6 Other Francophone figures, especially Herbrand 549

9.6.7 Polish logicians, especially Tarski 551

9.6.8 Southern Europe and its former colonies 553

CHAPTER 10 The Fate of the Search

10.1 Influences on Russell, negative and positive 556

10.1.1 Symbolic logics: living together and living apart 556

10.1.2 The timing and origins of Russell's logicism 557

10.1.3 (Why) was Frege (so) little read in his lifetime? 558

10.2 The content and impact of logicism 559

10.2.1 Russell's obsession with reductionist logic and epistemology 560

10.2.2 The logic and its metalogic 562

10.2.3 The fate of logicism 563

10.2.4 Educational aspects, especially Piaget 566

10.2.5 The role of the U.S.A.: judgements in the Schi1pp series 567

10.3 The panoply of foundations 569

10.4 Sallies 573

CHAPTER 11 Transcription of Manuscripts

11.1 Couturat to Russell, 18 December 1904 574

11.2 Veblen to Russell, 13 May 1906 577

11.3 Russell to Hawtrey, 22 January 1907 (or 1909?) 579

11.4 Jourdain's notes on Wittgenstein's first views on Russell's paradox, April 1909 580

11.5 The application of Whitehead and Russell to the Royal Society, late 1909 581

11.6 Whitehead to Russell, 19 January 1911 584

11.7 Oliver Strachey to Russell, 4 January 1912 585

11.8 Quine and Russell, June-July 1935 586

11.8.1 Russell to Quine, 6 June 1935 587

11.8.2 Quine to Russell, 4 July 1935 588

11.9 Russell to Henkin, 1 April 1963 592

BIBLIOGRAPHY 594

INDEX 671