内容

目次

CHAPTER 1 Explanations 1.1 Sallies 3 1.2 Scope and limits of the book3 1.2.1 An outline history 3 1.2.2 Mathematical aspects 4 1.2.3 Historicalpresentation 6 1.2.4 Other logics, mathematics and philosophies 7 1.3Citations, terminology and notations 1.3.1 References and the bibliography 91.3.2 Translations, quotations and notations 10 1.4 Permissions andacknowledgements 11 CHAPTER 2 Preludes: Algebraic Logic and MathematicalAnalysis up to 1870 2.1 Plan of the chapter 14 2.2 'Logique' and algebrasin French mathematics 14 2.2.1 The 'logique' and clarity of 'ideologie' 142.2.2 Lagrange's algebraic philosophy 15 2.2.3 The many senses of 'analysis'17 2.2.4 Two Lagrangian algebras: functional equations and differentialoperators 17 2.2.5 Autonomy for the new algebras 19 2.3 Some Englishalgebraists and logicians 20 2.3.1 A Cambridge revival: the 'AnalyticalSociety, Lacroix, and the professing of algebras 20 2.3.2 The advocacy ofalgebras 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 onalgebras and logic 25 2.4.1 Summary of his life 25 2.4.2 De Morgan'sphilosophies of algebra 25 2.4.3 De Morgan's logical career 26 2.4.4 DeMorgan's contributions to the foundations of logic 27 2.4.5 Beyond thesyllogism 29 2.4.6 Contretemps over 'the quantification of the predicate' 302.4.7 The logic of two place relations, 1860 32 2.4.8 Analogies betweenlogic and mathematics 35 2.4.9 De Morgan's theory of collections 36 2.5 ALincoln outsider: Boole on logic as applied mathematics 37 2.5.1 Summary ofhis career 37 2.5.2 Boole's 'general method in analysis' 1844 39 2.5.3 Themathematical analysis of logic, 1847. 'elective symbols' and laws 40 2.5.4'Nothing' and the 'Universe' 42 2.5.5 Propositions, expansion theorems, andsolutions 43 2.5.6 The laws of thought, 1854: modified principles andextended methods 46 2.5.7 Boole's new theory of propositions 49 2.5.8 Thecharacter of Boole's system 50 2.5.9 Boole's search for mathematical roots53 2.6 The semi-followers of Boole 54 2.6.1 Some initial reactions toBoole's theory 54 2.6.2 The reformulation by Jevons 56 2.6.3 Jevons versusBoole 59 2.6.4 Followers of Boole and/or Jevons 60 2.7 Cauchy, Weierstrassand the rise of mathematical analysis 63 2.7.1 Different traditions in thecalculus 63 2.7.2 Cauchy and the Ecole Polytechnique 64 2.7.3 The gradualadoption and adaptation of Cauchy's new tradition 67 2.7.4 The refinementsof Weierstrass and his followers 68 2.8 Judgement and supplement 70 2.8.1Mathematical analysis versus algebraic logic 70 2.8.2 The places of Kant andBolzano 71 CHAPTER 3 Cantor: Mathematics as Mengenlehre 3.1 Prefaces 753.1.1 Plan of the chapter 75 3.1.2 Cantor's career 75 3.2 The launching ofthe Mengenlehre, 1870-1883 79 3.2.1 Riemann's thesis: the realm ofdiscontinuous functions 79 3.2.2 Heine on trigonometric series and the realline, 1870-1872 81 3.2.3 Cantor's extension of Heine's findings, 1870-187283 3.2.4 Dedekind on irrational numbers, 1872 85 3.2.5 Cantor on line andplane, 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-classes92 3.2.8 The Grundlagen: the definition of continuity 95 3.2.9 Thesuccessor 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 3.3.2Unpublished and published 'communications' 1884-1885 98 3.3.3 Order-typesand partial derivatives in the 'communications' 100 3.3.4 Commentators onCantor, 1883-1885 102 3.4 The extension of the Mengenlehre, 1886-1897 1033.4.1 Dedekind's developing set theory, 1888 103 3.4.2 Dedekind's chains ofintegers 105 3.4.3 Dedekind's philosophy of arithmetic 107 3.4.4 Cantor'sphilosophy of the infinite, 1886-1888 109 3.4.5 Cantor's new definitions ofnumbers 110 3.4.6 Cardinal exponentiation: Cantor's diagonal argument, 1891110 3.4.7 Transfinite cardinal arithmetic and simply ordered sets, 1895 1123.4.8 Transfinite ordinal arithmetic and well-ordered sets, 1897 114 3.5Open and hidden questions in Cantor's Mengenlehre 114 3.5.1 Well-orderingand the axioms of choice 114 3.5.2 What was Cantor's 'Cantor's continuumproblem'? 116 3.5.3 "Paradoxes" and the absolute infinite 117 3.6 Cantor'sphilosophy of mathematics 119 3.6.1 A mixed position 119 3.6.2 (No) logicand metamathematics 120 3.6.3 The supposed impossibility of infinitesimals121 3.6.4 A contrast with Kronecker 122 3.7 Concluding comments: thecharacter of Cantor's achievements 124 CHAPTER 4 Parallel Processes in SetTheory, Logics and Axiomatics, 1870s-1900s 4.1 Plans for the chapter 1264.2 The splitting and selling of Cantor's Mengenlehre 126 4.2.1 National andinternational support 126 4.2.2 French initiatives, especially from Borel127 4.2.3 Couturat outlining the infinite, 1896 129 4.2.4 Germaninitiatives from Mein 130 4.2.5 German proofs of the Schroder-Bernsteintheorem 132 4.2.6 Publicity from Hilbert, 1900 134 4.2.7 Integral equationsand functional analysis 135 4.2.8 Kempe on 'mathematical form' 137 4.2.9Kempe-who? 139 4.3 American algebraic logic: Peirce and his followers 1404.3.1 Peirce, published and unpublished 141 4.3.2 Influences on Peirre'slogic: father's algebras 142 4.3.3 Peirce's first phase: Boolean logic andthe 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 1474.3.6 Peirre's second phase, 1881: finite and infinite 149 4.3.7 Peirce'sstudents, 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 inthe 1890s 154 4.4 German algebraic logic: from the Grassmanns to Schr6der156 4.4.1 The Grassmanns on duality 156 4.4.2 Schroder's Grassmannian phase159 4.4.3 Schroder's Peirrean 'lectures' on logic 161 4.4.4 Schrrider'sfirst volume, 1890 161 4.4.5 Part of the second volume, 1891 167 4.4.6Schroder's third volume, 1895: the 'logic of relatives' 170 4.4.7 Peirce onand against Schroder in The monist, 1896-1897 172 4.4.8 Schroder onCantorian themes, 1898 174 4.4.9 The reception and publication of Schroderin 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 1794.5.3 Frege's arguments for logicising arithmetic, 1884 183 4.5.4 Keny'sconception of Fregean concepts in the mid 1880s 187 4.5.5 Important newdistinctions in the early 1890s 187 4.5.6 The 'fundamental laws' oflogicised arithmetic, 1893 191 4.5.7 Frege's reactions to others in thelater 1890s 194 4.5.8 More 'fundamental laws' of arithmetic, 1903 195 4.5.9Frege, 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 1994.6.2 The phenomenological 'philosophy of arithmetic; 1891 201 4.6.3 Reviewsby Frege and others 203 4.6.4 Husserl's 'logical investigations; 1900-1901204 4.6.5 Husserl's early talks in Gottingen, 1901 206 4.7 Hilbert: earlyproof and model theory, 1899-1905 207 4.7.1 Hilbert's growing concern withaxiomatics 207 4.7.2 Hilbert's diferent axiom systems for Euclideangeometry, 1899-1902 208 4.7.3 From German completeness to American modeltheory 209 4.7.4 Frege, Hilbert and Korselt on the foundations of geometries212 4.7.5 Hilbert's logic and proof theory, 1904-1905 213 4.7.6 Zermelo'slogic and set theory, 1904-1909 216 CHAPTER 5 Peano: the Formulary ofMathematics 5.1 Prefaces 219 5.1.1 Plan of the chapter 219 5.1.2 Peano'scareer 219 5.2 Formalising mathematical analysis 221 5.2.1 ImprovingGenocchi, 1884 221 5.2.2 Developing Grassmann's 'geometrical calculus; 1888223 5.2.3 The logistic of arithmetic, 1889 225 5.2.4 The logistic ofgeometry, 1889 229 5.2.5 The logistic of analysis, 1890 230 5.2.6 Bettazzion magnitudes, 1890 232 5.3 The Rivista: Peano and his school, 1890-1895 2325.3.1 The 'society of mathematicians' 232 5.3.2 'Mathematical logic, 1891234 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 ondefinition by equivalence classes 239 5.3.7 Burali-Forti's textbook, 1894240 5.3.8 Burali-Forti's research, 1896-1897 241 5.4 The Formulaire and theRivista, 1895-1900 242 5.4.1 The first edition of the Formulaire, 1895 2425.4.2 Towards the second edition of the Formulaire, 1897 244 5.4.3 Peano onthe eliminability of 'the' 246 5.4.4 Frege versus Peano on logic anddefinitions 247 5.4.5 Schroder's steamships versus Peano's sailing boats 2495.4.6 New presentations of arithmetic, 1898 251 5.4.7 - Padoa on classhoody1899 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 ondefinitions 256 5.5.3 Burali-Forti on definitions of numbers 257 5.5.4Padoa on definability and independence 259 5.5.5 Pieri on the logic ofgeometry 261 5.6 Concluding comments: the character of Peano's achievements262 5.6.1 Peano's little dictionary, 1901 262 5.6.2 Partly graspedopportunities 264 5.6.3 Logic without relations 266 CHAPTER 6 Russell's WayIn: From Certainty to Paradoxes, 1895-1903 6.1 Prefaces 268 6.1.1 Plans fortwo chapters 268 6.1.2 Principal sources 269 6.1.3 Russell as a Cambridgeundergraduate, 1891-1894 271 6.1.4 Cambridge philosophy in the 1890s 2736.2 Three philosophical phases in the foundation of mathematics, 1895-1899274 6.2.1 Russell's idealist axiomatic geometries 275 6.2.2 The importanceof axioms and relations 276 6.2.3 A pair of pas de deux with Paris: Couturatand Poincare on geometries 278 6.2.4 The emergence of "itehead, 1898 2806.2.5 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, 1896-1899285 6.3 From neo-Hegelianism towards 'Principles', 1899-1901 286 6.3.1Changing 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 2906.4.1 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 Datingthe 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 andmagnitude 298 6.4.6 Part 4, November 1900: order and ordinals 299 6.4.7Part 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 ofsymbolic logic, 1900 302 6.5 Convoluting towards logicism, 1900-1901 3036.5.1 Logicism as generalised metageometry, January 1901 303 6.5.2 The firstpaper for Peano, February 1901: relations and numbers 305 6.5.3 Cardinalarithmetic with "itehead and Russell, June 1901 307 6.5.4 The second paperfor 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 6.6.3Other paradoxes: three too large numbers 312 6.6.4 Three passions and threecalamities, 1901-1902 314 6.7 Refining logicism, 1901-1902 315 6.7.1Attempting 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: theimplicational logicism 316 6.7.4 Part 1: discussing the indefinables 3186.7.5 Part 7, June 1902: dynamics without statics; and within logic? 3226.7.6 Sort-of finishing the book 323 6.7.7 The first impact of Frege, 1902323 6.7.8 AppendixA on Frege 326 6.7.9 Appendix B: Russell's first attemptto solve the paradoxes 327 6.8 The roots of pure mathematics? Publishing Theprinciples at last, 1903 328 6.8.1 Appearance and appraisal 328 6.8.2 Agradual collaboration with Whitehead 331 CHAPTER 7 Russell and WhiteheadSeek the Principia Mathematica, 1903-1913 7.1 Plan of the chapter 333 7.2Paradoxes and axioms in set theory, 1903-1906 333 7.2.1 Uniting theparadoxes of sets and numbers 333 7.2.2 New paradoxes, mostly of naming 3347.2.3 The paradox that got away: heterology 336 7.2.4 Russell as cataloguerof the paradoxes 337 7.2.5 Controversies over axioms of choice, 1904 3397.2.6 Uncovering Russell's 'multiplicative axiom, 1904 340 7.2.7 Keyserversus Russell over infinite classes, 1903-1905 342 7.3 The perplexities ofdenoting, 1903-1906 342 7.3.1 First attempts at a general system, 1903-1905342 7.3.2 Propositional functions, reducible and identical 344 7.3.3 Themathematical importance of definite denoting functions 346 7.3.4 'Ondenoting' and the complex, 1905 348 7.3.5 Denoting, quantification and themysteries 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 pasde deux with Paris: Boutroux and Poincare on logicism 355 7.4.3 Poincare onthe 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, 1906358 7.4.6 The rise of the substitutional theory, 1905-1906 360 7.4.7 Thefall of the substitutional theory, 1906-1907 362 7.4.8 Russell'ssubstitutional propositional calculus 364 7.5 Reactions to mathematicallogic and logicism, 1904-1907 366 7.5.1 The International Congress ofPhilosophy, 1904 366 7.5.2 German philosophers and mathematicians,especially Schonflies 368 7.5.3 Activities among the Peanists 370 7.5.4American philosophers: Royce and Dewey 371 7.5.5 American mathematicians onclasses 373 7.5.6 Huntington on logic and orders 375 7.5.7 Judgements fiomShearman 376 7.6 Whitehead's role and activities, 1905-1907 377 7.6.1Whitehead's construal of the 'material world' 377 7.6.2 The axioms ofgeometries 379 7.6.3 Whitehead's lecture course, 1906-1907 379 7.7 The sadcompromise: 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'soutline of 'mathematical logic, 1908 383 7.8 The forming of Principiamathematica 384 7.8.1 Completing and funding Principia mathematica 3847.8.2 The Organisation of Principia mathematica 386 7.8.3 The propositionalcalculus, and logicism 388 7.8.4 The predicate calculus, and descriptions391 7.8.5 Classes and relations, relative to propositional functions 3927.8.6 The multiplicative axiom: some uses and avoidance 395 7.9 Types andthe treatment of mathematics in Principia mathematica 396 7.9.1 7~pes inorders 396 7.9.2 Reducing the edifice 397 7.9.3 Individuals, their natureand number 399 7.9.4 Cardinals and their finite arithmetic 401 7.9.5 Thegeneralised ordinals 403 7.9.6 The ordinals and the alephs 404 7.9.7 Theodd small ordinals 406 7.9.8 Series and continuity 406 7.9.9 Quantity withratios 408 CHAPTER 8 The Influence and Place of Logicism, 1910-1930 8.1Plans for two chapters 411 8.2 Whitehead's and Russell's transitions fromlogic 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 the1910s 413 8.2.3 British reviews of Principia mathematica 415 8.2.4 Russelland Peano on logic, 1911-1913 416 8.2.5 Russell's initial problems withepistemology, 1911-1912 417 8.2.6 Russell's first interactions withWittgenstein, 1911-1913 418 8.2.7 Russell's confrontation with Wiener, 1913419 8.3 Logicism and epistemology in America and with Russell, 1914-1921 4218.3.1 Russell on logic and epistemology at Harvard, 1914 421 8.3.2 Two longAmerican reviews 424 8.3.3 Reactions from Royce students: Sheffer and Lewis424 8.3.4 Reactions to logicism in New York 428 8.3.5 OtherAmericanestimations 429 8.3.6 Russell's 'logical atomism' and psychology, 1917-1921430 8.3.7 Russell's 'introduction'to logicism, 1918-1919 432 8.4 Revisinglogic 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 4368.4.3 The limitations of Wittgenstein's logic 437 8.4.4 Towards extensionallogicism: Russell's revision of Principia mathematica, 1923-1924 440 8.4.5Ramsey's entry into logic and philosophy, 1920-1923 443 8.4.6 Ramsey'srecasting of the theory of types, 1926 444 8.4.7 Ramsey on identity andcomprehensive extensionality 446 8.5 Logicism and epistemology in Britainand America, 1921-1930 448 8.5.1 Johnson on logic, 1921-1924 448 8.5.2Other Cambridge authors, 1923-1929 450 8.5.3 American reactions to logicismin mid decade 452 8.5.4 Groping towards metalogic 454 8.5.5 Reactions inand around Columbia 456 8.6 Peripherals: Italy and France 458 8.6.1 Theoccasional Italian survey 458 8.6.2 New French attitudes in the Revue 4598.6.3 Commentaries in French, 1918-1930 461 8.7 German-speaking reactions tologicism, 1910-1928 463 8.7.1 (Neo-)Kantians in the 1910s 463 8.7.2Phenomenologists in the 1910s 467 8.7.3 Frege's positive and then negativethoughts 468 8.7.4 Hilbert's definitive 'metamathematics; 1917-1930 4708.7.5 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 8.7.7Intuitionistic 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 8.8The rise of Poland in the 1920s: the Lvnv-Warsaw school 489 8.8.1 From Lv6vto Warsaw: students of Twardowski 489 8.8.2 Logics with Lukasiewicz andTarski 490 8.8.3 Russell's paradox and Lesniewski's three systems 492 8.8.4Pole apart: Chwistek's 'semantic' logicism at Cracov 495 8.9 The rise ofAustria in the 1920s: the Schlick circle 497 8.9.1 Formation and influence497 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 5028.9.5 Intuitionism and proof theory: Brouwer and Godel, 1928-1930 504CHAPTER 9 Postludes: Mathematical Logic and Logicism in the 1930s 9.1 Planof the chapter 506 9.2 Godel's incompletability theorem and its immediatereception 507 9.2.1 The consolidation of Schlick's 'Vienna' Circle 5079.2.2 News from G6del: the Konigsberg lectures, September 1930 508 9.2.3G6del's incompletability theorem, 1931 509 9.2.4 Effects and reviews ofG6del'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 5139.3.1 Carnap for 'metalogic' and against metaphysics 513 9.3.2 Carnap'stransformed metalogic: the 'logical syntax of language; 1934 515 9.3.3Carnap on incompleteness and truth in mathematical theories, 1934-1935 5179.3.4 Dubislav on definitions and the competing philosophies of mathematics519 9.3.5 Behmann's new diagnosis of the paradoxes 520 9.3.6 Kaufmann andWaismann 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 5259.4.3 Whitehead's new attempt to ground logicism, 1934 527 9.4.4 The debutof Quine 529 9.4.5 Two journals and an encyclopaedia, 1934-1938 531 9.4.6Carnap's acceptance of the autonomy of semantics 533 9.5 The battle ofBritain 535 9.5.1 The campaign of Stebbing for Russell and Carnap 535 9.5.2Commentary from Black and Ayer 538 9.5.3 Mathematicians-and biologists 5399.5.4 Retiring into philosophy: Russell's return, 1936-1937 542 9.6European, mostly northern 543 9.6.1 Dingler and Burkamp again 543 9.6.2German proof theory after Godel 544 9.6.3 Scholz's little circle at Munster546 9.6.4 Historical studies, especially by Jorgensen 547 9.6.5 Historyphilosophy, especially Cavailles 548 9.6.6 Other Francophone figures,especially Herbrand 549 9.6.7 Polish logicians, especially Tarski 551 9.6.8Southern Europe and its former colonies 553 CHAPTER 10 The Fate of theSearch 10.1 Influences on Russell, negative and positive 556 10.1.1Symbolic logics: living together and living apart 556 10.1.2 The timing andorigins of Russell's logicism 557 10.1.3 (Why) was Frege (so) little read inhis lifetime? 558 10.2 The content and impact of logicism 559 10.2.1Russell's obsession with reductionist logic and epistemology 560 10.2.2 Thelogic and its metalogic 562 10.2.3 The fate of logicism 563 10.2.4Educational 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 56910.4 Sallies 573 CHAPTER 11 Transcription of Manuscripts 11.1 Couturat toRussell, 18 December 1904 574 11.2 Veblen to Russell, 13 May 1906 577 11.3Russell to Hawtrey, 22 January 1907 (or 1909?) 579 11.4 Jourdain's notes onWittgenstein's first views on Russell's paradox, April 1909 580 11.5 Theapplication of Whitehead and Russell to the Royal Society, late 1909 58111.6 Whitehead to Russell, 19 January 1911 584 11.7 Oliver Strachey toRussell, 4 January 1912 585 11.8 Quine and Russell, June-July 1935 58611.8.1 Russell to Quine, 6 June 1935 587 11.8.2 Quine to Russell, 4 July1935 588 11.9 Russell to Henkin, 1 April 1963 592 BIBLIOGRAPHY 594 INDEX671