While many books have been written about Bertrand Russell's philosophy and some on his logic, I. Grattan-Guinness has written the first comprehensive history of the mathematical background, content, and impact of the mathematical logic and philosophy of mathematics that Russell developed with A. N. Whitehead in their Principia mathematica (1910-1913).
This definitive history of a critical period in mathematics includes detailed accounts of the two principal influences upon Russell around 1900: the set theory of Cantor and the mathematical logic of Peano and his followers. Substantial surveys are provided of many related topics and figures of the late nineteenth century: the foundations of mathematical analysis under Weierstrass; the creation of algebraic logic by De Morgan, Boole, Peirce, Schrder, and Jevons; the contributions of Dedekind and Frege; the phenomenology of Husserl; and the proof theory of Hilbert. The many-sided story of the reception is recorded up to 1940, including the rise of logic in Poland and the impact on Vienna Circle philosophers Carnap and Gdel. A strong American theme runs though the story, beginning with the mathematician E. H. Moore and the philosopher Josiah Royce, and stretching through the emergence of Church and Quine, and the 1930s immigration of Carnap and GdeI.
Grattan-Guinness draws on around fifty manuscript collections, including the Russell Archives, as well as many original reviews. The bibliography comprises around 1,900 items, bringing to light a wealth of primary materials.
Written for mathematicians, logicians, historians, and philosophers--especially those interested in the historical interaction between these disciplines--this authoritative account tells an important story from its most neglected point of view. Whitehead and Russell hoped to show that (much of) mathematics was expressible within their logic; they failed in various ways, but no definitive alternative position emerged then or since.
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"Grattan-Guiness's uniformly interesting and valuable account of the interwoven development of logic and related fields of mathematics ... between 1870 and 1940 presents a significantly revised analysis of the history of the period... [ His] book is important because it supplies what has been lacking: a full account of the period from a primary mathematical perspective."--James W. Van Evra, Isis
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I know of no comparably comprehensive treatment of the history of this important period in modern logic. There is a large body of historical literature that is in need of just the kind of synthesis and masterly overview that this work provides. Though most people recognize mathematics as a principal motivating force behind the development of modern logic, the influences on and from mathematics have been largely ignored or minimized. The Search for Mathematical Roots acts as a guide through that challenging mathematical thicket. -- Albert C. Lewis, chief editor of "The History of Mathematics from Antiquity to the Present" Ivor Grattan-Guinness provides a marvelous, comprehensive overview of the history of efforts to come to an understanding of mathematical logic and its relation to mathematics in the period 1870-1940. Given its rich detail and inclusion of under-appreciated figures who deserve to be better known, this is an especially important and useful book. -- Joseph Dauben, author of "George Cantor: His Mathematics and Philosophy of the Infinite"
Explanations Sallies 3(1) Scope and limits of the book 3(6) An outline history 3(1) Mathematical aspects 4(2) Historical presentation 6(1) Other logics, mathematics and philosophies 7(2) Citations, terminology and notations 9(2) References and the bibliography 9(1) Translations, quotations and notations 10(1) Permissions and acknowledgements 11(3) Preludes: Algebraic Logic and Mathematical Analysis up to 1870 Plan of the chapter 14(1) `Logique and algebras in French mathematics 14(6) The `logique and clarity of `ideologie 14(1) Lagranges algebraic philosophy 15(2) The many senses of `analysis 17(1) Two Lagrangian algebras: functional equations and differential operators 17(2) Autonomy for the new algebras 19(1) Some English algebraists and logicians 20(5) A Cambridge revival: the `Analytical Society, Lacroix, and the professing of algebras 20(1) The advocacy of algebras by Babbage, Herschel and Peacock 20(2) An Oxford movement: Whatley and the professing of logic 22(3) A London pioneer: De Morgan on algebras and logic 25(12) Summary of his life 25(1) De Morgans philosophies of algebra 25(1) De Morgans logical career 26(1) De Morgans contributions to the foundations of logic 27(2) Beyond the syllogism 29(1) Contretemps over `the quantification of the predicate 30(2) The logic of two-place relations, 1860 32(3) Analogies between logic and mathematics 35(1) De Morgans theory of collections 36(1) A Lincoln outsider: Boole on logic as applied mathematics 37(17) Summary of his career 37(2) Booles `general method in analysis, 1844 39(1) The mathematical analysis of logic, 1847: `elective symbols and laws 40(2) `Nothing and the `Universe 42(1) Propositions, expansion theorems, and solutions 43(3) The laws of thought, 1854: modified principles and extended methods 46(3) Booles new theory of propositions 49(1) The character of Booles system 50(3) Booles search for mathematical roots 53(1) The semi-followers of Boole 54(9) Some initial reactions to Booles theory 54(2) The reformulation 56(3) Jevons Jevons versus Boole 59(1) Followers of Boole and/or Jevons 60(3) Cauchy, Weierstrass and the rise of mathematical analysis 63(7) Different traditions in the calculus 63(1) Cauchy and the Ecole Polytechnique 64(3) The gradual adoption and adaptation of Cauchys new tradition 67(1) The refinements of Weierstrass and his followers 68(2) Judgement and supplement 70(5) Mathematical analysis versus algebraic logic 70(1) The places of Kant and Bolzano 71(4) Cantor: Mathematics as Mengenlehre Prefaces 75(4) Plan of the chapter 75(1) Cantors career 75(4) The launching of the Mengenlehre, 1870-1883 79(18) Riemanns thesis: the realm of discontinuous functions 79(2) Heine on trigonometric series and the real line, 1870-1872 81(2) Cantors extension of Heines findings, 1870-1872 83(2) Dedekind on irrational numbers, 1872 85(3) Cantor on line and plane, 1874-1877 88(1) Infinite numbers and the topology of linear sets, 1878-1883 89(3) The Grundlagen, 1883: the construction of number-classes 92(3) The Grundlagen: the definition of continuity 95(1) The successor to the Grundlagen, 1884 96(1) Cantors Acta mathematica phase, 1883-1885 97(6) Mittag-Leffler and the French translations, 1883 97(1) Unpublished and published `communications, 1884-1885 98(2) Order-types and partial derivatives in the `communications 100(2) Commentators on Cantor, 1883-1885 102(1) The extension of the Mengenlehre, 1886-1897 103(11) Dedekinds developing set theory, 1888 103(2) Dedekinds chains of integers 105(2) Dedekinds philosophy of arithmetic 107(2) Cantors philosophy of the infinite, 1886-1888 109(1) Cantors new definitions of numbers 110(1) Cardinal exponentiation: Cantors diagonal argument, 1891 110(2) Transfinite cardinal arithmetic and simply ordered sets, 1895 112(2) Transfinite ordinal arithmetic and well-ordered sets, 1897 114(1) Open and hidden questions in Cantors Mengenlehre 114(5) Well-ordering and the axioms of choice 114(2) What was Cantors `Cantors continuum problem? 116(1) ``Paradoxes and the absolute infinite 117(2) Cantors philosophy of mathematics 119(5) A mixed position 119(1) (No) logic and metamathematics 120(1) The supposed impossibility of infinitesimals 121(1) A contrast with Kronecker 122(2) Concluding comments: the character of Cantors achievements 124(2) Parallel Processes in Set Theory, Logics and Axiomatics, 1870s-1900s Plans for the chapter 126(1) The splitting and selling of Cantors Mengenlehre 126(14) National and international support 126(1) French initiatives, especially from Borel 127(2) Couturat outlining the infinite, 1896 129(1) German initiatives from Klein 130(2) German proofs of the Schroder-Bernstein theorem 132(2) Publicity from Hilbert, 1900 134(1) Integral equations and functional analysis 135(2) Kempe on `mathematical form 137(2) Kempe-who? 139(1) American algebraic logic: Peirce and his followers 140(16) Peirce, published and unpublished 141(1) Influences on Peirces logic: fathers algebras 142(2) Peirces first phase: Boolean logic and the categories, 1867-1868 144(1) Peirces virtuoso theory of relatives, 1870 145(2) Peirces second phase, 1880: the propositional calculus 147(2) Peirces second phase, 1881: finite and infinite 149(1) Peirces students, 1883: duality, and `Quantifying a proposition 150(3) Peirce on `icons and the order of `quantifiers, 1885 153(1) The Peirceans in the 1890s 154(2) German algebraic logic: from the Grassmanns to Schroder 156(21) The Grassmanns on duality 156(3) Schroders Grassmannian phase 159(2) Schroders Peircean `lectures on logic 161(1) Schroders first volume, 1890 161(6) Part of the second volume, 1891 167(3) Schroders third volume, 1895: the `logic of relatives 170(2) Peirce on and against Schroder in The monist, 1896-1897 172(2) Schroder on Cantorian themes, 1898 174(1) The reception and publication of Schroder in the 1900s 175(2) Frege: arithmetic as logic 177(22) Frege and Frege 177(2) The `concept-script calculus of Freges `pure thought, 1879 179(4) Fregers arguments for logicising arithmetic, 1884 183(4) Kerrys conception of Fregean concepts in the mid 1880s 187(1) Important new distictions in the early 1890s 187(4) The `fundamental laws of logicised arithmetic, 1893 191(3) Freges reactions to others in the later 1890s 194(1) More `fundamental laws of arithmetic, 1903 195(2) Frege, Korselt and Thomae on the foundations of arithmetic 197(2) Husserl: logic as phenomenology 199(8) A follower of Weierstrass and Cantor 199(2) The phenomenological `philosophy of arithmetic, 1891 201(2) Reviews by Frege and others 203(1) Husserls `logical investigations, 1900-1901 204(2) Husserls early talks in Gottingen, 1901 206(1) Hilbert: early proof and model theory, 1899-1905 207(12) Hilberts growing concern with axiomatics 207(1) Hilberts different axiom systems for Euclidean geometry, 1899-1902 208(1) From German completeness to American model theory 209(3) Frege, Hilbert and Korselt on the foundations of geometries 212(1) Hilberts logic and proof theory, 1904-1905 213(3) Zermelos logic and set theory, 1904-1909 216(3) Peano: the Formulary of Mathematics Prefaces 219(2) Plan of the chapter 219(1) Peanos career 219(2) Formalising mathematical analysis 221(11) Improving Genocchi, 1884 221(2) Developing Grassmanns `geometrical calculus, 1888 223(2) The logistic of arithmetic, 1889 225(4) The logistic of geometry, 1889 229(1) The logistic of analysis, 1890 230(2) Bettazzi on magnitudes, 1890 232(1) The Rivista: Peano and his school, 1890-1895 232(10) The `society of mathematicians 232(2) `Mathematical logic, 1891 234(1) Developing arithmetic, 1891 235(1) Infinitesimals and limits, 1892-1895 236(1) Notations and their range, 1894 237(2) Peano on definition by equivalence classes 239(1) Burali-Fortis textbook, 1894 240(1) Burali-Fortis research, 1896-1897 241(1) The Formulaire and the Rivista, 1895-1900 242(13) The first edition of the Formulaire, 1895 242(2) Towards the second edition of the Formulaire, 1897 244(2) Peano on the eliminability of `the 246(1) Frege versus Peano on logic and definitions 247(2) Schroders steamships versus Peanos sailing boats 249(2) New presentations of arithmetic, 1898 251(2) Padoa on classhood, 1899 253(1) Peanos new logical summary, 1900 254(1) Peanists in Paris, August 1900 255(7) An Italian Friday morning 255(1) Peano on definitions 256(1) Burali-Forti on definitions of numbers 257(2) Padoa on definability and independence 259(2) Pieri on the logic of geometry 261(1) Concluding comments: the character of Peanos achievements 262(6) Peanos little dictionary, 1901 262(2) Partly grasped opportunities 264(2) Logic without relations 266(2) Russells Way In: From Certainty to Paradoxes, 1895-1903 Prefaces 268(6) Plans for two chapters 268(1) Principal sources 269(2) Russell as a Cambridge undergraduate, 1891-1894 271(2) Cambridge philosophy in the 1890s 273(1) Three philosophical phases in the foundation of mathematics, 1895-1899 274(12) Russells idealist axiomatic geometries 275(1) The importance of axioms and relations 276(2) A pair of pas de deux with Paris: Couturat and Poincare on geometries 278(2) The emergence of Whitehead, 1898 280(2) The impact of G. E. Moore, 1899 282(1) Three attempted books, 1898-1899 283(2) Russells progress with Cantors Mengenlehre, 1896-1899 285(1) From neo-Hegelianism towards `Principles, 1899-1901 286(4) Changing relations 286(2) Space and time, absolutely 288(1) `Principles of Mathematics, 1899-1900 288(2) The first impact of Peano 290(13) The Paris Congress of Philosophy, August 1900: Schroder versus Peano on `the 290(1) Annotating and popularising in the autumn 291(1) Dating the origins of Russells logicism 292(4) Drafting the logic of relations, October 1900 296(2) Part 3 of The principles, November 1900: quantity and magnitude 298(1) Part 4, November 1900: order and ordinals 299(1) Part 5, November 1900: the transfinite and the continuous 300(1) Part 6, December 1900: geometries in space 301(1) Whitehead on `the algebra of symbolic logic, 1900 302(1) Convoluting towards logicism, 1900-1901 303(7) Logicism as generalised metageometry, January 1901 303(2) The first paper for Peano, February 1901: relations and numbers 305(2) Cardinal arithmetic with Whitehead and Russell, June 1901 307(1) The second paper for Peano, March-August 1901: set theory with series 308(2) From `fallacy to `contradiction, 1900-1901 310(5) Russell on Cantors `fallacy, November 1900 310(1) Russells switch to a `contradiction 311(1) Other paradoxes: three too large numbers 312(2) Three passions and three calamities, 1901-1902 314(1) Refining logicism, 1901-1902 315(13) Attempting part 1 of The principles, May 1901 315(1) Part 2, June 1901: cardinals and classes 316(1) Part 1 again, April-May 1902: the implicational logicism 316(2) Part 1: discussing the indefinables 318(4) Part 7, June 1902: dynamics without statics; and within logic? 322(1) Sort-of finishing the book 323(1) The first impact of Frege, 1902 323(3) Appendix A on Frege 326(1) Appendix B: Russells first attempt to solve the paradoxes 327(1) The roots of pure mathematics? Publishing The principles at last, 1903 328(5) Appearance and appraisal 328(3) A gradual collaboration with Whitehead 331(2) Russell and Whitehead Seek the Principia Mathematica, 1903-1913 Plan of the chapter 333(1) Paradoxes and axioms in set theory, 1903-1906 333(9) Uniting the paradoxes of sets and numbers 333(1) New paradoxes, mostly of naming 334(2) The paradox that got away: heterology 336(1) Russell as cataloguer of the paradoxes 337(2) Controversies over axioms of choice, 1904 339(1) Uncovering Russells `multiplicative axiom, 1904 340(2) Keyser versus Russell over infinite classes, 1903-1905 342(1) The perplexities of denoting, 1903-1906 342(12) First attempts at a general system, 1903-1905 342(2) Propositional functions, reducible and identical 344(2) The mathematical importance of definite denoting functions 346(2) On denoting and the complex, 1905 348(2) Denoting, quantification and the mysteries of existence 350(1) Russell versus MacColl on the possible, 1904-1908 351(3) From mathematical induction to logical substitution, 1905-1907 354(12) Couturats Russellian principles 354(1) A second pas de deux with Paris: Boutroux and Poincare on logicism 355(1) Poincare on the status of mathematical induction 356(1) Russells position paper, 1905 357(1) Poincare and Russell on the vicious circle principle, 1906 358(2) The rise of the substitutional theory, 1905-1906 360(2) The fall of the substitutional theory, 1906-1907 362(2) Russells substitutional propositional calculus 364(2) Reactions to mathematical logic and logicism, 1904-1907 366(11) The International Congress of Philosophy, 1904 366(2) German philosophers and mathematicians, especially Schonflies 368(2) Activities among the Peanists 370(1) American philosophers: Royce and Dewey 371(2) American mathematicians on classes 373(2) Huntington on logic and orders 375(1) Judgements from Shearman 376(1) Whiteheads role and activities, 1905-1907 377(3) Whiteheads construal of the `material world 377(2) The axioms of geometries 379(1) Whiteheads lecture course, 1906-1907 379(1) The sad compromise: logic in tiers 380(4) Rehabilitating propositional functions, 1906-1907 380(2) Two reflective pieces in 1907 382(1) Russells outline of `mathematical logic, 1908 383(1) The forming of Principia mathematica 384(12) Completing and funding Principia mathematica 384(2) The organisation of Principia mathematica 386(2) The propositional calculus, and logicism 388(3) The predicate calculus, and descriptions 391(1) Classes and relations, relative to propositional functions 392(3) The multiplicative axiom: some uses and avoidance 395(1) Types and the treatment of mathematics in Principia mathematica 396(15) Types in orders 396(1) Reducing the edifice 397(2) Individuals, their nature and number 399(2) Cardinals and their finite arithmetic 401(2) The generalised ordinals 403(1) The ordinals and the alephs 404(2) The odd small ordinals 406(1) Series and continuity 406(2) Quantity with ratios 408(3) The Influence and Place of Logicism, 1910-1930 Plans for two chapters 411(1) Whiteheads and Russells transitions from logic to philosophy, 1910-1916 412(9) The educational concerns of Whitehead, 1910-1916 412(1) Whitehead on the principles of geometry in the 1910s 413(2) British reviews of Principia mathematica 415(1) Russell and Peano on logic, 1911-1913 416(1) Russells initial problems with epistemology, 1911-1912 417(1) Russells first interactions with Wittgenstein, 1911-1913 418(1) Russells confrontation with Wiener, 1913 419(2) Logicism and epistemology in America and with Russell, 1914-1921 421(13) Russell on logic and epistemology at Harvard, 1914 421(3) Two long American reviews 424(1) Reactions from Royce students: Sheffer and Lewis 424(4) Reactions to logicism in New York 428(1) Other American estimations 429(1) Russells `logical atomism and psychology, 1917-1921 430(2) Russells `introduction to logicism, 1918-1919 432(2) Revising logic and logicism at Cambridge, 1917-1925 434(14) New Cambridge authors, 1917-1921 434(2) Wittgensteins Abhandlung and Tractatus, 1921-1922 436(1) The limitations of Wittgensteins logic 437(3) Towards extensional logicism: Russells revision of Principia mathematica, 1923-1924 440(3) Ramseys entry into logic and philosophy, 1920-1923 443(1) Ramseys recasting of the theory of types, 1926 444(2) Ramsey on identity and comprehensive extensionality 446(2) Logicism and epistemology in Britain and America, 1921-1930 448(10) Johnson on logic, 1921-1924 448(2) Other Cambridge authors, 1923-1929 450(2) American reactions to logicism in mid decade 452(2) Groping towards metalogic 454(2) Reactions in and around Columbia 456(2) Peripherals: Italy and France 458(5) The occasional Italian survey 458(1) New French attitudes in the Revue 459(2) Commentaries in French, 1918-1930 461(2) German-speaking reactions to logicism, 1910-1928 463(26) (Neo-)Kantians in the 1910s 463(4) Phenomenologists in the 1910s 467(1) Freges positive and then negative thoughts 468(2) Hilberts definitive `metamathematics, 1917-1930 470(5) Orders of logic and models of set theory: Lowenheim and skolem, 1915-1923 475(1) Set theory and Mengenlehre in various forms 476(4) Intuitionistic set theory and logic: Brouwer and Weyl, 1910-1928 480(4) (Neo-)Kantians in the 1920s 484(3) Phenomenologists in the 1920s 487(2) The rise of Poland in the 1920s: the Lvov-Warsaw school 489(8) From Lvov to Warsaw: students of Twardowski 489(1) Logics with Lukasiewicz and Tarski 490(2) Russells paradox and Lesniewskis three systems 492(3) Pole apart: Chwisteks `semantic logicism at Cracov 495(2) The rise of Austria in the 1920s: the Schlick circle 497(9) Formation and influence 497(2) The impact of Russell, especially upon Carnap 499(1) `Logicism in Carnaps Abriss, 1929 500(2) Epistemology in Carnaps Aufbau, 1928 502(2) Intuitionism and proof theory: Brouwer and Godel, 1928-1930 504(2) Postludes: Mathematical Logic and Logicism in the 1930s Plan of the chapter 506(1) Godels incompletability theorem and its immediate reception 507(6) The consolidation of Schlicks `Vienna Circle 507(1) News from Godel: the Konigsberg lectures, September 1930 508(1) Godels incompletability theorem, 1931 509(2) Effects and reviews of Godels theorem 511(1) Zermelo against Godel: the Bad Elster lectures, September 1931 512(1) Logic(ism) and epistemology in and around Vienna 513(10) Carnap for `metalogic and against metaphysics 513(2) Carnaps transformed metalogic: the `logical syntax of language, 1934 515(2) Carnap on incompleteness and truth in mathematical theories, 1934-1935 517(2) Dubislav on definitions and the competing philosophies of mathematics 519(1) Behmanns new diagnosis of the paradoxes 520(1) Kaufmann and Waismann on the philosophy of mathematics 521(2) Logic(ism) in the U.S.A. 523(12) Mainly Eaton and Lewis 523(2) Mainly Weiss and Langer 525(2) Whiteheads new attempt to ground logicism, 1934 527(2) The debut of Quine 529(2) Two journals and an encyclopaedia, 1934-1938 531(2) Carnaps acceptance of the autonomy of semantics 533(2) The battle of Britain 535(8) The campaign of Stebbing for Russell and Carnap 535(3) Commentary from Black and Ayer 538(1) Mathematicians-and biologists 539(3) Retiring into philosophy: Russells return, 1936-1937 542(1) European, mostly northern 543(13) Dingler and Burkamp again 543(1) German proof theory after Godel 544(2) Scholzs little circle at Munster 546(1) Historical studies, especially by Jorgensen 547(1) History-philosophy, especially Cavailles 548(1) Other Francophone figures, especially Herbrand 549(2) Polish logicians, especially Tarski 551(2) Southern Europe and its former colonies 553(3) The Fate of the Search Influences on Russell, negative and positive 556(3) Symbolic logics: living together and living apart 556(1) The timing and origins of Russells logicism 557(1) (Why) was Frege (so) little read in his lifetime? 558(1) The content and impact of logicism 559(10) Russells obsession with reductionist logic and epistemology 560(2) The logic and its metalogic 562(1) The fate of logicism 563(3) Educational aspects, especially Piaget 566(1) The role of the U.S.A.: judgements in the Schilpp series 567(2) The panoply of foundations 569(4) Sallies 573(1) Transcription of Manuscripts Couturat to Russell, 18 December 1904 574(3) Veblen to Russell, 13 May 1906 577(2) Russell to Hawtrey, 22 January 1907 (or 1909?) 579(1) Jourdains notes on Wittgnsteins first views on Russells paradox, April 1909 580(1) The application of Whitehead and Russell to the Royal Society, late 1909 581(3) Whitehead to Russell, 19 January 1911 584(1) Oliver Strachey to Russell, 4 January 1912 585(1) Quine and Russell, June-July 1935 586(6) Russell to Quine, 6 June 1935 587(1) Quine to Russell, 4 July 1935 588(4) Russell to Henkin, 1 April 1963 592(2) Bibliography 594(77) Index 671
I. Grattan-Guinness is Professor of the History of Mathematics and Logic at Middlesex University. Founder of the journal History and Philosophy of Logic and past President of the British Society for the History of Mathematics, he has authored or edited numerous books, including The Norton History of Mathematics, Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, and Convolutions in French Mathematics, 1800-1840.
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