The search for mathematical roots, 1870-1940 logics, set theories and the foundations of mathematics from Cantor through Russell to Gödel.
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...
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| Format: | E-Book Buch |
| Sprache: | Englisch |
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Princeton, N.J
Princeton University Press
2001
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| Ausgabe: | 1 |
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| ISBN: | 9780691058573, 0691058571, 069105858X, 9780691058580 |
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Inhaltsangabe:
- The search for mathematical roots, 1870-1940: logics, set theories and the foundations of mathematics from Cantor through Russell to Gödel -- Contents -- Chapter 1: Explanations -- Chapter 2: Preludes: Algebraic Logic and Mathematical Analysis up to 1870 -- Chapter 3: Cantor: Mathematics as Mengenlehre -- Chapter 4: Parallel Processes in set Theory, Logics and Axiomatics, 1870s-1900s -- Chapter 5: Peano: the Formulary of Mathematics -- Chapter 6: Russell’s Way In: From Certainty to Paradoxes, 1895-1903 -- Chapter 7: Russell and Whitehead Seek the Principia Mathematica, 1903-1913 -- Chapter 8: The Influence and Place of Logicism, 1910-1930 -- Chapter 9: Postludes: Mathematical Logic and Logicism in the 1930s -- Chapter 10: The Fate of the Search -- Bibliography -- Index
- Cover -- Title -- Copyright -- Contents -- CHAPTER 1 Explanations -- 1.1 Sallies -- 1.2 Scope and limits of the book -- 1.2.1 An outline history -- 1.2.2 Mathematical aspects -- 1.2.3 Historical presentation -- 1.2.4 Other logics, mathematics and philosophies -- 1.3 Citations, terminology and notations -- 1.3.1 References and the bibliography -- 1.3.2 Translations, quotations and notations -- 1.4 Permissions and acknowledgements -- CHAPTER 2 Preludes: Algebraic Logic and Mathematical Analysis up to 1870 -- 2.1 Plan of the chapter -- 2.2 'Logique' and algebras in French mathematics -- 2.2.1 The 'logique' and clarity of 'idéologie' -- 2.2.2 Lagrange's algebraic philosophy -- 2.2.3 The many senses of 'analysis' -- 2.2.4 Two Lagrangian algebras: functional equations and differential operators -- 2.2.5 Autonomy for the new algebras -- 2.3 Some English algebraists and logicians -- 2.3.1 A Cambridge revival: the 'Analytical Society', Lacroix, and the professing of algebras -- 2.3.2 The advocacy of algebras by Babbage, Herschel and Peacock -- 2.3.3 An Oxford movement: Whately and the professing of logic -- 2.4 A London pioneer: De Morgan on algebras and logic -- 2.4.1 Summary of his life -- 2.4.2 De Morgan's philosophies of algebra -- 2.4.3 De Morgan's logical career -- 2.4.4 De Morgan's contributions to the foundations of logic -- 2.4.5 Beyond the syllogism -- 2.4.6 Contretemps over 'the quantification of the predicate' -- 2.4.7 The logic of two-place relations, 1860 -- 2.4.8 Analogies between logic and mathematics -- 2.4.9 De Morgan's theory of collections -- 2.5 A Lincoln outsider: Boole on logic as applied mathematics -- 2.5.1 Summary of his career -- 2.5.2 Boole's 'general method in analysis', 1844 -- 2.5.3 The mathematical analysis of logic, 1847: 'elective symbols' and laws -- 2.5.4 'Nothing' and the 'Universe'
- 3.4 The extension of the Mengenlehre, 1886-1897 -- 3.4.1 Dedekind's developing set theory, 1888 -- 3.4.2 Dedekind's chains of integers -- 3.4.3 Dedekind's philosophy of arithmetic -- 3.4.4 Cantor's philosophy of the infinite, 1886-1888 -- 3.4.5 Cantor's new definitions of numbers -- 3.4.6 Cardinal exponentiation: Cantor's diagonal argument, 1891 -- 3.4.7 Transfinite cardinal arithmetic and simply ordered sets, 1895 -- 3.4.8 Transfinite ordinal arithmetic and well-ordered sets, 1897 -- 3.5 Open and hidden questions in Cantor's Mengenlehre -- 3.5.1 Well-ordering and the axioms of choice -- 3.5.2 What was Cantor's 'Cantor's continuum problem'? -- 3.5.3 "Paradoxes" and the absolute infinite -- 3.6 Cantor's philosophy of mathematics -- 3.6.1 A mixed position -- 3.6.2 (No) logic and metamathematics -- 3.6.3 The supposed impossibility of infinitesimals -- 3.6.4 A contrast with Kronecker -- 3.7 Concluding comments: the character of Cantor's achievements -- CHAPTER 4 Parallel Processes in Set Theory, Logics and Axiomatics, 1870s-1900s -- 4.1 Plans for the chapter -- 4.2 The splitting and selling of Cantor's Mengenlehre -- 4.2.1 National and international support -- 4.2.2 French initiatives, especially from Borel -- 4.2.3 Couturat outlining the infinite, 1896 -- 4.2.4 German initiatives from Klein -- 4.2.5 German proofs of the Schröder-Bernstein theorem -- 4.2.6 Publicity from Hilbert, 1900 -- 4.2.7 Integral equations and functional analysis -- 4.2.8 Kempe on 'mathematical form' -- 4.2.9 Kempe-who? -- 4.3 American algebraic logic: Peirce and his followers -- 4.3.1 Peirce, published and unpublished -- 4.3.2 Influences on Peirce's logic: father's algebras -- 4.3.3 Peirce's first phase: Boolean logic and the categories, 1867-1868 -- 4.3.4 Peirce's virtuoso theory of relatives, 1870 -- 4.3.5 Peirce's second phase, 1880: the propositional calculus
- 2.5.5 Propositions, expansion theorems, and solutions -- 2.5.6 The laws of thought, 1854: modified principles and extended methods -- 2.5.7 Boole's new theory of propositions -- 2.5.8 The character of Boole's system -- 2.5.9 Boole's search for mathematical roots -- 2.6 The semi-followers of Boole -- 2.6.1 Some initial reactions to Boole's theory -- 2.6.2 The reformulation by Jevons -- 2.6.3 Jevons versus Boole -- 2.6.4 Followers of Boole and/or Jevons -- 2.7 Cauchy, Weierstrass and the rise of mathematical analysis -- 2.7.1 Different traditions in the calculus -- 2.7.2 Cauchy and the Ecole Polytechnique -- 2.7.3 The gradual adoption and adaptation of Cauchy's new tradition -- 2.7.4 The refinements of Weierstrass and his followers -- 2.8 Judgement and supplement -- 2.8.1 Mathematical analysis versus algebraic logic -- 2.8.2 The places of Kant and Bolzano -- CHAPTER 3 Cantor: Mathematics as Mengenlehre -- 3.1 Prefaces -- 3.1.1 Plan of the chapter -- 3.1.2 Cantor's career -- 3.2 The launching of the Mengenlehre, 1870-1883 -- 3.2.1 Riemann's thesis: the realm of discontinuous functions -- 3.2.2 Heine on trigonometric series and the real line, 1870-1872 -- 3.2.3 Cantor's extension of Heine's findings, 1870-1872 -- 3.2.4 Dedekind on irrational numbers, 1872 -- 3.2.5 Cantor on line and plane, 1874-1877 -- 3.2.6 Infinite numbers and the topology of linear sets, 1878-1883 -- 3.2.7 The Grundlagen, 1883: the construction of number-classes -- 3.2.8 The Grundlagen : the definition of continuity -- 3.2.9 The successor to the Grundlagen, 1884 -- 3.3 Cantor's Acta mathematica phase, 1883-1885 -- 3.3.1 Mittag-Leffler and the French translations, 1883 -- 3.3.2 Unpublished and published 'communications', 1884-1885 -- 3.3.3 Order-types and partial derivatives in the 'communications' -- 3.3.4 Commentators on Cantor, 1883-1885
- 4.7.4 Frege, Hilbert and Korselt on the foundations of geometries -- 4.7.5 Hilbert's logic and proof theory, 1904-1905 -- 4.7.6 Zermelo's logic and set theory, 1904-1909 -- CHAPTER 5 Peano: the Formulary of Mathematics -- 5.1 Prefaces -- 5.1.1 Plan of the chapter -- 5.1.2 Peano's career -- 5.2 Formalising mathematical analysis -- 5.2.1 Improving Genocchi, 1884 -- 5.2.2 Developing Grassmann's 'geometrical calculus', 1888 -- 5.2.3 The logistic of arithmetic, 1889 -- 5.2.4 The logistic of geometry, 1889 -- 5.2.5 The logistic of analysis, 1890 -- 5.2.6 Bettazzi on magnitudes, 1890 -- 5.3 The Rivista: Peano and his school, 1890-1895 -- 5.3.1 The 'society of mathematicians' -- 5.3.2 'Mathematical logic', 1891 -- 5.3.3 Developing arithmetic, 1891 -- 5.3.4 Infinitesimals and limits, 1892-1895 -- 5.3.5 Notations and their range, 1894 -- 5.3.6 Peano on definition by equivalence classes -- 5.3.7 Burali-Forti's textbook, 1894 -- 5.3.8 Burali-Forti's research, 1896-1897 -- 5.4 The Formulaire and the Rivista, 1895-1900 -- 5.4.1 The first edition of the Formulaire, 1895 -- 5.4.2 Towards the second edition of the Formulaire, 1897 -- 5.4.3 Peano on the eliminability of 'the' -- 5.4.4 Frege versus Peano on logic and definitions -- 5.4.5 Schröder's steamships versus Peano's sailing boats -- 5.4.6 New presentations of arithmetic, 1898 -- 5.4.7 Padoa on classhood, 1899 -- 5.4.8 Peano's new logical summary, 1900 -- 5.5 Peanists in Paris, August 1900 -- 5.5.1 An Italian Friday morning -- 5.5.2 Peano on definitions -- 5.5.3 Burali-Forti on definitions of numbers -- 5.5.4 Padoa on definability and independence -- 5.5.5 Pieri on the logic of geometry -- 5.6 Concluding comments: the character of Peano's achievement -- 5.6.1 Peano's little dictionary, 1901 -- 5.6.2 Partly grasped opportunities -- 5.6.3 Logic without relations
- 4.3.6 Peirce's second phase, 1881: finite and infinite -- 4.3.7 Peirce's students, 1883: duality, and 'Quantifying' a proposition -- 4.3.8 Peirce on 'icons' and the order of 'quantifiers', 1885 -- 4.3.9 The Peirceans in the 1890s -- 4.4 German algebraic logic: from the Grassmanns to Schröder -- 4.4.1 The Grassmanns on duality -- 4.4.2 Schröder's Grassmannian phase -- 4.4.3 Schröder's Peircean 'lectures' on logic -- 4.4.4 Schröder's first volume, 1890 -- 4.4.5 Part of the second volume, 1891 -- 4.4.6 Schröder's third volume, 1895: the 'logic of relatives' -- 4.4.7 Peirce on and against Schröder in The monist, 1896-1897 -- 4.4.8 Schröder on Cantorian themes, 1898 -- 4.4.9 The reception and publication of Schröder in the 1900s -- 4.5 Frege: arithmetic as logic -- 4.5.1 Frege and Frege -- 4.5.2 The 'concept-script' calculus of Frege's 'pure thought', 1879 -- 4.5.3 Frege's arguments for logicising arithmetic, 1884 -- 4.5.4 Kerry's conception of Fregean concepts in the mid 1880s -- 4.5.5 Important new distinctions in the early 1890s -- 4.5.6 The 'fundamental laws' of logicised arithmetic, 1893 -- 4.5.7 Frege's reactions to others in the later 1890s -- 4.5.8 More 'fundamental laws' of arithmetic, 1903 -- 4.5.9 Frege, Korselt and Thomae on the foundations of arithmetic -- 4.6 Husserl: logic as phenomenology -- 4.6.1 A follower of Weierstrass and Cantor -- 4.6.2 The phenomenological 'philosophy of arithmetic', 1891 -- 4.6.3 Reviews by Frege and others -- 4.6.4 Husserl's 'logical investigations', 1900-1901 -- 4.6.5 Husserl's early talks in Göttingen, 1901 -- 4.7 Hilbert: early proof and model theory, 1899-1905 -- 4.7.1 Hilbert's growing concern with axiomatics -- 4.7.2 Hilbert's different axiom systems for Euclidean geometry, 1899-1902 -- 4.7.3 From German completeness to American model theory
- CHAPTER 6 Russell's Way In: From Certainty to Paradoxes, 1895-1903