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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 9 1.3.1 References and the bibliography 9 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 'ideologic' 2.2.2 Lagrange's algebraic philosophy 15 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 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 2.4.2 De Morgan's philosophies of algebra 25 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 quantificationof'the predicate' 2.4.7 The logic of two-place relations, 1860 32 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 39 2.5.3 The mathematical analysis of logic, 1847: 'elective symbols' and laws 2.5 .4 'Nothing' and the 'Universe' 42 2.5.5 Propositions, expansion theorems, and solutions VI CONTENTS 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

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