The Search for Mathematical Roots, 1870-1940

Abstract

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

The Monty Hall Problem and beyond: Digital-Mathematical and Cognitive Analysis in Boole’s Algebra, Including an Extension and Generalization to Related Cases

“…A recent voluminous book on the history of logistics from 1870 to 1940 documents all these contributions in detail. This account also reveals that it seems to be hardly the case that the ultimate roots of mathematics have been once and for all fully uncovered [5].…”

“…The structure of the problem is described as much as possible in the tradition and the spirit, and by means of the algebraic conventions, of George Boole's Investigation of the Laws of Thought (1854), the Magna Charta of the digital age, and of John Venn's Symbolic Logic (second edition, 1894), which is squarely based on Boole's Investigation and elucidates it in many ways [5].…”

“…And Equation (7) can be rewritten as Equation (5). As will be seen below, the deeper reason for this simplification is as follows.…”

“…In (1), (2), (3), (5), and (9), cars are picked in 3 or 4 of the 4 picks, but there are only 2 cars. In (8), (12), (14), (15), and (16), goats are picked i 3 or 4 of the 4 picks, but it is not possible to pick more than 2 in total.…”

El enfoque epistemológico de David Hilbert: el a priori del conocimiento y el papel de la lógica en la fundamentación de la ciencia

“…Antes de la axiomática de Zermelo y Fraenkel, Cantor había creído proporcionar uno de los métodos más efectivos para tratar el problema a partir de la axiomatización de los sistemas y concibiendo un conjunto como una sola entidad matemática que a su vez puede ser contenida por otro conjunto. El problema con este concepto era su imprecisión en la definición, aunque la idea intuitiva de "universo" o "dominio" de objetos, a partir del cual se construyen los conjuntos, tuvo siempre una plena aceptación entre los matemáticos (Grattan-Guinness 2000). De la misma manera era aceptado tratar esa entidad como un objeto singular.…”