Mathematics and Philosophy

Parrochia Daniel, Université Jean Moulin, Lyon.

• Introduction xiPart 1. The Contribution of Mathematician–Philosophers 1Introduction to Part 1 3Chapter 1. Irrational Quantities 71.1. The appearance of irrationals or the end of the Pythagorean dream 81.2. The first philosophical impact 91.3. Consequences of the discovery of irrationals 111.3.1. The end of the eternal return 111.3.2. Abandoning the golden ratio 111.3.3. The problem of di3.1sorder in medicine, morals and politics 121.4. Possible solutions 121.5. A famous example: the golden number 141.6. Plato and the dichotomic processes 161.7. The Platonic generalization of ancient Pythagoreanism 171.7.1. The Divided Line analogy 171.7.2. The algebraic interpretation 181.7.2.1. Impossibilities 191.7.2.2. The case where k = Ø 191.8. Epistemological consequences: the evolution of reason 20Chapter 2. All About the Doubling of the Cube 232.1. History of the question of doubling a cube 242.2. The non-rationality of the solution 242.2.1. Demonstration 242.2.2. The diagonal is not a solution 252.3. The theory proposed by Hippocrates of Chios 252.4. A philosophical application: platonic cosmology 272.5. The problem and its solutions 292.5.1. The future of the problem 292.5.2. Some solutions proposed by authors of the classical age 302.5.2.1. Mechanical solutions 302.5.2.2. Analytical solution 312.5.3. The doubling of the cube – going beyond Archytas: the evolution of mathematical methods 362.5.3.1. Menaechmus’ solution 372.5.3.2. A brief overview of the other solutions 392.6. The trisection of an angle 402.6.1. Bold mathematicians 402.6.2. Plato, the tripartition of the soul and self-propulsion 422.6.3. A very essential shell 442.6.4. A final excercus 462.7. Impossible problems and badly formulated problems 462.8. The modern demonstration 47Chapter 3. Quadratures, Trigonometry and Transcendance 513.1. π – the mysterious number 523.2. The error of the “squarers” 533.3. The explicit computation of π 553.4. Trigonometric considerations 573.5. The paradoxical philosophy of Nicholas of Cusa 593.5.1. An attempt at computing an approximate value for π 593.5.2. Philosophical extension 613.6. What came next and the conclusion to the history of π 633.6.1. The age of infinite products 643.6.2. Machin’s algorithm 643.6.3. The problem of the nature of π 653.6.4. Numerical and philosophical transcendance: Kant, Lambert and Legendre 66Part 2. Mathematics Becomes More Powerful 69Introduction to Part 2 71Chapter 4. Exploring Mathesis in the 17th Century 754.1. The innovations of Cartesian mathematics 764.2. The “plan” for Descartes’ Geometry 794.3. Studying the classification of curves 794.3.1. Possible explanations for the mistakes made by the Ancients 814.3.2. Conditions for the admissibility of curves in geometry 834.4. Legitimate constructions 854.5. Scientific consequences of Cartesian definitions 874.6. Metaphysical consequences of Cartesian mathematics 88Chapter 5. The Question of Infinitesimals 915.1. Antiquity – the prehistory of the infinite 925.1.1. Infinity as Anaximander saw it 925.1.2. The problem of irrationals and Zeno’s paradoxes 935.1.3. Aristotle and the dual nature of the Infinite 965.2. The birth of the infinitesimal calculus 985.2.1. Newton’s Writings 995.2.2. Leibniz’s contribution 1015.2.3. The impact of calculus on Leibnizian philosophy 1055.2.3.1. Small perceptions and differentials 1055.2.3.2. Matter and living beings 1095.2.3.3. The image of order 1105.2.4. The epistemological problem 117Chapter 6. Complexes, Logarithms and Exponentials 1216.1. The road to complex numbers 1226.2. Logarithms and exponentials 1256.3. De Moivre’s and Euler’s formulas 1286.4. Consequences on Hegelian philosophy 1306.5. Euler’s formula 1326.6. Euler, Diderot and the existence of God 1336.7. The approximation of functions 1346.7.1. Taylor’s formula 1356.7.2. MacLaurin’s formula 1356.8. Wronski’s philosophy and mathematics 1376.8.1. The Supreme Law of Mathematics 1386.8.2. Philosophical interpretation 1426.9. Historical positivism and spiritual metaphysics 1436.9.1. Comte’s vision of mathematics 1436.9.2. Renouvier’s reaction 1466.9.3. Spiritualist derivatives 1476.10. The physical interest of complex numbers 1486.11. Consequences on Bergsonian philosophy 150Part 3. Significant Advances 155Introduction to Part 3 157Chapter 7. Chance, Probability and Metaphysics 1617.1. Calculating probability: a brief history 1627.2. Pascal’s “wager” 1667.2.1. The Pensées passage 1667.2.2. The formal translation 1677.2.3. Criticism and commentary 1677.2.3.1. Laplace’s criticism 1677.2.3.2. Emile Borel’s observation 1697.2.3.3. Decision theory 1707.2.3.4. The non-standard analysis framework 1717.3. Social applications, from Condorcet to Musil 1727.4. Chance, coincidences and omniscience 174Chapter 8. The Geometric Revolution 1798.1. The limits of the Euclidean demonstrative ideal 1808.2. Contesting Euclidean geometry 1838.3. Bolyai’s and Lobatchevsky geometries 1848.4. Riemann’s elliptical geometry 1918.5. Bachelard and the philosophy of “non” 1948.6. The unification of Geometry by Beltrami and Klein 1968.7. Hilbert’s axiomatization 1988.8. The reception of non-Euclidean geometries 2008.9. A distant impact: Finsler’s philosophy 200Chapter 9. Fundamental Sets and Structures 2039.1. Controversies surrounding the infinitely large 2039.2. The concept of “the power of a set” 2079.2.1. The “countable” and the “continuous” 2089.2.2. The uniqueness of the continuum 2099.2.3. Continuum hypothesis and generalized continuum hypothesis . . .  2129.3. The development of set theory 2139.4. The epistemological route and others 2189.5. Analytical philosophy and its masters 2229.6. Husserl with Gödel? 2259.7. Appendix: Gödel’s ontological proof 226Part 4. The Advent of Mathematician-Philosophers 229Introduction to Part 4 231Chapter 10. The Rise of Algebra 23310.1. Boolean algebra and its consequences 23410.2. The birth of general algebra 23710.3. Group theory 23810.4. Linear algebra and non-commutative algebra 24110.5. Clifford: a philosopher-mathematician 245Chapter 11. Topology and Differential Geometry 25311.1. Topology 25311.1.1. Continuity and neighborhood 25411.1.2. Fundamental definitions and theorems 25511.1.3. Properties of topological spaces 25711.1.4. Philosophy of classifications versus topology of the being 26111.2. Models of differential geometry 26211.2.1. Space as a support to thought 26211.2.2. The general concept of manifold 26311.2.3. The formal concept of differential manifold 26411.2.4. The general theory of differential manifold 26511.2.5. G-structures and connections 26611.3. Some philosophical consequences 26811.3.1. Whitehead’s philosophy and relativity 26911.3.2. Lautman’s singular work 27011.3.3. Thom and the catastrophe theory 273Chapter 12. Mathematical Research and Philosophy 27912.1. The different domains 27912.2. The development of classical mathematics 28212.3. Number theory and algebra 28212.4. Geometry and algebraic topology 28412.5. Category and sheaves: tools that help in globalization 28612.5.1. Category theory 28612.5.2. The Sheaf theory 29212.5.3. Link to philosophy 29412.5.4. Philosophical impact 29512.6. Grothendieck’s unitary vision 29512.6.1. Schemes 29512.6.2. Topoi 29612.6.3. Motives 29812.6.4. Philosophical consequences of motives 301Conclusion 305Bibliography 311Index 327