Philosophy of Mathematics in Antiquity and in Modern Times

Description

»Philosophy of Mathematics« is understood, in this book, as an effort to clarify such questions that mathematics itself raises but cannot answer with its own methods. These include, for example, questions about the ontological status of mathematical objects (e.g., what is the nature of mathematical objects?) and the epistemological status of mathematical theorems (e.g., from what sources do we draw when we prove mathematical theorems?). The answers given by Plato, Aristotle, Euclid, Descartes, Locke, Leibniz, Kant, Cantor, Frege, Dedekind, Hilbert and others will be studied in detail. This will lead us to deep insights, not only into the history of mathematics, but also into the conception of mathematics as it is commonly held in the present time. The book is a translation from the German, however revised and considerably expanded. Various chapters have been completely rewritten. Preface Table of Contents Introduction Part I Philosophy of Mathematics in Antiquity Chapter 1 The Concept of Mathematics 1.1 The discovery of incommensurable quantities 1.2 The concept of ‘mathematics’ 1.3 The occurrence of ontological problems References Chapter 2 PLATO's Philosophy of Mathematics 2.1 PLATO's views on the teaching of mathematics: mathesis as anamnesis 2.2 The Platonic doctrine of ‘ideas’ 2.3 The world of mathematical objects 2.4 The construction of a mathematical theory according to PLATO 2.5 Of ideas, notions and concepts 2.6 Concluding remarks References Chapter 3 The Aristotelian Conception of Mathematics 3.1 The Aristotelian concept of a scientific theory 3.2 The Aristotelian Apodeixis 3.3 The ontological status of mathematical objects 3.4. Aphairesis (Αφαίρεσις ) 3.5. Chôrismós (Χωρισμός) 3.6 The foundation of Arithmetic according to ARISTOTLE 3.7 The foundation of geometry according to ARISTOTLE References Chapter 4 The Axiomatic Method of EUCLID 4.1 The The 'Elements' (Στοιχεῖα) of EUCLID 4.2 The terminology in the 'Elements' of EUCLID 4.3 What should the 'definitions' achieve? 4.4 What should the 'common notions' achieve? 4.5 What should the 'postulates' (αἰτήματα) achieve? 4.6 Axioms, postulates, hypotheses and lambanomena 4.7 The representation of Geometry in the 'Elements' of EUCLID 4.8 The arguments in the problems, resp. theorems I,1 and I,2 and I,4 4.9 Discussion References Chapter 5 Finitism in Greek Mathematics 5.1 Actual and potential infinity (ARISTOTLE) 5.2 Drawing perpendicular straight lines in the 'Elements' of EUCLID 5.3 The concept of parallelism (EUCLID) 5.4 The number of grains of sand (‘The Sand-Reckoner’ of ARCHIMEDES) 5.5 The existence of infinitely many prime numbers (EUCLID) 5.6 The exhaustion method (EUDOXOS) 5.7 Proofs of irrationality (HIPPASUS) 5.8 The exclusion of the 'unlimited' References Chapter 6 The Paradoxes of ZENO 6.1 The Zenonian Paradoxes 6.2 The effect of ZENO's paradoxes in the Middle Ages 6.3 The question of the existence of actual infinite quantities is critically examined 6.4 BURIDAN's treatment of the infinity problem according to the method of ‘sic et non’ 6.5 Concluding remarks References Part II Philosophy of Mathematics in the 16th, 17th and 18th Century Chapter 7 On Certainty in Mathematics 7.1 The publication of the works of EUCLID and PROCLUS in the original Greek 7.2 The differences between Aristotelian ‘apodeixis’ and Euclidean ‘demonstration’ 7.3 The dispute over the question as to whether Euclidean geometry is a science in the Aristotelian sense, or not 7.4 ARISTOTLE's own argument 7.5 Discussion References Chapter 8 The Cartesian Nativism 8.1 The Divine Origin of Mathematics 8.2 The Greek and Roman Stoics 8.3 The mathematical objects as thoughts of God (AUGUSTINUS) 8.4 RENÉ DESCARTES: Mathematical laws as edicts of a deity 8.5 DESCARTES’ nativism 8.6 The ideas of mathematical objects 8.7 DESCARTES' concept of ‘intuition’ 8.8 DESCARTES’ Essay 'La Géometrie' 8.9 Discussion References Chapter 9 JOHN LOCKE’s thoughts on Mathematics 9.1 LOCKE’s doctrine of ‘Ideas’ 9.2 Abstraction and general ideas 9.3 The abstract idea of a triangle 9.4 LOCKE's comments on the concept of the number 9.5 LOCKE's comments on some geometrical theorems 9.6 Psychologism in LOCKE's work 9.7 Discussion References Chapter 10 Rationalism 10.1 The problem of definitions in geometry 10.2 On refraining from defining the basic concepts (DESCARTES, PASCAL, ARNAULD) 10.3 The attempt to define the basic concepts with 'genetic definitions' 10.4 The contributions of HOBBES (1655) and BARROW (1664) 10.5 The contribution of LEIBNIZ (ca. 1676) 10.6 LEIBNIZ's 'Dialogue for an Introduction to Arithmetic and Algebra' (ca. 1676) 10.7 Proof of the axioms of equality 10.8 The concept of axiomatics in TSCHIRNHAUS (1687) 10.9 ‘The method of teaching Mathematics’ according to CHRISTIAN WOLFF 10.10 Discussion References Chapter 11 Empiricism in Mathematics 11.1. BERKELEY's critique 11.2 DAVID HUME's scepticism 11.3 JOHN STUART MILL's critique 11.4 Discussion References Chapter 12 KANT’s Conception of Mathematics 12.1 KANT's curriculum vitae 12.2 KANT's ‘Critique of Pure Reason’ 12.3 The distinction: a priori - a posteriori 12.4 The distinction: analytic - synthetic 12.5 The synthetic character of geometric propositions 12.6 The synthetic character of arithmetic theorems 12.7 Of ‘pure intuition’ and ‘empirical intuition’ 12.8 The a priori character of geometrical judgments 12.9 The a priori character of arithmetical judgments 12.10 Discussion References Part III Philosophy of Mathematics in the 19th and early 20th Century Chapter 13 Psychologism in Mathematics 13.1 Psyche, anima, mind and soul 13.2 The role of the psyche in ancient mathematics 13.3 The emergence of psychologism in the modern age 13.4 DEDEKIND’s creation of irrational numbers 13.5 On the creation of natural numbers 13.6 On definition by abstraction 13.7 Concluding remarks References Chapter 14 Logicism 14.1 FREGE’s logicism 14.2 FREGE’s foundation of arithmetic from the point of view of logicism 14.3 The appearance of antinomies References Chapter 15 The Concept of a ‘Set’ 15.1 The concept of a ‘set’ in classical antiquity 15.2 The BOLZANO concept of a ‘set’ (‘Menge’) 15.3 Cantorian set theory 15.4 The occurrence of set-theoretical antinomies 15.5 The Cantorian concept of a ‘set’ (‘Menge’) 15.6 An implicit definition of the concept of a 'set' (ZERMELO, QUINE, et al.) References Chapter 16 Contemporary Platonism 16.1 BOLZANO's Platonism 16.2 The usefulness of Platonism 16.3 The restricted (or weak) Platonism 16.4 GÖDEL's Platonism 16.5 GÖDEL's vindication of Platonism 16.6 Discussion References Chapter 17 The Problem of non-constructive Proofs of Existence 17.1 The ‘existence’ of transcendental real numbers 17.2 The ‘existence’ of roots of polynomials 17.3 Proofs of existence in ancient mathematics 17.4 GAUSS: ‘notio’ or ‘notatio’? 17.5 The Hilbertian basis theorem 17.6 Fast primality tests 17.7 Discussion References Chapter 18 The formal and the contentual Position 18.1 The contentual and the formal point of view 18.2 ‘Symbols’ and ‘empty signs’ 18.3 The dispute over the introduction of negative numbers 18.4 Combining the contentual and the formal standpoints 18.5 FREGE's polemics against HANKEL's formal standpoint 18.6 Résumé References Chapter 19 DEDEKIND and the emergence of Structuralism 19.1 The traditional concepts of the number 19.2 DEDEKIND's simply infinite systems 19.3 Properties of simply infinite systems 19.4 The different concepts of ‘abstraction’ 19.5 The problem of the existence of infinite systems 19.6 The axiomatization of Arithmetic (DEDEKIND, PEANO) 19.7 The emergence of Structuralism 19.8 A new approach to Abstraction 19.9 The 'abstract' direction in Algebra 19.10 Final considerations References Chapter 20 HILBERT's critical Philosophy 20.1 HILBERT's Philosophy of Mathematics 20.2 HILBERT's axiomatization of Geometry 20.3 HILBERT's critical study of Geometry 20.4 HILBERT's concept of an Axiomatic Theory and his Metamathematics References Epilogue E.1 Of Concepts and defining concepts by implicit definitions E.2 Mathematical theories are defined by the 'frameworks of their concepts' E.3 The Objects of a Mathematical Theory E.4 Deepening the level of the foundations Index of Names Index of Subjects Index of Abbreviations