Philosophy of Mathematics in Antiquity and in Modern Times

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