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| Thomas Bedürftig, Roman Murawski
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| Philosophy of Mathematics |
| Walter De Gruyter, Berlin/Boston 2018.
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| xii + 457 stron.
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Spis treści
Vorwort
Preface
Introduction
1 On the way to the reals
1.1 Irrationality
1.2 Incommensurability
1.3 Calculating with 2?
1.4 Procedure of approximating, nesting of intervals and completness
1.5 On the construction of the reals
1.6 On the handling of the infinite
1.7 Infinite non-periodic decimal fraction
2 On the history of the philosophy of mathematics
2.1 Pythagoras and Pythagoreans
2.2 Plato
2.3 Aristotle
2.4 Euclid
2.5 Proclus Diadochus
2.6 Nicholas of Cusa
2.7 Descartes
2.8 Pascal
2.9 Leibniz
2.10 Kant
2.11 Mill and the empirical conceptions
2.12 Bolzano
2.13 Gauß
2.14 Cantor
2.15 Dedekind
2.16 Poincaré
2.17 Peirce's pragmatism and the world of symbols
2.18 Husserl's phenomenology
2.19 Logicism
2.20 Intuitionism
2.21 Construktivism
2.22 Formalism
2.23 Philosophy of mathematics between 1931 and the end of
the 1950s
2.24 The evolutionary point of view - a new basic position in philosophy
Grundposition
2.24.1 Characterization
2.24.2 On studies of evolution of number concept
2.24.3 Concluding remarks
2.25 Philosophy of mathematic after 1960
2.25.1 Quasi-empirical concepyion
2.25.2 Realism and antirealism
3 On fundamental questions of the philosophy of mathematics
3.1 On the concept of number
3.1.1 Survey of some views
3.1.2 Resume
3.2 Infinities
3.2.1 On problems with the infinite
3.2.2 Conception of Aristotele
3.2.3 Idealistic approach
3.2.4 Empiricist point of view
3.2.5 Infinity by Kant
3.2.6 Intuitionistic infinity
3.2.7 Logicist hyphothesis of the infinite
3.2.8 Infinity and the new philosophy of mathematics
3.2.9 Formalistic approach and nowadays tendencies
3.3 The continuum and the infinitely small
3.3.1 Genaral problem
3.3.2 On the history of continuum
3.3.3 What is point?
3.3.4 On the history of the continuum - continuation
3.3.5 Survey of conceptions of the continuum
3.3.6 Notes on the arithemtization of the continuum
3.3.7 The end of infinitesimals and the rediscovery of them
3.3.8 Nonstandard numbers and the continuum
3.3.9 Consequences for the conception of the continuum
3.3.10 Medium of free evolution
3.3.11 The disapperance of magnitudes
3.3.12 Final remarks
3.4 On the problem of applications of mathematics
3.4.1 Aspects of the problem
3.4.2 The problem of applications of mathematics
3.4.3 Classical positions
3.4.4 New concepions
3.4.5 Retrospect
3.5 Conclusion
3.5.1 From natural to rational numbers
3.5.2 Incommensurability and irrationality
3.5.3 Adjunction
3.5.4 On the linear continuum
3.5.5 Infinitely small quantities
3.5.6 Construction, infinity, infinite non-periodic decimals
fractions
3.5.7 Concluding remarks
4 Sets and set theories
4.1 Paradoxes of the infinite
4.2 On the concept of a set
4.2.1 Collecting together versus putting together
4.2.2 Sets and the problem of universals
4.3 Two set theories
4.3.1 Set theory according to Zermelo and Fraenkel
4.3.2 Von Neumann, Bernays and Gödel set theory
4.3.3 Remarks
4.3.4 On modifications
4.4 The Axiom of Choice and the Continuum Hypothesis
4.4.1 Search for new axioms
4.4.2 Futher remarks and questions
4.5 Final remarks
5 Axiomatic approach and logic
5.1 Some elements of mathematical logic
5.1.1 Syntax
5.1.2 Semantics
5.1.3 Calculus
5.2 Historical remarks
5.2.1 From the history of logic
5.2.2 On the history of the axiomatic approach
5.3 Logical axioms and theories
5.3.1 Peano arithmetic
5.3.2 On the axioms for real numbers
5.4 On the arithmetic of natural numbers
5.4.1 Syntactical aspects
5.4.2 Semantical aspects
5.5 Truth and provability
5.5.1 Formal truth
5.5.2 Completness and truth
5.5.3 Syntactic reduction of truth
5.5.4 Truth is unequal to provability
5.5.5 Search for the way out
5.5.6 Concluding remarks
5.6 Final conclusions
5.6.1 Logic as the background of mathematics
5.6.2 Consequence for the shape of mathematics
6 Thinking and calculating infinitesimally - First nonstandard steps
6.1 Preliminary remark
6.2 The question about 0,999...
6.2.1 Empirical approach
6.2.2 The problem
6.2.3 Answer
6.2.4 Final remarks
6.3 A bit of infinitesimal calculus
6.3.1 Infinitesimal calculations
6.3.2 Continuity, differential quptient, derivative
6.4 On the construction of hyperreal numbers
6.4.1 Hypernatural numbers
6.4.2 Hyperreal numbers
6.4.3 How will *R become a model of R?
6.4.4 On the justification of the naive infinitesimal calculus
6.5 On the status of nonstandard numbers
7 Retrospection
7.1 Setting of real numbers
7.2 Axiomatic method
7.3 The concept of number
7.4 Infinity, Axiom of Choice and Continuum Hyphothesis
7.5 Infinitely small and the continuum
7.6 Aplicability
7.7 Theoretical limits
7.8 Usage of computers
7.9 What is the philosophy of mathematics and what does it provide?
7.10 Evidence and transcendence
Biographies
Bibliography
Index of names
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