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E-book: Essence of Numbers

  • Format: EPUB+DRM
  • Series: Lecture Notes in Mathematics 2278
  • Pub. Date: 06-Oct-2020
  • Publisher: Springer Nature Switzerland AG
  • Language: eng
  • ISBN-13: 9783030567002
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Essence of NumbersLarger Image
  • Format: EPUB+DRM
  • Series: Lecture Notes in Mathematics 2278
  • Pub. Date: 06-Oct-2020
  • Publisher: Springer Nature Switzerland AG
  • Language: eng
  • ISBN-13: 9783030567002
Other books in subject:

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This book considers the manifold possible approaches, past and present, to our understanding of the natural numbers. They are treated as epistemic objects: mathematical objects that have been subject to epistemological inquiry and attention throughout their history and whose conception has evolved accordingly. Although they are the simplest and most common mathematical objects, as this book reveals, they have a very complex nature whose study illuminates subtle features of the functioning of our thought.

Using jointly history, mathematics and philosophy to grasp the essence of numbers, the reader is led through their various interpretations, presenting the ways they have been involved in major theoretical projects from Thales onward. Some pertain primarily to philosophy (as in the works of Plato, Aristotle, Kant, Wittgenstein...), others to general mathematics (Euclid's Elements, Cartesian algebraic geometry, Cantorian infinities, set theory...).

Also serving as an introduction to the works and thought of major mathematicians and philosophers, from Plato and Aristotle to Cantor, Dedekind, Frege, Husserl and Weyl, this book will be of interest to a wide variety of readers, from scholars with a general interest in the philosophy or mathematics to philosophers and mathematicians themselves.

Reviews

Patras' book is an enjoyable read, highlighting the complexities of the familiar numbers, engaging a phenomenological approach which remains fairly rare in the philosophy of mathematics. (Emmylou Haffner, Mathematical Reviews, May, 2022)

1 Introduction
1(6)
1.1 Greek Origins
3(1)
1.2 The Contribution of Mathematics
3(1)
1.3 Gottlob Frege
4(1)
1.4 From Arithmetic to Algebra
5(2)
2 The Lasting Influence of Pythagorism
7(16)
2.1 Numbers in the Pythagorean School and Numerology
8(3)
2.2 Presocratic Philosophers and the Possibility of Scientific Knowledge
11(1)
2.3 Parmenides, the One, and the Birth of Metaphysics
12(2)
2.4 Platonic Ideal Numbers
14(1)
2.5 The Lasting Influence of Neo-Platonism
15(2)
2.6 Modern Structuralism and Pythagorism
17(3)
2.7 Aristotle on Numbers and Mathematics
20(3)
3 The One and the Multiple
23(12)
3.1 From Plato to Frege
25(1)
3.2 From Frege to Wittgenstein
26(2)
3.3 Aristotle: The Whole and the Parts
28(4)
3.4 The One, the Multiple, Man
32(3)
4 Mathematics and Reality
35(10)
4.1 The Emergence of the Problem of Participation
37(1)
4.2 The Logico-Semantic Approach
38(1)
4.3 Nominalism
39(1)
4.4 Schematism of Concepts
40(2)
4.5 Schematism and Pragmatism
42(3)
5 The Third Man Argument
45(12)
5.1 Any and Everything
46(4)
5.2 The Infinity of Ideas
50(2)
5.3 The Recursive Construction of Numbers
52(1)
5.4 Type Theory
52(5)
6 Numbers and Magnitudes
57(12)
6.1 The Notion of Quantity
58(1)
6.2 Number as Measure
58(2)
6.3 One Is Not a Number
60(2)
6.4 The Ontological Difference
62(1)
6.5 Arithmetic and Geometry in Euclid
63(2)
6.6 The Cartesian Revolution
65(2)
6.7 The Primacy of Arithmetic
67(1)
6.8 The Problem of Infinity
67(2)
7 Generalized Numbers I
69(10)
7.1 Zero
69(2)
7.2 Phenomenology of Zero and One
71(2)
7.3 Zero and One as Numbers
73(1)
7.4 Negative Numbers
74(1)
7.5 Methodological Clarifications
75(1)
7.6 Kant and the Enlightenment
76(3)
8 Generalized Numbers II
79(12)
8.1 Complex Numbers
79(1)
8.2 The Fundamental Theorem of Algebra
80(2)
8.3 The Geometrical Foundation
82(1)
8.4 Frege and Complex Numbers
83(2)
8.5 The Principle of Permanence of Formal Laws
85(2)
8.6 The Symbolic Approach
87(1)
8.7 Cauchy's Point of View
88(1)
8.8 The Algebraic Approach
89(2)
9 Cantor and Set Theory
91(10)
9.1 The Path of Analysis
92(1)
9.2 Measuring Infinity
93(2)
9.3 The Diagonal Argument
95(1)
9.4 Set Theory
96(1)
9.5 The Concept of Set
97(1)
9.6 The Legitimacy of Infinity
98(1)
9.7 The Activity of Thought
98(3)
10 Frege's Logicism
101(10)
10.1 The "Platonism" of Frege
103(1)
10.2 Frege's Relationalism
103(1)
10.3 Concept and Object
104(1)
10.4 A Priori Analysis and Synthesis
105(2)
10.5 Arithmetic Statements
107(1)
10.6 Unity
108(3)
11 Set Theory in Frege
111(10)
11.1 Mathematical Characterization of Number Statements
111(1)
11.2 The Logical Step of the Foundations of Arithmetic
112(1)
11.3 The Copernican Arithmetic Revolution
113(1)
11.4 Frege's Ontology: 0 and 1
114(1)
11.5 The Invention of the Empty Set Symbol
115(1)
11.6 Hermann Weyl's Analysis and the Later Frege
116(1)
11.7 Weyl on Natural Numbers
117(4)
12 Axioms and Formalisms
121(14)
12.1 Dedekind
123(2)
12.2 Peano
125(1)
12.3 Hilbert
126(3)
12.4 Godel
129(1)
12.5 Moderate Platonism
130(2)
12.6 Transfinite Numbers
132(3)
13 The Brain and Cognitive Processes
135(8)
13.1 The Real Distinction Between Soul and Body
135(1)
13.2 On Faculties, Time and Space
136(1)
13.3 Learning Arithmetic
137(1)
13.4 The Organ of Numbers
138(1)
13.5 Didactics
138(2)
13.6 Complexity of Numbers
140(1)
13.7 Malleability of the Brain
140(3)
14 Phenomenology of Numbers
143(12)
14.1 The Problem of Origins
144(1)
14.2 Three Points of View
145(1)
14.3 The Philosophy of Arithmetic
146(1)
14.4 Proper Representations and Intuitions
147(2)
14.5 Back to One
149(1)
14.6 The Transcendental Point of View
149(2)
14.7 The Problem of Symbolism
151(1)
14.8 Improper Representations
152(1)
14.9 Horizon Structures
153(2)
15 Universal Phenomena, Algebra, Categories
155(10)
15.1 Numbers as Invariants
157(3)
15.2 Universal Problems
160(2)
15.3 Phenomenology of Algebra
162(3)
Epilogue 165(2)
References 167(4)
Index 171
Frédéric Patras, alumnus of the École Normale Supérieure and research director at CNRS, is a mathematician who has long been committed to philosophical studies. Also the author of a book on contemporary mathematical thinking (La pensée mathématique contemporaine), he has published and edited over a hundred works on various subjects. He is interested in what the philosophical tradition can bring to our current understanding of science and mathematics.
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