Connecting Humans to Equations: A Reinterpretation of the Philosophy of Mathematics

197 Pages·2019·2.52 MB·English

by  Ole Ravn

History of Mathematics Education
Ole Ravn
Ole Skovsmose
Connecting
Humans to
Equations
A Reinterpretation of the Philosophy of
Mathematics
History of Mathematics Education
Series Editors
Nerida F. Ellerton
Illinois State University
Normal, IL, USA
M. A. (Ken) Clements
Illinois State University
Normal, IL, USA
More information about this series at http://www.springer.com/series/13545
Ole Ravn • Ole Skovsmose
Connecting Humans to Equations
A Reinterpretation of the Philosophy of Mathematics
Ole Ravn Ole Skovsmose
Department of Learning and Philosophy Department of Learning and Philosophy
Aalborg University Aalborg University
Aalborg, Denmark Aalborg, Denmark
Department of Mathematics Education
State University of São Paulo
(Universidade Estadual Paulista, Unesp)
São Paulo, Brazil
ISSN 2509-9736 ISSN 2509-9744 (electronic)
History of Mathematics Education
ISBN 978-3-030-01336-3 ISBN 978-3-030-01337-0 (eBook)
https://doi.org/10.1007/978-3-030-01337-0
Library of Congress Control Number: 2018959751
© Springer Nature Switzerland AG 2019
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Acknowledgements
On the following pages we have attempted to present some of the most important posi-
tions in the philosophy of mathematics. At the same time, we have tried to add new dimensions
to this special domain of philosophy. It is our hope that our presentation can help to broaden
the conception of mathematics as a general human concern. We see mathematics as an integral
part of human social life, meaning that a philosophy of mathematics turns relevant to any kind
of social theorising.
In 2011 we published the book Matematikfilosofi in Danish. The present book is not a
translation of the Danish version; it includes reformulations, specifications and additions.
During the process we realised that we were working with a new manuscript.
We have received help and comments from many people. We want to thank Ana Carolina
Faustino, Denival Biotto Filho, Peter Gates, Renato Marcone, Raquel Milani, Amanda Queiroz
Moura, João Luiz Muzinatti, Miriam Godoy Penteado, Marianne Ravn, Guilherme Henrique
Gomes da Silva, Lucas Penteado Lopes da Silva, Paola Valero, and Keiko Yasukawa for all
their support. Finally we want to thank Sarah Freire-Gibb and the good people at The Golden
Pen for preparing the manuscript for the publisher and Guilherme Francisco Ferreira for mak-
ing the figures and tables.
Aalborg, 2018 Ole Ravn and Ole Skovsmose
v
Contents
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Abstracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii
Part I Where is Mathematics?
1 Mathematics in Eternity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Platonism after Plato . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Platonism before Plato . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Plato’s Platonism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Axiomatisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Euclid’s Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 Mathematics in Nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Celestial Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Circles and Ellipses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Earthly Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Mathematics Everywhere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Unnatural Mathematics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Mathematics in Mind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Mathematics as Mental Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Mathematics as Associations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Mathematics as Forms of Understanding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Incomprehensible Mathematics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Part II How Certain is Mathematics?
4 A Solid House of Cards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Frege’s Criticism of Former Philosophy of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . 56
Begriffsschrift—A Philosophic Vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Begriffsschrift—A Logical System of Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Begriffsschrift—A Farewell to Subjects with Predicates . . . . . . . . . . . . . . . . . . . . . . . . . 61
Mathematics as Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
A Mind-Blowing Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Principia Mathematica—A Begriffsschrift after the Blow . . . . . . . . . . . . . . . . . . . . . . . 66
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
vii
viii Contents
5 The Magnificent Disaster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Tables, Chairs and Beer Mugs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Mathematics about Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
The Systems Are Not What They Seem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Gödel’s Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6 Mathematics Without Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Tractatus: What is Science? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Tractatus: What is a Tautology? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Tractatus: What Can be Said? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Logical Positivism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Part III How Social is Mathematics?
7 Mathematics as Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Brouwer’s Intuitionism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Beyond the Border of Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Intuitionism on a Fine Summer Day . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Implications for Doing Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
8 Mathematics as Dialogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Popper: Conjectures and Refutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Lakatos: Proofs and Refutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Euler’s Polyhedron Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
The Method of Proofs and Refutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
The Quasi-Empirical Social Turn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
9 Mathematics as Grammar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Wittgenstein Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Mathematics as Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
Mathematics as Rule-Following . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Mathematics as Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Proving as Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Certain or Fallible Knowledge? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Part IV How Good is Mathematics?
10 B eyond the Isolation of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Metaphysics: The Case of Geometric Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Metaphysics: The Case of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Technology: Computer-Based Investigations in Mathematics . . . . . . . . . . . . . . . . . . . . . 141
Technology: Computer-Based Imagination and Proving . . . . . . . . . . . . . . . . . . . . . . . . . 143
Contents ix
Anything But Isolated: Intertwined . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
11 Beyond the Neutrality of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Mathematics as Picturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Mathematics as Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Fabrication of Fictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Fabrication of Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Fabrication of Risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
Illusions of Objectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Anything but Neutral: Powerful . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
12 What is the Philosophy of Mathematics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Four Dimensions in a Philosophy of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
Beyond the Four Dimensions? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
Beyond the “Western” in a Philosophy of Mathematics? . . . . . . . . . . . . . . . . . . . . . . . . 167
Beyond “Philosophy” in a Philosophy of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . 170
Beyond Our Limited Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
Composite Reference List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
List of Figures
Figure 1.1 The most beautiful number according to Pythagoras. .................... 8
Figure 1.2 Illustration of 1 + 3 + 5 + … (2n + 1) = (n + 1)2. ........................ 9
Figure 1.3 Plato’s graduation of knowledge and opinion. .......................... 14
Figure 1.4 Construction of an equilateral triangle. ............................... 18
Figure 2.1 Kepler’s second law for the movements of the planets: The law of areas.. . . . . 25
Figure 3.1 Construction of the Cantor set, which is continued indefinitely. ............ 47
Figure 3.2 Construction of the Sierpinski triangle. ............................... 47
Figure 3.3 Construction of Peano’s curve. ..................................... 48
Figure 8.1 A box as an example of a polyhedron. ................................ 111
Figure 8.2 An unfolded box, Step 1 in the proof. ................................ 112
Figure 8.3 A triangulated box, Step 2 in the proof. ............................... 112
Figure 8.4 Removing triangles, Step 3 in the proof. .............................. 113
Figure 8.5 A hollow polyhedron. ............................................. 115
Figure 8.6 Two “deviant” polyhedrons. ........................................ 116
Figure 10.1 The plane, catenoid and helicoid. .................................... 144
Figure 10.2 Costa’s minimal surface as a computer picture. ......................... 145
Figure 11.1 The standard model of mathematical modelling. ........................ 153
xi
List of Tables
Table 3.1 Kant’s Knowledge Categories for Propositions .......................... 42
Table 5.1 A Gödel Numbering of Symbols by Primes ............................. 78
Table 6.1 The Truth-Table for Four Logical Connectives .......................... 87
Table 6.2 The Truth-Table for a Logical Valid Statement .......................... 88
Table 10.1 Number of Publications Relating to Wavelet- and Fourier-Transformations .... 143
xiii