Philosophy of Mathematics: Classic and Contemporary Studies

353 Pages·2021·7.144 MB·English

by  Ahmet Cevik

Philosophy of
Mathematics
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Philosophy of Mathematics
Classic and Contemporary Studies
Ahmet Çevik
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BOOMTH
Philosophy of
Mathematics
Classic and Contemporary
Studies
Ahmet Çevik
First edition published 2022
by CRC Press
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ISBN: 9781032121284 (hbk)
ISBN: 9781032022680 (pbk)
ISBN: 9781003223191 (ebk)
DOI: 10.1201/9781003223191
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Contents
Preface ix
1 Introduction 1
2 Mathematical Preliminaries 11
2.1 Summary of Propositional and Predicate Logic . . . . . . . . 11
2.1.1 Propositional logic . . . . . . . . . . . . . . . . . . . . 11
2.1.2 Predicate logic . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Methods of Proof . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.1 Direct proof . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.2 Proof by contrapositive . . . . . . . . . . . . . . . . . 20
2.2.3 Proof by contradiction . . . . . . . . . . . . . . . . . . 21
2.2.4 Proof by induction . . . . . . . . . . . . . . . . . . . . 23
2.2.5 Proof fallacies. . . . . . . . . . . . . . . . . . . . . . . 26
* 2.3 Basic Mathematical Notions . . . . . . . . . . . . . . . . . . 27
2.3.1 Axioms of ZFC set theory . . . . . . . . . . . . . . . . 28
2.3.2 Ordinal and cardinal numbers . . . . . . . . . . . . . . 36
3 Platonism 45
3.1 Theory of Forms . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 Plato’s Epistemological Philosophy . . . . . . . . . . . . . . 47
3.3 Aristotelian Realism . . . . . . . . . . . . . . . . . . . . . . . 51
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4 Intuitionism 57
4.1 Kant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 Brouwer and Constructivism . . . . . . . . . . . . . . . . . . 66
5 Logicism 81
5.1 Frege . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.2 Russell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.3 Carnap and Logical Positivism . . . . . . . . . . . . . . . . . 89
6 Formalism 95
6.1 Term vs. Game Formalism . . . . . . . . . . . . . . . . . . . 96
6.2 Hilbert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.3 G¨odel’s Impact . . . . . . . . . . . . . . . . . . . . . . . . . . 105
v
vi Contents
* 7 Go¨del’s Incompleteness Theorem and Computability 109
7.1 Arithmetisation of Syntax . . . . . . . . . . . . . . . . . . . 110
7.2 Primitive Recursive Functions . . . . . . . . . . . . . . . . . 112
7.3 Diagonalisation . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.4 Second Incompleteness Theorem . . . . . . . . . . . . . . . . 117
7.5 Speculations on G¨odel’s Theorems . . . . . . . . . . . . . . . 118
7.6 “Real” Mathematics vs. “Ideal” Mathematics . . . . . . . . . 120
7.7 Reasons Behind Incompleteness . . . . . . . . . . . . . . . . 122
7.7.1 Entscheidungsproblem . . . . . . . . . . . . . . . . . . 123
7.7.2 Irreducible information . . . . . . . . . . . . . . . . . 125
8 The Church-Turing Thesis 131
8.1 Minds and Machines . . . . . . . . . . . . . . . . . . . . . . . 131
8.2 Effective Computability . . . . . . . . . . . . . . . . . . . . . 133
8.3 The New Pythagoreanism . . . . . . . . . . . . . . . . . . . . 136
9 Infinity 141
9.1 Infinity in Ancient Greece . . . . . . . . . . . . . . . . . . . . 141
9.2 Middle Ages, the Renaissance, and the Age of Enlightenment 146
* 9.3 Cantor’s Set Theory . . . . . . . . . . . . . . . . . . . . . . . 152
* 10 Supertasks 159
10.1 Transfinite Computability and Continuity . . . . . . . . . . . 159
10.2 Infinite Time Turing Machines . . . . . . . . . . . . . . . . . 161
10.3 Physical Realisations . . . . . . . . . . . . . . . . . . . . . . 166
* 11 Models, Completeness, and Skolem’s Paradox 171
11.1 G¨odel’s Completeness Theorem . . . . . . . . . . . . . . . . . 176
11.2 Lo¨wenheim-Skolem Theorem . . . . . . . . . . . . . . . . . . 180
12 Axiom of Choice 185
12.1 Applications of the Axiom of Choice . . . . . . . . . . . . . . 186
12.1.1 Russell’s Example . . . . . . . . . . . . . . . . . . . . 186
12.1.2 K¨onig’s lemma . . . . . . . . . . . . . . . . . . . . . . 188
12.2 Which statements are obvious? . . . . . . . . . . . . . . . . . 189
12.2.1 Countable unions . . . . . . . . . . . . . . . . . . . . . 190
12.2.2 Countable pairs of “identical” objects . . . . . . . . . 191
12.3 Determining the Naturality or Otherwise . . . . . . . . . . . 193
* 12.3.1 Axiom of determinacy . . . . . . . . . . . . . . . . . . 200
12.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . 203
13 Naturalism 205
13.1 G¨odel’s Realism . . . . . . . . . . . . . . . . . . . . . . . . . 205
13.2 Quine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
13.3 Maddy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
13.3.1 The naturalist second philosopher . . . . . . . . . . . 217
Contents vii
14 Structuralism 223
14.1 Characteristic Properties . . . . . . . . . . . . . . . . . . . . 223
14.2 Identification Problem . . . . . . . . . . . . . . . . . . . . . . 228
14.3 Eliminative Structuralism . . . . . . . . . . . . . . . . . . . . 232
15 Yablo’s Paradox 239
15.1 Self-reference and Impredicative Definitions . . . . . . . . . . 239
15.2 What is Yablo’s Paradox? . . . . . . . . . . . . . . . . . . . . 241
* 15.3 Priest’s Inclosure Schema and ω-inconsistency . . . . . . . . 242
16 Mathematical Pluralism 247
16.1 Plurality of Models . . . . . . . . . . . . . . . . . . . . . . . 250
16.2 Multiverse Conception of Sets . . . . . . . . . . . . . . . . . 255
16.3 Liberating the Mathematical Ontology or Blurring the
Mathematical Truth? . . . . . . . . . . . . . . . . . . . . . . 262
* 17 Does Mathematics Need More Axioms? 265
17.1 Status of the Continuum Hypothesis . . . . . . . . . . . . . . 265
17.1.1 Axiom of Constructibility . . . . . . . . . . . . . . . . 269
17.2 Inner Model Programme . . . . . . . . . . . . . . . . . . . . 277
17.3 Hyperuniverse Programme . . . . . . . . . . . . . . . . . . . 281
18 Mathematical Nominalism 287
18.1 Problems of Realism . . . . . . . . . . . . . . . . . . . . . . . 288
18.2 Field’s Fictionalism . . . . . . . . . . . . . . . . . . . . . . . 289
18.3 Is Mathematics a “Subject with No Object”? . . . . . . . . . 298
18.4 Deflating Nominalism . . . . . . . . . . . . . . . . . . . . . . 302
Bibliography 305
Index 327

Preface
Philosophy of mathematics is an exciting subject studied by a small number
of philosophers today and even less by mathematicians. It is strongly related
to logic and foundations of mathematics. The Golden Age of the foundations
ofmathematicsbeganwiththefoundationalcrisis,whichisusuallyconsidered
to be the period between the late 19th century and mid-20th century. Unfor-
tunately, the foundations and philosophy of mathematics is not receiving the
attention it deservesby the mathematical community. I wrote this book with
thehopeofbringingbackthisintriguingsubjecttotheattentionofmathemat-
ical community to rekindle an interest in philosophical subjects surrounding
the foundations ofmathematics andintroduce variousphilosophicalpositions
ranging from the classic views to more contemporary ones, including those
which are more mathematical oriented. Ideally, as an outcome, I am hoping
toengageallphilosophicallyminded mathematiciansinphilosophicaldebates
and foundationaldiscussions.Another purpose in writing this book is to con-
tributetothephilosophicalliteraturefromtheperspectiveofamathematician
andencouragelike-mindedscholarstomakesimilarcontributions.Ihopethat
this book motivates mathematicians to argue about the foundations by get-
tinginvolvedinthetrendingphilosophicaldiscussionsandtocollaboratewith
philosophers, as this was happening in the Golden Age of the foundations of
mathematics.
Intended Audience
Thisbookisprimarilyforupperdivisionundergraduateandgraduatestu-
dents in mathematics or philosophy.Students in theoreticalcomputer science
canalsobenefitfromthismaterial.Ishouldemphasise,however,thatitispar-
ticularlyaimedforphilosophicallymindedmathematiciansduetotheselected
content and the presentation style. I would like to encourage young mathe-
maticians into thinking about the philosophical issues behind fundamental
concepts in mathematics and about different views one can have regarding
mathematical objects and mathematical knowledge. It is important to know
howanadoptedphilosophicalview maydramaticallyaffectthe mathematical
practice.Soacourseinphilosophyofmathematicsmayhelpthereaderrealise
how the methodology for mathematical practice changes in accordance with
thesupportedview.Italsoprovidesthephilosophicalbackgroundofmanyba-
sic notions used in mathematics, such as the concepts of set, number, space,
proof, computation, and so on. Philosophers, ipso facto, question all kinds of
philosophical problems about mathematics more often than mathematicians
ix