Truth, Proof and Gödelian Arguments: A Defence of Tarskian Truth in Mathematics

310 Pages·2009·1.87 MB·English

by  Markus Pantsar

Markus Pantsar

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Philosophical Studies from the University of Helsinki 23

Filosofisia tutkimuksia Helsingin yliopistosta
Filosofiska studier från Helsingfors universitet
Philosophical Studies from the University of Helsinki

Publishers:
Department of Philosophy
Department of Social and Moral Philosophy
P.O. Box 9 (Siltavuorenpenger 20 A)
00014 University of Helsinki
Finland

Editors:
Marjaana Kopperi
Panu Raatikainen
Petri Ylikoski
Bernt Österman
Markus Pantsar

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ISBN 978-952-10-5373-3 (paperback)
ISBN 978-952-10-5374-0 (PDF)
ISSN 1458-8331
Tampere 2009
Kopio Niini Finland Oy
Contents

CONTENTS ............................................................................................ 5

ACKNOWLEDGEMENTS ................................................................... 5

1. INTRODUCTION ........................................................................... 11

1.1 GENERAL BACKGROUND ................................................................... 11

1.2 ANOTHER APPROACH ....................................................................... 17

1.3 TRUTH AND PROOF ............................................................................ 20

1.4 TARSKIAN TRUTH .............................................................................. 22

1.5 REFERENCE ........................................................................................ 24

1.6 NON-CLASSICAL LANGUAGES ........................................................... 27

1.7 THE BASIC THEORY OF MATHEMATICS .............................................. 29

1.8 THE LIMITATIONS OF THE APPROACH HERE ..................................... 30

1.9 THE STRUCTURE OF THIS WORK ......................................................... 32

2. THE BACKGROUND ..................................................................... 37

2.1 THE PROBLEM OF TERMINOLOGY ...................................................... 37

2.2 PLATONISM ........................................................................................ 39

2.3 REALISM/OBJECTIVISM ...................................................................... 42

2.4 FORMALISM/NOMINALISM ............................................................... 46

2.5 SOUNDNESS AND COMPLETENESS ..................................................... 50

2.6 GÖDEL’S INCOMPLETENESS THEOREMS ............................................ 52

2.7 IS THE GÖDEL SENTENCE TRUE? ........................................................ 55

2.8 GÖDEL SENTENCES AND TARSKI ....................................................... 57

3. THE SEMANTICAL ARGUMENT .............................................. 63

3.1 FIELD’S NOMINALISM ........................................................................ 63

3.2 SHAPIRO’S SEMANTICAL ARGUMENT ................................................ 69

3.3 COUNTERARGUMENTS BEYOND CONSISTENCY ................................ 75

3.4 WHY NOT DEFLATIONARY TRUTH? ................................................... 85

3.5 TENNANT ........................................................................................... 89

3.6 WHY SOUNDNESS OVER TRUTH? ....................................................... 95

3.7 CONCLUSIONS ................................................................................... 98
6 Contents

4. FORMAL AND PRE-FORMAL MATHEMATICS .................. 104

4.1 ASSERTABILITY AND ARBITRARINESS .............................................. 104

4.2 UNDECIDABLE SENTENCES AND FORMALISM ................................. 112

4.3 TARSKIAN TRUTH AND MATHEMATICS ........................................... 117

4.4 ANOTHER APPROACH TO MATHEMATICAL THINKING ................... 134

4.5 PRE-FORMAL MATHEMATICS ........................................................... 138

4.6 PHILOSOPHICAL IMPORTANCE OF PRE-FORMAL MATHEMATICS .... 148

4.7 PRIORITY OF SEMANTICS OVER SYNTAX .......................................... 153

4.8 TRUTH, PROOF AND REFERENCE ..................................................... 154

5. TRUTH AND LOGIC.................................................................... 158

5.1 DIFFERENT LOGICS........................................................................... 158

5.2 HINTIKKA’S TRUTH .......................................................................... 160

5.3 WHY IF LOGIC? ................................................................................ 168

5.4 SECOND-ORDER LOGIC .................................................................... 176

5.5 KRIPKE’S TRUTH AND THE POTENTIAL OF MANY-VALUED LOGICS 184

5.6 COLLAPSING THE HIERARCHY WITH PRE-FORMAL LANGUAGES .... 189

5.7 WHY LOGICISM AND SINGLE TRUTH PREDICATE? ........................... 191

6. WHY NOT NOMINALISM? ....................................................... 195

6.1 SEMANTICAL ARGUMENTS AND THE TROUBLE WITH REFERENCE .. 195

6.2 MENO’S PARADOX AND THEORY CHOICE ....................................... 199

6.3 BENACERRAF’S DILEMMA AND NOMINALISM ................................. 203

6.4 FIELD’S NOMINALISM REVISITED ..................................................... 212

6.5 MODAL RECONSTRUCTIVISM ........................................................... 219

6.6 THE POWER OF OBJECTIVITY: PENROSE’S QUESTION ....................... 225

6.7 THE POWER OF NOMINALISM AND POTENTIAL WAYS OUT ............. 231

6.8 ONTOLOGY OF MATHEMATICS: AN ALTERNATIVE OUTLINE .......... 236

7. TRUTH AND REFERENCE ......................................................... 244

7.1 COUNTERFACTUALS ........................................................................ 244

7.2 TRUTH BEFORE REFERENCE OR VICE VERSA? ................................... 251

7.3 NEO-FREGEANISM ........................................................................... 254

7.4 BAD COMPANY AND NEO-FREGEAN EPISTEMOLOGY ..................... 262

7.5 TWO KINDS OF PRIORITY .................................................................. 268

7.6 NON-PLATONIST REFERENCE: LINNEBO ......................................... 271

7.7 NEO-FREGEANISM AND QUINE....................................................... 277
Contents 7

8. LOOSE ENDS ................................................................................. 280

8.1 NON-STANDARD MODELS ............................................................... 280

8.2 ANOTHER SEMANTICAL ARGUMENT ............................................... 282

8.3 GÖDELIAN FALLACIES ..................................................................... 285

8.4 CONCLUSION: WHAT DOES “SUBSTANTIAL” TRUTH MEAN? .......... 289

REFERENCES ..................................................................................... 293