Markus Pantsar Truth, PPProofProof and Gödelian AAArguments:Arguments: AAA DDDefenceDefence of Tarskian TTTruthTruth in MMMathematicsMathematics 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 Truth, Proof and Gödelian Arguments: AAA Defence of Tarskian Truth in Mathematics 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 Acknowledgements Although writing a monograph on philosophy of mathematics is largely a solitary pursuit, this project could not have been completed without the help of various people. I am very grateful to my supervisor Professor Gabriel Sandu, whose expertise was invaluable in leading my research down the right paths, and whose comments on various drafts of this work have been essential in giving it the current form. I also owe gratitude to various members of the Department of Philosophy in the University of Helsinki for their help along the years. Professors Matti Sintonen and Ilkka Niiniluoto in particular deserve special thanks. With great gratitude I want to acknowledge the help of everybody who has at one point or another used his time to answer my questions. These include Professor Jaakko Hintikka, Dr. Philippe de Rouilhan and Professor Jouko Väänänen. I also want to thank my pre-examiners Professors Jan Woleński and Sten Lindström for their valuable comments, which helped improve the book a great deal at the final stage. This project has been supported financially by the University of Helsinki, which is acknowledged with gratitude. I am thankful to all the personnel at the University of Helsinki who have helped me with the practical matters, in particular Dr. Ilpo Halonen, Dr. Panu Raatikainen, Auli Kaipainen, Kristiina Norlamo and Arto Kilpiö. Thanks also belong to my family and friends for showing interest and support in the project, but most of all for providing the settings that allowed me to forget the thesis every once in a while. My brother Tuomas and his wife Tytti deserve special thanks for their support. In addition, I want to thank Matt and Jessica Powell for their help in revising the language. Lastly, but in many ways most importantly, I want to thank my parents Lasse and Merja for their unconditional support in this project, and over my entire life. This thesis would never have been completed without their help. To show even some of the gratitude I owe to my parents, this book is dedicated to them. Helsinki, March 2009 Markus Pantsar 1. Introduction 1.1 General background Paul Benacerraf and Hilary Putnam (1983, p. 3) have divided philosophers of mathematics into those who “ promulgate certain mathematical methods as acceptable ” and those who “want to describe the accepted ones”. This work will without doubt fall into the latter category. In general, I think philosophers should be careful about telling mathematicians how to do their jobs. This is not to say that the accepted results and methods of mathematics should be considered sacrosanct. Nor is it to say that philosophy cannot offer anything of interest to mathematicians. I disagree on