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On Sober Platonism UNIVERSITÀ DEGLI STUDI DI MILANO Dipartimento di Filosofia UNIVERSITÉ PARIS 1 PANTHEON-SORBONNE IHPST - Insistut d’Histoire et de Philosophie des Sciences et des Techniques Costanza Brevini On Sober Platonism New Perspectives in Mathematical Platonism Beyond Strong Ontological Assumptions March, 4th 2016 SUPERVISORS: Marco Panza, Miriam Franchella MEMBERS OF THE JURY: Andrea Bottani Brice Halimi Sebastien Gandon Agustìn Rayo Andrea Sereni To you, the daring venturers and adventurers, and whoever hath embarked with cunning sails upon frightful seas; To you the enigma-intoxicated, the twilight-enjoyers, whose souls are allured by flutes to every treacherous gulf. NIETZSCHE, FRIEDRICH, Thus Spoke Zarathustra !2 TABLE OF CONTENTS: SINTESI 5 RÉSUMÉ 8 ABSTRACT 11 PRELIMINARY REMARKS 15 CHAPTER 1: PLATONISM 25 1.1 FROM PLATO TO CONTEMPORARY PLATONISM 25 1.2 INDEPENDENCE 30 1.3 EXISTENCE 35 1.4 EPISTEMOLOGY 40 1.4.1 THEORY OF KNOWLEDGE 41 1.4.2 TRUTH 45 1.4.3 REFERENCE 50 1.5 SOBER PLATONISM 54 CHAPTER 2: FULL-BLOODED PLATONISM 62 2.1 INTRODUCTION 62 2.1.1 THE PROBLEM OF FORMALIZATION 64 2.2 EPISTEMOLOGY 66 2.2.1 THE EPISTEMOLOGICAL ARGUMENT 66 2.2.2 THE FULL-BLOODED SOLUTION 70 2.3 CONSISTENCY 75 2.4 MATHEMATICAL CORRECTNESS AND MATHEMATICAL TRUTH 82 2.4.1 HOW TO DEAL WITH UNDECIDABLE SENTENCES IN MATHEMATICS 83 2.5 NON-UNIQUENESS 87 2.6 A DEFENSE OF FULL-BLOODED PLATONISM 93 CHAPTER 3: STRUCTURALISM 99 3.1 STRUCTURALISM 99 3.2 EXISTENCE 105 3.4 AXIOMS 114 3.5 EPISTEMOLOGY 118 3.6 REFERENCE 126 !3 3.7 STRUCTURALISM AND CAESAR’S PROBLEM 129 3.8 STRUCTURALISM AND FORMALISM 131 CHAPTER 4: OBJECTS THEORY 135 4.1 NAÏVE OBJECTS THEORY 135 4.2 ELEMENTARY OBJECTS THEORY 142 4.2.1 OBJECT CALCULUS 142 4.2.2 AXIOMS 149 4.3 MODAL THEORY OF ABSTRACT OBJECTS 153 4.3.1 MODAL OBJECT CALCULUS 153 4.3.2 AXIOMS 160 4.4 ONTOLOGY 162 4.5 EPISTEMOLOGY 171 CHAPTER 5: TRIVIALISM 176 5.1 METAPHYSICALISM AND COMPOSITIONALISM 176 5.2 CONSTRUCTING LOGICAL SPACE 183 5.3 ’JUST IS’-STATEMENTS 185 5.3.1 CONNECTIONS OF ‘JUST IS’-STATEMENTS 192 5.4 TRIVIALIST SEMANTICS 197 5.5 IS IT REALLY TRIVIAL? 202 5.6 TRIVIALISM, PLATONISM AND NOMINALISM 205 5.7 TRIVIALISM AND NEOFREGEANISM 209 CHAPTER 6: CONCLUDING REMARKS 213 6.1 SOBER PLATONISM 213 6.2 INDEPENDENCE 217 6.3 EXISTENCE 221 6.4 EPISTEMOLOGY 227 6.4.1 THEORY OF KNOWLEDGE 228 6.4.2 TRUTH 233 6.4.3 REFERENCE 236 6.5 CONCLUSION 240 BIBLIOGRAPHY 242 !4 Sintesi In questo lavoro ho voluto analizzare una tendenza che ha preso forma negli ultimi anni tra le teorie platoniste nell’ambito della filosofia della matematica. I modelli da cui sono partita sono quattro autorevoli proposte, che mi sono parse paradigmatiche: il platonismo purosangue di Mark Balaguer, lo strutturalismo ante rem di Stewart Shapiro, la teoria degli oggetti di Edward Zalta e il trivialismo di Agustìn Rayo. Queste quattro teorie hanno in comune tra loro un atteggiamento di tipo platonista nei confronti degli oggetti matematici, ovvero, assumono che gli oggetti di cui parlano gli enunciati matematici esistano effettivamente. Al contrario però del platonismo matematico classico, le loro assunzioni ontologiche, pur manifestando l’adesione alla teoria platonista, sono talmente misurate da dare l’impressione che non si tratti neanche di teorie platoniste. Propongo dunque di chiamare ‘platonismo sobrio’ questi approcci, che supportano il platonismo, pur rinunciando a forti assunzioni ontologiche. La caratteristica fondamentale di questa tendenza è che l’impegno nei confronti dell’esistenza degli oggetti matematici non costituisca più il punto di partenza di una teoria degli oggetti matematici, ma diventi condizione necessaria al verificarsi di un fatto: la mente umana attinge alla conoscenza matematica. Di conseguenza, gli oggetti matematici devono esistere e devono essere tali da rendere possibile un rapporto tra gli oggetti matematici e la mente umana. In realtà, ciò a cui queste teorie puntano è ottenere una descrizione della pratica matematica, che sia in grado di rispondere a domande !5 filosofiche, imponendo alla pratica matematica il minor numero possibile di assunzioni filosofiche. Il primo capitolo di questo lavoro è dedicato all’analisi del platonismo matematico classico. Propongo di considerare tale corrente di pensiero come la somma di tre assunti principali: Indipendenza (gli oggetti matematici sono indipendenti dal pensiero e dalle pratiche umane), Esistenza (gli oggetti matematici esistono) e Conoscenza (gli oggetti matematici sono conoscibili). Quest’ultima tesi è ulteriormente divisa in tre sotto-tesi: Teoria della Conoscenza, Riferimento e Verità. Nel secondo, terzo, quarto e quinto capitolo ho discusso le proposte dei quattro autori citati, accomunati, come si è visto, da un impegno ontologico solo implicito o sobrio nei confronti degli oggetti matematici. Le questioni filosofiche con oggetto l’esistenza degli oggetti matematici, la possibilità di accedere alla conoscenza matematica, il significato degli enunciati matematici e il riferimento dei termini in essi contenuti sono affrontate e considerate filosoficamente rilevanti da tali teorie. Ciononostante, il loro principale obiettivo è piuttosto l’elaborazione di una descrizione precisa della matematica nella sua autonomia. Nell’ultimo capitolo ho definito il platonismo sobrio attraverso la sua adesione alle stesse tesi cui aderisce il platonismo classico, Indipendenza, Esistenza e Conoscenza (anche qui analizzata nelle tre sotto-tesi già menzionate). Attraverso una valutazione comparativa, risulta evidente che il platonismo sobrio assume in buona parte ciò che assume il platonismo classico. Il vero elemento di distinzione risiede nel rapporto tra filosofia e matematica, in quanto nel platonismo sobrio l’autonomia e la dignità della matematica sono chiaramente affermate. La filosofia arriva solo dopo, a giochi fatti e, !6 piuttosto che una prescrizione normativa di come si deve fare matematica, si limita a fornire una descrizione metodologica di come si fa matematica. Di là dai risultati che può avere fino a oggi conseguito, il platonismo sobrio promette di essere in grado di ridurre molto rapidamente la portata se non altro di alcuni di quei problemi che sono rilevanti per la filosofia della matematica, ma non lo sono per la pratica matematica. In conclusione, il platonismo sobrio propone sia un innovativo approccio nella filosofia della matematica, sia un fecondo contributo alla riflessione della filosofia su discipline altre da sé: una proposta descrittiva, piuttosto che normativa, ma ricca di prospettive filosofiche. !7 Résumé Ce travail vise à identifier et définir une nouvelle tendance du platonisme mathématique. Je suis parti de l'analyse de quatre propositions fiable et paradigmatique: le platonisme pur-sang de Mark Balaguer, le structuralisme ante rem de Stewart Shapiro, la théorie de l’objet abstrait de Edward Zalta et le trivialisme de Agustìn Rayo. Ces quatre théories ont en commun une attitude platoniste, c’est à dire l’admission de l'existence d'objets mathématiques. Contrairement au platonisme mathématique classique, son engagement ontologique est si léger, ou bien sobre, qui donne l’impression de n’être même pas dans le platonism. Pour cette raison, je propose d’appeler ‘platonisme sobre’ ces propositions qui supportant le platonisme, en renonçant à de fortes engagements ontologiques. La principale caractéristique de cette tendance est que l’engagement de l'existence d'objets mathématiques ne est plus considéré comme le point de départ d'une théorie des objets mathématiques, mais devient une condition nécessaire d'un fait: l'esprit humain peu obtenir connaissance des mathématiques. Par conséquence, les objets mathématiques existent et doivent être tels qu’une relation entre les objets mathématiques et l'esprit humain soit possible et fiable. En fait, ces théories visent à obtenir une description de la pratique mathématique, qui est aussi capable de répondre à des questions philosophiques, sans imposer aucun argument philosophique aux pratiques mathématique. Dans le premier chapitre, on analyse le platonisme mathématique classique. Je propose d'examiner cette ligne de pensée comme la somme de trois thèses principales: Indépendance (les objets mathématiques sont indépendant de la pensée et de les !8 pratiques), Existence (les objets mathématiques existent) et Epistémologie (les objets mathématiques sont connaissables). Cette dernier thèse est encore divisée en trois sub- thèses: Théorie de la connaissance, Référence et Vérité. Le deuxième, troisième, quatrième et cinquième chapitre sont respectivement dévoués à l'examen des quatre théories cités, uni par un engagement ontologique seulement implicite ou sobre vers des objets mathématiques. Ces théories sont explicitement platonistes, mais seulement sobrement engagées dans l'existence d'objets mathématiques. Elles traitent l'existence d’objets mathématiques, la possibilité d'accéder à la connaissance mathématique, le sens des énoncés mathématiques et la référence de leur termes en tant que questions philosophiquement pertinentes. Cependant, leur objectif principal est l’élaboration d'une description précise des mathématiques en tant que telles. Dans le dernier chapitre, je définis le platonisme sobre à travers les façons dont il soutient la même thèse du platonisme classique, Indépendance, Existence et Epistémologie (encore analysée comme Théorie de la connaissance, Référence et Vérité). Grâce à une évaluation comparative, il est en train de devenir clair que le platonisme sobre assume en grande partie le même principe
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