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Reconsidering Wittgenstein's Philosophy of Mathematics

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Reconsidering Wittgenstein's Philosophy of Mathematics Universit`aCa’ Foscari Venezia DOTTORATO DI RICERCA IN FILOSOFIA, XXIII CICLO Scuola di Dottorato in Scienze Umanistiche Settore scientifico disciplinare di appartenenza: M-FIL/05 Reconsidering Wittgenstein’s Philosophy of Mathematics Tutore del Dottorando: Tesi di Dottorato di: Prof. Luigi Perissinotto Matteo Plebani Matricola 955448 Direttore della Scuola di Dottorato: Prof. Carlo Natali A.A. 2009-2010 Contents Aknowledgements . .5 Abbreviations . .6 Overview . .7 1 Wittgenstein’s Philosophy of Mathematics in a Nutshell 9 1.1 The Negative Claim: There are no mathematical propositions . 10 1.1.1 The Claim . 10 1.1.2 Platonism and Necessity . 13 1.1.3 Two Puzzles about Necessity . 14 1.1.4 The Shift From the Tractatus to the Intermediate Phase: mathematical propositions vindicated . 17 1.1.5 The Rule following considerations and the idea of Mathe- matical sentences as Rules of Grammar . 17 1.1.6 Dummett’s full blooded Conventionalism and its critics . 19 1.2 The positive Claim: Mathematical sentences are Rules of Gram- mar .................................. 23 1.2.1 What is a rule of Grammar? . 23 1.2.2 From Necessity to normativity . 24 1.2.3 Wittgenstein and Formalism . 25 1.2.4 The role of applications . 26 1.3 Classical Problems dissolved? . 29 1.3.1 Existence of mathematical object vs. Objectivity of Math- ematics . 29 1.3.2 Epistemology . 30 1.3.3 Applicability . 31 2 Wittgenstein and Mathematical Knowledge 33 2.1 The Problem . 33 2.1.1 Benacerraf’s Challenge . 33 2.1.2 Field’s Challenge . 34 2.1.3 Replies to Field (I) . 37 2.1.4 Replies to Field (II) . 39 2.2 And Wittgenstein . 40 2.3 The problem of consistency . 44 3 Wittgenstein on the existence of Mathematical Objects 47 3.1 Language and Reality . 47 3.1.1 Meaning and Referring . 48 3.1.2 Wittgenstein and Quine on Language and Reality . 49 3 4 Contents 3.1.3 Quine’s strategy (outline) . 51 3.1.4 The problem of Paraphrase . 52 3.1.5 Quine’s Strategy (Complete) . 54 3.1.6 Quine’s official menu . 54 3.1.7 Quine’s argument for Platonism . 55 3.1.8 Wittgenstein vs. Quine . 55 3.1.9 Way out (1) . 56 3.1.10 The problem of Applications . 57 3.1.11 Naturalism, Indispensability and Inference to the best explanation . 58 3.1.12 The Irrelevance Argument . 60 3.1.13 Aboutness , Applications and the problem of the content 62 3.2 Wittgenstein, Fictionalism and the problem of Aboutness . 64 3.2.1 The Problem . 64 3.2.2 Some Replies . 65 3.2.3 The project . 67 3.2.4 Figuralism . 67 3.2.5 Aboutness . 68 3.2.6 Mathematics and Generation Principles . 69 3.2.7 The problem of applicability . 70 3.2.8 Pure Mathematics . 72 3.2.9 Conclusions . 74 3.3 Two conceptions of Ontology . 74 3.3.1 The Neo-Fregeans’ Strategy . 74 3.3.2 Wright on Frege’s use of the context principle . 77 3.3.3 Objection to Neo-Fregeanism and Wittgenstein . 79 3.3.4 Quine and the Crystalline conception of Ontology . 81 3.3.5 Carnap and the internal/external distinction . 83 3.3.6 Context Principle, Meaning as use and the Irrelevance argument again . 86 3.3.7 Wittgenstein the meta-ontologist and the vacuity of Pla- tonism . 88 4 Ontology vs. Modality 91 4.1 The Problem Of The Objectivity of Mathematics In The Contem- porary Debate . 93 4.1.1 The Standard Analysis . 93 4.1.2 Field on Objectivity and Putnam’s Model-Theoretic Ar- gument . 95 4.1.3 A problem for extreme anti-objectivism . 97 4.2 Wittgenstein’s position . 99 4.3 Logical Objectivity and the Rule Following Considerations . 102 Bibliography 107 Acknowledgements I am very thankful to Prof. Victor Rodych for having discussed with me some of the issues addressed in Chapter Four. I am very grateful Prof. Pasquale Frascolla for having provided a lot of precious advice and constant encouragement. It was a pleasure to have Prof. Luigi Perissinotto as a tutor. Thanks to Ivan Kasa: discussing with him I learned a lot about the Neo-Fregean program and I shaped my ideas about epistemological objections to Platonism. Thanks to David Liggins, who read part of Chapter Three, providing very useful comments. Thanks to Vittorio Morato, Beppe Spolaore and Tommaso Gordini for LaTeX assistance. Thanks to Maria Baetzing for linguistic advice, and much more. Thanks to my beloved flatmates as well. 5 Abbreviations AWL Ambrose, A. (ed.) Wittgenstein’s Lectures, Cambridge 1932-35, Blackwell, Oxford, 1979. LB The Blue and Brown Books: Preliminary Studies for the ‘Philosophical Investigations’, ed. Rush Rees, Blackwell, Oxford, 1958. LFM Diamond C.(ed.) Wittgenstein’s Lectures on the Foundations of Mathe- matics, Cambridge 1939, Cornell University Press, Itacha, N.Y., 1976. PG Philosophical Grammar, ed. R. Rhees, trans. A. J. P. Kenny, Blackwell, Oxford, 1974. PI Philosophical Investigations, ed. G. E. M. Anscombe and R. Rhees, trans. G. E. M. Anscombe, 2nd edn, Blackwell, Oxford, 1958. PR Philosophical Remarks, ed. R. Rhees, 2nd edn, trans. R. Hargreaves and R. White, Blackwell, Oxford, 1958 RFM Remarks on the Foundations of Mathematics, ed. G.H. von Wright, R. Rhees and G. E. M. Anscombe, trans. G. E. M. Anscombe, 3rd edn, Blackwell, Oxford, 1978. T Tractatus Logico-Philosophicus, trans. D. F. Pears and B. F. McGuinness, Routledge & Kegan Paul, London, 1961. WVC McGuinness, B. F. (ed.) Wittgenstein and the Vienna Circle: Con- versations recorded by Friedrich Waismann, trans. J. Schulte and B. F. McGuinness, Blackwell, Oxford, 1979. Z Zettel, ed. G. E. M. Anscombe and G.H. von Wright, trans. G. E. M. Anscombe, 2nd edn, Blackwell, Oxford, 1981. 6 Overview The aim of the present work is to investigate whether Wittgenstein’s Philosophy of Mathematics turns out to be of some interest for the contemporary research in this field. Chapter 1 exposes Wittgenstein’s pivotal theses in the Philosophy of Mathematics and provides a brief summary of their possible relevance for the most global matters discussed by contemporary philosophers of Mathematics. The problems of mathematical knowledge, the ontological status of mathematical entities and the question about the Objectivity of Mathematics are analyzed in depth in the three following chapters. 7 8 Contents Chapter 1 Wittgenstein’s Philosophy of Mathematics in a Nutshell For we undoubtedly incline to think of the Peano axioms, informally ex- pressed as above, and their logical consequences, however that class is to be characterised, as being truths of some sort; so there has to be a philosophical question how we ought to conceive of the nature of the facts which make those statements true, and of how they fall within the human epistemological compass. At least, there have to be such questions if we may presuppose that it is philosophically respectable to think of those statements as truths of any sort. This presupposition has, of course, been questioned. It was questioned by the Formalists of Frege’s day [...] And it was questioned for quite different, and much more profound, reasons by the later Wittgenstein. [Wright, 1983, p. xiv] This is (W)right, to my mind: one of the key features of Wittgenstein’s account of Mathematics is a strong opposition to the idea that in Mathematics we are dealing with genuine truths. Given that propositions are usually taken to be truth-bearers, the same claim can be expressed by saying that there are no mathematical propositions. After a reconstruction of Wittgenstein’s motivation for this claim (paragraph 1.1) my main concern will be to point out how much Wittgenstein’s position is radical, comparing his mature position with a more traditional form of anti-realism he embraced during the intermediate phase of his reflections. The profound reason Wright mentions is in my opinion the difficulty that any Platonists account of necessity must face and which the normative conception of mathematical statements should solve. My idea is to summarize Wittgenstein’s position in two pivotal theses: that mathematical sentences do not depict facts (they do not express propositions), and that this is due to the fact that they are grammatical rules in disguise. Apart from the stress on the problem of necessity as the original source of Wittgenstein’s reflections, the key interpretative thesis of this section borrow heavily form [Rodych, 2008b], [Rodych, 1997] and [Frascolla, 1994]. 9 10 Wittgenstein’s Philosophy of Mathematics in a Nutshell 1.1 The Negative Claim: There are no mathe- matical propositions 1.1.1 The Claim Might we not do arithmetic without having the idea of uttering arithmetical propositions, and without ever being struck by the similarity between a multiplication and a proposition? Should we not shake our heads, though, when someone showed us a multiplication done wrong, as we do when someone tells us it is raining, if it is not raining? Yes; and here is a point of connexion. But we also make gestures to stop our dog, e.g., when he behaves as we do not wish. We are used to saying “2 times 2 is 4”, and the verb ‘is’ makes this into a proposition, and apparently establishes a close kinship with everything that we call a proposition. Whereas it is a matter only of a very superficial relationship. [RFM App. III 4] This is Wittgenstein in his mature phase and it is clear that in that context Wittgenstein’s answer to the rhetorical question raised in the first paragraph is “Yes”. The interesting fact is that the same conclusion was reached by Wittgenstein in the Tractatus. This provides ground for taking the idea that there are no mathematical propositions as a cornerstone of Wittgenstein’s overall philosophical reflection on Mathematics.
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