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TABLE OF CONTENTS Special Topic I WITTGENSTEIN Guest Editors Kai BÜTTNER, Florian DEMONT, David DOLBY, Anne-Katrin SCHLEGEL Introduction . 3 Pasquale FRASCOLLA: Realism, Anti-Realism, Quietism: Wittgen- stein’s Stance . 11 Severin SCHROEDER: Mathematical Propositions as Rules of Grammar . 23 Felix MÜHLHÖLZER: How Arithmetic is about Numbers. A Witt- gensteinian Perspective . 39 Mathieu MARION & Mitsuhiro OKADA: Wittgenstein on Equinu- merosity and Surveyability . 61 Esther RAMHARTER: Wittgenstein on Formulae . 79 Ian RUMFITT: Brouwer versus Wittgenstein on the Infi nite and the Law of Excluded Middle . 93 Special Topic II QUINE Guest Editor Dirk GREIMANN Introduction . 111 Peter HYLTON: Signifi cance in Quine . 113 Rogério Passos SEVERO: Are there Empirical Cases of Indetermi- nacy of Translation? . 135 Oswaldo CHATEAUBRIAND: Some Critical Remarks on Quine’s Th ought Experiment of Radical Translation . 153 Dirk GREIMANN: A Tension in Quine’s Naturalistic Ontology of Semantics . 161 Guido IMAGUIRE: In Defense of Quine’s Ostrich Nominalism . 185 Pedro SANTOS: Quinean Worlds: Possibilist Ontology in an Exten- sionalist Framework . 205 Obituary: Rudolf Haller (1929–2014) . 229 Rudolf HALLER: An Autobiographical Outline . 231 Special Topic WITTGENSTEIN Grazer Philosophische Studien 89 (2014), 3–10. INTRODUCTION Kai BÜTTNER, Florian DEMONT, David DOLBY, Anne-Katrin SCHLEGEL University of Zurich Mathematics occupied a central place in Wittgenstein’s work. He was led to philosophy by an interest in disputes about the foundations of mathematics. His refl ections resulted in the Tractatus’ short but insightful critique of the logicism of Frege and Russell. Moreover, it is reported that he decided to return to philosophy in 1929 after having attended a talk given by the intuitionist mathematician Brouwer. In the 1930s and early 1940s Wittgenstein wrote extensively on the philosophy of mathematics, covering a large variety of themes, including the notions of number and infi nity, the method of proof by induction, the role of contradictions and consistency proofs in mathematics, the application of mathematics, and the nature of mathematical proof and mathematical necessity. In 1944 he stated that his most signifi cant contribution had been in the philosophy of mathematics (Monk 1990, 466). However, the initial reception of Wittgenstein’s philosophy of math- ematics was predominantly negative (Dummett 1959, Kreisel 1958, Ber- nays 1959). Many of his remarks about mathematics were dismissed as either wrong or irrelevant; and some were alleged to contain technical errors. To a certain extent, this reaction can be explained by the radical nature of Wittgenstein’s ideas, which often challenge traditional assump- tions about mathematics. Another cause has been misunderstanding: the interpretation of Wittgenstein’s writings has proven diffi cult due to his unconventional style of writing, the fact that the material from his Nachlass is partly unfi nished and was not intended for publication, and the delayed publication of many relevant manuscripts from his middle period. Fortunately, the exegesis of Wittgenstein’s writings on mathematics has progressed considerably in recent years, with the appearance of such excellent monographs as Shanker 1987, Frascolla 1994, Marion 1998 and Mühlhölzer 2010. Th ese books clarify Wittgenstein’s claims and show how the accusations of technical incompetence were often the result of misinterpretation. Th is has led to a greater appreciation of Wittgenstein’s unorthodox views, which promise fresh perspectives on debates that had appeared to have reached deadlock. Th e essays in this volume are recent contributions to the newly invigorated debate about Wittgenstein’s phi- losophy of mathematics by leading scholars and specialists. According to the later Wittgenstein, many philosophical misconcep- tions and mythologies can be traced back to the assumption that any declarative sentence describes a corresponding portion of reality. Th us, he famously argues that self-ascriptions of mental predicates have an expres- sive function and do not describe an inner world of private experiences. And, similarly, mathematical propositions have a normative rather than a descriptive function. Instead of describing a realm of abstract objects, they are rules for the transformation of empirical descriptions. In his paper Pasquale Frascolla discusses a potential diffi culty for Wittgenstein’s conception, which arises from the intuitive assumption that for a sen- tence to have a descriptive function is equivalent with its being truth-apt. According to Frascolla, Wittgenstein did not mean to deny this principle in order to make room for sentences which are both non-descriptive and truth-apt. Instead he considered neither mathematical propositions nor self-ascriptions of mental predicates to be truth-apt. Mathematical proposi- tions, in particular, correspond to reality at best in the way in which rules correspond to social practices. And the proof of a mathematical proposition determines the latter’s sense rather than establishing its truth. In his contribution Severin Schroeder investigates how we should understand Wittgenstein’s claim that mathematical propositions are rules of grammar for non-mathematical language. In particular he explores how to resolve the tension between Wittgenstein’s claim that mathe- matical propositions are rules and his emphasis on the practical useful- ness of mathematics. As Wittgenstein himself asks: “How can the mere transformation of an expression be of practical consequence?” (RFM 357) According to Schroeder, Wittgenstein’s answer is that mathematical propositions are not merely stipulative defi nitional rules: they forge con- nections between independently comprehensible concepts. A proposition expressing an empirical correlation may be fi xed as a rule. It is in this way that mathematical propositions are “dependent on experience but made independent of it” (LFM 55). A problem for Wittgenstein’s view nevertheless remains: what violates a grammatical rule is nonsense and yet non-mathematical statements that are not in conformity with mathemati- 4 cal propositions would not be nonsensical but rather false (except in the most simple cases). Schroeder’s solution is to suggest that mathematical propositions do not determine what makes sense in natural language: they are norms for the activity and discourse in which we develop and apply a system for calculating quantities (rather than simply counting or measuring them). Felix Mühlhölzer addresses a problem which has occupied philosophers as well as mathematicians. It comes from elementary results in model theory to the eff ect that many formalized theories, e.g. fi rst-order Peano Arithmetic, have non-standard models. On the one hand, if one gives a precise formulation of what arithmetic is about, as is done by the model- theoretic notion of interpretation, then one cannot catch it uniquely: there is a multitude of non-intended interpretations that one cannot dismiss. If, on the other hand, one specifi es the standard model (i.e. the intended interpretation) in the meta-language in which model-theory is (often infor- mally) expressed, then what arithmetic is about is not made precise and transparent. Mühlhölzer calls this the aboutness dilemma. He argues that this dilemma can be seen as rooted in a blurring of the categorial diff erence between the notions of ‘interpretation’ and ‘reference’. Th e interpreta- tions mentioned in the fi rst horn are simply mathematical functions that do not involve any use of the so-called ‘signs’ that are interpreted. Th ese signs are ‘dead’, ‘petrifi ed’, and one should not think that petrifi cation is simply idealization: idealization abstracts from non-essential aspects; petrifi cation, by contrast, from the essential ones, namely aspects of use. Th e second horn of the aboutness dilemma concerns ‘reference’, a notion which Wittgenstein claims is essentially tied to the use of signs (see PI § 10). Th is view adopts what one might call the full use thesis, according to which there is nothing more to ‘reference’ than what can be seen in the use of our terms, and rejects the partial use thesis, usually adopted in the literature, which claims only that the use of our terms contributes to the necessary link between ourselves and the objects referred to. From the Wittgensteinian point of view, then, one can accept the fi rst horn of the aboutness dilemma, as asserting an uncontroversial mathematical result, and correct the second horn by presenting a perspective on ‘reference’ that is suffi ciently clear (even though imprecise). One of the constant targets of Wittgenstein’s criticism is Frege and Russell’s logicism. In their paper “Wittgenstein on equinumerosity and surveyability” Marion and Okada try to connect two of Wittgenstein’s objections, which they call the ‘modality argument’ and the ‘surveyability 5 argument’. Th e former is directed against the logicist’s defi nition of equi- numerosity according to which two concepts F and G are equinumerous if and only if there is a one-one relation between the Fs and Gs. Wittgen- stein objects to this defi nition that the possibility of a one-one correlation between the Fs and Gs is a consequence of, rather than a condition for, the equinumerosity of F and G. For it is only when the numbers of the F and G are comparatively small that one can determine whether F and G are equinumerous without counting the Fs and Gs and hence without identify- ing their respective numbers.
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