Frege On ‘I’, ‘Now’, ‘Today’ And Some Other Linguistic Devices

In this paper, I argue against an influential view of Frege's writings on indexical and other context-sensitive expressions, and in favour of an alternative. The centrepiece of the influential view, due to (among others) Evans and McDowell, is that according to Frege, context-sensitiveword-meaning plus context combine to express senses which are essentially first person, essentially present tense and so on, depending on the context-sensitive expression in question. Frege's treatment of indexicals thus fits smoothly with his Intuitive Criterion of difference of sense. On my view, by contrast, Frege stuck by the view which he held in his unpublished 1897 Logic, namely that the senses expressed by the combination of context-sensitive word-meaning and context could just as well be expressed by means of non-context-sensitive expressions: being first person, present tense and so on are properties, in Frege's view, only of language, not of thought. Given the irreducibility of indexicals – a phenomenon noticed by Castañeda, Perry and others – Frege's treatment of indexicals thus turns out to be inconsistent with the Intuitive Criterion. I argue that Frege was not aware of the inconsistency because he was not aware of the irreducibility of indexicals. This oversight was possible because the source of Frege's interest in indexicals, as inother context-sensitive expressions, differed from that of contemporary theorists. Whereas contemporary theorists are most often interested in indexicals (and in Frege's treatment of them) because they are interested in the indexical versions of Frege's Puzzle and their relation to psychological explanation, Frege himself was interested in them because they pose a prima facie threat to his general conception of thoughts. The only indexical expression Frege's view of which the above account does not cover is I insofar as it is associated with special and primitive senses, but Frege did not introduce such senses with a view to explaining theirreducibility of I his real reason for introducing them remains obscure.     

IV—Unsaturatedness: Wittgenstein's Challenge, Frege's Answer

Frege holds the distinction between complete (saturated) and incomplete (unsaturated) things to be a basic distinction of logic. Many disagree. In this paper I will argue that one can defend Frege's distinction against criticism if one takes, inspired by Frege, a wh -question to be the paradigm incomplete expression.     

Was Wittgenstein Frege's Heir

Dummett has claimed that Wittgenstein's views, as expressed in The Blue and Brown Books and Philosophical Investigations , build on the attack on psychologism initiated by Frege. Frege's rejection of psychologism led him to the view that the meanings of sentences are thoughts which objectively exist in a third realm. It was Wittgenstein, according to Dummett, who, inheriting Frege's insights, provided a genuinely anti-psychologistic account of understanding by insisting that we explain understanding a sentence in terms of the use that is made of it. I challenge this interpretation of the relationship between Wittgenstein and Frege. I argue that the analysis does not sufficiently distinguish anti-psychologism and anti-mentalism. In the light of this distinction we can see that Wittgenstein misrepresents Frege's views, and confuses concepts with ideas. By being more faithful to Frege's actual views concerning the objectivity of concepts we can give a robustly realist account of mathematical truth which does not involve any objectionable psychologism or mentalism.
email : Karen.Green@arts.monash.edu.au     

Realismbei Frege: Reply to Burge

Frege is celebrated as an arch-Platonist and arch-realist. He is renowned for claiming that truths of arithmetic are eternally true and independent of us, our judgments and our thoughts; that there is a third realm containing nonphysical objects that are not ideas. Until recently, there were few attempts to explicate these renowned claims, for most philosophers thought the clarity of Frege's prose rendered explication unnecessary. But the last ten years have seen the publication of several revisionist interpretations of Frege's writings — interpretations on which these claims receive a very different reading. In Frege on Knowing the Third Realm, Tyler Burge attempts to undermine this trend. Burge argues that Frege is the very Platonist most have thought him — that revisionist interpretations of Frege's Platonism, mine among them, run afoul of the words on Frege's pages. This paper is a response to Burge's criticisms. I argue that my interpretation is more faithful than Burge's to Frege's texts.     

Frege’s Unification

What makes certain definitions fruitful? And how can definitions play an explanatory role? The purpose of this paper is to examine these questions via an investigation of Frege’s treatment of definitions. Specifically, I pursue this issue via an examination of Frege’s views about the scientific unification of logic and arithmetic. In my view, what interpreters have failed to appreciate is that logicism is a project of unification, not reduction. For Frege, unification involves two separate steps: (1) an account of the content expressed by arithmetical claims and (2) the justification of that content. The distinction between these steps allows us to see that there are two notions of definition at play in Frege’s logicist work, viz., one concerned with conceptual analysis, the other concerned with the construction of gap-free proof. I then use this discussion to explain how Frege employs his definitions to defend an epistemological thesis about arithmetic, and to clarify Grundlagen’s fruitfulness condition of definitions, and thereby address two interpretive puzzles from the recent literature.     

OUT OF THE CLOSET—FREGE'S BOOTS

It is not obvious how one might reconcile Frege's claim that different numbers may not 'belong to the same thing' with his apparent identification of one pair with two boots, even if one grants his view of 'statements of number'. I suggest a way. It requires some revision of the semantic theory that is generally attributed to Frege.     

Paradoxien Und Die Vergegenst?ndlichung Von Begriffen – Zu Freges Unterscheidung Zwischen Begriff – Und Gegenstand

In this paper I discuss Frege's distinction between objects and concepts and suggest a solution of Frege's paradox of the concept horse. The expression 'the concept horse' is not eliminated and the concept is not identified with its extension, but the concept is identified with the sense of the corresponding predicate. This solution fits better into a fregean ontology and philosophy of language than alternative solutions and allows for a general answer to the question why Frege's system is infected with Russell's paradox. Russell's paradox is caused by the reification of a concept. Certain problems of modern set theory seem to have a similar cause.Eine weithin sichtbare Warnungstafel muss aufgerichtetwerden: niemand lasse sich einfallen, einen Begriff in einen Gegenstand zu verwandeln!Gottlob Frege     

Fregean Abstraction, Referential Indeterminacy and the Logical Foundations of Arithmetic

In Die Grundlagen der Arithmetik, Frege attempted to introduce cardinalnumbers as logical objects by means of a second-order abstraction principlewhich is now widely known as ``Hume's Principle' (HP): The number of Fsis identical with the number of Gs if and only if F and G are equinumerous.The attempt miscarried, because in its role as a contextual definition HP fails tofix uniquely the reference of the cardinality operator ``the number of Fs'. Thisproblem of referential indeterminacy is usually called ``the Julius Caesar problem'.In this paper, Frege's treatment of the problem in Grundlagen is critically assessed. In particular, I try to shed new light on it by paying special attention to the framework of his logicism in which it appears embedded. I argue, among other things, that the Caesar problem, which is supposed to stem from Frege's tentative inductive definition of the natural numbers, is only spurious, not genuine; that the genuine Caesar problem deriving from HP is a purely semantic one and that the prospects of removing it by explicitly defining cardinal numbers as objects which are not classes are presumably poor for Frege. I conclude by rejecting two closely connected theses concerning Caesar put forward by Richard Heck: (i) that Frege could not abandon Axiom V because he could not solve the Julius Caesar problem without it; (ii) that (by his own lights) his logicist programme in Grundgesetze der Arithmetik failed because he could not overcome that problem.     

Introduction

Frege attempted to provide arithmetic with a foundation in logic. But his attempt to do so was confounded by Russell's discovery of paradox at the heart of Frege's system. The papers collected in this special issue contribute to the on-going investigation into the foundations of mathematics and logic. After sketching the historical background, this introduction provides an overview of the papers collected here, tracing some of the themes that connect them.     

Frege: two theses,two senses

One particular topic in the literature on Frege's conception of sense relates to two apparently contradictory theses held by Frege: the isomorphism of thought and language on one hand and the expressibility of a thought by different sentences on the other. I will divide the paper into five sections. In (1) I introduce the problem of the tension in Frege's thought. In (2) I discuss the main attempts to resolve the conflict between Frege's two contradictory claims, showing what is wrong with some of them. In (3), I analyse where, in Frege'ps writings and discussions on sense identity, one can find grounds for two different conceptions of sense. In (4) I show how the two contradictory theses held by Frege are connected with different concerns, compelling Frege to a constant oscillation in terminology. In (5) I summarize two further reasons that prevented Frege from making the distinction between two conceptions of sense clear: (i) the antipsychologism problem and (ii) the overlap of traditions in German literature contemporary to Frege about the concept of value. I conclude with a hint for a reconstruction of the Fregean notion of ‘thought’ which resolves the contradiction between his two theses.     

Geometry and generality in Frege's philosophy of arithmetic

This paper develops some respects in which the philosophy of mathematics can fruitfully be informed by mathematical practice, through examining Frege'sGrundlagen in its historical setting. The first sections of the paper are devoted to elaborating some aspects of nineteenth century mathematics which informed Frege's early work. (These events are of considerable philosophical significance even apart from the connection with Frege.) In the middle sections, some minor themes ofGrundlagen are developed: the relationship Frege envisions between arithmetic and geometry and the way in which the study of reasoning is to illuminate this. In the final section, it is argued that the sorts of issues Frege attempted to address concerning the character of mathematical reasoning are still in need of a satisfying answer.I am indebted to many people for helpful conversations and comments on this paper, notably Stephen Glaister, Phil Kremer, Madeline Larson, John McDowell, Jim Conant, Charles Chihara, William Craig, Jan Alnes, Joan Weiner, Leon Henkin, Paul Benacerraf, Juliet Floyd, Bill Demopoulos, Jose Ferreiros, Tom Hawkins, Gideon Rosen. Two superb papers on Frege — Bill Demopoulos' (1994) and Mark Wilson (1992) played a significant role in the early stages of composition. Special thanks are due to Hans Sluga, Mark Wilson, Bob Brandom, and Ken Manders for comments, encouragement, information and advice.     

Erkenntnistheorie der Zahldefinition und philosophische Grundlegung der Arithmetik unter Bezugnahme auf einen Vergleich von Gottlob Freges Logizismus und platonischer Philosophie (Syrian, Theon von Smyrna u.a.)

The epistomology of the definition of number and the philosophical foundation of arithmetic based on a comparison between Gottlob Frege's logicism and Platonic philosophy (Syrianus, Theo Smyrnaeus, and others). The intention of this article is to provide arithmetic with a logically and methodologically valid definition of number for construing a consistent philosophical foundation of arithmetic. The – surely astonishing – main thesis is that instead of the modern and contemporary attempts, especially in Gottlob Frege's Foundations of Arithmetic, such a definition is found in the arithmetic in Euclid's Elements. To draw this conclusion a profound reflection on the role of epistemology for the foundation of mathematics, especially for the method of definition of number, is indispensable; a reflection not to be found in the contemporary debate (the predominate ‘pragmaticformalism’ in current mathematics just shirks from trying to solve the epistemological problems raised by the debate between logicism, intuitionism, and formalism). Frege's definition of number, ‘The number of the concept F is the extension of the concept ‘numerically equal to the concept F”, which is still substantial for contemporary mathematics, does not fulfil the requirements of logical and methodological correctness because the definiens in a double way (in the concepts ‘extension of a concept’ and ‘numerically equal’) implicitly presupposes the definiendum, i.e. number itself. Number itself, on the contrary, is defined adequately by Euclid as ‘multitude composed of units’, a definition which is even, though never mentioned, an implicit presupposition of the modern concept ofset. But Frege rejects this definition and construes his own - for epistemological reasons: Frege's definition exactly fits the needs of modern epistemology, namely that for to know something like the number of a concept one must become conscious of a multitude of acts of producing units of ‘given’ representations under the condition of a 1:1 relationship to obtain between the acts of counting and the counted ‘objects’. According to this view, which has existed at least since the Renaissance stoicism and is maintained not only by Frege but also by Descartes, Kant, Husserl, Dummett, and others, there is no such thing as a number of pure units itself because the intellect or pure reason, by itself empty, must become conscious of different units of representation in order to know a multitude, a condition not fulfilled by Euclid's conception. As this is Frege's main reason to reject Euclid's definition of number (others are discussed in detail), the paper shows that the epistemological reflection in Neoplatonic mathematical philosophy, which agrees with Euclid's definition of number, provides a consistent basement for it. Therefore it is not progress in the history of science which hasled to the a poretic contemporary state of affairs but an arbitrary change of epistemology in early modern times, which is of great influence even today. This revised version was published online in August 2006 with corrections to the Cover Date.     

SYMBOLIC FOODS: PREGNANCY CRAVINGS AND THE ENVIOUS FEMALE

In my article I intend to show that the pregnancy cravings by Sri Lankan women constitute a system of personal symbols that must be understood in relation to the female role and the psychological problems engendered by it. I define personal symbols as those operating simultaneously at the level of both personality and culture. Through the notion of personal symbols, I intend to bridge the conventional distinction between private and public symbols and between culture and emotion. I show that pregnancy desires, which may be constituted as symptoms in the West, have been converted into personal symbols in South Asia since antiquity and given public meaning and validation.     

Frege and propositional unity

This paper identifies a tension in Frege’s philosophy and offers a diagnosis of its origins. Frege’s Context Principle can be used to dissolve the problem of propositional unity. However, Frege’s official response to the problem does not invoke the Context Principle, but the distinction between ‘saturated’ and ‘unsaturated’ propositional constituents. I argue that such a response involves assumptions that clash with the Context Principle. I suggest, however, that this tension is not generated by deep-seated philosophical commitments, but by Frege’s occasional attempt to take a dubious shortcut in the justification of his conception of propositional structure.     

Meaning and Truth-Conditions: a Reply to Kemp

In his 'Meaning and Truth-Conditions', Gary Kemp offers a reconstruction of Frege's infamous 'regress argument', which purports to rely only upon the premises that the meaning of a sentence is its truth-condition and that each sentence expresses a unique proposition. If cogent, the argument would show that only someone who accepts a form of semantic holism can use the notion of truth to explain that of meaning. I respond that Kemp relies heavily upon what he himself styles 'a literal, rather wooden' understanding of truth-conditions. I explore alternatives, and say a few words about how Frege's regress argument might best be understood.     

Thought's Social Nature

Abstract: Wittgenstein, throughout his career, was deeply Fregean. Frege thought of thought as essentially social, in this sense: whatever I can think is what others could think, deny, debate, investigate. Such, for him, was one central part of judgement's objectivity. Another was that truths are discovered, not invented: what is true is so, whether recognised as such or not. (Later) Wittgenstein developed Frege's idea of thought as social compatibly with that second part. In this he exploits some further Fregean ideas: of a certain generality intrinsic to a thought; of lack of that generality in that which a thought represents as instancing some such generality. (I refer to this below as the ‘conceptual‐nonconceptual’ distinction.) Seeing Wittgenstein as thus building on Frege helps clarify (inter alia) his worries, in the Blue Book, and the Investigations, about meaning, intending, and understanding, and the point of the rule following discussion.     

The Three Quines

This paper concerns Quine's stance on the issue of meaning normativity. I argue that three distinct and not obviously compatible positions on meaning normativity can be extracted from his philosophy of language - eliminative ]naturalism (Quine I), deflationary pragmatism (Quine II), and (restricted) strong normativism (Quine III) - which result from Quine's failure to separate adequately four different questions that surround the issue: the reality, source, sense, and scope of the normative dimension. In addition to the incompatibility of the views taken together, I argue on the basis of considerations due to Wittgenstein, Dummett, and Davidson that each view taken separately has self-standing problems. The first two fail to appreciate the ineliminability of the strong normativity of logic and so face a dilemma: they either smuggle it in illicitly, or insofar as they do not, fail to give an account of anything like a language. The third position's mixture of a universalism about logical concepts with a thorough-going relativism about non-logical concepts can be challenged once a distinction is drawn between the universalist and contextualist readings of strong normativity, a distinction inspired by Wittgenstein's distinction between grammatical and empirical judgements.     

Extensions As Representative Objects In Frege's Logic

Matthias Schirn has argued on a number of occasions against the interpretation of Frege's ``objects of a quite special kind' (i.e., the objects referred to by names like `the concept F') as extensions of concepts. According to Schirn, not only are these objects not extensions, but also the idea that `the concept F' refers to objects leads to some conclusions that are counter-intuitive and incompatible with Frege's thought. In this paper, I challenge Schirn's conclusion: I want to try and argue that the assumption that `the concept F' refers to the extension of F is entirely consistent with Frege's broader views on logic and language. I shall examine each of Schirn's main arguments and show that they do not support his claim.     

FREGE'S CONCEPT PARADOX AND THE MIRRORING PRINCIPLE

Frege held that singular terms can refer only to objects, not to concepts. I argue that the counter-intuitive consequences of this claim ('the concept paradox') arise from Frege's mirroring principle that an incomplete expression can only express an incomplete sense and stand for an incomplete reference. This is not, as is sometimes thought, merely because predicates and singular terms cannot be intersubstituted salva veritate ( congruitate ). The concept paradox, properly understood, poses therefore a different, harder, challenge. An investigation of the foundations of the mirroring principle also sheds light on the role which language plays in Frege's epistemology of logic.