Three clusters of philosophically significant issues arise from Frege's discussions of definitions. First, Frege criticizes the definitions of mathematicians of his day, especially those of Weierstrass and Hilbert. Second, central to Frege's philosophical discussion and technical execution of logicism is the so-called Hume's Principle, considered in The Foundations of Arithmetic . Some varieties of neo-Fregean logicism are based on taking this principle as a contextual definition of the operator 'the number of …', and criticisms of such neo-Fregean programs sometimes appeal (...) to Frege's objections to contextual definitions in later writings. Finally, a critical question about the definitions on which Frege's proofs of the laws of arithmetic depend is whether the logical structures of the definientia reflect our pre-Fregean understanding of arithmetical terms. It seems that unless they do, it is unclear how Frege's proofs demonstrate the analyticity of the arithmetic in use before logicism. Yet, especially in late writings, Frege characterizes the definitions as arbitrary stipulations of the senses or references of expressions unrelated to pre-definitional understanding. One or more of these topics may be studied in a survey course in the philosophy of mathematics or a course on Frege's philosophy. The latter two topics are obviously central in a seminar in the philosophy of mathematics in general or more specialized seminars on logicism, or on mathematical definitions and concept formation. Author Recommends: 1. Kant, Immanuel. Critique of Pure Reason . Trans. P. Guyer and A. Wood. Cambridge: Cambridge University Press, 1999 [1781, 1787], A7-10/B11-14, A151/B190. In the first Critique , Kant appears to give four distinct accounts of analytic judgments. The initial famous account explains analyticity in terms of the predicate-concept belonging to the subject-concept (A6–7/B11). In this passage, we also find an account of establishing analytic judgments on the basis of conceptual containments and the principle of non-contradiction. (The other accounts are in terms of 'identity' (A7/B1l), in terms of the explicative–ampliative contrast (A7/B11), and by reference to the notion of 'cognizability in accordance with the principle of contradiction' (A151/B190).) 2. Frege, Gottlob. The Foundations of Arithmetic . Trans. J. L. Austin. 2nd ed. Evanston, IL: Northwestern University Press, 1980 , especially sections 1–4, 87–91. Frege here criticizes and reformulates Kant's account of analyticity. Central to Frege's account is the provability of an analytic statement on the basis of (Frege's) logic and definitions that express analyses of (mathematical, especially arithmetical) concepts. 3. Frege, Gottlob. Review of E. G. Husserl. 'Philosophie der Arithmetik I ,' in Frege, Collected Papers . Ed. B. McGuinness. Trans. M. Black et al. Oxford: Blackwell, 1984. 195–209. In this review, Frege responds to Husserl's charge that Frege's definitions fail to capture our intuitive pre-analytic arithmetical concepts by claiming that the adequacy of mathematical definitions is measured, not by their expressing the same senses, but merely by their having the same references, as pre-definitional vocabulary. It follows not only that Husserl's criticism is unfounded, but also that there can be alternative, equally legitimate, definitions of mathematical terms. 4. Frege, 'Logic in Mathematics,' in Frege, Posthumous Writings . Trans. P. Long and R. White. Oxford: Blackwell, 1979 . 203–50. These are a set of lecture notes including, among other things, an account of proper definitions as mere abbreviation of complex signs by simple ones, in contrast to definitions which purport to express the analyses of existing concepts. Frege here claims that if there is any doubt whether a definition purporting to express an analysis succeeds in capturing the senses of the pre-definitional expressions, then the definition fails as an analysis, and should be regarded as the introduction of an entirely new expression abbreviating the definiens . 5. Picardi, Eva. 'Frege on Definition and Logical Proof,' Temi e Prospettive della Logica e della Filosofia della Scienza Contemporanee . i vol. Eds. C. Cellucci and G. Sambin. Bologna: Cooperativa Libraria Universitaria Editrice Bologna, 1988. 227–30. Picardi sets out forcefully the view that unless Frege's definitions capture the meanings of existing arithmetical terms, his logicism cannot have the epistemological significance he takes it to have. 6. Dummett, Michael. 'Frege and the Paradox of Analysis,' in Dummett, Frege and other Philosophers . Oxford: Oxford University Press, 1991. 17–52. Dummett agrees with Picardi's view and analyzes the philosophical pressures that led Frege to the account of definition in 'Logic in Mathematics.' Especially significant is Dummett's claim of the centrality of the transparency of sense – that if one grasps the senses of any two expressions, one must know whether they have the same sense – in Frege's account. 7. Benacerraf, Paul. 'Frege: The Last Logicist,' Midwest Studies in Philosophy . vol. 6. Eds. P. French, T. Uehling, and H. Wettstein. Minneapolis: University of Minnesota Press, 1981. 17–35. Frege's aims, on Benacerraf's reading, are primarily mathematical. Frege was interested in traditional philosophical issues such as the analyticity of arithmetic only to the extent that they can be exploited for the mathematical goal of proving previously unproven arithmetical statements. Hence, Frege never had any serious interest in or need for showing that his definitions of arithmetical terms reflect existing arithmetical conceptions. 8. Weiner, Joan. 'The Philosopher Behind the Last Logicist,' in Frege: Tradition and Influence . Ed. C. Wright. Oxford: Blackwell, 1984. 57–79. Weiner argues that on Frege's view, prior to his definitions of arithmetical terms the references of such expressions are in fact not known by those who use arithmetical vocabulary. Thus, in Foundations , Frege operated with a 'hidden agenda' (263) namely, replacing existing arithmetic with a new science based on stipulative definitions that assign new senses to key arithmetical terms. 9. Tappenden, Jamie. 'Extending Knowledge and 'Fruitful Concepts': Fregean Themes in the Foundations of Mathematics.' Noûs 29 (1995): 427–67. Tappenden argues that Frege takes his crucial innovation over previous practices and accounts of mathematical concept formation to be the role of quantificational structure made possible by his logical discoveries. 10. Horty, John. Frege on Definitions: A Case Study of Semantic Content . Oxford: Oxford University Press, 2007. A useful interpretation of Frege's views of definition, together with suggestive extensions for resolving the issues framing Frege's views. 11. Shieh, Sanford. 'Frege on Definitions,' Philosophy Compass 3/5 (2008): 992–1012. A more detailed account of Frege's views on definition and the philosophical issues they raise, surveying and discussing critically the main substantive and interpretive issues. Online Materials On Frege http://plato.stanford.edu/entries/frege/ On the Paradox of Analysis http://plato.stanford.edu/entries/analysis/ Sample Syllabus The following is a 3-week module that can be incorporated into fairly focused historically oriented graduate-level seminars on logicism or on the paradox of analysis. It is also possible to compress the material into 2 weeks in an undergraduate or graduate class Frege's thought in general. Week I: Background, Kant on Analyticity; Definition in Foundations , Review of Husserl, and 'Logic in Mathematics' Readings Kant, Immanuel. Critique of Pure Reason , A7–10/B11–14. Frege, Gottlob. The Foundations of Arithmetic , sections 1–4, 87–91. Frege, Gottlob. Review of E. G. Husserl, Philosophie der Arithmetik I. Frege, Gottlob. 'Logic in Mathematics.' Optional Proops, Ian. 'Kant's Conception of Analytic Judgment,' Philosophy and Phenomenological Research LXX, 3 (2005): 588–612. Week II: The Supposed Paradox of Analysis, Picardi and Dummett; Bypassing Traditional Epistemological Issues About Mathematics, Benacerraf Readings Picardi, Eva. 'Frege on Definition and Logical Proof.' Dummett, Michael. 'Frege and the Paradox of Analysis.' Benacerraf, Paul. 'Frege: The Last Logicist.' Optional Tappenden, Jamie. 'Extending Knowledge and 'Fruitful Concepts': Fregean Themes in the Foundations of Mathematics.' Week III: Weiner's Hidden Agenda Interpretation Readings Weiner, Joan. 'The Philosopher Behind the Last Logicist.' Optional Weiner, Joan. Frege in Perspective . Ithaca, NY: Cornell University Press, 1990. Focus Questions 1. To what extent is Frege's account of analyticity in Foundations a rejection, and to what extent an updating, of Kant's view of analyticity? 2. According to Picardi it 'would be incomprehensible' how Frege's proofs tells us anything about the arithmetic we already have unless his 'definitions [are] somehow responsible to the meaning of [arithmetical] sentences as these are understood' (228). Why does she hold this? Why does Dummett agree with her? Do you think Frege's logicism needs to address this worry? 3. What are the major differences and continuities in Frege's discussions of definition in mathematics in Foundations , the review of Husserl and 'Logic in Mathematics'? 4. Frege writes that definitions must prove their worth by being fruitful. He also says that nothing can be proven using a proper definition that cannot be proven without it. Are these claims consistent? Why or why not? 5. Weiner held that in Foundations Frege had 'hidden agenda.' What, according to her, is this agenda? How does this fit with Frege's later views of definition? 6. What are Frege's main complaints about Weierstrass's definitions in 'Logic in Mathematics'? Are these criticisms consistent with Frege's account of 'definition proper' in the same text? Seminar/Project Ideas What, if anything, is the relation between Frege's critique of Hilbert's use of definitions and Frege's later views of definitions? (shrink)
This article treats three aspects of Frege's discussions of definitions. First, I survey Frege's main criticisms of definitions in mathematics. Second, I consider Frege's apparent change of mind on the legitimacy of contextual definitions and its significance for recent neo-Fregean logicism. In the remainder of the article I discuss a critical question about the definitions on which Frege's proofs of the laws of arithmetic depend: do the logical structures of the definientia reflect the understanding of arithmetical terms prevailing prior to (...) Frege's analyses? Unless they do, it is unclear how Frege's proofs demonstrate the analyticity of the arithmetic in use before logicism. Yet, especially in late writings, Frege characterizes definitions as arbitrary stipulations of the senses or references of expressions unrelated to pre-definitional understanding. I conclude by examining some options for conceiving of the status of Frege's logicism in light of this apparent tension, and outline a suggestion for a philosophically fruitful way of resolving this tension. (shrink)
Alongside Richard Rorty, Hilary Putnam and Jacques Derrida, Stanley Cavell is arguably one of the best-known philosophers in the world. In this state-of-the-art collection, Alice Crary explores the work of this original and interesting figure who has already been the subject of a number of books, conferences and Phd theses. A philosopher whose work encompasses a broad range of interests, such as Wittgenstein, scepticism in philosophy, the philosophy of art and film, Shakespeare, and philosophy of mind and language, Cavell has (...) also written much about Henry Thoreau and Ralph Waldo Emerson. Including contributions from Hilary Putnam, Cora Diamond, Jim Conant and Stephen Mulhall, this book is a must-have for libraries and students alike. (shrink)
This collection of previously unpublished essays presents a new approach to the history of analytic philosophy--one that does not assume at the outset a general characterization of the distinguishing elements of the analytic tradition. Drawing together a venerable group of contributors, including John Rawls and Hilary Putnam, this volume explores the historical contexts in which analytic philosophers have worked, revealing multiple discontinuities and misunderstandings as well as a complex interaction between science and philosophical reflection.
The central premise of Michael Dummett's global argument for anti-realism is the thesis that a speaker's grasp of the meaning of a declarative, indexical-free sentence must be manifested in her uses of that sentence. This enigmatic thesis has been the subject of a great deal of discussion, and something of a consensus has emerged about its content and justification. The received view is that the manifestation thesis expresses a behaviorist and reductive theory of meaning, essentially in agreement with Quine's view (...) of language, and motivated by worries about the epistemology of communication. In the present paper I begin by arguing that this standard interpretation of the manifestation thesis is neither particularly faithful to Dummett's writings nor philosophically compelling. I then continue by reconstructing, from Dummett's texts, an account of the manifestation thesis, and of its justification, that differ sharply from the received view. On my reading, the thesis is motivated not epistemologically, but conceptually. I argue that connections among our conceptions of meaning, assertion, and justification lead to a conclusion about the metaphysics of meaning: we cannot form a clearly coherent conception of how two speakers can attach different meanings to a sentence without at the same time differing in what they count as justifying assertions made with that sentence. I conclude with some suggestions about how Dummett's argument for global anti-realism should be understood, given my account of the manifestation thesis. (shrink)
In this paper I attempt to clarify a relatively little-studied aspect of Michael Dummett's argument for intuitionism: its use of the notion of ‘undecidable’ sentence. I give a new analysis of this concept in epistemic terms, with which I resolve some puzzles and questions about how it works in the anti-realist critique of classical logic.