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- Aldo Antonelli, Frege's New Science.In this paper, we explore Fregean metatheory, what Frege called the New Science. The New Science arises in the context of Frege’s debate with Hilbert over independence proofs in geometry and we begin by considering their dispute. We propose that Frege’s critique rests on his view that language is a set of propositions, each immutably equipped with a truth value (as determined by the thought it expresses), so to Frege it was inconceivable that axioms could even be considered to be other than true. Because of his adherence to this view, Frege was precluded from the sort of metatheoretical considerations that were available to Hilbert; but from this, we shall argue, it does not follow that Frege was blocked from metatheory in toto. Indeed, Frege suggests in Die Grundlagen der Geometrie a metatheoretical method for establishing independence proofs in the context of the New Science. Frege had reservations about the method, however, primarily because of the apparent need to stipulate the logical terms, those terms that must be held invariant to obtain such proofs. We argue that Frege’s skepticism on this score is not warranted, by showing that within the New Science a characterization of logical truth and logical constant can be obtained by a suitable adaptation of the permutation argument Frege employs in indicating how to prove independence. This establishes a foundation for Frege’s metatheoretical method of which he himself was unsure, and allows us to obtain a clearer understanding of Frege’s conception of logic, especially in relation to contemporary conceptions.Reading list | Discuss | Edit | Categorize |My bibliography
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Frege's account of indirect proof has been thought to be problematic. This thought seems to rest on the supposition that some notion of logical consequence ? which Frege did not have ? is indispensable for a satisfactory account of indirect proof. It is not so. Frege's account is no less workable than the account predominant today. Indeed, Frege's account may be best understood as a restatement of the latter, although from a higher order point of view. I argue that this ascent is motivated by Frege's conception of logic.
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| | Other links: dx.doi.org | Scholar | At my library | More options ... I examine Frege’s explanation of how Hilbert ought to have presented his proofs of the independence of the axioms of geometry: in terms of mappings between (what we would call) fully interpreted statements. This helps make sense of Frege’s objections to the notion of different interpretations, which many have found puzzling. (The paper is the text of a talk presented in October 1994.).
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| | Scholar | More options ... Contemporary semantical discussions make mention of the traditional approach to semantics represented by Frege and/or Russell--even sometimes by Frege-Russell. Is there a Frege-Russell view in the philosophy of language? How much of a common semantical perspective did Frege and Russell share? The matter bears exploration. I begin with Frege and Russell on propositions.
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| | Scholar | More options ... Gottlob Frege famously rejects the methodology for consistency and independence proofs offered by David Hilbert in the latter's Foundations of Geometry. The present essay defends against recent criticism the view that this rejection turns on Frege's understanding of logical entailment, on which the entailment relation is sensitive to the contents of non-logical terminology. The goals are (a) to clarify further Frege's understanding of logic and of the role of conceptual analysis in logical investigation, and (b) to point out the extent to which his understanding of logic differs importantly from that of the model-theoretic tradition that grows out of Hilbert's work. Many thanks for helpful comments to Wilfrid Hodges, to Bob Hale, and to an anonymous referee.
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| | Other links: philmat.oxfordjournals.org dx.doi.org | Scholar | At my library | More options ... 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.
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| | Other links: blackwell-synergy.com dx.doi.org | Scholar | At my library | More options ... It is widely believed that some puzzling and provocative remarks that Frege makes in his late writings indicate he rejected independence arguments in geometry, particularly arguments for the independence of the parallels axiom. I show that this is mistaken: Frege distinguished two approaches to independence arguments and his puzzling remarks apply only to one of them. Not only did Frege not reject independence arguments across the board, but also he had an interesting positive proposal about the logical structure of correct independence arguments, deriving from the geometrical principle of duality and the associated idea of substitution invariance. The discussion also serves as a useful focal point for independently interesting details of Frege’s mathematical environment. This feeds into a currently active scholarly debate because Frege’s supposed attitude to independence arguments has been taken to support a widely accepted thesis (proposed by Ricketts among others) concerning Frege’s attitude toward metatheory in general. I show that this thesis gains no support from Frege’s puzzling remarks about independence arguments.
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| | Other links: projecteuclid.org:80 dx.doi.org | Scholar | At my library | More options ... A cluster of recent papers on Frege have urged variations on the theme that Frege’s conception of logic is in some crucial way incompatible with ‘metatheoretic’ investigation. From this observation, significant consequences for our interpretation of Frege’s understanding of his enterprise are taken to follow. This chapter aims to critically examine this view, and to isolate what I take to be the core of truth in it. However, I will also argue that once we have isolated the defensible kernel, the sense in which Frege was committed to rejecting ‘metatheory’ is too narrow and uninteresting to support the con-.
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| | Other links: pdcnet.org | Scholar | At my library | More options ... In a letter to Frege of 29 December 1899, Hilbert advances his formalist doctrine, according to which consistency of an arbitrary set of mathematical sentences is a sufficient condition for its truth and for the existence of the concepts described by it. This paper discusses Frege's analysis, as carried out in the context of the Frege-Hilbert correspondence, of the formalist approach in particular and the axiomatic method in general. We close with a speculation about Frege's influence on Hilbert's later work in foundations, which we consider to have been greater than previously assumed. This conjecture is based on a hitherto neglected revision of Hilbert's talk "Über den Zahlbegriff".
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| | Other links: tandfonline.com dx.doi.org | Scholar | At my library | More options ... This paper attempts to confine the preconceptions that prevented Frege from appreciating Hilbert?s Grundlagen der Geometrie to two: (i) Frege?s reliance on what, following Wilfrid Hodges, I call a Frege?Peano language, and (ii) Frege?s view that the sense of an expression wholly determines its reference.I argue that these two preconceptions prevented Frege from achieving the conceptual structure of model theory, whereas Hilbert, at least in his practice, was quite close to the model?theoretic point of view.Moreover, the issues that divided Frege and Hilbert did not revolve around whether one or the other allowed metalogical notions.Frege, e.g., succeeded in formulating the notion of logical consequence, at least to the extent that Bolzano did; the point is rather that even though Frege had certain semantic concepts, he did not articulate them model?theoretically, whereas, in some limited sense, Hilbert did.
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| | Other links: tandfonline.com dx.doi.org | Scholar | At my library | More options ... In this paper, we explore Fregean metatheory, what Frege called the New Science. The New Science arises in the context of Frege’s debate with Hilbert over independence proofs in geometry and we begin by considering their dispute. We propose that Frege’s critique rests on his view that language is a set of propositions, each immutably equipped with a truth value (as determined by the thought it expresses), so to Frege it was inconceivable that axioms could even be considered to be other than true. Because of his adherence to this view, Frege was precluded from the sort of metatheoretical considerations that were available to Hilbert; but from this, we shall argue, it does not follow that Frege was blocked from metatheory in toto. Indeed, Frege suggests in Die Grundlagen der Geometrie a metatheoretical method for establishing independence proofs in the context of the New Science. Frege had reservations about the method, however, primarily because of the apparent need to stipulate the logical terms, those terms that must be held invariant to obtain such proofs. We argue that Frege’s skepticism on this score is not warranted, by showing that within the New Science a characterization of logical truth and logical constant can be obtained by a suitable adaptation of the permutation argument Frege employs in indicating how to prove independence. This establishes a foundation for Frege’s metatheoretical method of which he himself was unsure, and allows us to obtain a clearer understanding of Frege’s conception of logic, especially in relation to contemporary conceptions.
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| | Scholar | More options ... Discussion of Aldo Antonelli, Frege's new science
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