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- Joan Weiner (2007). What's in a Numeral? Frege's Answer. Mind 116 (463):677 - 716.Frege wanted to define the number 1 and the concept of number. What is required of a satisfactory definition? A truly arbitrary definition will not do: to stipulate that the number one is Julius Caesar is to change the subject. One might expect Frege to define the number 1 by giving a description that picks out the object that the numeral '1' already names; to define the concept of number by giving a description that picks out precisely those objects that are numbers. Yet Frege appears to do no such thing. Indeed, when he defends his definitions, he does not argue that they pick out objects that we have been talking about all along-the issue never comes up. The aim of this paper is to explain why. I argue that, on Frege's view, our numerals do not, antecedent to his work, name particular objects. This raises an obvious question: If (like 'Odysseus') the numerals do not name particular objects, how can Frege write (as he does) as if sentences in which numerals appear state truths? One central concern of this paper is exegetical-to answer these questions. But my aim is not solely exegetical. For these questions direct us to something that, I believe, creates only an apparent problem for Frege but an actual problem for many contemporary philosophers: the assumption that singular terms appearing in statements about the world must actually have referents. Another aim of this paper is to suggest that the problem-as well as a solution that can be found in Frege's writings-should be of import to contemporary philosophers.Reading list | Discuss | Edit | Categorize |My bibliography
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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.
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| | Other links: doi.wiley.com dx.doi.org | Scholar | At my library | More options ... In this paper, the author derives the Dedekind–Peano axioms for number theory from a consistent and general metaphysical theory of abstract objects. The derivation makes no appeal to primitive mathematical notions, implicit definitions, or a principle of infinity. The theorems proved constitute an important subset of the numbered propositions found in Frege"s Grundgesetze. The proofs of the theorems reconstruct Frege"s derivations, with the exception of the claim that every number has a successor, which is derived from a modal axiom that (philosophical) logicians implicitly accept. In the final section of the paper, there is a brief philosophical discussion of how the present theory relates to the work of other philosophers attempting to reconstruct Frege"s conception of numbers and logical objects.
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| | Other links: springerlink.com dx.doi.org jstor.org | Scholar | At my library | More options ... In this paper, I seek to clarify an aspect of Frege's thought that has been only insufficiently explained in the literature, namely, his notion of logical objects. I adduce some elements of Frege's philosophy that elucidate why he saw extensions as natural candidates for paradigmatic cases of logical objects. Moreover, I argue (against the suggestion of some contemporary scholars, in particular, Wright and Boolos) that Frege could not have taken Hume's Principle instead of Axiom V as a fundamental law of arithmetic. This would be inconsistent with his views on logical objects. Finally, I shall argue that there is a connection between Frege's view on this topic and the famous thesis first formulated in ‘Über Begriff und Gegenstand’ that ‘the concept horse is not a concept’. As far as I know, no due attention has been given to this connection in the scholarly literature so far.
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| | Other links: mind.oupjournals.org jstor.org dx.doi.org | Scholar | At my library | More options ... 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.
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| | Other links: indiana.edu jstor.org | Scholar | At my library | More options ... The aim of this paper is to present the Nyya concept of number in the light of contemporary philosophy and to show that the Frege-Russell concept of number does not contradict the Nyya concept of number but rather supplements it.
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| | Other links: jstor.org | Scholar | At my library | More options ... In his Grundgesetze, Frege hints that prior to his theory that cardinal numbers are objects (courses-of-values) he had an “almost completed” manuscript on cardinals. Taking this early theory to have been an account of cardinals as second-level functions, this paper works out the significance of the fact that Frege’s cardinal numbers (as objects) is a theory of concept-correlates. Frege held that, where n>2, there is a one–one correlation between each n-level function and an n−1 level function, and a one–one correlation between each first-level function and an object (a course-of-values of the function). Applied to cardinals, the correlation offers new answers to some perplexing features of Frege’s philosophy. It is shown that within Frege’s concept-script, a generalized form of Hume’s Principle is equivalent to Russell’s Principle of Abstraction – a principle Russell employed to demonstrate the inadequacy of definition by abstraction. Accordingly, Frege’s rejection of definition of cardinal number by Hume’s Principle parallels Russell’s objection to definition by abstraction. Frege’s correlation thesis reveals that he has a way of meeting the structuralist challenge (later revived by Benacerraf, 1965) that it is arithmetic, and not a privileged progression of objects, that matters to the finite cardinals.
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| | Other links: jstor.org | Scholar | At my library | More options ... Although the notion of logical object plays a key role in Frege's foundational project, it has hardly been analyzed in depth so far. I argue that Marco Ruffino's attempt to fill this gap by establishing a close link between Frege's treatment of expressions of the form ‘the concept F’ and the privileged status Frege assigns to extensions of concepts as logical objects is bound to fail. I argue, in particular, that Frege's principal motive for introducing extensions into his logical theory is not to be able to make in-direct statements about concepts, but rather to define all numbers as logical objects of a fundamental kind in order to ensure that we have the right cognitive access to them qua logical objects via Axiom V. Contrary to what Ruffino claims, reducibility to extensions cannot be the ‘ultimate criterion’ for Frege of what is to be regarded as a logical object.
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| | Other links: mind.oxfordjournals.org jstor.org dx.doi.org | Scholar | At my library | More options ... 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.
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| | Other links: jstor.org | Scholar | At my library | More options ... 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.
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| | Other links: springerlink.com dx.doi.org jstor.org | Scholar | At my library | More options ... The question whether numbers are objects is a central question in the philosophy of mathematics. Frege made use of a syntactic criterion for objethood: numbers are objects because there are singular terms that stand for them, and not just singular terms in some formal language, but in natural language in particular. In particular, Frege (1884) thought that both noun phrases like the number of planets and simple numerals like eight as in (1) are singular terms referring to numbers as abstract objects.
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| | Other links: semantics.univ-paris1.fr | Scholar | At my library | More options ... Discussion of Joan Weiner, What's in a Numeral? Frege's Answer
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