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- Friederike Moltmann (forthcoming). Reference to Numbers in Natural Language. Philosophical Studies:-.Abstract A common view is that natural language treats numbers as abstract objects, with expressions like the number of planets , eight , as well as the number eight acting as referential terms referring to numbers. In this paper I will argue that this view about reference to numbers in natural language is fundamentally mistaken. A more thorough look at natural language reveals a very different view of the ontological status of natural numbers. On this view, numbers are not primarily treated abstract objects, but rather ‘aspects’ of pluralities of ordinary objects, namely number tropes, a view that in fact appears to have been the Aristotelian view of numbers. Natural language moreover provides support for another view of the ontological status of numbers, on which natural numbers do not act as entities, but rather have the status of plural properties, the meaning of numerals when acting like adjectives. This view matches contemporary approaches in the philosophy of mathematics of what Dummett called the Adjectival Strategy, the view on which number terms in arithmetical sentences are not terms referring to numbers, but rather make contributions to generalizations about ordinary (and possible) objects. It is only with complex expressions somewhat at the periphery of language such as the number eight that reference to pure numbers is permitted. Content Type Journal Article Pages 1-38 DOI 10.1007/s11098-011-9779-1 Authors Friederike Moltmann, IHPST (Paris1/ENS/CNRS), Paris, France Journal Philosophical Studies Online ISSN 1573-0883 Print ISSN 0031-8116.Reading list | Discuss | Edit | Categorize |My bibliography
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A number is the number of a class which is an objective, nonactual, mathematical object. The concept of class is analyzed and it is concluded that a number is the number of a pure founded class. A tempting strategy of explaining numbers away is rejected. Some well-known definitions of numbers are analyzed and it is concluded that this analysis purports the thesis that the unique notion of number does not exist. Numbers are conventional. Nevertheless, an argument is offered purporting the thesis that von Neumann's ordinal numbers are the ordinal numbers. Accordingly, the corresponding von Neumann's cardinal numbers are the numbers.
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| | Other links: tandfonline.com dx.doi.org | Scholar | At my library | More options ... The internal language of a monoidal category yields simple proofs of results about a natural numbers object therein.
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| | Other links: jstor.org | Scholar | At my library | More options ... Abstract A number is the number of a class which is an objective, nonactual, mathematical object. The concept of class is analyzed and it is concluded that a number is the number of a pure founded class. A tempting strategy of explaining numbers away is rejected. Some well?known definitions of numbers are analyzed and it is concluded that this analysis purports the thesis that the unique notion of number does not exist. Numbers are conventional. Nevertheless, an argument is offered purporting the thesis that von Neumann's ordinal numbers are the ordinal numbers. Accordingly, the corresponding von Neumann's cardinal numbers are the numbers.
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| | Other links: tandfonline.com dx.doi.org | Scholar | At my library | More options ... This book presents a comprehensive treatment of basic mathematical logic. The author's aim is to make exact the vague, intuitive notions of natural number, preciseness, and correctness, and to invent a method whereby these notions can be communicated to others and stored in the memory. He adopts a symbolic language in which ideas about natural numbers can be stated precisely and meaningfully, and then investigates the properties and limitations of this language. The treatment of mathematical concepts in the main body of the text is rigorous, but, a section of 'historical remarks' traces the evolution of the ideas presented in each chapter. Sources of the original accounts of these developments are listed in the bibliography.
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| | Scholar | At my library | More options ... I follow standard mathematical practice and theory to argue that the natural numbers are the finite von Neumann ordinals. I present the reasons standardly given for identifying the natural numbers with the finite von Neumann's (e.g., recursiveness; well-ordering principles; continuity at transfinite limits; minimality; and identification of n with the set of all numbers less than n). I give a detailed mathematical demonstration that 0 is { } and for every natural number n, n is the set of all natural numbers less than n. Natural numbers are sets. They are the finite von Neumann ordinals.
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| | Other links: jstor.org | Scholar | At my library | More options ... In this paper, I suggest that infinite numbers are large finite numbers, and that infinite numbers, properly understood, are 1) of the structure omega + (omega* + omega)Ө + omega*, and 2) the part is smaller than the whole. I present an explanation of these claims in terms of epistemic limitations. I then consider the importance, part of which is demonstrating the contradiction that lies at the heart of Cantorian set theory: the natural numbers are too large to be counted by any finite number, but too small to be counted by any infinite number – there is no number of natural numbers.
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| | Other links: gc-cuny.academia.edu | Scholar | More options ... Early sections of the paper develop a view of the natural numbers and a view of counting which are suggested by the remarks of several modern philosophers. Further investigation of these views leads to one of the main theses of the paper: a special kind of quantifier, the "numerical quantifier" is essential to counting. The remainder of the paper suggests the rudiments of a new view of the natural numbers, a view which maintains that numerical quantifiers are one kind of natural number.
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| | Other links: jstor.org journals.uchicago.edu dx.doi.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 ... Abstract A common view is that natural language treats numbers as abstract objects, with expressions like the number of planets , eight , as well as the number eight acting as referential terms referring to numbers. In this paper I will argue that this view about reference to numbers in natural language is fundamentally mistaken. A more thorough look at natural language reveals a very different view of the ontological status of natural numbers. On this view, numbers are not primarily treated abstract objects, but rather ‘aspects’ of pluralities of ordinary objects, namely number tropes, a view that in fact appears to have been the Aristotelian view of numbers. Natural language moreover provides support for another view of the ontological status of numbers, on which natural numbers do not act as entities, but rather have the status of plural properties, the meaning of numerals when acting like adjectives. This view matches contemporary approaches in the philosophy of mathematics of what Dummett called the Adjectival Strategy, the view on which number terms in arithmetical sentences are not terms referring to numbers, but rather make contributions to generalizations about ordinary (and possible) objects. It is only with complex expressions somewhat at the periphery of language such as the number eight that reference to pure numbers is permitted. Content Type Journal Article Pages 1-38 DOI 10.1007/s11098-011-9779-1 Authors Friederike Moltmann, IHPST (Paris1/ENS/CNRS), Paris, France Journal Philosophical Studies Online ISSN 1573-0883 Print ISSN 0031-8116.
Reading list
| Discuss | Edit | Categorize | My bibliography | Export citation | Other links: springerlink.com semantics.univ-paris1.fr dx.doi.org | Scholar | At my library | More options ...
| | Other links: springerlink.com semantics.univ-paris1.fr dx.doi.org | Scholar | At my library | More options ... The notion of a natural numbers object in a monoidal category is defined and it is shown that the theory of primitive recursive functions can be developed. This is done by considering the category of cocommutative comonoids which is cartesian, and where the theory of natural numbers objects is well developed. A number of examples illustrate the usefulness of the concept.
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| | Other links: jstor.org | Scholar | At my library | More options ... Discussion of Friederike Moltmann, Reference to numbers in natural language
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