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Properties and kinds of tropes: New linguistics facts and old philosophical insights
 Mind
, 2004
"... Terms like ‘wisdom ’ are commonly held to refer to abstract objects that are properties. On the basis of a greater range of linguistic data and with the support of some ancient and medieval philosophical views, I argue that such terms do not stand for objects, but rather for kinds of tropes, entitie ..."
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Terms like ‘wisdom ’ are commonly held to refer to abstract objects that are properties. On the basis of a greater range of linguistic data and with the support of some ancient and medieval philosophical views, I argue that such terms do not stand for objects, but rather for kinds of tropes, entities that do not have the status of objects, but only play a role as semantic values of terms and as arguments of predicates. Such ‘nonobjects ’ crucially differ from objects in that they are not potential bearers of properties. 1.
Categories, structures, and the fregehilbert controversy: The status of metamathematics
 Philosophia Mathematica, 13:61–77. Pagenumbers in
, 2005
"... There is a parallel between the debate between Gottlob Frege and David Hilbert at the turn of the twentieth century and at least some aspects of the current controversy over whether category theory provides the proper framework for structuralism in the philosophy of mathematics. The main issue, I th ..."
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There is a parallel between the debate between Gottlob Frege and David Hilbert at the turn of the twentieth century and at least some aspects of the current controversy over whether category theory provides the proper framework for structuralism in the philosophy of mathematics. The main issue, I think, concerns the place and interpretation of metamathematics in an algebraic or structuralist approach to mathematics. Can metamathematics itself be understood in algebraic or structural terms? Or is it an exception to the slogan that mathematics is the science of structure? The slogan of structuralism is that mathematics is the science of structure. Rather than focusing on the nature of individual mathematical objects, such as natural numbers, the structuralist contends that the subject matter of arithmetic, for example, is the structure of any collection of objects that has a designated, initial object and a successor relation that satisfies the induction principle. In the contemporary scene, Paul Benacerraf’s classic
Induction and Indefinite Extensibility: The Gödel Sentence is True, but Did Someone Change the Subject?
"... Over the last few decades Michael Dummett developed a rich program for assessing logic and the meaning of the terms of a language. He is also a major exponent of Frege’s version of logicism in the philosophy of mathematics. Over the last decade, Neil Tennant developed an extensive version of logicis ..."
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Over the last few decades Michael Dummett developed a rich program for assessing logic and the meaning of the terms of a language. He is also a major exponent of Frege’s version of logicism in the philosophy of mathematics. Over the last decade, Neil Tennant developed an extensive version of logicism in Dummettian terms, and Dummett influenced other contemporary logicists such as Crispin Wright and Bob Hale. The purpose of this paper is to explore the prospects for Fregean logicism within a broadly Dummettian framework. The conclusions are mostly negative: Dummett’s views on analyticity and the logical/nonlogical boundary leave little room for logicism. Dummett’s considerations concerning manifestation and separability lead to a conservative extension requirement: if a sentence S is logically true, then there is a proof of S which uses only the introduction and elimination rules of the logical terms that occur in S. If basic arithmetic propositions are logically true—as the logicist contends—then there is tension between this conservation requirement and the ontological commitments
Frege, Boolos, and Logical Objects
"... Objects In this section, we discuss the following kinds of logical object: natural cardinals, extensions, directions, shapes, and truth values. The material concerning the latter four kinds of logical objects are presented here as new results of OT. However, before introducing those results, we fir ..."
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Objects In this section, we discuss the following kinds of logical object: natural cardinals, extensions, directions, shapes, and truth values. The material concerning the latter four kinds of logical objects are presented here as new results of OT. However, before introducing those results, we first briefly rehearse the development of number theory in Zalta [1999].
Logic and Metaphysics ∗
"... In this article, we canvass a few of the interesting topics that philosophers can pursue as part of the simultaneous study of logic and metaphysics. To keep the discussion to a manageable length, we limit our survey to deductive, as opposed to inductive, logic. Though most of this article ..."
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In this article, we canvass a few of the interesting topics that philosophers can pursue as part of the simultaneous study of logic and metaphysics. To keep the discussion to a manageable length, we limit our survey to deductive, as opposed to inductive, logic. Though most of this article
1 PYTHAGOREAN POWERS or A CHALLENGE TO PLATONISM
"... I have tried to apprehend the Pythagorean power by which number holds sway above the flux. Bertrand Russell, Autobiography, vol. 1, Prologue. The Quine/Putnam indispensability argument is regarded by many as the chief argument for the existence of platonic objects. We argue that this argument cannot ..."
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I have tried to apprehend the Pythagorean power by which number holds sway above the flux. Bertrand Russell, Autobiography, vol. 1, Prologue. The Quine/Putnam indispensability argument is regarded by many as the chief argument for the existence of platonic objects. We argue that this argument cannot establish what its proponents intend. The form of our argument is simple. Suppose indispensability to science is the only good reason for believing in the existence of platonic objects. Either the dispensability of mathematical objects to science can be demonstrated and, hence, there is no good reason for believing in the existence of platonic objects, or their dispensability cannot be demonstrated and, hence, there is no good reason for believing in the existence of mathematical objects which are genuinely platonic. Therefore, indispensability, whether true or false, does not support platonism. Mathematical platonists claim that at least some of the objects
226 CRITICAL STUDIES / BOOK REVIEWS
"... Bob Hale and Crispin Wright have gathered together fifteen of their papers, two of them jointly authored, on abstraction, logicism, neoFregeanism and Hume's Principle (HP): 1 #xF{x) = #xG{x) f * 3R(R maps the Fs 11 onto the Gs). ..."
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Bob Hale and Crispin Wright have gathered together fifteen of their papers, two of them jointly authored, on abstraction, logicism, neoFregeanism and Hume's Principle (HP): 1 #xF{x) = #xG{x) f * 3R(R maps the Fs 11 onto the Gs).
DOI 10.1007/s1109800508984
"... ABSTRACT. Nominalizations are expressions that are particularly challenging philosophically in that they help form singular terms that seem to refer to abstract or derived objects often considered controversial. The three standard views about the semantics of nominalizations are [1] that they map me ..."
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ABSTRACT. Nominalizations are expressions that are particularly challenging philosophically in that they help form singular terms that seem to refer to abstract or derived objects often considered controversial. The three standard views about the semantics of nominalizations are [1] that they map mere meanings onto objects, [2] that they refer to implicit arguments, and [3] that they introduce new objects, in virtue of their compositional semantics. In the second case, nominalizations do not add anything new but pick up objects that would be present anyway in the semantic structure of a corresponding sentence without a nominalization. In the first and third case, nominalizations in a sense ‘create ’ new objects’, enriching the ontology on the basis of the meaning of expressions. I will argue that there is a fourth kind of nominalization which requires a quite different treatment. These are nominalizations that introduce ‘new ’ objects, but only partially characterize them. Such nominalizations generally refer to events or tropes. I will explore an account according on which such nominalizations refer to
Philos Stud DOI 10.1007/s1109801197791 Reference to numbers in natural language
"... 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 reve ..."
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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.