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63Developments in Logic: Carnap, Gödel, and TarskiIn Michael Beaney (ed.), Oxford Handbook of the History of Analytic Philosophy, Oxford University Press. pp. 546-571. 2013.Analytic philosophy and modern logic are intimately connected, both historically and systematically. Thinkers such as Frege, Russell, and Wittgenstein were major contributors to the early development of both; and the fruitful use of modern logic in addressing philosophical problems was, and still is, definitive for large parts of the analytic tradition. More specifically, Frege's analysis of the concept of number, Russell's theory of descriptions, and Wittgenstein's notion of tautology have long…Read more
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232Structures and structuralism in contemporary philosophy of mathematicsSynthese 125 (3): 341-383. 2000.In recent philosophy of mathematics avariety of writers have presented ``structuralist''views and arguments. There are, however, a number ofsubstantive differences in what their proponents take``structuralism'' to be. In this paper we make explicitthese differences, as well as some underlyingsimilarities and common roots. We thus identifysystematically and in detail, several main variants ofstructuralism, including some not often recognized assuch. As a result the relations between thesevariants…Read more
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61Frege or Dedekind? Towards a reevalaution of their legaciesIn The Historical turn in Analytic Philosophy, Palgrave-macmillan. pp. 139-170. 2013.The philosophy of mathematics has long been an important part of philosophy in the analytic tradition, ever since the pioneering works of Frege and Russell. Richard Dedekind was roughly Frege's contemporary, and his contributions to the foundations of mathematics are widely acknowledged as well. The philosophical aspects of those contributions have been received more critically, however. In the present essay, Dedekind's philosophical reception is reconsidered. At the essay’s core lies a comparis…Read more
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104From Frege to Wittgenstein: perspectives on early analytic philosophy (edited book)Oxford University Press. 2002.Analytic philosophy--arguably one of the most important philosophical movements in the twentieth century--has gained a new historical self-consciousness, particularly about its own origins. Between 1880 and 1930, the most important work of its founding figures (Frege, Russell, Moore, Wittgenstein) not only gained attention but flourished. In this collection, fifteen previously unpublished essays explore different facets of this period, with an emphasis on the vital intellectual relationship betw…Read more
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60Completeness and categoricty, part II: 20th century metalogic to 21st century semanticsHistory and Philosophy of Logic 23 (1): 77-92. 2002.This paper is the second in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully ext…Read more
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69Logic in the 1930s: type theory and model theoryBulletin of Symbolic Logic 19 (4): 433-472. 2013.In historical discussions of twentieth-century logic, it is typically assumed that model theory emerged within the tradition that adopted first-order logic as the standard framework. Work within the type-theoretic tradition, in the style of Principia Mathematica, tends to be downplayed or ignored in this connection. Indeed, the shift from type theory to first-order logic is sometimes seen as involving a radical break that first made possible the rise of modern model theory. While comparing sever…Read more
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147Hempel, Carnap, and the Covering Law ModelIn Nikolay Milkov & Volker Peckhaus (eds.), The Berlin Group and the Philosophy of Logical Empiricism, Springer. pp. 311--324. 2013.
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96Frege-Russell numbers: Analysis or explication?In The Analytic Turn, Routledge. pp. 33-50. 2007.For both Gottlob Frege and Bertrand Russell, providing a philosophical account of the concept of number was a central goal, pursued along similar logicist lines. In the present paper, I want to focus on a particular aspect of their accounts: their definitions, or reconstructions, of the natural numbers as equivalence classes of equinumerous classes. In other words, I want to examine what is often called the "Frege-Russell conception of the natural numbers" or, more briefly, the Frege-Russell num…Read more
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135Dedekind's structuralism: An interpretation and partial defenseSynthese 137 (3). 2003.Various contributors to recent philosophy of mathematics havetaken Richard Dedekind to be the founder of structuralismin mathematics. In this paper I examine whether Dedekind did, in fact, hold structuralist views and, insofar as that is the case, how they relate to the main contemporary variants. In addition, I argue that his writings contain philosophical insights that are worth reexamining and reviving. The discussion focusses on Dedekind''s classic essay Was sind und was sollen die Zahlen?, …Read more
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27The Historical turn in Analytic Philosophy (edited book)Palgrave-Macmillan. 2013.During the last 25 years, a large number of publications on the history of analytic philosophy have appeared, significantly more than in the preceding period. As most of these works are by analytically trained authors, it is tempting to speak of a 'historical turn' in analytic philosophy. The present volume constitutes both a contribution to this body of work and a reflection on what is, or might be, achieved in it. The twelve new essays, by an international group of contributors, range from cas…Read more
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224Frege on Numbers: Beyond the Platonist PictureThe Harvard Review of Philosophy 13 (2): 25-40. 2005.Gottlob Frege is often called a "platonist". In connection with his philosophy we can talk about platonism concerning three kinds of entities: numbers, or logical objects more generally; concepts, or functions more generally; thoughts, or senses more generally. I will only be concerned about the first of these three kinds here, in particular about the natural numbers. I will also focus mostly on Frege's corresponding remarks in The Foundations of Arithmetic (1884), supplemented by a few asides o…Read more
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From Frege to Wittgenstein: Essays on Early Analytic Philosophy (edited book)Oxford University Press. 2002.
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23Gottlob Frege: Critical Assessments of Leading Philosophers, Vol. II (edited book)Routledge. 2005.
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Aesthetics |
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Philosophy of Computing and Information |
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History: Philosophy of Mathematics |
The Infinite |