•  1
    In a career that spans 60 years so far, W.W. Tait has made many contributions to logic, the philosophy of mathematics, and their history. The present collection of essays—contributed by former students, colleagues, and friends—is a Festschrift, i.e., a celebration of his life and work. Contributors include: Steve Awodey, Solomon Feferman, Michael Friedman, Warren Goldfarb, Geoffrey Hellman, William Howard, Steven Menn, Rebecca Morris, Charles Parsons, Erich Reck, Thomas Ricketts, and Wilfried…Read more
  •  16
    This book presents a series of case studies and reflections on the historiographical assumptions, methods, and approaches that shape the way in which philosophers construct their own past. The chapters in the volume advance discussion of the methods of historians of philosophy, while at the same time illustrating the various ways in which philosophical canons come into existence, debunking the myth of analytical philosophy's ahistoricism, and providing a deeper understanding of the roles histori…Read more
  • It is well known that Frege and his writings were an important influence on Wittgenstein. There is no agreement, however, on the nature and scope of this influence. In this paper, I clarify the situation in three related ways: by tracing Frege's and Wittgenstein's actual interactions, i.e., their face‐to‐face meetings and their correspondence between 1911 and 1920; by documenting Wittgenstein's continued study of Frege's writings, until the very end of his life in 1951; and by constructing, on t…Read more
  •  9
    In a career that spans 60 years so far, W.W. Tait has made many highly influential contributions to logic, the philosophy of mathematics, and their history. The present collection of new essays - contributed by former students, colleagues, and friends - is a Festschrift, i.e., a celebration of his life and work. The essays address a variety of themes prominent in his work or related to it. The collection starts with an introduction in which Tait's contributions are sketched and put into context.…Read more
  •  23
    The Pre-History of Mathematical Structuralism (edited book)
    Oxford University Press. 2020.
    This edited volume explores the previously underacknowledged 'pre-history' of mathematical structuralism, showing that structuralism has deep roots in the history of modern mathematics. The contributors explore this history along two distinct but interconnected dimensions. First, they reconsider the methodological contributions of major figures in the history of mathematics. Second, they re-examine a range of philosophical reflections from mathematically-inclinded philosophers like Russell, Carn…Read more
  •  27
    Logic in the 1930s: Type Theory and Model Theory
    Bulletin 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 ofPrincipia 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 severa…Read more
  •  17
    Introduction to Special Issue: Dedekind and the Philosophy of Mathematics
    Philosophia Mathematica 25 (3): 287-291. 2017.
    © The Author [2017]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected] Dedekind was a contemporary of Bernhard Riemann, Georg Cantor, and Gottlob Frege, among others. Together, they revolutionized mathematics and logic in the second half of the nineteenth century. Dedekind had an especially strong influence on David Hilbert, Ernst Zermelo, Emmy Noether, and Nicolas Bourbaki, who completed that revolution in the twentiet…Read more
  •  203
    Carnapian explication, formalisms as cognitive tools, and the paradox of adequate formalization
    with Catarina Dutilh Novaes
    Synthese 194 (1): 195-215. 2017.
    Explication is the conceptual cornerstone of Carnap’s approach to the methodology of scientific analysis. From a philosophical point of view, it gives rise to a number of questions that need to be addressed, but which do not seem to have been fully addressed by Carnap himself. This paper reconsiders Carnapian explication by comparing it to a different approach: the ‘formalisms as cognitive tools’ conception. The comparison allows us to discuss a number of aspects of the Carnapian methodology, as…Read more
  •  180
    This paper is the first 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 exte…Read more
  •  64
    Frege's natural numbers: Motivations and modifications
    In Michael Beaney & Erich Reck (eds.), Gottlob Frege: Critical Assessments of Leading Philosophers, Vol. III, Routledge. pp. 270-301. 2005.
    Frege's main contributions to logic and the philosophy of mathematics are, on the one hand, his introduction of modern relational and quantificational logic and, on the other, his analysis of the concept of number. My focus in this paper will be on the latter, although the two are closely related, of course, in ways that will also play a role. More specifically, I will discuss Frege's logicist reconceptualization of the natural numbers with the goal of clarifying two aspects: the motivations for…Read more
  •  176
    Frege on truth, judgment, and objectivity
    Grazer Philosophische Studien 75 (1): 149-173. 2007.
    In Frege's writings, the notions of truth, judgment, and objectivity are all prominent and important. This paper explores the close connections between them, together with their ties to further cognate notions, such as those of thought, assertion, inference, logical law, and reason. It is argued that, according to Frege, these notions can only be understood properly together, in their inter-relations. Along the way, interpretations of some especially cryptic Fregean remarks, about objectivity, l…Read more
  •  401
    Frege's influence on Wittgenstein: Reversing metaphysics via the context principle
    In Gottlob Frege: Critical Assessments of Leading Philosophers, Vol. I, Routledge. pp. 241-289. 2005.
    Gottlob Frege and Ludwig Wittgenstein (the later Wittgenstein) are often seen as polar opposites with respect to their fundamental philosophical outlooks: Frege as a paradigmatic "realist", Wittgenstein as a paradigmatic "anti-realist". This opposition is supposed to find its clearest expression with respect to mathematics: Frege is seen as the "arch-platonist", Wittgenstein as some sort of "radical anti-platonist". Furthermore, seeing them as such fits nicely with a widely shared view about the…Read more
  •  78
    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
  •  23
    Frege, natural numbers, and arithmetic's umbilical cord
    Manuscrito 26 (2): 427-70. 2003.
    A central part of Frege's logicism is his reconstruction of the natural numbers as equivalence classes of equinumerous concepts or classes. In this paper, I examine the relationship of this reconstruction both to earlier views, from Mill all the way back to Plato, and to later formalist and structuralist views; I thus situate Frege within what may be called the “rise of pure mathematics” in the nineteenth century. Doing so allows us to acknowledge continuities between Frege's and other approache…Read more
  •  81
    Frege, Dedekind, and the Origins of Logicism
    History and Philosophy of Logic 34 (3): 242-265. 2013.
    This paper has a two-fold objective: to provide a balanced, multi-faceted account of the origins of logicism; to rehabilitate Richard Dedekind as a main logicist. Logicism should be seen as more deeply rooted in the development of modern mathematics than typically assumed, and this becomes evident by reconsidering Dedekind's writings in relation to Frege's. Especially in its Dedekindian and Fregean versions, logicism constitutes the culmination of the rise of ?pure mathematics? in the nineteenth…Read more
  •  100
    This paper is the first 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 exte…Read more
  •  97
    In Untersuchungen zur allgemeinen Axiomatik and Abriss der Logistik, Carnap attempted to formulate the metatheory of axiomatic theories within a single, fully interpreted type-theoretic framework and to investigate a number of meta-logical notions in it, such as those of model, consequence, consistency, completeness, and decidability. These attempts were largely unsuccessful, also in his own considered judgment. A detailed assessment of Carnap’s attempt shows, nevertheless, that his approach is …Read more