CHRISTOPHER PINCOCK, Department of Philosophy, Purdue University, West Lafayette, IN 47907, USA The volume under review contains fifteen new essays by some of the most influential scholars of the history of early analytic philosophy. The focus of the essays is, as the editor says in the preface, ‘the work of Gottlob Frege and of Ludwig Wittgenstein (mostly the early Wittgenstein), as well as various ties between them’ (p. x). The essays are divided into four parts. The first part, ‘Background and (...) General Themes’, contains essays by E. Reck, G. Gabriel and S. Gerrard. The second part on Frege has contributions by H. Sluga, S. Shieh, M. Ruffino and J. Weiner. Essays on the relation between Frege and the early Wittgenstein by W. Goldfarb, D. Macbeth, T. Ricketts and C. Diamond comprise the third part. The volume concludes with essays by I. Proops, J. Floyd, M. Ostrow and J. Conant on the early Wittgenstein. This volume is an important contribution to our understanding of Frege and the early Wittgenstein and should prove a help to specialists in the history of analytic philosophy. I have chosen to briefly discuss seven of these essays with an emphasis on topics in the history and philosophy of logic. Reck’s opening essay, ‘Wittgenstein’s “Great Debt” to Frege: Biographical Traces and Philosophical Themes’, gives a helpful overview of our current knowledge of the contacts between Frege and Wittgenstein. Reck argues quite persuasively for the conclusion that Wittgenstein engaged with Frege’s work throughout his philosophical career. The depth of this engagement is in-. (shrink)
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 been seen as paradigmatic pieces of philosophy in this tradition. (...) This close connection remained beyond what is now often called "early analytic philosophy", i.e., the tradition's first phase. In the present chapter I will consider three thinkers who played equally important and formative roles in analytic philosophy's second phase, the period from the 1920s to the 1950s: Rudolf Carnap, Kurt Gödel, and Alfred Tarski. (shrink)
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 comparison of Frege's and Dedekind's legacies, within and outside of (...) analytic philosophy. While the comparison proceeds historically, it is shaped by current philosophical concerns, especially by debates about neo-logicist and neo-structuralist views. In fact, philosophical and historical considerations are intertwined thoroughly, to the benefit of both. The underlying motivation is to rehabilitate Dedekind as a major philosopher of mathematics, in relation, but not necessarily in opposition, to Frege. (shrink)
Carl Gustav Hempel was one of the most influential figures in the development of “scientific philosophy” in the twentieth century, particularly in the English-speaking world. While he made a variety of contributions to the philosophy of science, he is perhaps most remembered for his careful formulation and detailed elaboration of the “Covering Law model” for scientific explanation. In this essay I consider why the CL model was, and still is, so influential, in spite of the fact that it has been (...) subjected to many criticisms and is usually seen as superseded by alternative models. Answering this question involves a reexamination of Hempel’s relationship to other influential “scientific philosophers”, especially Rudolf Carnap. It also sheds new light on issues concerning the notions of analysis, explication, and modeling that remain relevant today. (shrink)
The last few decades have witnessed a broadening of the philosophy of mathematics, beyond narrowly foundational and metaphysical issues, and towards the inclusion of more general questions concerning "mathematical methodology" and "mathematical practice" (a development parallel to an earlier broadening of the philosophy of science). There is now widespread, and growing, interest in topics such as: concept formation and conceptual change in mathematics, the use of heuristics in mathematical research, the applicability of mathematics, and even sociological or anthropological questions concerning (...) the mathematical community. Part of this broadening, although a part that remains relatively close to foundational and metaphysical issues, is the turn towards a "new epistemology" for mathematics. The latter includes the study of topics such as: the role of visualization in mathematics, the use of computers in proving mathematical theorems, and the notion of explanation as applied to mathematics.1 The present paper is a contribution to this new epistemology. More particularly, it is an attempt to bring into sharper focus, and to argue for the relevance of, two related themes: "structural reasoning" and "mathematical understanding". As the notion of understanding is vague and slippery in general, as well as very loaded in philosophical discussions of the sciences, the latter label has to be handled with care, though. It will have to be clarified what, if anything (or anything reasonably precise), is to be meant by "understanding" in connection with mathematics. Similarly, while talking about "structural" reasoning in mathematics may be suggestive, that term too requires further elaboration. My clarifications and elaborations will be tied to a specific historical figure and period: Richard Dedekind and his contributions to algebraic number theory in the nineteenth century. This is not an incidental choice; Dedekind's case is particularly pertinent in this context, as I also hope to establish in this paper. I will proceed as follows: In the first section, I will provide a brief summary of Dedekind's work on the foundations of mathematics, as well as of its usual perception in.... (shrink)
In recent philosophy of mathematics a variety of writers have presented "structuralist" views and arguments. There are, however, a number of substantive differences in what their proponents take "structuralism" to be. In this paper we make explicit these differences, as well as some underlying similarities and common roots. We thus identify, systematically and in detail, several main variants of structuralism, including some not often recognized as such. As a result the relations between these variants, and between the respective problems they (...) face, become manifest. Throughout our focus is on semantic and metaphysical issues, including what is or could be meant by "structure" in this connection. (shrink)
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 their relation: the later Wittgenstein is supposed to have developed his (...) ideas in direct opposition to Frege. The purpose of this paper is to challenge these standard assumptions. I will argue that Frege's and Wittgenstein's basic outlooks have something crucial in common; and I will argue that this is the result of the positive influence Frege had on Wittgenstein. (shrink)
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 numbers. My main concern (...) will be to determine the precise sense in which this conception was, or could be, meant to constitute an analysis.1 I will be mostly concerned with Frege’s views on the matter; but Russell will come up along the way, for illustration and comparison, as will some recent neo-Fregean proposals and results. The structure of the paper is as follows: In the first section, I sketch Frege's general approach. Next, I differentiate several kinds, or modes, of analysis, as further background. In the third section, I zero in on the equivalence class construction, raising the question of why it might, from a Fregean point of view, be seen as 'the right' construction, thus as an analysis in a strong sense. In the fourth section, I provide a contrasting, more conventionalist view of the matter, often associated with the Carnapian notion of explication, and expressed in some remarks by Russell. I then discuss the motivation for the Frege-Russell numbers in more depth. In the sixth section, I introduce a neo-Fregean alternative, to be examined along similar lines. Finally, I reflect further on the significance of the kinds of arguments available in this connection. (shrink)
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 its core ideas; the step-by-step development of these ideas, from Begriffsschrift through Die Grundlagen der Arithmetik and Grundgesetze der Arithmetik to Frege's very last writings, indeed even beyond those, to a number of recent "neo-Fregean" proposals for how to update them. One main development, or break, in Frege's views occurred after he was informed of Russell's antinomy. His attempt to come to terms with this antinomy has found some attention in the literature already. It has seldom been analyzed in connection with earlier changes in his views, however, partly because those changes themselves have been largely ignored. Nor has it been discussed much in connection with Frege's basic motivations, as formed in reaction to earlier positions. Doing both in this paper will not only shed new light on his response to Russell's antinomy, but also on other aspects of his views. In addition, it will provide us with a framework for comparing recent updates of these views, thus for assessing the remaining attraction of Frege's general approach. I will proceed as follows: In the first part of the paper (§1.1 and §1.2), I will consider the relationship of Frege's conception of the natural numbers to earlier conceptions, in particular to what I will call the "pluralities conception", thus bringing into sharper focus his core ideas and their motivations. In the next part (§2.1 and §2.2), I will trace the order in which these ideas come up in Frege's writings, as well as the ways in which his position gets modified along the way, both before and after Russell's antinomy.. (shrink)
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 on Basic Laws of (...) Arithmetic (1893/1903) and "Thoughts" (1918). My goal is to clarify in which sense the Frege of Foundations and Basic Laws is a platonist concerning the natural numbers.1.. (shrink)
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, laws of truth, and reason, are offered, and seemingly (...) opposed "realist" and "idealist" strands in his position reconciled. (shrink)
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?, supplemented by evidence from Stetigkeit und (...) irrationale Zahlen, his scientific correspondence, and his Nachlaß. (shrink)
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 extend the usual set-theoretic semantics (...) so as to shed new light on the relevant strengths and limits of higher-order logic. (shrink)
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 extend the usual set-theoretic semantics (...) so as to shed new light on the relevant strengths and limits of higher-order logic. (shrink)
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 extend the usual set-theoretic semantics (...) so as to shed new light on the relevant strengths and limits of higher-order logic. (shrink)
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 extend the usual set-theoretic semantics (...) so as to shed new light on the relevant strengths and limits of higher-order logic. (shrink)
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 between Frege and the early Wittgenstein.
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, and between the respective problems theyface, become manifest. Throughout our focus is onsemantic and metaphysical issues, (...) including what is orcould be meant by ``structure'' in this connection. (shrink)
