Erich Reck

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.

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 several early attempts to develop the semantics of axiomatic theories in the 1930s, by two proponents of the type-theoretic tradition and two proponents of the first-order tradition, we argue that, instead, the move from type theory to first-order logic is better understood as a gradual transformation, and further, that the contributions to semantics made in the type-theoretic tradition should be seen as central to the evolution of model theory.

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.

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.

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 case studies on individual philosophers through discussions of broader themes in the history of analytic philosophy to related methodological reflections.

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..

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

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..