Erich Reck

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.

Gottlob Frege (1848-1925) taught at the University of Jena for thirty years, and was scarcely known outside a small circle of professional mathematicians and philosophers. However, later in the twentieth century he came to be recognized as someone who, in demonstrating the affinity of logic with mathematics, laid the foundations for modern philosophy of language and modern logic. Frege regarded logic as the foundation for philosophy. In doing so, he instigated a radical change in the stance of the majority of Western philosophers whose main pre-occupation since Descartes had been the nature of knowledge rather than logic. His influence can be clearly seen in the work of local positivists of the early twentieth century, as well as in much of Ludwig Wittgenstein’s philosophy. This collection brings together important scholarship on Frege, including new translations of German material, made available to Anglophone scholars for the first time. Volume II focuses on Frege's contribution to the philosophy of logic.

Gottlob Frege (1848-1925) taught at the University of Jena for thirty years, and was scarcely known outside a small circle of professional mathematicians and philosophers. However, later in the twentieth century he came to be recognized as someone who, in demonstrating the affinity of logic with mathematics, laid the foundations for modern philosophy of language and modern logic. Frege regarded logic as the foundation for philosophy. In doing so, he instigated a radical change in the stance of the majority of Western philosophers whose main pre-occupation since Descartes had been the nature of knowledge rather than logic. His influence can be clearly seen in the work of local positivists of the early twentieth century, as well as in much of Ludwig Wittgenstein’s philosophy. This collection brings together important scholarship on Frege, including new translations of German material, made available to Anglophone scholars for the first time. Volume IV focuses on Frege's contribution to the philosophy of mathematics.

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.

Among all of Frege’s contemporaries, Richard Dedekind is arguably the thinker closest to him in terms of their general backgrounds and core projects. This essay provides a reexamination of Frege’s critical reactions to Dedekind, in _Grundgesetze_ and some related texts. The reexamination includes documenting their interaction in some detail and putting it into a broader context, both philosophically and systematically. It also involves separating Frege’s less compelling criticisms of Dedekind from those that have deeper, more lasting significance. The essay ends with a suggestion for how to reconcile Fregean and Dedekindian forms of logicism, based on distinguishing two distinct but complementary kinds of abstraction principles.

Ernst Cassirer was a keen observer of development in the mathematical sciences, especially in the 19th and early 20th centuries. In this essay, the focus is on his reception of Dedekind’s contributions to the foundations of mathematics, and with it, on Dedekind’s mathematical structuralism. Cassirer adopts that structuralism early on, defends it against a number of criticisms, and embeds it into a rich historical account of the structuralist transformation of modern mathematical science. He also adds some original elements to our understanding of structuralism, e.g., by relating it to the Kantian notion of the “construction of concepts” in mathematics, by introducing a basic distinction between “substance concepts” and “function concepts”, and by tracing the “unfolding” of structuralist aspects far back in the history of thought. Overall, Cassirer’s approach is guided by the conviction that the metaphysics of modern mathematics should be approached by way of its distinctive methodology.

This essay concerns Dedekind’s “mathematical structuralism,”by which we mean methodological features characteristic for the approach to mathematics in his mature writings. The discussion starts with some background on forerunners, especially Gauss, Dirichlet, and Riemann, whose “conceptual” style of work influenced him strongly. But Dedekind went further than them, by making methodological choices that are more distinctly and fully “structuralist”. This includes his resolute acceptance of actually infinite systems, understood within a “logical” framework, and studied not just axiomatically, but also in terms of isomorphisms and related notions (since 1871). As an illustration, the essay discusses his early adoption and strikingly modern transformation of Galois theory, together with his contributions to algebraic number theory. After that, the essayturns to Dedekind’s more “foundational” contributions, i.e., his writings on the real numbers, the natural numbers, and set theory. It shows that the same methodological choices inform them. Yet his approach kept evolving in subtle ways too, especially in terms of the centrality of functions (_Abbildungen_, mappings).

The core idea of mathematical structuralism is that mathematical theories, always or at least in many central cases, are meant to characterize abstract structures (as opposed to more concrete, individual objects). As such, structuralism is a general position about the subject matter of mathematics, namely abstract structures; but it also includes, or is intimately connected with, views about its methodology, since studying such structures involves distinctive tools and procedures. The goal of the present collection of essays is to discuss mathematical structuralism with respect to both aspects. This is done by examining contributions by a number of mathematicians and philosophers of mathematics from the second half of the 19th and the early 20th centuries.

One of Gottlob Frege’s most original contributions to logic and philosophy was his logical notation, his ‘Begriffsschrift’. While long criticized, dismissed, or simply ignored, the recent secondary literature contains some helpful re-evaluations and partial defenses of it. These rely largely on technical, pragmatic, or cognitive-psychological considerations. In this paper, we reconsider Frege’s own reasons for valuing his notation highly. We argue that there is a further semiotic dimension, one that matters epistemologically. This dimension becomes evident once one takes seriously, partly also literally, some striking visual metaphors Frege uses. The result is an interpretation highlighting a side of Frege’s position that is not widely known. It involves his views about the indispensable role that language, or sign systems more generally, play for human thought, and especially, for logic and mathematics. More particularly and as we read Frege, a good logical notation allows for expressing conceptual/inferential relations in a ‘visually inscribed’, thus ‘perspicuous’ way, and this leads to a special kind of ‘insight’ in mathematics. Or as we elaborate this point further, it involves a kind of articulation and understanding hardly possible without it. Frege designed his Begriffsschrift with that specific goal in mind.

Gottlob Frege (1848-1925) taught at the University of Jena for thirty years, and was scarcely known outside a small circle of professional mathematicians and philosophers. However, later in the twentieth century he came to be recognized as someone who, in demonstrating the affinity of logic with mathematics, laid the foundations for modern philosophy of language and modern logic. Frege regarded logic as the foundation for philosophy. In doing so, he instigated a radical change in the stance of the majority of Western philosophers whose main pre-occupation since Descartes had been the nature of knowledge rather than logic. His influence can be clearly seen in the work of local positivists of the early twentieth century, as well as in much of Ludwig Wittgenstein’s philosophy. This collection brings together important scholarship on Frege, including new translations of German material, made available to Anglophone scholars for the first time. Volume III focuses on Frege's contribution to the philosophy of language.

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 historiographical devices play in philosophical thought. More importantly, the contributors attempt to understand history of philosophy in connection with other historical and historiographical approaches: contributors engage classical history of science, sociology of knowledge, history of psychology and historiography, in dialogue with historiographical practices in philosophy more narrowly construed. Additionally, select chapters adopt a more diverse perspective, by making place for non-Western approaches and for efforts to construe new philosophical narratives that do justice to the voice of women across the centuries. Historiography and the Formation of Philosophical Canons will be of interest to researchers and advanced students working in history of philosophy, meta-philosophy, philosophy of history, historiography, intellectual history, and sociology of knowledge.

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 that basis, a new framework for understanding the themes that connect Wittgenstein to Frege, both in his early and his later works.

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. The eleven essays that follow are arranged in three parts: Part I. Proof Theory and its History; Part II. Logic and Philosophy of Mathematics; and Part III. History of Logic and Philosophy of Mathematics. Each of the essays contributes substantially to one or several of these areas. The authors included are: Steve Awodey, Solomon Feferman, Michael Friedman, Warren Goldfarb, Geoffrey Hellman, William Howard, Stephen Menn, Rebecca Morris, Charles Parsons, Erich Reck, Thomas Ricketts, and Wilfried Sieg. The editor, Erich H. Reck is Professor of Philosophy at the University of California at Riverside.

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, Carnap, and Quine, whose work led to profound conclusions about logical, epistemological, and metaphysic.

Gottlob Frege (1848-1925) taught at the University of Jena for thirty years, and was scarcely known outside a small circle of professional mathematicians and philosophers. However, later in the twentieth century he came to be recognized as someone who, in demonstrating the affinity of logic with mathematics, laid the foundations for modern philosophy of language and modern logic. Frege regarded logic as the foundation for philosophy. In doing so, he instigated a radical change in the stance of the majority of Western philosophers whose main pre-occupation since Descartes had been the nature of knowledge rather than logic. His influence can be clearly seen in the work of local positivists of the early twentieth century, as well as in much of Ludwig Wittgenstein’s philosophy. This collection brings together important scholarship on Frege, including new translations of German material, made available to Anglophone scholars for the first time. Volume I focuses on Frege as seen in his historical context.