Erich H. Reck

Completeness and Categoricity. Part I: Nineteenth-century Axiomatics to Twentieth-century Metalogic

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.

DEDEKIND'S STRUCTURALISM: AN INTERPRETATION AND PARTIAL DEFENSE

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 Dedekinds classic essay “Was sind und was sollen die Zahlen?”, supplemented by evidence from “Stetigkeit und irrationale Zahlen”, his scientific correspondence, and his Nachlaß.

Completeness and Categoricity, Part II: Twentieth-Century Metalogic to Twenty-first-Century Semantics

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.

Carnapian explication, formalisms as cognitive tools, and the paradox of adequate formalization

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 (Formal languages in logic. Cambridge University Press, Cambridge 2012a). The comparison allows us to discuss a number of aspects of the Carnapian methodology, as well as issues pertaining to formalization in general. We start by introducing Carnap’s conception of explication, arguing that there is a tension between his proposed criteria of fruitfulness and similarity; we also argue that his further desideratum of exactness is less crucial than might appear at first. We then bring in the general idea of formalisms as cognitive tools, mainly by discussing the reliability of so-called statistical prediction rules (SPRs), i.e. simple algorithms used to make predictions across a range of areas. SPRs allow for a concrete instantiation of Carnap’s fruitfulness desideratum, which is arguably the most important desideratum for him. Finally, we elaborate on what we call the ‘paradox of adequate formalization’, which for the Carnapian corresponds to the tension between similarity and fruitfulness. We conclude by noting that formalization is an inherently paradoxical enterprise in general, but one worth engaging in given the ‘cognitive boost’ it affords as a tool for discovery.

Carnapian Explication: A Case Study and Critique

During the last two decades, there has been a revival of interest in Rudolf Carnap. This has led to the revision of various stereotypes about him, e.g., of the view that his first major work, Der Logische Aufbau der Welt (1928), represents a crude form of positivism or the culmination of classical empiricisms. Careful historical studies of the Aufbau have, instead, brought to light its neo-Kantian and Husserlian roots, thus leading to a much subtler, more complicated story about the book’s origins and goals. Another result of recent scholarship has been a shift of focus from the early Aufbau to Carnap’s middle and later writings. Thus, his Logische Syntax der Sprache (1934d/1937) has been rediscovered as a major contribution to the philosophy of mathematics and logic, and Carnap’s writings from the 1940s and 1950s have come into focus as well. One result of the latter shift is a renewed interest in his notion of explication, together with pragmatist elements in Carnap’s mature philosophy.

The historical turn in analytic philosophy

Series Editors Foreword Acknowledgements Notes on Contributors Introduction: Analytic Philosophy and Philosophical History Erich H. Reck PART I: CASE STUDIES 1. Philosophy and the Tide of History: Bertrand Russells Role in the Rise of Analytic Philosophy Stewart Candlish 2. Taking the Measure of Carnaps Philosophical Engineering: Metalogic as Metrology Alan Richardson 3. Quine and the Aufbau : The Possibility of Objective Knowledge Peter Hylton 4. Ryles Conceptual Cartography Julia Tanney PART II: BROADER THEMES 5. Frege, Lotze, and Boole Jeremy Heis 6. Frege or Dedekind? Towards a Reevalution of Their Legacies Erich H. Reck 7. Psychology, Epistemology, and the Problem of the External World: Russell and Before Gary Hatfield 8. C. I. Lewis and the Analyticity Debate Thomas Baldwin PART III: METHODOLOGICAL REFLECTIONS 9. Analytic Philosophy and History of Philosophy: The Development of the Idea of Rational Reconstruction Michael Beaney 10. History and the Future of Logical Empiricism A. W. Carus 11. What is the Good of Philosophical History? Michael Kremer 12. The Owl of Minerva: Is Analytic Philosophy Moribund? Hans-Johann Glock Index

Frege, Dedekind, and the Origins of Logicism

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 Dedekinds writings in relation to Freges. Especially in its Dedekindian and Fregean versions, logicism constitutes the culmination of the rise of ‘pure mathematics’ in the nineteenth century; and this rise brought with it an inter-weaving of methodological and epistemological considerations. The latter aspect illustrates how philosophical concerns can grow out of mathematical practice, as opposed to being imposed on it from outside. It also sheds new light on the legacy and the lasting significance of logicism today.

Frege or Dedekind? Towards a Reevaluation of Their Legacies

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 chapter, Dedekind’s philosophical reception is reconsidered. At the chapter’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.

FREGE ON TRUTH, JUDGMENT, AND OBJECTIVITY

In Frege’s writings, the notions of truth, judgment, and objectivity are all prominent and important. Th is 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 off ered, and seemingly opposed “realist” and “idealist” strands in his position reconciled.

Hempel, Carnap, and the Covering Law Model

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.