Since the 1960s, there has been a vigorous and ongoing debate about structuralism in English-spea... more Since the 1960s, there has been a vigorous and ongoing debate about structuralism in English-speaking philosophy of mathematics. But structuralist ideas and methods go back further in time, i.e., there is a rich pre-history to this debate, also in the German-and French-speaking literature. In the present collection of essays this pre-history is explored in a two-fold way: first, by reconsidering various mathematicians in the 19th and early 20th centuries (from Grassmann, Dedekind, Pasch, and Klein to Hilbert, Noether, Bourbaki, and Mac Lane) who contributed to structuralism in a methodological sense; second, by reexamining a range of philosophical reflections on such contributions during the same period (also by Peirce, Poincaré, Russell, Cassirer, Bernays, Carnap, and Quine), including suggestions about logical, epistemological, and metaphysical aspects that remain relevant today. Overall, the collection makes evident that structuralism has deep roots in the history of modern mathematics, that mathematical and philosophical views about it are often closely intertwined, and that the range of philosophical options available in this context is significantly richer than a narrow focus on current debates may make one believe.
In his Untersuchungen zur allgemeinen Axiomatik (1928) and Abriss der Logistik (1929), Rudolf Car... more In his Untersuchungen zur allgemeinen Axiomatik (1928) and Abriss der Logistik (1929), Rudolf 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 much less confused and hopeless than it has often been made out to be. By providing such a reassessment, the paper contributes to a reevaluation of Carnap's contributions to the development of modern logic.
This essay concerns Dedekind’s “mathematical structuralism,”by which we mean methodological featu... more 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 s...
Analytic philosophy and modern logic are intimately connected, both historically and systematical... more 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.
Among all of Frege’s contemporaries, Richard Dedekind is arguably the thinker closest to him in t... more 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.
Richard Dedekind (1831–1916) was one of the greatest mathematicians of the nineteenth-century, as... more Richard Dedekind (1831–1916) was one of the greatest mathematicians of the nineteenth-century, as well as one of the most important contributors to algebra and number theory of all time. Any comprehensive history of mathematics will mention him for his investigation of the notions of algebraic number, field, group, module, lattice, etc., and especially for the invention of his theory of ideals (see, e.g., Dieudonné 1985, Boyer & Merzbach 1991, Stillwell 2000, Kolmogorov & Yushkevich 2001, Wussing 2012). Dedekind’s more foundational work in mathematics is also widely known. Often acknowledged in that connection are: his analysis of the notion of continuity, his introduction of the real numbers by means of Dedekind cuts, his formulation of the Dedekind-Peano axioms for the natural numbers, his proof of the categoricity of these axioms, and his contributions to the early development of set theory (Grattan-Guinness 1980, Ferreirós 1996, 1999, 2016b, Jahnke 2003, Corry 2015). While many ...
Any history of logic from Kant to the 20th century needs to take into account the emergence of lo... more Any history of logic from Kant to the 20th century needs to take into account the emergence of logicism, since it is largely in this context that logic as we know it became prominent. Usually Gottlob Frege and Bertrand Russell are seen as its two main representatives, especially early on, and the more recent rise of neo-logicism, in writings by Crispin Wright, Bob Hale and others, proceeds along Fregean lines as well. In this chapter, I will focus on Richard Dedekind instead. Today Dedekind is sometimes mentioned in connection with logicism; yet at the end of the 19th century, he was the most prominent logicist, ahead of Frege. This alone invites further reflection. Reconsidering Dedekind also leads naturally to questions about what was, or could be, understood by "logic" in this context. Addressing them will involve developments in 19thcentury mathematics that are relevant in two respects: by forming core parts of the background for the emergence of modern logic; and by pointing towards a distinction between two conceptions of logic that deserve more attention, both historically and philosophically. 1 The chapter proceeds as follows. First, I will document Dedekind's characterization of his project as showing that arithmetic, understood in an inclusive sense, is "part of logic", and I will put that project into the context of broader developments in 19th-century mathematics. In the second section, I will look in more detail at Dedekind's procedure, including the fact that it involves set-theoretic "constructions" and a certain kind of "abstraction". Third, a brief summary of later appropriations and developments of Dedekind's contributions will be provided, especially in axiomatic set theory and category theory, although these are usually not seen as forms of "logicism". This will, in the fourth section, provide the background for the question of how Dedekind-and parallel to him, Frege-must have understood "logic" for their logicist projects to make sense, namely in a wider sense than the one dominant in the 20th century. The essay will conclude with some general observations, about the contested status of the notion of "logic", i.e. a lack of consensus about its nature that is still often underestimated, and about its historical and philosophical relationship to modern mathematics.
Carnap’s Ideal of Explication and Naturalism, 2012
During the last two decades, there has been a revival of interest in Rudolf Carnap. This has led ... more 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.
In Frege's writings, the notions of truth, judgment, and objectivity are all prominent and import... more 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.
2005) Gottlob Frege: Critical Assessments of Leading …, 2005
... numbers as "pluralities", "multitudes", or "groups of things". ... more ... numbers as "pluralities", "multitudes", or "groups of things". That general conception has a long history, from Mill and Weierstrass in the nineteenth century back to Aristotle, Euclid, and Ancient Greek thought. The basic idea behind it is this: Consider an ...
Ernst Cassirer was a keen observer of development in the mathematical sciences, especially in the... more 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 methodol...
Various contributors to recent philosophy of mathematics have taken Richard Dedekind to be the fo... more Various contributors to recent philosophy of mathematics have taken Richard Dedekind to be the founder of structuralism in 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. A direct forerunner of ante rem structuralism is [.. . ] Dedekind. His development of the notion of continuity and the real numbers, in ["Stetigkeit und irrationale Zahlen"], his presentation of the natural numbers via the notion of Dedekind infinity, in ["Was sind und was sollen die Zahlen?"], and some of his correspondence constitute a structuralist manifesto. (Shapiro 1997, p. 14)
... EDITORIAL BOARD James Lennox University of Pittsburgh Richard Creath Arizona State University... more ... EDITORIAL BOARD James Lennox University of Pittsburgh Richard Creath Arizona State University John Earman University of Pittsburgh Michael Friedman Stanford University GottfriedGabriel University of Jena Dana Scott Carnegie Mellon University Wilfried Sieg Carnegie ...
Rudolf Carnap was a remarkably productive thinker and writer, making contributions on a variety o... more Rudolf Carnap was a remarkably productive thinker and writer, making contributions on a variety of philosophical topics. As a result, he is a different philosopher for different audiences: For the general public (insofar as it knows him), he is the author of Schein- probleme in der Philosophie (1928a), Wissenschaftliche Weltauffassung (Hahn, Neurath & Carnap 1929), and "Überwindung der Metaphysik durch
... Part I: 19th Century Axiomatics to 20th Century Metalogic Steve Awodey Department of Philosop... more ... Part I: 19th Century Axiomatics to 20th Century Metalogic Steve Awodey Department of Philosophy Carnegie Mellon University Pittsburgh, PA 15213 [email protected] ... 3T he secondpart is(Aw odey andReck 2002),referred to hereafteras"the sequel" . ...
This paper is the second in a two-part series in which we discuss several notions of completeness... more 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.
Take-down policy If you believe that this document breaches copyright please contact us providing... more Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.
Since the 1960s, there has been a vigorous and ongoing debate about structuralism in English-spea... more Since the 1960s, there has been a vigorous and ongoing debate about structuralism in English-speaking philosophy of mathematics. But structuralist ideas and methods go back further in time, i.e., there is a rich pre-history to this debate, also in the German-and French-speaking literature. In the present collection of essays this pre-history is explored in a two-fold way: first, by reconsidering various mathematicians in the 19th and early 20th centuries (from Grassmann, Dedekind, Pasch, and Klein to Hilbert, Noether, Bourbaki, and Mac Lane) who contributed to structuralism in a methodological sense; second, by reexamining a range of philosophical reflections on such contributions during the same period (also by Peirce, Poincaré, Russell, Cassirer, Bernays, Carnap, and Quine), including suggestions about logical, epistemological, and metaphysical aspects that remain relevant today. Overall, the collection makes evident that structuralism has deep roots in the history of modern mathematics, that mathematical and philosophical views about it are often closely intertwined, and that the range of philosophical options available in this context is significantly richer than a narrow focus on current debates may make one believe.
In his Untersuchungen zur allgemeinen Axiomatik (1928) and Abriss der Logistik (1929), Rudolf Car... more In his Untersuchungen zur allgemeinen Axiomatik (1928) and Abriss der Logistik (1929), Rudolf 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 much less confused and hopeless than it has often been made out to be. By providing such a reassessment, the paper contributes to a reevaluation of Carnap's contributions to the development of modern logic.
This essay concerns Dedekind’s “mathematical structuralism,”by which we mean methodological featu... more 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 s...
Analytic philosophy and modern logic are intimately connected, both historically and systematical... more 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.
Among all of Frege’s contemporaries, Richard Dedekind is arguably the thinker closest to him in t... more 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.
Richard Dedekind (1831–1916) was one of the greatest mathematicians of the nineteenth-century, as... more Richard Dedekind (1831–1916) was one of the greatest mathematicians of the nineteenth-century, as well as one of the most important contributors to algebra and number theory of all time. Any comprehensive history of mathematics will mention him for his investigation of the notions of algebraic number, field, group, module, lattice, etc., and especially for the invention of his theory of ideals (see, e.g., Dieudonné 1985, Boyer & Merzbach 1991, Stillwell 2000, Kolmogorov & Yushkevich 2001, Wussing 2012). Dedekind’s more foundational work in mathematics is also widely known. Often acknowledged in that connection are: his analysis of the notion of continuity, his introduction of the real numbers by means of Dedekind cuts, his formulation of the Dedekind-Peano axioms for the natural numbers, his proof of the categoricity of these axioms, and his contributions to the early development of set theory (Grattan-Guinness 1980, Ferreirós 1996, 1999, 2016b, Jahnke 2003, Corry 2015). While many ...
Any history of logic from Kant to the 20th century needs to take into account the emergence of lo... more Any history of logic from Kant to the 20th century needs to take into account the emergence of logicism, since it is largely in this context that logic as we know it became prominent. Usually Gottlob Frege and Bertrand Russell are seen as its two main representatives, especially early on, and the more recent rise of neo-logicism, in writings by Crispin Wright, Bob Hale and others, proceeds along Fregean lines as well. In this chapter, I will focus on Richard Dedekind instead. Today Dedekind is sometimes mentioned in connection with logicism; yet at the end of the 19th century, he was the most prominent logicist, ahead of Frege. This alone invites further reflection. Reconsidering Dedekind also leads naturally to questions about what was, or could be, understood by "logic" in this context. Addressing them will involve developments in 19thcentury mathematics that are relevant in two respects: by forming core parts of the background for the emergence of modern logic; and by pointing towards a distinction between two conceptions of logic that deserve more attention, both historically and philosophically. 1 The chapter proceeds as follows. First, I will document Dedekind's characterization of his project as showing that arithmetic, understood in an inclusive sense, is "part of logic", and I will put that project into the context of broader developments in 19th-century mathematics. In the second section, I will look in more detail at Dedekind's procedure, including the fact that it involves set-theoretic "constructions" and a certain kind of "abstraction". Third, a brief summary of later appropriations and developments of Dedekind's contributions will be provided, especially in axiomatic set theory and category theory, although these are usually not seen as forms of "logicism". This will, in the fourth section, provide the background for the question of how Dedekind-and parallel to him, Frege-must have understood "logic" for their logicist projects to make sense, namely in a wider sense than the one dominant in the 20th century. The essay will conclude with some general observations, about the contested status of the notion of "logic", i.e. a lack of consensus about its nature that is still often underestimated, and about its historical and philosophical relationship to modern mathematics.
Carnap’s Ideal of Explication and Naturalism, 2012
During the last two decades, there has been a revival of interest in Rudolf Carnap. This has led ... more 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.
In Frege's writings, the notions of truth, judgment, and objectivity are all prominent and import... more 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.
2005) Gottlob Frege: Critical Assessments of Leading …, 2005
... numbers as "pluralities", "multitudes", or "groups of things". ... more ... numbers as "pluralities", "multitudes", or "groups of things". That general conception has a long history, from Mill and Weierstrass in the nineteenth century back to Aristotle, Euclid, and Ancient Greek thought. The basic idea behind it is this: Consider an ...
Ernst Cassirer was a keen observer of development in the mathematical sciences, especially in the... more 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 methodol...
Various contributors to recent philosophy of mathematics have taken Richard Dedekind to be the fo... more Various contributors to recent philosophy of mathematics have taken Richard Dedekind to be the founder of structuralism in 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. A direct forerunner of ante rem structuralism is [.. . ] Dedekind. His development of the notion of continuity and the real numbers, in ["Stetigkeit und irrationale Zahlen"], his presentation of the natural numbers via the notion of Dedekind infinity, in ["Was sind und was sollen die Zahlen?"], and some of his correspondence constitute a structuralist manifesto. (Shapiro 1997, p. 14)
... EDITORIAL BOARD James Lennox University of Pittsburgh Richard Creath Arizona State University... more ... EDITORIAL BOARD James Lennox University of Pittsburgh Richard Creath Arizona State University John Earman University of Pittsburgh Michael Friedman Stanford University GottfriedGabriel University of Jena Dana Scott Carnegie Mellon University Wilfried Sieg Carnegie ...
Rudolf Carnap was a remarkably productive thinker and writer, making contributions on a variety o... more Rudolf Carnap was a remarkably productive thinker and writer, making contributions on a variety of philosophical topics. As a result, he is a different philosopher for different audiences: For the general public (insofar as it knows him), he is the author of Schein- probleme in der Philosophie (1928a), Wissenschaftliche Weltauffassung (Hahn, Neurath & Carnap 1929), and "Überwindung der Metaphysik durch
... Part I: 19th Century Axiomatics to 20th Century Metalogic Steve Awodey Department of Philosop... more ... Part I: 19th Century Axiomatics to 20th Century Metalogic Steve Awodey Department of Philosophy Carnegie Mellon University Pittsburgh, PA 15213 [email protected] ... 3T he secondpart is(Aw odey andReck 2002),referred to hereafteras"the sequel" . ...
This paper is the second in a two-part series in which we discuss several notions of completeness... more 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.
Take-down policy If you believe that this document breaches copyright please contact us providing... more Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.
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