The Prehistory of Mathematical Structuralism, 2020
Bertrand Russell was one of the first philosophers to recognize clearly the philosophically innov... more Bertrand Russell was one of the first philosophers to recognize clearly the philosophically innovative nature of Richard Dedekind’s philosophy of arithmetic: a position we now describe as non-eliminative structuralism. But Russell’s response was deeply negative: “If [numbers] are to be anything at all, they must be intrinsically something” (Principles of Mathematics, §242). Nevertheless, Russell also played a significant positive role in making possible the emergence of structuralist philosophy of mathematics. This chapter explains Russell’s double role, identifying three positive contributions to structuralism, while laying out Russell’s objections to Dedekind’s non-eliminative structuralism.
The fifth postulate in Euclid's Elements is the notorious Axiom of Parallels: That, if a straight... more The fifth postulate in Euclid's Elements is the notorious Axiom of Parallels: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles. 1 This axiom was criticized already in the ancient world. By the late 19 th century, Beltrami and Klein showed that it is consistent to deny the axiom, and of course Einstein showed that it is not even true of physical space. Kant believed that the axioms of geometry are synthetic a priori truths. This view sits uncomfortably with the now consensus view among geometers that both Euclidean and non-Euclidean geometries are equally legitimate subjects of mathematical study, and it sits very uncomfortably with the consensus among physicists that the axiom of parallels is false of physical space. Not surprisingly, then, some philosophers have argued that Kant's philosophy of mathematics (indeed, his entire theoretical philosophy) has been refuted by the real possibility of non-Euclidean physical spaces. 2 Moreover, some of Kant's critics, surprised that Kant says nothing about the wellknown problems with Euclid's axiom of parallels, have viewed Kant's silence on this topic as blameworthy. Here, for instance, is William Ewald, in his well-known collection From Kant to Hilbert: 1 Euclid, The Thirteen Books of the Elements, vol.1, trans. and ed. Heath, 202. This is the fifth "postulate" in Euclid's Elements; for reasons that will become clear below, I will refer to the principle as Euclid's "axiom" of parallels. 2 See, for example, Reichenbach, "Logical Empiricism in Germany," 145.
This new "Kantian" theory of modern mathematics, Cassirer argues, is inconsistent with the tradit... more This new "Kantian" theory of modern mathematics, Cassirer argues, is inconsistent with the traditional theory of concept formation by abstraction. Drawing on earlier Neo-Kantian interpretations, Cassirer argues that Kant's theory of concepts as rules undermines the traditional theory of concept formation, and he gives a "transcendental" defense of the new logic of Frege and Russell. (In an appendix, I discuss the contemporaneous accounts of concept formation in Gottlob Frege and Hermann Lotze.) vi TABLE OF CONTENTS
This paper gives a contextualized reading of Kant’s theory of real definitions in geometry. Thoug... more This paper gives a contextualized reading of Kant’s theory of real definitions in geometry. Though Leibniz, Wolff, Lambert and Kant all believe that definitions in geometry must be ‘real’, they disagree about what a real definition is. These disagreements are made vivid by looking at two of Euclid’s definitions. I argue that Kant accepted Euclid’s definition of circle and rejected his definition of parallel lines because his conception of mathematics placed uniquely stringent requirements on real definitions in geometry. Leibniz, Wolff and Lambert thus accept definitions that Kant rejects because they assign weaker roles to real definitions.
Studies in History and Philosophy of Science Part A, 2014
This paper presents the main ideas of Cassirer's general philosophy of science, focusing on the t... more This paper presents the main ideas of Cassirer's general philosophy of science, focusing on the two aspects of his thought thatdin addition to being the most central ideas in his philosophy of sciencedhave received the most attention from contemporary philosophers of science: his theory of the a priori aspects of physical theory, and his relation to scientific realism.
The Berlin Group and the Philosophy of Logical Empiricism, 2013
This chapter concerns the philosophical and personal interactions between the Neo-Kantian philoso... more This chapter concerns the philosophical and personal interactions between the Neo-Kantian philosopher Ernst Cassirer and two members of the Berlin Group, Kurt Lewin and Hans Reichenbach. Both Lewin and Reichenbach were students of Cassirer’s in Berlin, and both explicitly discuss Cassirer’s work in their writings. After some biographical material drawn from their published and unpublished works and correspondence (Sect. 4.1), Sect. 4.2 of the paper argues that each of the three shared a broad conception of philosophy as the “logical analysis of science,” though each carried out this project in different ways. Lewin used philosophical ideas from Cassirer to motivate his original and consequential research program in experimental psychology (Sect. 4.3). Reichenbach, on the other hand, initially drew on Cassirer’s theory of the a priori in isolating a priori “coordinating principles” in general relativity, and then later rejected Cassirer’s theory by relabeling these principles as conventions (Sect. 4.4). That Cassirer did not follow Reichenbach into conventionalism springs, I argue, from Cassirer’s rejection of Reichenbach’s view that theory-neutral experimental measurements could ground physical theories in an objective way.
In the ‘analytic tradition’, Hans Sluga wrote thirty years ago in his book Gottlob Frege, there h... more In the ‘analytic tradition’, Hans Sluga wrote thirty years ago in his book Gottlob Frege, there has been a ‘lack of interest in historical questions — even in the question of its own roots. Anti-historicism has been the baggage of the tradition since Frege’ (Sluga, 1980, p. 2). The state of the discussion of Frege among analytic philosophers, Sluga claimed, illustrated well this indifference. Despite the numbers of pages devoted to Frege, there was still, Sluga claimed, little understanding of the sources of Frege’s ideas, his intellectual debts, and the historical circumstances of his thought. Sluga singled out Michael Dummett’s Frege: Philosophy of Language ‘as a paradigm for the failure of analytic philosophers to come to grips with the actual, historical Frege’. In that book, Dummett famously wrote that the logical system that Frege put forward in his 1879 Begriffsschrift ‘is astonishing because it has no predecessors: it appears to have been born from Frege’s brain unfertilized by external influences’ (Dummett, 1981a, p. xxxv) — and Dummett devoted almost none of its 700 pages to the relationship between Frege and his contemporaries.
The being of a concept depends on the being of the judgment (as the being of a part in an organic... more The being of a concept depends on the being of the judgment (as the being of a part in an organic whole depends on the being of the whole). Weak Semantic Priority: A concept (or concept-word) has meaning because it is part of a judgment (or sentence) that has meaning. Strong Semantic Priority: The meaning of a concept (or concept-word) consists in the contribution it makes to the meaning of a judgment (or sentence). Epistemic Priority: A subject can come to know a concept only through coming to know a judgment containing that concept. (Or: a subject can form a concept only through forming a judgment containing it.) KANT'S AND FREGE'S PRIORITY PRINCIPLES (AS I READ THEM) Frege's (Epistemic) Priority Thesis: Judgments are prior to concepts because the formation of compound concepts requires first grasping a complete judgment and then replacing some of its components. Kant's (Semantic) Priority Thesis: Judgments are prior to concepts because a concept is a mediate representation of an object, and it is in virtue of being combined in judgments that concepts relate to objects. 'Aristotelianism' about conceptual structure: concepts are either simple or are composed of simple concepts by conjunction, addition, or exclusion. Abstractionism about conceptual formation: concepts are formed by noticing similarities or differences among particulars and abstracting the concept, as the common element, from these similarities or differences; compound concepts can arise from simpler concepts either by conjunction, disjunction, and exclusion, or by noticing similarities or differences among concepts and abstracting from these similarities or differences.
British Journal for the History of Philosophy, 2011
One of the most important philosophical topics in the early twentieth century-and a topic that wa... more One of the most important philosophical topics in the early twentieth century-and a topic that was seminal in the emergence of analytic philosophy-was the relationship between Kantian philosophy and modern geometry. This paper discusses how this question was tackled by the Neo-Kantian trained philosopher Ernst Cassirer. Surprisingly, Cassirer does not affirm the theses that contemporary philosophers often associate with Kantian philosophy of mathematics. He does not defend the necessary truth of Euclidean geometry but instead develops a kind of logicism modeled on Richard Dedekind's foundations of arithmetic. Further, because he shared with other Neo-Kantians an appreciation of the developmental and historical nature of mathematics, Cassirer developed a philosophical account of the unity and methodology of mathematics over time. With its impressive attention to the detail of contemporary mathematics and its exploration of philosophical questions to which other philosophers paid scant attention, Cassirer's philosophy of mathematics surely deserves a place among the classic works of twentieth century philosophy of mathematics. Though focused on Cassirer's philosophy of geometry, this paper also addresses both Cassirer's general philosophical orientation and his reading of Kant.
Bertrand Russell was one of the first philosophers to recognize clearly the philosophically innov... more Bertrand Russell was one of the first philosophers to recognize clearly the philosophically innovative nature of Richard Dedekind’s philosophy of arithmetic: a position we now describe as non-eliminative structuralism. But Russell’s response was deeply negative: “If [numbers] are to be anything at all, they must be intrinsically something” (Principles of Mathematics, §242). Nevertheless, Russell also played a significant positive role in making possible the emergence of structuralist philosophy of mathematics. This chapter explains Russell’s double role, identifying three positive contributions to structuralism, while laying out Russell’s objections to Dedekind’s non-eliminative structuralism.
Innovations in the History of Analytical Philosophy, 2017
One of the most significant events in the emergence of analytic philosophy was Bertrand Russell's... more One of the most significant events in the emergence of analytic philosophy was Bertrand Russell's rejection of the idealist philosophy of mathematics contained in his An Essay on the Foundations of Geometry [EFG] (1897) in favor of the logicism that he developed and defended in his Principles of Mathematics [POM] (1903). Russell's language in POMand in the sharp and provocative paper, "Recent Work on the Principles of Mathematics" (1901a), which is the first public presentation of many of the doctrines of POM-is deliberately revolutionary. "The proof that all pure mathematics, including Geometry, is nothing but formal logic, is a fatal blow to Kantian philosophy" (1901a, 379)-including, of course, the Kantian philosophy that he defended just four years earlier in EFG. "The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age" (POM, §4), a discovery that will initiate a new era in philosophy: "[T]here is every reason to hope that the near
The Prehistory of Mathematical Structuralism, 2020
Bertrand Russell was one of the first philosophers to recognize clearly the philosophically innov... more Bertrand Russell was one of the first philosophers to recognize clearly the philosophically innovative nature of Richard Dedekind’s philosophy of arithmetic: a position we now describe as non-eliminative structuralism. But Russell’s response was deeply negative: “If [numbers] are to be anything at all, they must be intrinsically something” (Principles of Mathematics, §242). Nevertheless, Russell also played a significant positive role in making possible the emergence of structuralist philosophy of mathematics. This chapter explains Russell’s double role, identifying three positive contributions to structuralism, while laying out Russell’s objections to Dedekind’s non-eliminative structuralism.
The fifth postulate in Euclid's Elements is the notorious Axiom of Parallels: That, if a straight... more The fifth postulate in Euclid's Elements is the notorious Axiom of Parallels: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles. 1 This axiom was criticized already in the ancient world. By the late 19 th century, Beltrami and Klein showed that it is consistent to deny the axiom, and of course Einstein showed that it is not even true of physical space. Kant believed that the axioms of geometry are synthetic a priori truths. This view sits uncomfortably with the now consensus view among geometers that both Euclidean and non-Euclidean geometries are equally legitimate subjects of mathematical study, and it sits very uncomfortably with the consensus among physicists that the axiom of parallels is false of physical space. Not surprisingly, then, some philosophers have argued that Kant's philosophy of mathematics (indeed, his entire theoretical philosophy) has been refuted by the real possibility of non-Euclidean physical spaces. 2 Moreover, some of Kant's critics, surprised that Kant says nothing about the wellknown problems with Euclid's axiom of parallels, have viewed Kant's silence on this topic as blameworthy. Here, for instance, is William Ewald, in his well-known collection From Kant to Hilbert: 1 Euclid, The Thirteen Books of the Elements, vol.1, trans. and ed. Heath, 202. This is the fifth "postulate" in Euclid's Elements; for reasons that will become clear below, I will refer to the principle as Euclid's "axiom" of parallels. 2 See, for example, Reichenbach, "Logical Empiricism in Germany," 145.
This new "Kantian" theory of modern mathematics, Cassirer argues, is inconsistent with the tradit... more This new "Kantian" theory of modern mathematics, Cassirer argues, is inconsistent with the traditional theory of concept formation by abstraction. Drawing on earlier Neo-Kantian interpretations, Cassirer argues that Kant's theory of concepts as rules undermines the traditional theory of concept formation, and he gives a "transcendental" defense of the new logic of Frege and Russell. (In an appendix, I discuss the contemporaneous accounts of concept formation in Gottlob Frege and Hermann Lotze.) vi TABLE OF CONTENTS
This paper gives a contextualized reading of Kant’s theory of real definitions in geometry. Thoug... more This paper gives a contextualized reading of Kant’s theory of real definitions in geometry. Though Leibniz, Wolff, Lambert and Kant all believe that definitions in geometry must be ‘real’, they disagree about what a real definition is. These disagreements are made vivid by looking at two of Euclid’s definitions. I argue that Kant accepted Euclid’s definition of circle and rejected his definition of parallel lines because his conception of mathematics placed uniquely stringent requirements on real definitions in geometry. Leibniz, Wolff and Lambert thus accept definitions that Kant rejects because they assign weaker roles to real definitions.
Studies in History and Philosophy of Science Part A, 2014
This paper presents the main ideas of Cassirer's general philosophy of science, focusing on the t... more This paper presents the main ideas of Cassirer's general philosophy of science, focusing on the two aspects of his thought thatdin addition to being the most central ideas in his philosophy of sciencedhave received the most attention from contemporary philosophers of science: his theory of the a priori aspects of physical theory, and his relation to scientific realism.
The Berlin Group and the Philosophy of Logical Empiricism, 2013
This chapter concerns the philosophical and personal interactions between the Neo-Kantian philoso... more This chapter concerns the philosophical and personal interactions between the Neo-Kantian philosopher Ernst Cassirer and two members of the Berlin Group, Kurt Lewin and Hans Reichenbach. Both Lewin and Reichenbach were students of Cassirer’s in Berlin, and both explicitly discuss Cassirer’s work in their writings. After some biographical material drawn from their published and unpublished works and correspondence (Sect. 4.1), Sect. 4.2 of the paper argues that each of the three shared a broad conception of philosophy as the “logical analysis of science,” though each carried out this project in different ways. Lewin used philosophical ideas from Cassirer to motivate his original and consequential research program in experimental psychology (Sect. 4.3). Reichenbach, on the other hand, initially drew on Cassirer’s theory of the a priori in isolating a priori “coordinating principles” in general relativity, and then later rejected Cassirer’s theory by relabeling these principles as conventions (Sect. 4.4). That Cassirer did not follow Reichenbach into conventionalism springs, I argue, from Cassirer’s rejection of Reichenbach’s view that theory-neutral experimental measurements could ground physical theories in an objective way.
In the ‘analytic tradition’, Hans Sluga wrote thirty years ago in his book Gottlob Frege, there h... more In the ‘analytic tradition’, Hans Sluga wrote thirty years ago in his book Gottlob Frege, there has been a ‘lack of interest in historical questions — even in the question of its own roots. Anti-historicism has been the baggage of the tradition since Frege’ (Sluga, 1980, p. 2). The state of the discussion of Frege among analytic philosophers, Sluga claimed, illustrated well this indifference. Despite the numbers of pages devoted to Frege, there was still, Sluga claimed, little understanding of the sources of Frege’s ideas, his intellectual debts, and the historical circumstances of his thought. Sluga singled out Michael Dummett’s Frege: Philosophy of Language ‘as a paradigm for the failure of analytic philosophers to come to grips with the actual, historical Frege’. In that book, Dummett famously wrote that the logical system that Frege put forward in his 1879 Begriffsschrift ‘is astonishing because it has no predecessors: it appears to have been born from Frege’s brain unfertilized by external influences’ (Dummett, 1981a, p. xxxv) — and Dummett devoted almost none of its 700 pages to the relationship between Frege and his contemporaries.
The being of a concept depends on the being of the judgment (as the being of a part in an organic... more The being of a concept depends on the being of the judgment (as the being of a part in an organic whole depends on the being of the whole). Weak Semantic Priority: A concept (or concept-word) has meaning because it is part of a judgment (or sentence) that has meaning. Strong Semantic Priority: The meaning of a concept (or concept-word) consists in the contribution it makes to the meaning of a judgment (or sentence). Epistemic Priority: A subject can come to know a concept only through coming to know a judgment containing that concept. (Or: a subject can form a concept only through forming a judgment containing it.) KANT'S AND FREGE'S PRIORITY PRINCIPLES (AS I READ THEM) Frege's (Epistemic) Priority Thesis: Judgments are prior to concepts because the formation of compound concepts requires first grasping a complete judgment and then replacing some of its components. Kant's (Semantic) Priority Thesis: Judgments are prior to concepts because a concept is a mediate representation of an object, and it is in virtue of being combined in judgments that concepts relate to objects. 'Aristotelianism' about conceptual structure: concepts are either simple or are composed of simple concepts by conjunction, addition, or exclusion. Abstractionism about conceptual formation: concepts are formed by noticing similarities or differences among particulars and abstracting the concept, as the common element, from these similarities or differences; compound concepts can arise from simpler concepts either by conjunction, disjunction, and exclusion, or by noticing similarities or differences among concepts and abstracting from these similarities or differences.
British Journal for the History of Philosophy, 2011
One of the most important philosophical topics in the early twentieth century-and a topic that wa... more One of the most important philosophical topics in the early twentieth century-and a topic that was seminal in the emergence of analytic philosophy-was the relationship between Kantian philosophy and modern geometry. This paper discusses how this question was tackled by the Neo-Kantian trained philosopher Ernst Cassirer. Surprisingly, Cassirer does not affirm the theses that contemporary philosophers often associate with Kantian philosophy of mathematics. He does not defend the necessary truth of Euclidean geometry but instead develops a kind of logicism modeled on Richard Dedekind's foundations of arithmetic. Further, because he shared with other Neo-Kantians an appreciation of the developmental and historical nature of mathematics, Cassirer developed a philosophical account of the unity and methodology of mathematics over time. With its impressive attention to the detail of contemporary mathematics and its exploration of philosophical questions to which other philosophers paid scant attention, Cassirer's philosophy of mathematics surely deserves a place among the classic works of twentieth century philosophy of mathematics. Though focused on Cassirer's philosophy of geometry, this paper also addresses both Cassirer's general philosophical orientation and his reading of Kant.
Bertrand Russell was one of the first philosophers to recognize clearly the philosophically innov... more Bertrand Russell was one of the first philosophers to recognize clearly the philosophically innovative nature of Richard Dedekind’s philosophy of arithmetic: a position we now describe as non-eliminative structuralism. But Russell’s response was deeply negative: “If [numbers] are to be anything at all, they must be intrinsically something” (Principles of Mathematics, §242). Nevertheless, Russell also played a significant positive role in making possible the emergence of structuralist philosophy of mathematics. This chapter explains Russell’s double role, identifying three positive contributions to structuralism, while laying out Russell’s objections to Dedekind’s non-eliminative structuralism.
Innovations in the History of Analytical Philosophy, 2017
One of the most significant events in the emergence of analytic philosophy was Bertrand Russell's... more One of the most significant events in the emergence of analytic philosophy was Bertrand Russell's rejection of the idealist philosophy of mathematics contained in his An Essay on the Foundations of Geometry [EFG] (1897) in favor of the logicism that he developed and defended in his Principles of Mathematics [POM] (1903). Russell's language in POMand in the sharp and provocative paper, "Recent Work on the Principles of Mathematics" (1901a), which is the first public presentation of many of the doctrines of POM-is deliberately revolutionary. "The proof that all pure mathematics, including Geometry, is nothing but formal logic, is a fatal blow to Kantian philosophy" (1901a, 379)-including, of course, the Kantian philosophy that he defended just four years earlier in EFG. "The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age" (POM, §4), a discovery that will initiate a new era in philosophy: "[T]here is every reason to hope that the near
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