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The Fact of Modern Mathematics: Geometry, Logic, and Concept Formation in Kant and Cassirer

Heis, Jeremy Richard (2008) The Fact of Modern Mathematics: Geometry, Logic, and Concept Formation in Kant and Cassirer. Doctoral Dissertation, University of Pittsburgh.

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    Abstract

    It is now commonly accepted that any adequate history of late nineteenth and early twentieth century philosophy - and thus of the origins of analytic philosophy - must take seriously the role of Neo-Kantianism and Kant interpretation in the period. My dissertation is a contribution to our understanding of this interesting but poorly understood stage in the history of philosophy. Kant's theory of the concepts, postulates, and proofs of geometry was informed by philosophical reflection on diagram-based geometry in the Greek synthetic tradition. However, even before the widespread acceptance of non-Euclidean geometry, the projective revolution in nineteenth century geometry eliminated diagrams from proofs and introduced "ideal" elements that could not be given a straightforward interpretation in empirical space. A Kantian like the very early Russell felt forced to regard the ideal elements as convenient fictions. The Marburg Neo-Kantians—the philosophical school that included Ernst Cassirer (1874-1945)—thought that philosophy, as "transcendental logic," needed to take the results of established pure mathematics as a "fact," not a fiction. Cassirer therefore updates Kant by rejecting the "Transcendental Aesthetic" and by using elements in Richard Dedekind's foundations of arithmetic to rework Kant's idea that the geometrical method is the "construction of concepts". He further argues that geometry is "synthetic" because it progresses when mathematicians introduce new structures (like the complex projective plane) that are not contained in the old structures, but unify them under a new point-of-view. 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 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.)


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    Item Type: University of Pittsburgh ETD
    Creators/Authors:
    CreatorsEmailORCID
    Heis, Jeremy Richardjeremyheis@gmail.com
    ETD Committee:
    ETD Committee TypeCommittee MemberEmailORCID
    Committee ChairWilson, Mark
    Committee MemberGupta, Anil
    Committee MemberAvigad, Jeremy
    Committee MemberManders, Kenneth
    Committee MemberEngstrom, Stephen
    UNSPECIFIEDRicketts, Thomas
    Title: The Fact of Modern Mathematics: Geometry, Logic, and Concept Formation in Kant and Cassirer
    Status: Unpublished
    Abstract: It is now commonly accepted that any adequate history of late nineteenth and early twentieth century philosophy - and thus of the origins of analytic philosophy - must take seriously the role of Neo-Kantianism and Kant interpretation in the period. My dissertation is a contribution to our understanding of this interesting but poorly understood stage in the history of philosophy. Kant's theory of the concepts, postulates, and proofs of geometry was informed by philosophical reflection on diagram-based geometry in the Greek synthetic tradition. However, even before the widespread acceptance of non-Euclidean geometry, the projective revolution in nineteenth century geometry eliminated diagrams from proofs and introduced "ideal" elements that could not be given a straightforward interpretation in empirical space. A Kantian like the very early Russell felt forced to regard the ideal elements as convenient fictions. The Marburg Neo-Kantians—the philosophical school that included Ernst Cassirer (1874-1945)—thought that philosophy, as "transcendental logic," needed to take the results of established pure mathematics as a "fact," not a fiction. Cassirer therefore updates Kant by rejecting the "Transcendental Aesthetic" and by using elements in Richard Dedekind's foundations of arithmetic to rework Kant's idea that the geometrical method is the "construction of concepts". He further argues that geometry is "synthetic" because it progresses when mathematicians introduce new structures (like the complex projective plane) that are not contained in the old structures, but unify them under a new point-of-view. 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 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.)
    Date: 24 January 2008
    Date Type: Completion
    Defense Date: 05 September 2007
    Approval Date: 24 January 2008
    Submission Date: 23 August 2007
    Access Restriction: No restriction; The work is available for access worldwide immediately.
    Patent pending: No
    Institution: University of Pittsburgh
    Thesis Type: Doctoral Dissertation
    Refereed: Yes
    Degree: PhD - Doctor of Philosophy
    URN: etd-08232007-132755
    Uncontrolled Keywords: Ernst Cassirer; Gottlob Frege; Hermann Lotze; Immanuel Kant; Neo-Kantianism; Projective Geometry
    Schools and Programs: Dietrich School of Arts and Sciences > Philosophy
    Date Deposited: 10 Nov 2011 15:01
    Last Modified: 30 Apr 2012 13:07
    Other ID: http://etd.library.pitt.edu/ETD/available/etd-08232007-132755/, etd-08232007-132755

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