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Practice-dependent realism and mathematics

Cole, Julian C.
http://rave.ohiolink.edu/etdc/view?acc_num=osu1124122328

Abstract Details

2005, Doctor of Philosophy, Ohio State University, Philosophy.
I present a solution to a puzzle concerning the interpretation of mathematical practices. Mathematical claims seem to be objective and about objects or structures. Furthermore, these objects or structures do not appear to be located in the spatio-temporal world. For these reasons, Platonists have suggested that mathematical claims concern domains that are abstract and independent of our intellectual activities. Yet if Platonists are correct, then something very peculiar is going on. Items that are abstract and independent of our intellectual activities cannot influence spatio-temporally instantiated activities like mathematical practices. Contemporary authors have responded to this observation in two distinct ways. Some have argued that mathematics should be understood as a fiction. Others hold that the surface grammar of mathematical claims is misleading; they are really claims about what is logically possible or necessary. A philosopher who accepts either suggestion must pay a high price. Specifically, (s)he expresses a problematic lack of respect for actual mathematical practice. As a solution to this puzzle, I articulate and defend a new metaphysical interpretation of mathematics. According to this interpretation, mathematical domains are constituted by the mathematical activities of rational beings in a way analogous to the constitution of laws and legal borders by the legislative activities of rational beings. The presence of appropriate types of mathematical activity is both necessary and sufficient for the existence of particular mathematical domains, just as the presence of appropriate types of legislative activity is both necessary and sufficient for the existence of particular laws. Yet while laws are constituted by explicit stipulation, mathematicians constitute mathematical domains by providing coherent and adequate characterizations of those domains. My interpretation of mathematics offers an authentic solution to this puzzle, since my solution takes mathematics to be about the very things it seems to be about. It interprets mathematicians as making literal claims. And, it is not affected by the peculiarity associated with Platonism. Finally, this view, which I call practice-dependent realism, contains the resources for a novel account of the objectivity, necessity, and applicability of mathematical truth.
Stewart Shapiro (Advisor)
248 p.
Philosophy
Philosophy of mathematics; Mathematics; Metaphysics; Ontology; Social construction; Metaphysical dependence; Mathematics; Practice-dependent realism; Platonism; Fictionalism; Hermeneutic fictionalism; The epistemological argument against platonism; Natura

Recommended Citations

Citations

  • Cole, J. C. (2005). Practice-dependent realism and mathematics [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1124122328

    APA Style (7th edition)

  • Cole, Julian. Practice-dependent realism and mathematics. 2005. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1124122328.

    MLA Style (8th edition)

  • Cole, Julian. "Practice-dependent realism and mathematics." Doctoral dissertation, Ohio State University, 2005. http://rave.ohiolink.edu/etdc/view?acc_num=osu1124122328

    Chicago Manual of Style (17th edition)

osu1124122328
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© 2005, all rights reserved.
This open access ETD is published by The Ohio State University and OhioLINK.