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Why Did Weyl Think that Dedekind’s Norm of Belief in Mathematics is Perverse?

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Early Analytic Philosophy - New Perspectives on the Tradition

Part of the book series: The Western Ontario Series in Philosophy of Science ((WONS,volume 80))

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Abstract

This paper discusses an intriguing, though rather overlooked case of normative disagreement in the history of philosophy of mathematics: Weyl's criticism of Dedekind’s famous principle that “In science, what is provable ought not to be believed without proof.” This criticism, as I see it, challenges not only a logicist norm of belief in mathematics, but also a realist view about whether there is a fact of the matter as to what norms of belief are correct.

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Notes

  1. 1.

    The expression “probative completeness” has been introduced in Detlefsen (2010) to characterize an epistemic ideal shared, before Dedekind, by Bolzano. See also Detlefsen (2011).

  2. 2.

    See Dedekind’s letter to H. Weber from November 19, 1878; in Dedekind (1932), 486.

  3. 3.

    It is, of course, difficult to give a compelling delineation of the class of unprovable propositions. As Frege duly noted, Dedekind himself offered no “inventory of the logical or other laws taken by him as basic.” Cf. Frege (1893, viii).

  4. 4.

    Cf. Metaphysics, Book IV, Chap. 4.

  5. 5.

    Cf. Dedekind (1888). Eng. tr. in Ewald (1996, 792).

  6. 6.

    For the view of belief as doxastic performance, see Sosa (2011). My adopting it here without discussion will be remedied in a fuller version of the paper.

  7. 7.

    For a more general discussion of Peirce’s criticism of Cartesian epistemology, see Haack (1982). For a discussion of his view on logicism, see Haack (1993).

  8. 8.

    My understanding of Weyl’s philosophy of mathematics benefited from numerous discussions, during my graduate studies, with Paddy Blanchette, Mic Detlefsen, Chris Porter, and Sean Walsh. Since then, I presented this material at several conferences, most recently in Cracow, Rome, and Vienna. Many thanks to my audiences there, and to Hanoch Ben-Yami, Anandi Hattiangadi, and Dirk Schlimm, for their helpful comments and suggestions.

References

  • Bell, J. (2000). Hermann Weyl on intuition and the continuum. Philosophia Mathematica III, 8, 259–273.

    Article  Google Scholar 

  • Dedekind, R. (1888). Was sind und was sollen die Zahlen? Vieweg, Braunschweig. Repr. in Dedekind 1932. Eng. tr. in Ewald 1996.

    Google Scholar 

  • Dedekind, R. (1932). Gesammelte mathematische Werke (vol. 3). In R. Fricke et al. (Ed.), Braunschweig: Vieweg & Sohn.

    Google Scholar 

  • Detlefsen, M. (2010). Rigor, re-proof and Bolzano’s critical program. In P.-E. Bour et al. (Ed.), Construction (pp. 171–184). UK: King’s College Publications.

    Google Scholar 

  • Detlefsen, M. (2011). Dedekind against intuition: Rigor, scope and the motives of his logicism. In C. Cellucci et al. (Ed.), Logic and knowledge (pp. 205–217). UK: Cambridge Scholars Publishing.

    Google Scholar 

  • Ewald, W. B. (1996). From Kant to Hilbert, vol. 2, Oxford University Press.

    Google Scholar 

  • Field, H. (2015). What is Logical Validity? In Colin Caret and Ole Hjortland (Eds.), Foundations of Logical Consequence. Oxford: Oxford University Press (forthcoming).

    Google Scholar 

  • Frege, G. (1893). Die Grundgesetze der Arithmetik, Jena.

    Google Scholar 

  • Haack, S. (1982). Descartes, Peirce and the cognitive community. The Monist, 65, 156–181.

    Article  Google Scholar 

  • Haack, S. (1993). Peirce and logicism: Notes towards an exposition. Transactions of the Charles S. Peirce Society, 29, 33–56.

    Google Scholar 

  • Peirce, C. S. (1877). “The fixation of belief” in Peirce 1955, 5–21.

    Google Scholar 

  • Peirce, C. S. (1878). “How to make our ideas clear” in Peirce 1955, 23--41.

    Google Scholar 

  • Peirce, C. S. (1955). Philosophical Writings of Peirce, (Ed.), by J. Buchler, Dover.

    Google Scholar 

  • Ryckman, T. (2005). The reign of relativity. Oxford: Oxford University Press.

    Google Scholar 

  • Sosa, E. (2011). Knowing full well. Princeton: Princeton University Press.

    Google Scholar 

  • Weyl, H. (1918). Das Kontinuum. In Das Kontinuum und andere Monographien, Chelsea Publishing Company, 1960. Eng. tr. as The Continuum, Thomas Jefferson University Press, 1987.

    Google Scholar 

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Authors and Affiliations

  1. University of Bucharest, Bucharest, Romania

    Iulian D. Toader

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  1. Iulian D. Toader
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Correspondence to Iulian D. Toader .

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  1. Philosophy Department, University of Bucharest, Bucharest, Romania

    Sorin Costreie

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Toader, I.D. (2016). Why Did Weyl Think that Dedekind’s Norm of Belief in Mathematics is Perverse?. In: Costreie, S. (eds) Early Analytic Philosophy - New Perspectives on the Tradition. The Western Ontario Series in Philosophy of Science, vol 80. Springer, Cham. https://doi.org/10.1007/978-3-319-24214-9_19

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