Advertisement

Synthese

, Volume 196, Issue 3, pp 929–971 | Cite as

Frege’s philosophy of geometry

  • Matthias SchirnEmail author
Article

Abstract

In this paper, I critically discuss Frege’s philosophy of geometry with special emphasis on his position in The Foundations of Arithmetic of 1884. In Sect. 2, I argue that that what Frege calls facultyof intuition in his dissertation (1873) is probably meant to refer to a capacity of visualizing geometrical configurations structurally in a way which is essentially the same for most Western educated human beings. I further suggest that according to his Habilitationsschrift (1874) it is through spatial intuition that we come to know the axioms of Euclidean geometry. In Sect. 3, I argue that Frege seems right in claiming in The Foundations, §14 that the synthetic nature of the Euclidean axioms follows from the fact that they are independent of one another and of the primitive laws of logic. If the former were dependent on (provable from) the latter, they would be analytic in Frege’s sense of analyticity. But then they would not be independent of one another and due to their mutual provability would lose their status as axioms of Euclidean geometry, since according to Frege an axiom of a theory T is per definitionen unprovable in T. I further argue that only by invoking pure spatial intuition can Frege “explain” the (alleged) epistemological status of the axioms of Euclidean geometry completely: their synthetic a priori nature. Finally, I argue that his view about independence in The Foundations, §14 seems to clash with his conception of independence in his mature period. In Sect. 4, I scrutinize Frege’s somewhat vague, but unduly neglected remarks in The Foundations, §26 on space, spatial intuition and the axioms of Euclidean geometry. I argue that for the sake of coherence Frege should have said unambiguously that space is objective, that it is independedent not only of our spatial intuition, but of our mental life altogether including our judgements about space, instead of encouraging the possible conjecture that in his view it contains an objective and a subjective component. I further argue that for Frege the objectivity of both space and the axioms of Euclidean geometry manifests itself in our universal and compulsory acknowledgement of the Euclidean axioms as true. I conclude Sect. 4 by arguing that there is a conflict between the subjectivity of our spatial intuitions as stressed in The Foundations, §26 and Frege’s thesis in his dissertation that the axioms of Euclidean geometry derive their validity from the nature of our faculty of intuition. To resolve this conflict, I propose that in the light of his avowed realism in The Foundations Frege could have replaced his early thesis by saying that although we come to know the axioms of Euclidean geometry through spatial intuition and are justified in acknowledging them as true on the basis of geometrical intuition, their truth is independent not only of the nature of our faculty of intuition and singular acts of intuition, but of our mental processes and activities in general, including the inner mental act of judging. In Sect. 5, I argue that Frege most likely did not adopt Kant’s method of acquiring geometrical knowledge via the ostensive construction of concepts in spatial intuition. In contrast to Kant, Frege holds that the axioms of three-dimensional Euclidean geometry express state of affairs about space obtaining independently of our spatial intuition. In Sect. 6, I conclude with a summarized assessment of Frege’s philosophy of geometry.

Keywords

Imaginary forms in the plane Euclidean geometry Non-Euclidean geometry Projective geometry Mathematical geometry Physical geometry Geometrical axioms Geometrical source of knowledge Ranges of application Independence of the geometrical axioms Synthetic (a priori) Spatial intuition Space Objectivity 

Notes

Acknowledgements

I presented previous and shorter versions of this paper at the International Workshop “The Imaginary, the Ideal and the Infinite in Mathematics” in Pont-à-Mousson (France), 24.06.–27.06.2009; Munich Center for Mathematical Philosophy; University of Oxford (as one of my lectures on Frege’s philosophy of mathematics in Trinity Term 2014); Kyoto University; Hokkaido University (Sapporo). Thanks to the audiences, especially to Godehard Link, Hannes Leitgeb, Jamie Tappenden, Paolo Mancosu, Andrei Rodin, Daniel Isaacson, Yasuo Deguchi and Koji Nakatogawa. I am most grateful to Roberto Torretti for inspiring discussion of issues in the philosophy of geometry over thirty years, not least at his home in Santiago de Chile in the Spring of 2011. In our subsequent correspondence (2011-2017), he helped me, among other things, to clarify an important problem arising from Frege’s geometrical treatment of imaginary forms in the plane. I dedicate the present paper to Roberto Torretti. I am also most grateful to Patricia Blanchette for our recent discussion of the notion of independence in Frege, with special emphasis on his view in The Foundations of Arithmetic, §14. Thanks to this stimulating discussion, I managed to write an essential part of Sect. 3 of my essay. Thanks are also due to Mark Wilson for his interesting comments on an earlier draft of my paper and to Robert Thomas for his comments on the geometrical construction which I discuss in Sect. 2 of my paper. I am very grateful to three anonymous referees for their substantial and critical reports. Their reports helped me to improve my paper considerably and make it richer in content. Special thanks are due to referee #1 for checking meticulously the revised penultimate version and for his final list of typos and minor errors which I probably would not have detected before submitting the final version of my paper for publication in Synthese. My thanks go also to Daniel Mook for his help with this paper and to the editor of Synthese, Wiebe van der Hoek, for his special interest in the improvement of the paper, his encouragement and advice. Finally, I would like to thank Catherine Murphy and Palanimuthu Athimoolam (Production, Springer) for their help and Adittya Iyer for his advice and technical help in his capacity as JEO assistant for Synthese.

References

  1. Adeleke, S., Dummett, M., & Neumann, P. (1987). On a question of Frege’s about right-ordered groups. Bulletin of the London Mathematical Society, 19, 513–521.CrossRefGoogle Scholar
  2. Beltrami, E. (1868). Saggio di interpretazione della geometria non-euclidea. Giornale di matematiche 6, 284–312; reprinted in Beltrami, Opere matematiche, Ulrico Hoepli, Vol. I, Milan, 1902, 374–405.Google Scholar
  3. Blanchette, P. (1996). Frege and Hilbert on consistency. Journal of Philosophy, 93, 317–336.CrossRefGoogle Scholar
  4. Blanchette, P. (2007). Frege on consistency and conceptual analysis. Philosophia Mathematica, 15, 321–346.CrossRefGoogle Scholar
  5. Blanchette, P. (2012). Frege’s conception of logic. New York: Oxford University Press.CrossRefGoogle Scholar
  6. Blanchette, P. (2014). Frege on formality and the 1906 independence test. In G. Link (Ed.), Formalism and beyond. On the nature of mathematical discourse (pp. 97–118). Boston & Berlin: Walter de Gruyter/Ontos.Google Scholar
  7. Blanchette, P. (2016). The breadth of the paradox. Philosophia Mathematica, 24, 30–49.CrossRefGoogle Scholar
  8. Blanchette, P. (2017). Frege’s understanding of the role of axioms. In P. Ebert & M. Rossberg (Eds.), Essays on Frege’s basic laws of arithmetic. Oxford: Oxford University Press (forthcoming).Google Scholar
  9. Bonola, R. (1955). Non-Euclidean geometry. A critical and historical study of its development; english translation of H. S. Carslaw, New York.Google Scholar
  10. Burge, T. (2000). Frege on apriority. In P. Boghossian & C. Peacocke (Eds.), New essays on the a priori (pp. 11–42). Oxford: Oxford University Press.CrossRefGoogle Scholar
  11. Cayley, A. (1889-1897). The collected mathematical papers of Arthur Cayley (Vol. 13). Cambridge: Cambrige University Press.Google Scholar
  12. Cohen, H. (1871). Kants Theorie der Erfahrung. Berlin: Ferd. Dümmlers Verlagsbuchhandlung.Google Scholar
  13. Cohen, H. (1883). Das Prinzip der Infinitesimal-Methode und seine Geschichte. Ein Kapitel zur Grundlegung der Erkenntniskritik. Ferd. Dümmlers Verlagsbuchhandlung, Berlin: New edition with an introduction of W. Flach, Suhrkamp, Frankfurt/M. 1968.Google Scholar
  14. Dummett, M. (1976). Frege on the consistency of mathematical theories, in Schirn 1976, Vol. I, 229–242.Google Scholar
  15. Dummett, M. (1981). The interpretation of Frege’s philosophy. London: Duckworth.Google Scholar
  16. Dummett, M. (1982). Frege and Kant on geometry. Inquiry, 25, 233–254.CrossRefGoogle Scholar
  17. Frege, G. (1873). Über eine geometrische Darstellung der imaginären Gebilde in der Ebene, in Frege 1967, 1–49.Google Scholar
  18. Frege, G. (1874). Rechnungsmethoden, die sich auf eine Erweiterung des Größenbegriffs gründen, in Frege 1967, 50–84.Google Scholar
  19. Frege, G. (1879). Begriffsschrift. Eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle a. S.: L. Nebert.Google Scholar
  20. Frege, G. (1884). Die Grundlagen der Arithmetik. Eine logisch mathematische Untersuchung über den Begriff der Zahl. Breslau: W. Koebner.Google Scholar
  21. Frege, G. (1885). Über formale Theorien der Arithmetik, in Frege 1967, 103–111.Google Scholar
  22. Frege, G. (1893). Grundgesetze der Arithmetik. Begriffsschriftlich abgeleitet (Vol. I). Jena: H. Pohle.Google Scholar
  23. Frege, G. (1899). Letter to Hilbert of 27th December, 1899, in Frege 1976, 60–64.Google Scholar
  24. Frege, G. (1899-1906). Über Euklidische Geometrie, in Frege 1969, 182–184.Google Scholar
  25. Frege, G. (1903). Grundgesetze der Arithmetik. Begriffsschriftlich abgeleitet (Vol. II). Jena: H. Pohle.Google Scholar
  26. Frege, G. (1914). Logik in der Mathematik, in Frege 1969, 219–270.Google Scholar
  27. Frege, G. (1924/1925). Erkenntnisquellen der Mathematik und der mathematischen Naturwissenschaften, in Frege 1969, 286–294.Google Scholar
  28. Frege, G. (1964). Begriffsschrift und andere Aufsätze. Edited by I. Angelelli. Darmstadt & Hildesheim: Wissenschaftliche Buchgesellschaft Darmstadt.Google Scholar
  29. Frege, G. (1967). Kleine Schriften. Edited by I. Angelelli, Hildesheim: G. Olms.Google Scholar
  30. Frege, G. (1969). Nachgelassene Schriften. Edited by H. Hermes, F. Kambartel & F. Kaulbach. Hamburg: F. Meiner.Google Scholar
  31. Frege, G. (1976). Wissenschaftlicher Briefwechsel. Edited by G. Gabriel, H. Hermes, F. Kambartel, C. Thiel & A. Veraart. Hamburg: F. Meiner.Google Scholar
  32. Friedman, M. (1992). Kant’s theory of geometry. In C. J. Posy (Ed.), Kant’s philosophy of mathematics. Dordrecht: Kluwer.Google Scholar
  33. Gauss, C. F. (1900). Werke (Vol. VIII). Edited by Königliche Gesellschaft der Wissenschaften in Göttingen, Leipzig.Google Scholar
  34. Giaquinto, M. (2007). Visual thinking in mathematics. Oxford: Oxford University Press.CrossRefGoogle Scholar
  35. Giaquinto, M. (2011). Crossing curves: A limit to the use of diagrams in proofs. Philosophia Mathematica, 19, 281–307.CrossRefGoogle Scholar
  36. Heck, R. G. (2011). Frege’s theorem. Oxford: Oxford University Press.Google Scholar
  37. Hilbert, D. (1899). Grundlagen der Geometrie. Leipzig.Google Scholar
  38. Kant, I. (1781/1787). Kritik der reinen Vernunft. Edited by R. Schmidt. Hamburg: F. Meiner, 1956.Google Scholar
  39. Kitcher, P. (1979). Frege’s epistemology. The Philosophical Review, 66, 235–262.CrossRefGoogle Scholar
  40. Klein, F. (1921). Gesammelte mathematische Abhandlungen. Vol. I: Liniengeometrie. Grundlegung der Geometrie. Zum Erlanger Programm. Edited by R. Fricke & A. Ostrowski, Berlin. Reprint Springer, Berlin, Heidelberg, New York 1973.Google Scholar
  41. Klein, F. (1926). Vorlesungen über die Mathematik im 19. Jahrhundert. Berlin: Teil I.Google Scholar
  42. Kratzsch, I. (1979). Liste der von Frege zwischen 1874 und 1918 an der Universität Jena angekündigten Lehrveranstaltungen. In “Begriffsschrift”. Jenaer FREGE-conference, 7–11 May 1979, pp. 534–546.Google Scholar
  43. Kvasz, L. (2011). Kant’s philosophy of geometry—On the road to a final assessment. Philosophia Mathematica, 19, 139–166.CrossRefGoogle Scholar
  44. Manders, K. (2008a). Diagram-based geometric practice. In P. Mancosu (Ed.), The philosophy of mathematical practice (pp. 65–79). Oxford: Oxford University Press.CrossRefGoogle Scholar
  45. Manders, K. (2008b). The Euclidean diagram. In P. Mancosu (Ed.), The philosophy of mathematical practice (pp. 80–133). Oxford: Oxford University Press.CrossRefGoogle Scholar
  46. Merrick, T. (2006). What Frege meant when he said: Kant is right about geometry. Philosophia Mathematica, 14, 344–375.CrossRefGoogle Scholar
  47. Newton, I. (1726). Philosophiae Naturalis Principia Mathematica. Edited by A. Koyré & I. B. Cohen (3rd ed.). Cambridge, MA: Harvard University Press.Google Scholar
  48. Pasch, M. (1882). Vorlesungen über neuere Geometrie. Leipzig: B. G. Teubner.Google Scholar
  49. Reichenbach, H. (1928). Philosophie der Raum-Zeit-Lehre, Berlin; Vol. 2 of H. Reichenbach, collected works in 9 volumes. Edited by A. Kamlah & M. Reichenbach, Braunschweig 1977. English translation: The philosophy of space & time, translated by M. Reichenbach and J. Freund, with introductory remarks by Rudolf Carnap, Dover Publications, New York 1957.Google Scholar
  50. Riemann, B. (1868). Über die Hypothesen, welche der Geometrie zugrunde liegen. Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 13, 133–152.Google Scholar
  51. Schirn, M. (Ed.). (1976). Studien zu Frege—Studies on Frege (Vol. I–III). Stuttgart-Bad Cannstatt: Frommann-Holzboog.Google Scholar
  52. Schirn, M. (1991). Kants Theorie der geometrischen Erkenntnis und die nichteuklidische Geometrie. Kantstudien, 82, 1–28.Google Scholar
  53. Schirn, M. (2010). Consistency, models, and soundness. Axiomathes, 20, 153–207.CrossRefGoogle Scholar
  54. Schirn, M. (2013). Frege’s approach to the foundations of analysis (1873–1903). History and Philosophy of Logic, 34, 266–292.CrossRefGoogle Scholar
  55. Schirn, M. (2014). Frege on quantities and real numbers in consideration of the theories of Cantor, Russell and others. In G. Link (Ed.), Formalism and beyond. On the nature of mathematical discourse (pp. 25–95). Boston & Berlin: Walter de Gruyter.Google Scholar
  56. Schirn, M. (2018a). Frege on the Foundations of Mathematics. Synthese Library: Studies in epistemology, logic, methodology, and philosophy of science, editor-in-chief: O. Bueno, New York, London: Springer (forthcoming).Google Scholar
  57. Schirn, M. (2018b). Funktion, Gegenstand, Bedeutung. Freges Philosophie und Logik im Kontext. Münster: Mentis (forthcoming).Google Scholar
  58. Schlimm, D. (2010). Pasch’s philosophy of mathematics. Review of Symbolic Logic, 3, 93–118.CrossRefGoogle Scholar
  59. Shabel, L. (2006). Kant’s philosophy of mathematics. In P. Guyer (Ed.), The Cambridge companion to Kant and modern philosophy. Cambridge: Cambridge University Press.Google Scholar
  60. Tappenden, J. (1995). Geometry and generality in Frege’s philosophy of arithmetic. Synthese, 102, 319–361.CrossRefGoogle Scholar
  61. Tappenden, J. (2000). Frege on axioms, indirect proof, and independence arguments in geometry: Did Frege reject independence arguments? Notre Dame Journal of Formal Logic, 41, 271–315.CrossRefGoogle Scholar
  62. Torretti, R. (1978). Philosophy of geometry. From Riemann to Poincaré. Dordrecht: D. Reidel.CrossRefGoogle Scholar
  63. von Helmholtz, H. (1968). Über Geometrie. Darmstadt: Wissenschaftliche Buchgesellschaft.Google Scholar
  64. von Staudt, G. K. C. (1847). Geometrie der Lage. Nürnberg: F. Korn.Google Scholar
  65. von Staudt, G. K. C. (1856-1860). Beiträge zur Geometrie der Lage. Nürnberg: Bauer & Raspe.Google Scholar
  66. Wilson, M. (1992). The Royal Road from Geometry, Noûs 26, 149–180; reprinted with a Postscript in W. Demopoulos (Ed.), Frege’s philosophy of mathematics. Harvard University Press, Cambridge, MA, 1995, 108–159.Google Scholar
  67. Wilson, M. (2011). Frege’s mathematical setting. In M. Potter & T. Ricketts (Eds.), The Cambridge Companion to Frege (pp. 379–412). Cambridge: Cambridge University Press.Google Scholar

Copyright information

© Springer Nature B.V. 2017

Authors and Affiliations

  1. 1.Munich Center for Mathematical PhilosophyLudwig-Maximilians-Universität MünchenMunichGermany

Personalised recommendations