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Frege’s philosophy of geometry. (English) Zbl 1474.03044

Summary: 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 faculty of 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 independent 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.

MSC:

03A05 Philosophical and critical aspects of logic and foundations
03B30 Foundations of classical theories (including reverse mathematics)
00A30 Philosophy of mathematics
51M04 Elementary problems in Euclidean geometries

Citations:

Zbl 0784.01047
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[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. · Zbl 0632.06022 · doi:10.1112/blms/19.6.513
[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.
[3] Blanchette, P. (1996). Frege and Hilbert on consistency. Journal of Philosophy, 93, 317-336. · doi:10.2307/2941124
[4] Blanchette, P. (2007). Frege on consistency and conceptual analysis. Philosophia Mathematica, 15, 321-346. · Zbl 1126.03007 · doi:10.1093/philmat/nkm028
[5] Blanchette, P. (2012). Frege’s conception of logic. New York: Oxford University Press. · Zbl 1251.03005 · doi:10.1093/acprof:oso/9780199891610.001.0001
[6] Blanchette, P.; Link, G. (ed.), Frege on formality and the 1906 independence test, 97-118 (2014), Boston & Berlin · Zbl 1329.03002
[7] Blanchette, P. (2016). The breadth of the paradox. Philosophia Mathematica, 24, 30-49. · Zbl 1356.03009 · doi:10.1093/philmat/nkv038
[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).
[9] Bonola, R. (1955). Non-Euclidean geometry. A critical and historical study of its development; english translation of H. S. Carslaw, New York. · JFM 43.0557.02
[10] Burge, T.; Boghossian, P. (ed.); Peacocke, C. (ed.), Frege on apriority, 11-42 (2000), Oxford · doi:10.1093/0199241279.003.0002
[11] Cayley, A. (1889-1897). The collected mathematical papers of Arthur Cayley (Vol. 13). Cambridge: Cambrige University Press.
[12] Cohen, H. (1871). Kants Theorie der Erfahrung. Berlin: Ferd. Dümmlers Verlagsbuchhandlung.
[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. · JFM 15.0021.05
[14] Dummett, M. (1976). Frege on the consistency of mathematical theories, in Schirn 1976, Vol. I, 229-242.
[15] Dummett, M. (1981). The interpretation of Frege’s philosophy. London: Duckworth.
[16] Dummett, M. (1982). Frege and Kant on geometry. Inquiry, 25, 233-254. · doi:10.1080/00201748208601964
[17] Frege, G. (1873). Über eine geometrische Darstellung der imaginären Gebilde in der Ebene, in Frege 1967, 1-49.
[18] Frege, G. (1874). Rechnungsmethoden, die sich auf eine Erweiterung des Größenbegriffs gründen, in Frege 1967, 50-84.
[19] Frege, G. (1879). Begriffsschrift. Eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle a. S.: L. Nebert.
[20] Frege, G. (1884). Die Grundlagen der Arithmetik. Eine logisch mathematische Untersuchung über den Begriff der Zahl. Breslau: W. Koebner. · Zbl 0654.03005
[21] Frege, G. (1885). Über formale Theorien der Arithmetik, in Frege 1967, 103-111.
[22] Frege, G. (1893). Grundgesetze der Arithmetik. Begriffsschriftlich abgeleitet (Vol. I). Jena: H. Pohle. · JFM 25.0101.02
[23] Frege, G. (1899). Letter to Hilbert of 27th December, 1899, in Frege 1976, 60-64.
[24] Frege, G. (1899-1906). Über Euklidische Geometrie, in Frege 1969, 182-184.
[25] Frege, G. (1903). Grundgesetze der Arithmetik. Begriffsschriftlich abgeleitet (Vol. II). Jena: H. Pohle. · JFM 34.0071.05
[26] Frege, G. (1914). Logik in der Mathematik, in Frege 1969, 219-270.
[27] Frege, G. (1924/1925). Erkenntnisquellen der Mathematik und der mathematischen Naturwissenschaften, in Frege 1969, 286-294.
[28] Frege, G. (1964). Begriffsschrift und andere Aufsätze. Edited by I. Angelelli. Darmstadt & Hildesheim: Wissenschaftliche Buchgesellschaft Darmstadt.
[29] Frege, G. (1967). Kleine Schriften. Edited by I. Angelelli, Hildesheim: G. Olms. · Zbl 0193.28203
[30] Frege, G. (1969). Nachgelassene Schriften. Edited by H. Hermes, F. Kambartel & F. Kaulbach. Hamburg: F. Meiner. · Zbl 0185.00503
[31] Frege, G. (1976). Wissenschaftlicher Briefwechsel. Edited by G. Gabriel, H. Hermes, F. Kambartel, C. Thiel & A. Veraart. Hamburg: F. Meiner. · Zbl 0341.01019
[32] Friedman, M. (1992). Kant’s theory of geometry. In C. J. Posy (Ed.), Kant’s philosophy of mathematics. Dordrecht: Kluwer.
[33] Gauss, C. F. (1900). Werke (Vol. VIII). Edited by Königliche Gesellschaft der Wissenschaften in Göttingen, Leipzig. · JFM 31.0012.02
[34] Giaquinto, M. (2007). Visual thinking in mathematics. Oxford: Oxford University Press. · Zbl 1175.00023 · doi:10.1093/acprof:oso/9780199285945.001.0001
[35] Giaquinto, M. (2011). Crossing curves: A limit to the use of diagrams in proofs. Philosophia Mathematica, 19, 281-307. · Zbl 1272.03019 · doi:10.1093/philmat/nkr023
[36] Heck, R. G. (2011). Frege’s theorem. Oxford: Oxford University Press. · Zbl 1245.00015
[37] Hilbert, D. (1899). Grundlagen der Geometrie. Leipzig. · JFM 30.0424.01
[38] Kant, I. (1781/1787). Kritik der reinen Vernunft. Edited by R. Schmidt. Hamburg: F. Meiner, 1956.
[39] Kitcher, P. (1979). Frege’s epistemology. The Philosophical Review, 66, 235-262. · doi:10.2307/2184508
[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. · JFM 48.0012.02
[41] Klein, F. (1926). Vorlesungen über die Mathematik im 19. Jahrhundert. Berlin: Teil I. · JFM 52.0022.05
[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.
[43] Kvasz, L. (2011). Kant’s philosophy of geometry—On the road to a final assessment. Philosophia Mathematica, 19, 139-166. · Zbl 1259.00007 · doi:10.1093/philmat/nkr007
[44] Manders, K.; Mancosu, P. (ed.), Diagram-based geometric practice, 65-79 (2008), Oxford · doi:10.1093/acprof:oso/9780199296453.003.0004
[45] Manders, K.; Mancosu, P. (ed.), The Euclidean diagram, 80-133 (2008), Oxford · doi:10.1093/acprof:oso/9780199296453.003.0005
[46] Merrick, T. (2006). What Frege meant when he said: Kant is right about geometry. Philosophia Mathematica, 14, 344-375. · Zbl 1110.03001 · doi:10.1093/philmat/nkj013
[47] Newton, I. (1726). Philosophiae Naturalis Principia Mathematica. Edited by A. Koyré & I. B. Cohen (3rd ed.). Cambridge, MA: Harvard University Press. · Zbl 0050.00201
[48] Pasch, M. (1882). Vorlesungen über neuere Geometrie. Leipzig: B. G. Teubner. · JFM 14.0498.01
[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.
[50] Riemann, B. (1868). Über die Hypothesen, welche der Geometrie zugrunde liegen. Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 13, 133-152. · JFM 01.0022.02
[51] Schirn, M. (Ed.). (1976). Studien zu Frege—Studies on Frege (Vol. I-III). Stuttgart-Bad Cannstatt: Frommann-Holzboog. · Zbl 0352.02003
[52] Schirn, M. (1991). Kants Theorie der geometrischen Erkenntnis und die nichteuklidische Geometrie. Kantstudien, 82, 1-28.
[53] Schirn, M. (2010). Consistency, models, and soundness. Axiomathes, 20, 153-207. · Zbl 1230.03034 · doi:10.1007/s10516-010-9110-3
[54] Schirn, M. (2013). Frege’s approach to the foundations of analysis (1873-1903). History and Philosophy of Logic, 34, 266-292. · Zbl 1345.03007 · doi:10.1080/01445340.2013.806398
[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. · Zbl 1329.03003
[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).
[57] Schirn, M. (2018b). Funktion, Gegenstand, Bedeutung. Freges Philosophie und Logik im Kontext. Münster: Mentis (forthcoming).
[58] Schlimm, D. (2010). Pasch’s philosophy of mathematics. Review of Symbolic Logic, 3, 93-118. · Zbl 1193.03002 · doi:10.1017/S1755020309990311
[59] Shabel, L.; Guyer, P. (ed.), Kant’s philosophy of mathematics (2006), Cambridge
[60] Tappenden, J. (1995). Geometry and generality in Frege’s philosophy of arithmetic. Synthese, 102, 319-361. · Zbl 1059.03512 · doi:10.1007/BF01064120
[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. · Zbl 1009.03003 · doi:10.1305/ndjfl/1038336845
[62] Torretti, R. (1978). Philosophy of geometry. From Riemann to Poincaré. Dordrecht: D. Reidel. · Zbl 0415.01007 · doi:10.1007/978-94-009-9909-1
[63] von Helmholtz, H. (1968). Über Geometrie. Darmstadt: Wissenschaftliche Buchgesellschaft.
[64] von Staudt, G. K. C. (1847). Geometrie der Lage. Nürnberg: F. Korn.
[65] von Staudt, G. K. C. (1856-1860). Beiträge zur Geometrie der Lage. Nürnberg: Bauer & Raspe.
[66] Wilson, M. (1992). The Royal Road from Geometry, Noûs26, 149-180; reprinted with a Postscript in W. Demopoulos (Ed.), Frege’s philosophy of mathematics. Harvard University Press, Cambridge, MA, 1995, 108-159. · Zbl 0900.00009
[67] Wilson, M.; Potter, M. (ed.); Ricketts, T. (ed.), Frege’s mathematical setting, 379-412 (2011), Cambridge
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