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

Frege’s philosophy of geometry

  • Matthias SchirnEmail author


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.


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 



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.


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© Springer Nature B.V. 2017

Authors and Affiliations

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

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