Abstract
Tatiana Alexeyevna Afanassjewa (from 1904 Tatiana Ehrenfest-Afanassjewa) was a mathematician, a physicist, and a teacher. All three of these vocations come together in her philosophy of geometry, which bases a novel approach to the teaching of geometry on her understanding of the proper roles of intuition and logical reasoning in geometry, grounded in our experience of concrete objects occupying physical space. Having been a student at Göttingen during the time of its greatest flowering as a centre of mathematical research, and a member of the physics community during the revolutionary period from Einstein’s annus mirabilis of 1905, she was close to the centre of some of the most exciting developments in science of her time. Since early on she was also deeply invested in teaching, and in developing new and better ways to communicate her subject to her students. Afanassjewa’s reflections on the teaching of geometry are thus those of a mathematician and a theoretical physicist who was passionate about scientific discussion and teaching: her ideas originate in her own experience as a student, researcher, and teacher, and in the debates with her scientific contemporaries—debates in which she played an active and important role.
The writing of Sects. 2.1–2.3 of this paper was supported by funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 709265. The writing of Sects. 2.3–2.6 was supported by the German Research Foundation (Gefördert durch die Deutsche Forschungsgemeinschaft (DFG)—Projektnummer 390218268). Their support is gratefully acknowledged. My thanks to Benedict Eastaugh and two anonymous referees for comments on a previous draft of this paper, and to Paolo Bussotti for helpful discussions about the history of geometry. I also thank the ETH Zurich University Archives for making available to me the correspondence between Tatiana Afanassjewa and Paul Bernays. The visit to the ETH Zurich University Archives was funded by LMUexcellent within the framework of the German Excellence Strategy.
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Notes
- 1.
Unless otherwise noted, all page numbers given for the manifesto refer to the translation in the present volume, and all page numbers given for the Übungensammlung refer to the translation by Hoechsmann (2011).
- 2.
In this context, it would also be interesting to analyse the similarities and the differences between Afanassjewa’s view and that of Ferdinand Gonseth. It is plausible to think that she was acquainted with Gonseth’s work, as he is mentioned in the correspondence between Afanassjewa and Bernays in the early 1950s. However, since the first philosophical work of Gonseth dates to 1926, 2 years after the publication of Afanassjewa’s manifesto, such an analysis is beyond the scope of this paper.
- 3.
It is surprising that there is no correspondence between Klein and Afanassjewa, since Klein in particular was an avid correspondent (for an idea of the volume of Klein’s correspondence, see e.g. Schlimm 2013, p. 184). Neither the Göttingen University Library nor the Rijksmuseum Boerhaave, which holds the Ehrenfest–Afanassjewa archive, hold letters between her and other Göttingen mathematicians on topics related to her philosophy of geometry (though there is some correspondence with Klein about the encyclopaedia entry on the foundations of statistical mechanics which she authored together with her husband Paul Ehrenfest). The Kalliope Verbundkatalog Nachlässe only lists correspondence between Afanassjewa and the physicist Gustav Herglotz. There is, however, extant correspondence between Afanassjewa and Paul Bernays from the late 1940s and early 1950s, mostly about theoretical physics, which can be accessed in the Bernays Nachlass at the ETH Zurich University Archives. For a detailed list of courses of Klein’s that Afanassjewa attended while in Göttingen, see the recently published Tobies (2020).
- 4.
More recent research in formalised geometry suggests that Hilbert’s axiomatic treatment in the Grundlagen still contained gaps that had to be filled by diagrammatic reasoning and geometrical intuition: see Meikle and Fleuriot (2003).
- 5.
It should be noted that on Afanassjewa’s view, not only is knowledge of geometrical theorems not necessary for knowledge of spatial relations, it is also not sufficient. Her objections to the teaching of geometry by means of an axiomatic presentation of Euclidean geometry, and the relation of such a course to the development of spatial imagination, will be discussed below.
- 6.
- 7.
A sample exercise of this kind is the following: “What form does a surface of water have in a cylindrical glass held at different angles?” (Ehrenfest-Afanassjewa 1931, p. 32). Such an activity is clearly aimed at developing projective imagination. Even though Afanassjewa does not discuss it as explicitly as Klein, training this aspect of spatial imagination is nevertheless important in her view.
- 8.
Afanassjewa was fascinated by the discovery of non-Euclidean geometries, their respective axiom systems and associated models, which she saw as a possible subject of study in high school.
- 9.
- 10.
- 11.
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Antonutti Marfori, M. (2021). Intuition, Understanding, and Proof: Tatiana Afanassjewa on Teaching and Learning Geometry. In: Uffink, J., Valente, G., Werndl, C., Zuchowski, L. (eds) The Legacy of Tatjana Afanassjewa. Women in the History of Philosophy and Sciences, vol 7. Springer, Cham. https://doi.org/10.1007/978-3-030-47971-8_2
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