Fragments of a History of the Concept of Ideal. Poncelet's and Chasles's Reflections on Generality in Geometry and Their Impact on Kummer's Work with Ideal Divisors
In this essay, I argue for the following theses. First, Kummer's concept of ''ideal prime factors of a complex number'' was inspired by Poncelet's introduction of ideal elements in geometry as well as by the reconceptualization that Michel Chasles put forward for them i...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2024 |
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Інститут математики НАН України
2024
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| Zitieren: | Fragments of a History of the Concept of Ideal. Poncelet's and Chasles's Reflections on Generality in Geometry and Their Impact on Kummer's Work with Ideal Divisors. Karine Chemla. SIGMA 20 (2024), 066, 37 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862683516789063680 |
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| author | Chemla, Karine |
| author_facet | Chemla, Karine |
| citation_txt | Fragments of a History of the Concept of Ideal. Poncelet's and Chasles's Reflections on Generality in Geometry and Their Impact on Kummer's Work with Ideal Divisors. Karine Chemla. SIGMA 20 (2024), 066, 37 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In this essay, I argue for the following theses. First, Kummer's concept of ''ideal prime factors of a complex number'' was inspired by Poncelet's introduction of ideal elements in geometry as well as by the reconceptualization that Michel Chasles put forward for them in 1837. In other words, the idea of ideal divisors in Kummer's ''theory of complex numbers'' derives from the introduction of ideal elements in the new geometry. This is where the term ''ideal'' comes from. Second, the introduction of ideal elements into geometry and the subsequent reconceptualization of what was in play with these elements were linked to philosophical reflections on generality that practitioners of geometry in France developed in the first half of the 19th century to devise a new approach to geometry, which would eventually become projective geometry. These philosophical reflections circulated as such and played a key part in the advance of other domains, including Kummer's major innovation in the context of number theory.
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| first_indexed | 2026-03-17T08:49:06Z |
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| id | nasplib_isofts_kiev_ua-123456789-212354 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-17T08:49:06Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
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| spelling | Chemla, Karine 2026-02-05T09:56:14Z 2024 Fragments of a History of the Concept of Ideal. Poncelet's and Chasles's Reflections on Generality in Geometry and Their Impact on Kummer's Work with Ideal Divisors. Karine Chemla. SIGMA 20 (2024), 066, 37 pages 1815-0659 2020 Mathematics Subject Classification: 01A55; 00A30 arXiv:2307.16234 https://nasplib.isofts.kiev.ua/handle/123456789/212354 https://doi.org/10.3842/SIGMA.2024.066 In this essay, I argue for the following theses. First, Kummer's concept of ''ideal prime factors of a complex number'' was inspired by Poncelet's introduction of ideal elements in geometry as well as by the reconceptualization that Michel Chasles put forward for them in 1837. In other words, the idea of ideal divisors in Kummer's ''theory of complex numbers'' derives from the introduction of ideal elements in the new geometry. This is where the term ''ideal'' comes from. Second, the introduction of ideal elements into geometry and the subsequent reconceptualization of what was in play with these elements were linked to philosophical reflections on generality that practitioners of geometry in France developed in the first half of the 19th century to devise a new approach to geometry, which would eventually become projective geometry. These philosophical reflections circulated as such and played a key part in the advance of other domains, including Kummer's major innovation in the context of number theory. It is a pleasure to dedicate this article to Jean-Pierre Bourguignon, without whom I might never have known that the mathematics section of CNRS intended to open hirings to the history of mathematics. Without this piece of information, I might never have continued research in this domain. Given this context, I find it appropriate to dedicate to him an article showing how the historical and philosophical reflections developed by mathematicians led to significant advancements in mathematics. I have been interested in the history of geometry in France in the first half of the 19th century since the 1980s, thanks to Serge Pahaut’s (1945–2023) invitation to work with him on duality. His passing away just as I was completing this article infuses great sadness with some of its pages. My interest in the notions of ideality was spurred by the ANR project “Ideals of proof” that Mic Detlefsen (1948–2019) developed at the University of Paris Diderot between 2007 and 2011. In this context, I gave a joint talk with Bruno Belhoste on Poncelet’s ideal elements (titled: “Poncelet’s ideal elements in geometry: between Carnot and Chasles”, March 5, 2009), which we intend to publish. The inspiration for the present article derives from my first encounter with Ernst Kummer’s writings on number theory, in the context of a talk given at our research group (REHSEIS at the time, SPHERE, now) by Jacqueline Boniface on the topic. As we instructed speakers to do at the time, she handed out pages of Kummer’s original text. I immediately recognized that these pages had been inspired by Michel Chasles’s “principle of contingent relationships”. This formed part of the results that we published in [11], which concluded the research project carried out between 2004 and 2016, about the epistemic and epistemological value of generality (about these expressions, I refer the reader to the introduction of the book). In the book [9], I analyzed Chasles’s understanding of generality in geometry and mentioned the impact of his reflections, notably those relating to the aforementioned principle, on Kummer’s work in number theory. In [4], Boniface described Kummer’s creation of ideal numbers, mentioning the analogy with geometry. The prologue to the book [10] took this case as an example of the circulation of scientists’ reflections about an epistemological value from one context to another context. The present article builds on all this previous research, and I am in particular grateful to Jacqueline Boniface for having introduced me to Kummer’s work in number theory. I am also immensely grateful to Harold Edwards, since, without his book [18], I might not have been able to enter into the maze of Kummer’s mathematical world. Last, but not least, since 2022, Bruno Belhoste and I have begun a joint project on Poncelet’s works. Our common work has helped me refine my understanding of Poncelet’s philosophical and geometrical reflections. Since I completed the first version of this article, I was fortunate to receive remarks from Christian Houzel, Max Lindh, Colin McLarty, Patrick Popescu-Pampu, Jean-Pierre Serre, as well as from three referees. I thank them all for their comments, which have enabled me to significantly improve the first version of the article. The contribution of Jeremy Gray and Oussama Hijazi to the final version of this article was of crucial importance. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Fragments of a History of the Concept of Ideal. Poncelet's and Chasles's Reflections on Generality in Geometry and Their Impact on Kummer's Work with Ideal Divisors Article published earlier |
| spellingShingle | Fragments of a History of the Concept of Ideal. Poncelet's and Chasles's Reflections on Generality in Geometry and Their Impact on Kummer's Work with Ideal Divisors Chemla, Karine |
| title | Fragments of a History of the Concept of Ideal. Poncelet's and Chasles's Reflections on Generality in Geometry and Their Impact on Kummer's Work with Ideal Divisors |
| title_full | Fragments of a History of the Concept of Ideal. Poncelet's and Chasles's Reflections on Generality in Geometry and Their Impact on Kummer's Work with Ideal Divisors |
| title_fullStr | Fragments of a History of the Concept of Ideal. Poncelet's and Chasles's Reflections on Generality in Geometry and Their Impact on Kummer's Work with Ideal Divisors |
| title_full_unstemmed | Fragments of a History of the Concept of Ideal. Poncelet's and Chasles's Reflections on Generality in Geometry and Their Impact on Kummer's Work with Ideal Divisors |
| title_short | Fragments of a History of the Concept of Ideal. Poncelet's and Chasles's Reflections on Generality in Geometry and Their Impact on Kummer's Work with Ideal Divisors |
| title_sort | fragments of a history of the concept of ideal. poncelet's and chasles's reflections on generality in geometry and their impact on kummer's work with ideal divisors |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212354 |
| work_keys_str_mv | AT chemlakarine fragmentsofahistoryoftheconceptofidealponceletsandchaslessreflectionsongeneralityingeometryandtheirimpactonkummersworkwithidealdivisors |