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From Summetria to Symmetry: The Making of a Revolutionary Scientific Concept

From Summetria to Symmetry: The Making of a Revolutionary Scientific Concept is a 2008 book written by historians of science Giora Hon and Bernard R. Goldstein. The book documents the history of the various concepts represented by the word symmetry starting from Plato the ancient Greek philosopher and Vitruvius the Roman architect, and ending with the German philosopher Immanuel Kant, the Swiss mathematician Leonhard Euler, and the French mathematician Adrien-Marie Legendre at the end of the 18th century.

From Summetria to Symmetry: The Making of a Revolutionary Scientific Concept
Cover image of book
Cover page of the first edition
AuthorGiora Hon and Bernard R. Goldstein
SeriesArchimedes: New Studies in the History of Science and Technology
Release number
20
Subject
PublisherSpringer Science+Business Media
Publication date
2008
Publication placeDordrecht
Media typePrint
Pages335
ISBN978-1-4020-8447-8

The book argues that the key figure in revolutionizing the concept of symmetry was the mathematician, Adrien-Marie Legendre. The book makes clear the changing meaning of symmetry through the centuries and its historical analysis has been much cited, but its initial reception was mixed, particularly its claim for the importance of Legendre's role in defining the modern meaning of symmetry.

Structure and topics

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The book is an academic work in the history of science and is based on the analysis of primary sources. It begins with a 65-page introduction covering the historical, philosophical and historiographical background to the concept of symmetry.[1] Following the introduction the book is split into two parts. Part I is entitled "Tradition: Ancient Perspectives and Their Survival in the Early Modern Era" and covers:

Part II is entitled "The Path to Revolution: Symmetry as a Modern Scientific Concept" and covers:

  • The treatment of symmetry in natural history including progress in botany, crystallography and zoology 1738–1815
  • The contributions of Leonhard Euler and Immanuel Kant
  • The contribution of Adrien-Marie Legendre in 1794 (termed "revolutionary" by the authors)
  • New applications of symmetry in mathematics and physics 1788–1815

The authors describe the development of the use of the Greek term symmetria which expressed, over time, at least three different concepts: proportionality or commensurability in ancient sources, equivalence of sides in the seventeenth and eighteenth centuries, and the modern concept of symmetry, acquired at the end of the 18th century.[2]: 305  In the introduction Hon and Goldstein criticised other historians and philosophers of science, specifically John J. Roche,[3] Klaus Mainzer,[4] and Katherine Brading and Elena Castellani,[5] for taking an anachronistic approach to the history of symmetry.

Hon and Goldstein followed up the book with further research papers bolstering their arguments and responding to their critics.[6][7]

Audience

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The book is aimed at historians and philosophers of science, mathematics, architecture, and aesthetics.

Reception

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Branko Mitrovic in a review in the Journal of the Society of Architectural Historians accepted the authors' argument that the development of the modern concept of symmetry can be attributed to Legendre, but he was sceptical of the evidence for the authors' view that this version of the concept (the equivalence between the left and right sides of an object) had not been articulated by the ancient philosophers or architects.[8]

Hardy Grant in an extensive review in Historia Mathematica described the book as "flawed but fascinating". While commending their command of the sources and their "superb" scholarship, Grant criticises as unrealistic the authors' requirement for explicit textual references to the word symmetry to justify thinking that only then could the concept of symmetry have been known to the researchers of previous times. Grant also criticised in detail their identification of Legendre as the first to define symmetry in modern scientific terms. In conclusion, he stated that the book has "substantial virtues on the one hand, grave weaknesses on the other".[9]

Joseph Dauben in a review for Aestimatio stated that "Undoubtedly the one aspect that will cause the greatest concern amongst historians of mathematics is the extent to which (or whether) the concept of symmetry can be considered 'revolutionary';" however Dauben did not express his personal view on the correctness of the main thesis of the book.[10] Adhemar Bultheel in a review for the newsletter of the Belgian Mathematical Society stated that the authors make a "somewhat controversial and surprising claim: The modern (mathematical) notion of symmetry was not known to men until 1794", however Bultheel did not state his personal view of this proposition in the review.[11]

György Darvas in a review for Centaurus took a more negative than positive view of the book. While acknowledging its "value as a handbook for finding written historical sources on the usage of the term symmetry" Darvas, who authored a monograph on symmetry in 2007,[12] fundamentally disagreed with the authors' conclusions regarding the revolutionary nature of Legendre's use of the word symmetry.[13]

Katherine Brading in a review for Metascience stated "I disagree with the framework of both the historical and the critical narrative offered by Hon and Goldstein" and gave detailed reasons for where she took issue with the authors' arguments, including her own contributions to the subject.[14]

Michael Otte in a negative review for MathSciNet stated "The authors try everything to provide some plausibility to this enthusiastic exclamation [Legendre's revolutionary use of symmetry] sometimes in doubtful and wrongheaded ways". Otte specifically took issue with the authors' treatment of Kant's contribution.[15]

Influence

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Later writers, whilst not necessarily accepting Hon and Goldstein's view that Legendre was the first to use the term symmetry in the modern sense,[16]: 711–712 [17]: 240  nevertheless accepted Hon and Goldstein's thesis that the word symmetry had the meaning of commensurability in ancient times[18]: 403  and did not acquire its modern meaning until the latter part of the 18th century.[19][20]: 93 [21]: 60 

Čelkytė considers Hon and Goldstein's book to be a monograph on functional aesthetics.[22]: 104  O'Loughlin and McCallum cite Hon and Goldstein to support their statement that symmetry is relatively unimportant as an aesthetic criterion.[23]: 332 

The book has become a standard reference for how the word symmetry was used before the 19th century.[24]

References

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  1. ↑ Hon, Giora; Goldstein, Bernard R. (2008). From Summetria to Symmetry: The Making of a Revolutionary Scientific Concept. Archimedes New Studies in the History and Philosophy of Science and Technology. Vol. 20. Dordrecht: Springer. doi:10.1007/978-1-4020-8448-5. ISBN 978-1-4020-8447-8. Retrieved 27 April 2026.
  2. ↑ Mitrović, Branko (October 2011). "Attribution of concepts and problems with anachronism". History and Theory. 50 (3): 303–327. doi:10.1111/j.1468-2303.2011.00587.x. JSTOR 41300097.
  3. ↑ Roche, John J. (1987). "A critical study of symmetry in physics from Galileo to Newton". In Doncel, Manuel G.; Hermann, Armin; Michel, Louis; Pais, Abraham (eds.). Symmetries in Physics (1600–1980). Bellaterra: Universitat Autònoma de Barcelona. pp. 1–30. ISBN 9788474881486. Retrieved 29 April 2026.
  4. ↑ Mainzer, Klaus (1996). Symmetries of nature: a handbook for philosophy of nature and science. Berlin New York: W. de Gruyter. ISBN 3-11-012990-6.
  5. ↑ Brading, Katherine; Castellani, Elena, eds. (2003). Symmetries in physics: philosophical reflections. Cambridge, U.K New York: Cambridge University Press. ISBN 978-0-521-52889-4.
  6. ↑ Hon, Giora; Goldstein, Bernard R. (November 2009). "In Pursuit of Conceptual Change: the Case of Legendre and Symmetry". Centaurus. 51 (4): 288–293. doi:10.1111/j.1600-0498.2009.00152.x.
  7. ↑ Hon, Giora; Goldstein, Bernard R. (2016). "The Double-Face of Symmetry: A Conceptual History". Symmetrie und Asymmetrie in Wissenschaft und Kunst: Vorträge anlässlich der Jahresversammlung am 18. und 19. September 2015 in Halle (Saale) (PDF). Halle (Saale): Deutsche Akademie der Naturforscher Leopoldina - Nationale Akademie der Wissenschaften. pp. 45–74. ISBN 978-3-8047-3610-8. Retrieved 1 May 2026.
  8. ↑ Mitrović, Branko (1 December 2009). "Review: From Summetria to Symmetry: The Making of a Revolutionary Scientific Concept, by Giora Hon and Bernard R. Goldstein". Journal of the Society of Architectural Historians. 68 (4): 576–577. doi:10.1525/jsah.2009.68.4.576.
  9. ↑ Grant, Hardy (May 2009). "What's in a word? Symmetry through the centuries". Historia Mathematica. 36 (2): 171–177. doi:10.1016/j.hm.2008.10.002.
  10. ↑ Dauben, Joseph W. (31 December 2011). "From Summetria to Symmetry by G. Hon and B. R. Goldstein". Aestimatio: Critical Reviews in the History of Science (Volume 6) (review): 188–190. doi:10.31826/9781463232436-025. Retrieved 29 April 2026.
  11. ↑ Bultheel, Adhemar (January 2009). "From Summetria to Symmetry: The making of a revolutionary scientific concept Giora Hon and Bernard R. Goldstein. Archimedes: new studies in the history of science and technology Springer, 2008 (xvi+335) hard cover, ISBN 978-1-4020-8447-8" (PDF). Newsletter of the Belgian Mathematical Society (review) (71): 6–7. Retrieved 1 May 2026.
  12. ↑ Darvas, György (2007). Symmetry: Cultural-historical and Ontological Aspects of Science-Arts Relations; the Natural and Man-made World in an Interdisciplinary Approach. Basel: Birkhäuser Verlag AG. ISBN 978-3-7643-7554-6. Retrieved 2 May 2026.
  13. ↑ Darvas, György (May 2010). "From Summetria to Symmetry: The Making of a Revolutionary Scientific Concept - by Giora Hon and Bernard R. Goldstein". Centaurus (review). 52 (2): 160–162. doi:10.1111/j.1600-0498.2010.00171.x.
  14. ↑ Brading, Katherine (July 2010). "Mathematical and aesthetic aspects of symmetry: G. Hon, B. R. Goldstein: From summetria to symmetry: the making of a revolutionary scientific concept. Springer, Dordrecht, 2008, xvi + 335 pp, £135.00 HB". Metascience (review). 19 (2): 277–280. doi:10.1007/s11016-010-9363-x.
  15. ↑ Otte, Michael (2012). "From summetria to symmetry: the making of a revolutionary scientific concept". MathSciNet (review). American Mathematical Society. MR 2724356. Retrieved 2 May 2026.
  16. ↑ Bhatta, Varun S.; Sarukkai, Sundar (10 March 2020). "Duality in science". Current Science. 118 (5): 705–713. doi:10.18520/cs/v118/i5/705-713. JSTOR 27226350. For him [Adrien-Marie Legendre], two polyhedra are equal only if they are equal in magnitude and are congruent (i.e. superposable). In contrast, the figures which are not superposable, Legendre argued, are still equal, but the principle at work is equality by symmetry.
  17. ↑ Drago, Antonino (2023). "The other side of the history of symmetries. Their link with intuitionist logic". Proceedings of the SISFA 42nd Annual Conference. Perugia, 26-29 September 2022: 239–245. doi:10.12871/978883339843332. Hon and Goldstein 2008 wrote a book attributing to a book by Legendre book [sic] of 1794 (concerning geometrical symmetries) the birth of symmetries in modern science. The authors ignored Haüy and L. Carnot.
  18. ↑ Lehoux, Daryn (September 2019). "Why does Aristotle think bees are divine? Proportion, triplicity and order in the natural world". The British Journal for the History of Science. 52 (3): 383–403. doi:10.1017/S0007087419000165. JSTOR 10.2307/26851468.
  19. ↑ Čelkytė, Aistė (2020). "The Stoic Definition of Beauty as Summetria". The Stoic theory of beauty. Edinburgh: Edinburgh University Press. p. 144. doi:10.3366/edinburgh/9781474461610.003.0006. ISBN 9781474461641. Archived from the original on 30 May 2024. Retrieved 27 April 2026. As [the authors show] the contemporary understanding of symmetry as either bilateralism or, in mathematics and logic, a certain stability in mathematical properties that undergo changes originated in the work of the eighteenth-century French mathematician Adrien-Marie Legendre. In antiquity, summetria referred to, broadly speaking, a property of being well proportioned.
  20. ↑ Gent, Lucy (2014). "Elizabethan Architecture: a View from Rhetoric". Architectural History. 57: 73–108. doi:10.1017/s0066622x00001386. JSTOR 43489746. Recent studies on how the modern term symmetry evolved show that only in the eighteenth century did the word acquire today's sense.
  21. ↑ Borrelli, Arianna (April 2015). "The making of an intrinsic property: "Symmetry heuristics" in early particle physics". Studies in History and Philosophy of Science Part A. 50: 59–70. doi:10.1016/j.shpsa.2014.09.009.
  22. ↑ Čelkytė, Aistė (May 2017). "The Stoic definition of beauty as Summetria". The Classical Quarterly. 67 (1): 88–105. doi:10.1017/s0009838817000386. JSTOR 26292407.
  23. ↑ O’Loughlin, Ian; McCallum, Kate (April 2019). "The Aesthetics of Theory Selection and the Logics of Art". Philosophy of Science. 86 (2): 325–343. doi:10.1086/701953. [Hon and Goldstein] have argued that the concept of symmetry as it is used in modern mathematics and physics is an invention of the modern age and has little to do with the visual sense of symmetry we might employ in evaluating a painting. If [they] are right, then this is already one reason for skepticism about symmetry's place as a primary candidate for genuinely aesthetic criteria in science
  24. ↑
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