Editing Vladimir Arnold (section)

==Popular mathematical writings==
Arnold is well known for his lucid writing style, combining mathematical rigour with physical intuition, and an easy conversational style of teaching and education. His writings present a fresh, often [[geometric]] approach to traditional mathematical topics like [[ordinary differential equation]]s, and his many textbooks have proved influential in the development of new areas of mathematics. The standard criticism about Arnold's pedagogy is that his books "are beautiful treatments of their subjects that are appreciated by experts, but too many details are omitted for students to learn the mathematics required to prove the statements that he so effortlessly justifies." His defense was that his books are meant to teach the subject to "those who truly wish to understand it" (Chicone, 2007).<ref>Carmen Chicone (2007), Book review of "Ordinary Differential Equations", by Vladimir I. Arnold. Springer-Verlag, Berlin, 2006. ''SIAM Review'' '''49'''(2):335–336. ''(Chicone mentions the criticism but does not agree with it.)''</ref>

Arnold was an outspoken critic of the trend towards high levels of abstraction in mathematics during the middle of the last century. He had very strong opinions on how this approach—which was most popularly implemented by the [[Nicolas Bourbaki|Bourbaki]] school in France—initially had a negative impact on French [[mathematical education]], and then later on that of other countries as well.<ref>See [https://iopscience.iop.org/article/10.1070/RM1998v053n01ABEH000005/https://archive.today/20210331201831/https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html] (archived from [http://pauli.uni-muenster.de/~munsteg/arnold.html] {{Webarchive|url=https://web.archive.org/web/20170428233041/http://pauli.uni-muenster.de/~munsteg/arnold.html |date=28 April 2017 }}) and other essays in [http://www.pdmi.ras.ru/~arnsem/Arnold/].</ref><ref name="interview1">[http://www.ams.org/notices/199704/arnold.pdf An Interview with Vladimir Arnol'd], by S. H. Lui, ''[[AMS Notices]]'', 1991.</ref> He was very concerned about what he saw as the divorce of mathematics from the [[natural science]]s in the 20th century.<ref>{{cite journal | last1=Ezra | first1=Gregory S. | last2=Wiggins | first2=Stephen | title=Vladimir Igorevich Arnold | journal=[[Physics Today]] | volume=63 | issue=12 | date=1 December 2010 | issn=0031-9228 | doi=10.1063/1.3529010 | pages=74–76| bibcode=2010PhT....63l..74E }}</ref> Arnold was very interested in the [[history of mathematics]].<ref>[https://arxiv.org/abs/1007.0688 Oleg Karpenkov. "Vladimir Igorevich Arnold"]</ref> In an interview,<ref name="interview1" /> he said he had learned much of what he knew about mathematics through the study of [[Felix Klein]]'s book ''Development of Mathematics in the 19th Century'' —a book he often recommended to his students.<ref>[[Boris Khesin|B. Khesin]] and [[Sergei Tabachnikov|S. Tabachnikov]], Tribute to Vladimir Arnold, ''Notices of the AMS'', '''59''':3 (2012) 378–399.</ref> He studied the classics, most notably the works of [[Christiaan Huygens|Huygens]], [[Isaac Newton|Newton]] and [[Henri Poincaré|Poincaré]],<ref>{{citation|last1=Goryunov|first1=V.|last2=Zakalyukin|first2=V.|title=Vladimir I. Arnold|journal=[[Moscow Mathematical Journal]]|volume=11|issue=3|url=http://www.ams.org/distribution/mmj/vol11-3-2011/vladimir-arnold.html|year=2011}}.</ref> and many times he reported to have found in their works ideas that had not been explored yet.<ref>See for example: Arnold, V. I.; Vasilev, V. A. (1989), "Newton's Principia read 300 years later" and Arnold, V. I. (2006); "Forgotten and neglected theories of Poincaré".</ref>