{
"type": "Article",
"authors": [
{
"type": "Person",
"familyNames": [
"Haffner"
],
"givenNames": [
"Emmylou"
]
}
],
"description": "Bernhard Riemann’s collected works were published for the first time in 1876 by\nRichard Dedekind and Heinrich Weber. The editors’ correspondence and the\navailable archive tell us that the process of editing Riemann’s collected works\nwas a hands-on process, which is itself of historical and mathematical\nsignificance. In this paper, we show how the editors shaped the published\ntexts, and how this can influence our reading of them.",
"identifiers": [],
"references": [
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}
],
"title": "The edition of Bernhard Riemann’s collected works: Then and now",
"meta": {},
"content": [
{
"type": "Heading",
"id": "Sx1",
"depth": 1,
"content": [
"A complex history and a wealth of archive"
]
},
{
"type": "Paragraph",
"id": "Sx1.p1",
"content": [
"In 1876 were published Bernhard Riemann’s (1826–1866) ",
{
"type": "Emphasis",
"content": [
"Gesammelte\nmathematische Werke und wissenschaftlicher Nachlass"
]
},
" (Collected mathematical\nworks and scientific archive). These collected works were edited by Heinrich\nWeber (1842–1913) and Richard Dedekind (1831–1916) and published by B. G.\nTeubner.",
{
"type": "Note",
"id": "idm12",
"noteType": "Footnote",
"content": [
{
"type": "Paragraph",
"id": "footnote1",
"content": [
"At this time, a considerable number of projects of publishing\ncollected works were launched in France, Germany, Italy, the\nUnited Kingdom … The publisher B. G. Teubner, created in 1811 in\nLeipzig, which specialised in scientific editions (broadly construed, i.e.,\nphilology, history, mathematics, physics, etc.), was one of the leading\npublishers for this type of book in Germany. – Steven W. Rockey from Cornell\nUniversity published a very complete list of collected works in mathematics:\n",
{
"type": "Link",
"target": "https://mathematics.library.cornell.edu/about-collected-works/",
"content": [
"mathematics.library.cornell.edu/about-collected-works/"
]
},
"."
]
}
]
}
]
},
{
"type": "Paragraph",
"id": "Sx1.p2",
"content": [
"Riemann and Dedekind met while they were Gauss’ students in Göttingen. They\ndefended their doctoral dissertation within a year of each other (Riemann in\n1851 and Dedekind in 1852), and their respective ",
{
"type": "Emphasis",
"content": [
"Habilitation"
]
},
" with only\na few days difference in 1854. Following this, they both worked as\n",
{
"type": "Emphasis",
"content": [
"Privatdozenten"
]
},
" in Göttingen, during which time Dedekind followed\nRiemann’s classes. In 1858 Dedekind was offered a position in Zürich and\nRiemann a post in Göttingen, and they remained friends until Riemann’s untimely\ndeath in 1866.\n"
]
},
{
"type": "Paragraph",
"id": "Sx1.p3",
"content": [
"It was Riemann’s wish that Dedekind would be the editor of his collected works\nand in charge of his scientific archive after his death. Struggling with this\ndifficult editorial enterprise, in early 1872, Dedekind accepted to work with\nAlfred Clebsch (1833–1872), who had taken Riemann’s chair in Göttingen. Seven\nof the most complete of Riemann’s unpublished works were first published\nposthumously in various mathematical\njournals.",
{
"type": "Note",
"id": "idm25",
"noteType": "Footnote",
"content": [
{
"type": "Paragraph",
"id": "footnote2",
"content": [
"[",
{
"type": "Cite",
"target": "bib-bib22",
"content": [
"22"
]
},
"] was edited by Karl Hattendorff\n(1834–1882), [",
{
"type": "Cite",
"target": "bib-bib21",
"content": [
"21"
]
},
"] by Ernst Schering (1824–1889) and Friedrich\nHenle (1809–1885), the other texts presumably by Dedekind. – A note on the\ndates of the publications: when it is possible to date Riemann’s texts, these\nare the given dates; when it is not, the dates are that of the first\npublication."
]
}
]
},
" Clebsch did not wish to publish more of Riemann’s manuscripts as\nhe felt the edition, as he wanted it to be, was nearing completion (according\nto his letters to Dedekind, published in [",
{
"type": "Cite",
"target": "bib-bib8",
"content": [
"8"
]
},
"], and to Dedekind’s\nfirst letter to Weber in [",
{
"type": "Cite",
"target": "bib-bib32",
"content": [
"32"
]
},
"]). Clebsch’s sudden death in 1872\nput the edition in some difficulty. Dedekind’s teaching duties kept him from\nhandling the project by himself. Eventually, upon meeting Heinrich Weber in\nZürich in 1873,",
{
"type": "Note",
"id": "idm39",
"noteType": "Footnote",
"content": [
{
"type": "Paragraph",
"id": "footnote3",
"content": [
"Maybe a less famous name than Riemann, Clebsch and\nDedekind, Heinrich Weber was a prominent mathematician throughout his career.\nHe studied in Heidelberg, Leipzig and Königsberg. He taught in Heidelberg,\nZürich, Königsberg (where he taught number theory to Hilbert and Minkowski),\nBerlin, Marburg, Göttingen, and Strasbourg. He worked extensively on complex\nfunction theory, number theory, and algebra. Among several important\ncontributions to the latter, his ",
{
"type": "Emphasis",
"content": [
"Lehrbuch der Algebra"
]
},
" was to be the\nmain reference for teaching algebra in the German speaking world until the\npublication of Van der Waerden’s ",
{
"type": "Emphasis",
"content": [
"Moderne Algebra"
]
},
" in 1930. He also made\ncontributions to mathematical physics, and published ",
{
"type": "Emphasis",
"content": [
"Die partiellen\nDifferentialgleichungen der mathematischen Physik nach Riemann’s\nVorlesungen"
]
},
", which was, for a long time, the only reference for Riemann’s\nmathematical physics. Weber was also actively involved in the mathematical\ncommunity, for example he was a member of the editorial committee of the\n",
{
"type": "Emphasis",
"content": [
"Mathematische Annalen"
]
},
" and a founding member of the ",
{
"type": "Emphasis",
"content": [
"Deutsche\nMathematiker-Vereinigung"
]
},
"."
]
}
]
},
" Dedekind offered him the responsibility of the\nedition, which he accepted. At this stage, Dedekind wished to retreat from the\nproject, but eventually became more involved in the edition of some of the\nmanuscripts. Both Weber and Dedekind wished to publish more of Riemann’s\nunpublished archive, and it took them two additional years to complete the\nedition, during which time they also had help from Hermann Schwarz (1843–1921)\nin working on [",
{
"type": "Cite",
"target": "bib-bib20",
"content": [
"20"
]
},
"]."
]
},
{
"type": "Paragraph",
"id": "Sx1.p4",
"content": [
"The final product of this ten-year editorial endeavour, Riemann’s\n",
{
"type": "Emphasis",
"content": [
"Gesammelte mathematische Werke und wissenschaftlicher Nachlass"
]
},
", is one\nvolume divided into three parts and two appendices: the first part contains the\n11 papers published by Riemann in his lifetime; the second part contains the\n7 papers published posthumously in journals as mentioned above; and the third\npart contains 12 unpublished texts from Riemann’s archive. The two appendices\nare a selection of Riemann’s philosophical writings, and a biography written by\nDedekind on the basis of letters from Riemann’s widow, Elise Riemann."
]
},
{
"type": "Paragraph",
"id": "Sx1.p5",
"content": [
"Riemann’s collected works were republished in 1892, by Weber. In the preface,\nhe explained that Riemann’s texts were still very relevant in 1892. Two\nimportant changes in the edition should be mentioned. Firstly, the text\n",
{
"type": "Emphasis",
"content": [
"Verbreitung der Wärme im Ellipsoid"
]
},
" (Diffusion of heat in an ellipsoid)\n[",
{
"type": "Cite",
"target": "bib-bib28",
"content": [
"28"
]
},
"], which was briefly discussed and eventually excluded\nfrom the 1876 edition, was published. There are no indications or\ncorrespondence that indicate why it was initially excluded (in fact, the\nletters suggest that it was going to be published in 1876), nor why it was\nfinally published in 1892. Secondly, the notes and commentaries by the editors\nwere revised (following feedback on the first edition) and completed. In 1876,\n4 texts were commented (30 pages of commentaries), while in 1892, 10 texts were\ncommented (for a total of 60 pages of commentaries). A third edition was\npublished in 1902 by Max Noether and Wilhelm Wirtinger. The sole but very\nnotable change here is the addition of over a hundred pages of notes from\nRiemann’s lectures (on Abelian, elliptic, hyperelliptic functions,\nhypergeometric series, etc.) which had only recently become known.",
{
"type": "Note",
"id": "idm60",
"noteType": "Footnote",
"content": [
{
"type": "Paragraph",
"id": "footnote4",
"content": [
"In\n1990, the 1902 edition was reprinted along with additions. A French\ntranslation appeared in 1898, translated by Léonce Laugel and published by\nGauthiers-Villars (see p. ",
{
"type": "Cite",
"target": "Sx3",
"content": [
"Shaping the book and shaping the image of the editee"
]
},
"). The first English translation\nappeared in 2004 [",
{
"type": "Cite",
"target": "bib-bib29",
"content": [
"29"
]
},
"]."
]
}
]
}
]
},
{
"type": "Paragraph",
"id": "Sx1.p6",
"content": [
"Only for the 1876 edition do we have, rather exceptionally, extensive\ndocumentation on the process of editing Riemann’s collected works. This is one\nreason why my focus in this paper will be this first edition.",
{
"type": "Note",
"id": "idm72",
"noteType": "Footnote",
"content": [
{
"type": "Paragraph",
"id": "footnote5",
"content": [
"Unless\nstated otherwise, “edition” will refer to the first edition, from now on."
]
}
]
},
" A\nsecond reason is that a core interest, here, is how the editorial work shaped\nRiemann’s text, which was largely accomplished in the first edition."
]
},
{
"type": "Paragraph",
"id": "Sx1.p7",
"content": [
"Dedekind and Weber’s editorial work was meticulous, mindful and even devoted,\naccording to Elise Riemann. Their collaboration for this publication, which\nmarked the beginning of almost forty years of friendship, was largely carried\nout in letters written from November 1st 1874 to the end of 1876. These letters\nhave been preserved in Riemann’s archive (Cod. Ms. Bernhard Riemann,\nNiedersächsische Staats- und Universitätsbibliothek Göttingen) and in\nDedekind’s (Cod. Ms. Richard Dedekind, Niedersächsische Staats- und\nUniversitätsbibliothek Göttingen, and G 98:11–13, Archiv der\nUniversitätsbibliothek Braunschweig),",
{
"type": "Note",
"id": "idm80",
"noteType": "Footnote",
"content": [
{
"type": "Paragraph",
"id": "footnote6",
"content": [
"Heinrich Weber’s archive seem,\nhowever, to have been lost [",
{
"type": "Cite",
"target": "bib-bib32",
"content": [
"32"
]
},
", p. 16]."
]
}
]
},
" and published in\n2014 [",
{
"type": "Cite",
"target": "bib-bib32",
"content": [
"32"
]
},
"]. As most of their discussions appear in these letters, we\nhave an extensive and detailed vision of the editorial process. Weber and\nDedekind discussed every aspect of the edition, from the practical (e.g., the\ncontract with Teubner, the copyrights, the advertisement of the book) to the\nscientific and philological (e.g., the choice of which texts to publish, their\ndifficulties in understanding Riemann’s manuscripts, what kind of corrections\nor completions should be made before the publication). Indeed, a number of\nmodifications were made to Riemann’s texts, from orthographical and\ntypographical changes to the redaction of missing passages."
]
},
{
"type": "Paragraph",
"id": "Sx1.p8",
"content": [
"The process of editing Riemann’s ",
{
"type": "Emphasis",
"content": [
"Werke"
]
},
" was thus a hands-on process, in\nwhich the editors were deeply involved in both the mathematical and\nphilological aspects. Weber and Dedekind – and Hattendorff and Schwarz for some\ntexts – engaged in a systematic verification of each and every one of Riemann’s\ntexts, including those that had already been published. Some texts were, in\nfact, written by several hands: Riemann’s and the editor’s (for example, some\nparts of [",
{
"type": "Cite",
"target": "bib-bib20",
"content": [
"20"
]
},
"] are marked as being explicitly written by\nSchwarz). This raises questions on the genesis of the text and on the\nauthorship."
]
},
{
"type": "Figure",
"id": "Sx1-F1",
"caption": [
{
"type": "Paragraph",
"content": [
"Cod. Ms. Riemann 34 I, p. 4r: extract from the manuscript on minimal\nsurfaces (Niedersächsische Staats- und Universitätsbibliothek Göttingen)"
]
}
],
"licenses": [
{
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{
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"All rights reserved"
]
}
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"id": "Sx1.p9",
"content": [
"After the publication of the Dedekind-Weber correspondence, it became clear to\nme that there was, here, material to study how the edition of Riemann’s\ncollected works was crafted. It also provides an opportunity to unfold parts of\ntheir mathematical activity which have been largely overlooked (until now!),\nand indeed to understand important aspects of Riemann’s influence on both\nmathematicians.",
{
"type": "Note",
"id": "idm112",
"noteType": "Footnote",
"content": [
{
"type": "Paragraph",
"id": "footnote7",
"content": [
"In 1882, Dedekind and Weber published ",
{
"type": "Emphasis",
"content": [
"Theorie\nder algebraischen Funktionen einer Veränderlichen"
]
},
" (Theory of algebraic\nfunctions of one complex variable) in which they transfer Dedekind’s concepts\nof field, module and ideal from number theory to function theory to give a new\ndefinition of the Riemann surface and related notion, such as the genus."
]
}
]
},
" It\nalso allows us to make connections with research in the history of text – how\ndid the editing process shape the texts published? how did it shape the\nbook itself? – and with the history of mathematical publishing."
]
},
{
"type": "Paragraph",
"id": "Sx1.p10",
"content": [
"Common interests in these questions led to collaboration between the History of\nScience, History of Text research group in the Laboratoire SPHERE (Université\nde Paris) and the Interdisziplinäre Zentrum für Wissenschafts- und\nTechnikforschung (Bergische Universität Wuppertal) with the organisation of an\nongoing series of workshops and seminars on the history of collected works as\nan editorial and scientific practice. Among our observations (some of which I\nwill return to towards the end of this paper), the most relevant to the case of\nthe Riemann edition are the following: texts published in collected works often\nbear the traces of the editorial work – maybe in more ways than we would expect\n– and for the editors, this was not solely an editorial or philological\nundertaking, but also a scholarly endeavour, and indeed one we seem to have\noverlooked so far."
]
},
{
"type": "Paragraph",
"id": "Sx1.p11",
"content": [
"The ongoing analysis of the edition of Riemann’s collected works is made\npossible by the documents available in Riemann’s archive,",
{
"type": "Note",
"id": "idm123",
"noteType": "Footnote",
"content": [
{
"type": "Paragraph",
"id": "footnote8",
"content": [
"The catalog\nis available here:\n",
{
"type": "Link",
"target": "http://hans.sub.uni-goettingen.de/nachlaesse/Riemann.pdf",
"content": [
"hans.sub.uni-goettingen.de/nachlaesse/Riemann.pdf"
]
}
]
}
]
},
"\nwhose origins are described in [",
{
"type": "Cite",
"target": "bib-bib14",
"content": [
"14"
]
},
"], and in which most of\nthe documents used by Weber and Dedekind are\navailable.",
{
"type": "Note",
"id": "idm132",
"noteType": "Footnote",
"content": [
{
"type": "Paragraph",
"id": "footnote9",
"content": [
"See [",
{
"type": "Cite",
"target": "bib-bib30",
"content": [
"30"
]
},
"] for details on the development of\nGöttingen as an archive center."
]
}
]
},
" The only exception are the manuscripts that\nwere the basis for [",
{
"type": "Cite",
"target": "bib-bib24",
"content": [
"24"
]
},
", ",
{
"type": "Cite",
"target": "bib-bib27",
"content": [
"27"
]
},
"] which are in Schwarz’ archive at\nthe ",
{
"type": "Emphasis",
"content": [
"Archiv der Berlin-Brandenburgischen Akademie der Wissenschaften"
]
},
".\nDedekind’s archive also contains interesting material on his work as an editor\n(see below)."
]
},
{
"type": "Paragraph",
"id": "Sx1.p12",
"content": [
"Since the 1970s and the great work done by Erwin Neuenschwander, many\ninteresting historical works have been published using Riemann’s archive, a\nnumber of which will certainly be useful to the present project. The goal of\nthis project is solely a critical analysis of the process of ",
{
"type": "Emphasis",
"content": [
"editing"
]
},
"\nRiemann’s collected works, which comes along with a comparison of the original\nmanuscripts and the published texts. The files in Riemann’s archive relating to\nthe published texts contain thousands of pages (and around 500 pages in\nDedekind’s archive). Most of the files contain several copies of the texts\n(usually by the editors, more rarely by Riemann), Riemann’s original texts and\nmany of his drafts. Using a (semi-)automated approach\nto the transcription and comparison of the manuscripts with digital tools for\nhandwritten text recognition and the tools developed by the CollEx-Persée\nproject\nAMOr (",
{
"type": "Link",
"target": "https://www.collexpersee.eu/projet/amor/",
"content": [
"www.collexpersee.eu/projet/amor/"
]
},
")\nshould help manage these relatively large files.",
{
"type": "Note",
"id": "idm148",
"noteType": "Footnote",
"content": [
{
"type": "Paragraph",
"id": "footnote10",
"content": [
"Of course, for parts\nof this archive, in particular the letters, transcriptions are already\navailable."
]
}
]
},
" In some of these files, the most challenging task might be to\nidentify which documents were indeed used by the editors to produce the\npublished text."
]
},
{
"type": "Heading",
"id": "Sx2",
"depth": 1,
"content": [
"Shaping the individual texts"
]
},
{
"type": "Paragraph",
"id": "Sx2.p1",
"content": [
"Heinrich Weber wrote an announcement of Riemann’s collected works for\nKoenigsberger and Zeuner’s ",
{
"type": "Emphasis",
"content": [
"Repertorium der literarischen Arbeiten aus dem\nGebiete der reinen und angewandten Mathematik"
]
},
", in which he mentions the\nextent of the editorial work:"
]
},
{
"type": "QuoteBlock",
"content": [
{
"type": "Paragraph",
"content": [
"We only corrected some slight inaccuracies which were made known to the editor\nand could be seen as certain. Some additions, written according to Riemann’s\nmanuscripts, and some necessary clarifications were placed in final notes. […]\n[T]he majority of [Riemann’s] posthumous writings contain only formulae with\nvery little indications to find what link them. Hence, a lot of passages\nwritten only in a very fragmentary form had to be established as well as we\ncould, and many others are still buried in his archive, for want of being\ndeciphered. [",
{
"type": "Cite",
"target": "bib-bib36",
"content": [
"36"
]
},
", pp. 7–8]"
]
}
]
},
{
"type": "Paragraph",
"content": [
"A similar statement can also be found in Weber’s preface\nin [",
{
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"target": "bib-bib25",
"content": [
"25"
]
},
", p. iv]."
]
},
{
"type": "Paragraph",
"id": "Sx2.p3",
"content": [
"There are several types of modifications of Riemann’s original texts: the\nlocal, more or less significant changes to the texts, e.g., correcting an\nerror, which are mentioned in notes; a number of such local changes, which are\n",
{
"type": "Emphasis",
"content": [
"not"
]
},
" mentioned in notes; and texts extracted from Riemann’s archive\nwhich are completed to a greater or lesser extent by the editor.",
{
"type": "Note",
"id": "idm172",
"noteType": "Footnote",
"content": [
{
"type": "Paragraph",
"id": "footnote11",
"content": [
"I\nhave considered these questions in [",
{
"type": "Cite",
"target": "bib-bib11",
"content": [
"11"
]
},
", ",
{
"type": "Cite",
"target": "bib-bib13",
"content": [
"13"
]
},
"]. An in-depth\nanalysis of the edition of [",
{
"type": "Cite",
"target": "bib-bib17",
"content": [
"17"
]
},
"] is in progress and, as mentioned,\nso is a critical edition of Riemann’s texts."
]
}
]
},
" While the reader could expect to\nbe able to identify clearly what was changed or added by the editors, this is\nnot always the case. A number of changes are not clearly identified in any way,\nand can only be recognised as such by reading the editors’ correspondence or\ncomparing the published texts with the manuscripts.",
{
"type": "Note",
"id": "idm183",
"noteType": "Footnote",
"content": [
{
"type": "Paragraph",
"id": "footnote12",
"content": [
"It is the case\nwith [",
{
"type": "Cite",
"target": "bib-bib18",
"content": [
"18"
]
},
"], whose edition I presented in [",
{
"type": "Cite",
"target": "bib-bib12",
"content": [
"12"
]
},
"]."
]
}
]
}
]
},
{
"type": "Paragraph",
"id": "Sx2.p4",
"content": [
"Of course, Weber and Dedekind were cautious with their corrections. In a letter\nfrom July 8, 1875, as he was proofreading Riemann’s famous ",
{
"type": "Emphasis",
"content": [
"Ueber die\nAnzahl der Primzahlen unter einer gegebenen Grösse"
]
},
" (On the number of primes\nless than a given magnitude) [",
{
"type": "Cite",
"target": "bib-bib19",
"content": [
"19"
]
},
"], Weber wrote to Dedekind:"
]
},
{
"type": "QuoteBlock",
"content": [
{
"type": "Paragraph",
"content": [
"Do you have any remarks on the work on primes? I have come to a difference from\nRiemann’s formula in the calculation, namely to the same one which Scheibner\nalready noticed in his analysis of this work in Schlömilch’s\njournal. [[",
{
"type": "Cite",
"target": "bib-bib33",
"content": [
"33"
]
},
"]] Despite this, I am far from taking\nRiemann’s result to be incorrect, whose actual proof, as can be seen from a\nfragment of a letter, is not contained in the work at all. I do not dare to\nmake any changes or additions. [",
{
"type": "Cite",
"target": "bib-bib32",
"content": [
"32"
]
},
", p. 71]"
]
}
]
},
{
"type": "Paragraph",
"content": [
"We don’t have any answer from Dedekind, but Weber later wrote again that he was\nfinding “",
{
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"meta": {
"altText": "-\\log 2"
}
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" instead of ",
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"meta": {
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"” (as Scheibner had) but still\ndidn’t dare to make any change or note, assuming that “it is probable that\nRiemann is right” but that he was missing the proof. The 1876 edition does not\ncontain any correction or note, but there is a note by Weber in the 1892\nedition, stating that"
]
},
{
"type": "QuoteBlock",
"content": [
{
"type": "Paragraph",
"content": [
"If one continues the computation indicated by Riemann, one finds in the formula\n",
{
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"altText": "\\log{\\frac{1}{2}}"
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"mathLanguage": "mathml",
"text": "logξ(0)",
"meta": {
"altText": "\\log\\xi(0)"
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{
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"meta": {
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{
"type": "Note",
"id": "idm287",
"noteType": "Footnote",
"content": [
{
"type": "Paragraph",
"id": "footnote13",
"content": [
"This seems to be a typo\ncorrecting the typo, as ",
{
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"mathLanguage": "mathml",
"text": "",
"meta": {
"altText": "\\log\\zeta(0)=-\\log 2+\\pi i=\\log\\frac{1}{2}+\\pi i"
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",\nWeber meant to write “indeed ",
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"\n[",
{
"type": "Cite",
"target": "bib-bib25",
"content": [
"25"
]
},
", p. 155, 2nd edition 1892]"
]
}
]
},
{
"type": "Paragraph",
"content": [
"Changes can be even more important in texts extracted from Riemann’s archive.\nFor some of them, the editors decided to write entire paragraphs themselves to\ncomplete Riemann’s original text before publication. Such changes raise\nquestions as to the authorship of the texts, and the extent to which some of\ntheir content could be a result of edition as a collaborative enterprise. Some\nmathematicians in the years following the publication of Riemann’s collected\nworks seemed to keep this aspect in mind, as suggested by a letter from Felix\nKlein to Henri Poincaré, sent on April 3rd, 1882, following a discussion on\nRiemann’s possible anticipation of some of Friedrich Schottky’s\nresults:",
{
"type": "Note",
"id": "idm355",
"noteType": "Footnote",
"content": [
{
"type": "Paragraph",
"id": "footnote14",
"content": [
"Klein is, here, referring to [",
{
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"34"
]
},
"] and which was\npublished in 1877 in the ",
{
"type": "Emphasis",
"content": [
"Journal für die reine und angewandte\nMathematik"
]
},
", 83: 300–351, in which he studied conformal mappings of multiply\nconnected domains, which he was the first to analyse systematically."
]
}
]
}
]
},
{
"type": "Figure",
"id": "Sx2-F2",
"caption": [
{
"type": "Paragraph",
"content": [
"Pages 12r, 14r and 15v of Cod. Ms. Riemann 5 (Niedersächsische\nStaats- und Universitätsbibliothek Göttingen)"
]
}
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"[Regarding] Schottky, I would like to draw your attention to a posthumous essay\nin Riemann’s collected works, p. 413, where exactly corresponding ideas are\ndeveloped. However, it will be difficult to establish how much the editor,\nProf. Weber, has put into it. Riemann’s collected works appeared in 1876,\nSchottky’s dissertation in 1870, later as an essay in Borchardt’s Journal,\n1877. (Letter from Klein, in [",
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{
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"Gleichgewicht der Elektrizität auf Zylindern mit\nkreisförmigem Querschnitt und parallelen Axen"
]
},
" (Equilibrium of electricity\non cylinders with circular crosssection and parallel\naxes) [",
{
"type": "Cite",
"target": "bib-bib26",
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},
"], which indeed deals with conformal mappings on a\nmultiply connected surface. In a footnote, Weber states that"
]
},
{
"type": "QuoteBlock",
"content": [
{
"type": "Paragraph",
"content": [
"[t]here are no completed manuscripts of this and the following works by\nRiemann. They are composed of pages which, apart from a few hints, contain only\nformulae. [",
{
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"25"
]
},
", p. 413]"
]
}
]
},
{
"type": "Paragraph",
"content": [
"Early in his correspondence with Dedekind, Weber mentioned that he would be\n“very interested” in being able to “decipher” the manuscripts on “the\ndistribution of electricity on three spheres” [sic], which he hoped to be able\nto achieve since “on one of the sheets the results seem to be essentially in\nplace” [",
{
"type": "Cite",
"target": "bib-bib32",
"content": [
"32"
]
},
", p. 62, letter from March 22, 1875]. As this last remark\nsuggests, Riemann’s manuscripts in Cod. Ms. Riemann 5 contain many sheets with\nvarious states of development of his investigation. There are 27 pages by\nRiemann’s hand, for 4 pages of text by Weber, and certainly the material\ndifferences of each mathematician’s handwriting and use of paper do not account\nfor such a large difference. In fact, many of Riemann’s notes contain similar\ncomputations, see Figure ",
{
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"target": "Sx2-F2",
"content": [
"2"
]
},
"."
]
},
{
"type": "Paragraph",
"id": "Sx2.p8",
"content": [
"In addition, to put it bluntly, Weber’s version of Riemann’s research contains\na lot more sentences and far fewer calculations. It is fairly easy to identify\nwhich formulae Weber included in his text. However, the ",
{
"type": "Emphasis",
"content": [
"sentences"
]
},
"\npresent in the published text are quite difficult to find in Riemann’s\nmanuscripts. Thus, it seems that most of the redaction is by Weber, who\ncompleted and clarified Riemann’s text. He did not, here, correct or complete\nRiemann’s formulae – rather, he selected the relevant ones. It is, without a\ndoubt, a text written by both Riemann and Weber."
]
},
{
"type": "Paragraph",
"id": "Sx2.p9",
"content": [
"In the available correspondence, Weber did not himself mention Schottky’s\nworks. However, Schwarz wrote to Weber about Schottky’s dissertation on\nNovember 11, 1875 [",
{
"type": "Cite",
"target": "bib-bib32",
"content": [
"32"
]
},
", p. 362].",
{
"type": "Note",
"id": "idm405",
"noteType": "Footnote",
"content": [
{
"type": "Paragraph",
"id": "footnote15",
"content": [
"Schwarz wrote: “On Saturday and\nSunday of last week, I was in Berlin and learned from Prof. Weierstrass of\nthe dissertation of one of his students, a certain Schottky: ‘",
{
"type": "Emphasis",
"content": [
"Über die\nconforme Abbildung mehrfach zusammenhängender Flächen"
]
},
"’; if you do not not\nalready know about this dissertation, please allow me to draw your attention\nto it. The results which are presented in this essay, are of great interest\nand scientific value; I myself will seek to obtain the dissertation in order to\npossess it.” "
]
}
]
},
" Weber’s letters to Schwarz have been lost, and we do not know\nwhat he answered to this mention of Schottky’s paper. In January 1876, this\ntext had, with seven others, already been sent to Teubner [",
{
"type": "Cite",
"target": "bib-bib32",
"content": [
"32"
]
},
", p. 95]."
]
},
{
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"id": "Sx2.p10",
"content": [
"Another – and one of the most striking – examples of an extensive\nmathematical and editorial investment is the work done by Dedekind on\n“",
{
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"content": [
"Fragmente über die Grenzfälle der elliptischen Modulfunctionen"
]
},
"”\n(Fragments on the limit-cases of elliptic modular\nfunctions) [",
{
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"content": [
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},
"]."
]
},
{
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"caption": [
{
"type": "Paragraph",
"content": [
"Cod. Ms. Riemann 14, p. 18v: Excerpt from Riemann’s “very pale\nmanuscript” (Niedersächsische Staats- und Universitätsbibliothek Göttingen)"
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"Dedekind started working on these manuscripts in February 1876. The lack of\nclarity of the notes, both from a material and a mathematical viewpoint, was so\nbad that editing them took Dedekind several weeks and led him to fear having\nnightmares (see [",
{
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"target": "bib-bib32",
"content": [
"32"
]
},
", pp. 101–104]). Over a dozen letters were\nexchanged between Dedekind and Weber from December 1875 to April 1876. Dedekind\nconfided to Weber his difficulties in understanding and editing Riemann’s text\n(which he nicknamed, in his letters and in his own archive, “",
{
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"sehr blasses\nManuskript von Riemann"
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"content": [
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" consider properties of Jacobi series in elliptic\nfunction theory. Without entering into any detail, Dedekind interpreted\nRiemann’s formulae as the study of the logarithm of some modular functions at\nthe limits of their domain of definition. In the collected works, he stated:"
]
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"content": [
"The time of writing of the first of the two fragments (September 1852) makes it\nlikely that Riemann, while working on his memoir ",
{
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"On the representation of\na function by a trigonometric series"
]
},
", was looking for examples of functions\nwith infinitely many discontinuities in each interval. Perhaps the second\ninvestigation, which occurs on the barely legible sheet, has the same\nobject.",
{
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"noteType": "Footnote",
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]
}
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},
{
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"content": [
"In Cod. Ms. Riemann 14, we find the 15 pages of Riemann’s original manuscript,\ntwo handwritten transcriptions, the handwritten text for Dedekind’s 1876\ncommentary and the version sent to the editor, some notes written by Weber, the\n1876 letters between Dedekind and Weber relating to that text, and one of\nDedekind’s early works on elliptic functions, which he intended to use to\nunderstand Riemann’s ideas and likely sent to Weber with one of his letters."
]
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{
"type": "Paragraph",
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"The most exceptional documents can be found in Dedekind’s archive. In Cod.\nMs. Dedekind XI 11-1, XI 11-2, XII 4,",
{
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"noteType": "Footnote",
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"Cod. Ms. Dedekind XII 4 is\nmistakenly listed as referring to [",
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"] in the Göttingen catalog,\nbecause Dedekind refers to the text using the numbering in the table of\ncontents in the 1892 reedition of Riemann’s collected works. The contents of\nthe file are, however, undoubtedly related to [",
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"]. In exploring\nthese documents, I have greatly benefited from Walter Strobl’s help."
]
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" we find\nseveral hundred pages of notes written solely by Dedekind. There, we see the\nprogression in his understanding of Riemann’s texts and of the writing of his\n1876 and 1892 commentaries, as well as continuations of his research on the\nsubject. These pages show the breadth and depth of the mathematical reflections\ndeveloped by Dedekind for his editorial work. In addition to computations\nfollowing Riemann’s manuscripts and trying to obtain again Riemann’s results,\nDedekind developed his own approach to the subject, which ended up being his\nonly way to verify Riemann’s results. For this, he drew comparisons between\nboth approaches, at some points relying only on the correspondences between\nnumerical examples, and eventually systematically exploring the correspondences\nbetween his and Riemann’s results. This research was also the basis for his\ncommentaries, of which we find several drafts in the archive. Both Dedekind’s\ncommentaries, although entitled “Explanation on the preceding fragments” do not\nactually explain what Riemann was trying to do, rather they present:"
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"a very interesting application related to the so-called theory of the\ninfinitely many forms of the theta-functions, namely the determination of the\nconstants appearing via transformations of first degree, which as is known,\nwere reduced by Jacobi and Hermite to Gauss sums, and thus to the theory of\nquadratic residues. The following commentary illustrates these relationships.\n[",
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"4"
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", p. 438]"
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"In particular, it is there that Dedekind introduced what we today call the\nDedekind eta function.",
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"The Dedekind eta function is a modular form\ndefined on the upper-half part of the complex plane by\n",
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", p. 438]."
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"Cod. Ms. R. Dedekind XI 11-1, p. 19r: Summary of Dedekind’s comparison of Riemann’s results with his own (Niedersächsische Staats- und Universitätsbibliothek Göttingen)"
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"content": [
"Shaping the book and shaping the image of the editee"
]
},
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"id": "Sx3.p1",
"content": [
"The way in which a book such as a mathematician’s collected works is\nconstructed – which texts are chosen to be in this publication; whether\nunpublished manuscripts are selected and if so, which ones, and how they are\npublished; whether a critical apparatus is added and which one; how texts are\norganised and, when applicable, how the multiple volumes are themselves\norganised, etc. – shapes the image of the editee presented to the readers.\nIndeed, such choices are a reflection of the editor’s own idea of the editee’s\nwork, and of what they want to showcase of it.",
{
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"noteType": "Footnote",
"content": [
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"id": "footnote19",
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"Note that this is also\nan important point regarding the role that individuals play in editing their\nown work or in supervising such an edition (e.g., Poncelet, Weierstrass)."
]
}
]
},
" The\nshaping of the book is, in fact, the shaping of the vector of circulation of\nthe editee’s works. Without undue generalization on the possibility of biases\non the part of editors, the history of mathematics gives us several examples in\nwhich mathematicians works were largely reconstructed by the editors."
]
},
{
"type": "Paragraph",
"id": "Sx3.p2",
"content": [
"The selection of which texts are deemed suitable for publication plays a\nsignificant role in such a reconstruction of the works of the editee. Through\nthese choices, the editors impose their own criteria and their own values on\nthe editee’s texts. And it is all the more pregnant regarding the choice of\nexcerpts from the author’s archives, as there are few ways of knowing whether\nthe author had any intention of publishing these texts, or why they didn’t. As\nsuch, our vision of the editee’s work can be restricted to the editors’ reading\nof it. And this contributes, to a certain extent, to a mythologised history of\nmathematics."
]
},
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"id": "Sx3.p3",
"content": [
"Let me give three examples, which are not Riemann’s collected works, in which\nthis happened. A first, and very striking, example is the edition of Leibniz’s\nworks, which was mentioned in David Rabouin’s recent paper in the archive\nseries on Leibniz’s archives [",
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"16"
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},
"], in which he explains how “some\ntexts edited by Gerhardt and Couturat have turned out to be mere artefacts”. A\nsecond example is the edition of Gauss’ collected works, a gigantic enterprise\nthat took several decades, first directed by Ernst Schering, then by Felix\nKlein (see [",
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"10"
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", pp. 67–68] and [",
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"]). Maarten Bullynck\nshowed, in a talk at the Laboratoire SPHERE (Université de Paris), how the\nedition of Gauss’ collected works was one of the elements of Klein’s\nretrospective reconstruction of the so-called Göttingen tradition in\nmathematics. My third example is the edition of Dedekind’s collected works by\nEmmy Noether, Øystein Ore and Robert Fricke in 1930–1932. The three volumes are\norganized as follows: the first two contain Dedekind’s mathematical papers\narranged in chronological order with some extracts from his archive, the third\ncontains his foundational essays on real and natural numbers, partial\nreproductions of his algebraic number theory",
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"[",
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"] which\nwere respectively published as Supplements to the 1871 and 1894 editions of\nLejeune-Dirichlet’s ",
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"Vorlesungen über Zahlentheorie"
]
},
" and [",
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"] which\nwas published in French and later as a Supplement to the 1879 edition of\nLejeune-Dirichlet’s ",
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"Vorlesungen über Zahlentheorie"
]
},
". In these papers,\nDedekind introduced and developed the concepts of field and ideal."
]
}
]
},
" and more\nextracts from his archive. This arrangement creates two illusions. First, that\nof a difference of status between Dedekind’s ‘mathematical’ and his\n‘foundational’ papers, a distinction he did not make himself. Second, the\n",
{
"type": "Emphasis",
"content": [
"partial"
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},
" reproduction of his algebraic number theory completely\ndisconnects this research from its number-theoretical context and, in fact,\nexcludes its more traditional parts. These choices were likely guided by the\neditors considering Dedekind as a precursor of the modern structural algebra\nand certainly participated in perpetuating this retrospective reading of his\nwork."
]
},
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"id": "Sx3.p4",
"content": [
"While in many of these cases we can only observe the choices made by the\neditors and make assumptions about their intentions, for Riemann’s collected\nworks, the letters exchanged between Dedekind and Weber offer us a considerable\namount of information on these questions, making it a rather exceptional case\nstudy. Their exchanges indeed tell us which texts were dismissed as not\n‘worthy’ of being published, and the criteria that presided on their choices.\nLet me sum up the main criteria for Dedekind and Weber’s choices:"
]
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"A text had to be (of course) scientifically sound and generally correct\n– as correct as possible but the scientific interest came first;"
]
}
]
},
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{
"type": "Paragraph",
"id": "Sx3.I1.i2.p1",
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"it had to be understandable – even if this sometimes meant that the\neditors had to make the text more understandable than the way it was left by\nRiemann;"
]
}
]
},
{
"type": "ListItem",
"content": [
{
"type": "Paragraph",
"id": "Sx3.I1.i3.p1",
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"it had to be representative of Riemann’s research, it had to have a\nrecognisable place in his overall intellectual production;"
]
}
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},
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"content": [
{
"type": "Paragraph",
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"it had to (of course) give a flattering image of Riemann;"
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"content": [
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"id": "Sx3.I1.i5.p1",
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"it had to fit into the scientific and philosophical context of the time,\nto ensure that it would be well received by the scientific community."
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"content": [
"Any process of choice is subjective – it would be difficult to think of any\neditions that are completely unbiased. But in the case of Riemann’s collected\nworks, we can pinpoint some of the effects that the editors’ choices had. The\nquestion of whether the texts in Riemann’s collected works can be attributed\nsolely to him, raised by Klein, is one of them.\n"
]
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"content": [
"Another issue is the extent to which the image of Riemann provided by the\ncollected works might have been shaped by what the editors thought it should\nbe. Dedekind was very vocal about seeing Riemann as the best representative of\nhow mathematical definitions and proofs should be grounded on conceptual,\nfundamental characteristics rather than on computations and notations. He\nconsidered himself as following these methodological guidelines. Thus, was born\nthe narrative of a tradition of “conceptual mathematics” in Göttingen, which\nwas later largely continued by Klein and Hilbert’s group. These highly\ninfluential mathematicians developed a culture in Göttingen which has been\ndescribed as largely relying on “nostrification” (see\n[",
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"content": [
"2"
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},
"]), a tendency to reinterpret other people’s thoughts so that\nthey would fit their own current picture of the domain. The desire, strongly\nexpressed by Klein, to create a new kind of scientific institution might have\nled to the reconstruction of a history, an inheritance, which selected and\noveremphasized some isolated ideas (see [",
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"10"
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"] and\n[",
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"31"
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"])."
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"content": [
"This goes, of course, beyond the mere publication of collected works. It is\nhowever tangible in the French translation of Riemann’s collected works,\npublished in 1898, edited and translated by Léonce Laugel. He\nchose to exclude not only the papers published in French and Latin, but also\nmost of the papers not related to mathematics (i.e., all papers on physics and\nthe philosophical fragments). He replaced Weber’s preface with a preface by\nCharles Hermite (1822–1901) and added the translation of a talk given by Klein,\nboth of which embrace the idea of Riemann as avoiding computations and relying\nsolely on concepts and a “brilliant power of thought and [an] anticipatory\nimagination [which] led him frequently to take very great steps that others\ncould not so easily follow”, as Dedekind wrote of Riemann in his biography."
]
},
{
"type": "Paragraph",
"id": "Sx3.p8",
"content": [
"Later commentators did not all agree with the image of Riemann that this\nnarrative participated in popularizing among mathematicians. Carl Siegel, who\nfamously discovered the Riemann–Siegel formula in Riemann’s archive wrote:"
]
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"The legend according to which Riemann found his mathematical results through\ngrand general ideas without requiring the formal tools of analysis, is not as\nwidely believed today as it was during Felix Klein’s lifetime. Just how strong\nRiemann’s analytic technique was is especially clearly shown by the derivation\nand transformation of his asymptotic series for ",
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", p. 276]\n(translated in [",
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"This was also defended by the historian Harold M. Edwards,\nin [",
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"9"
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},
"], who argued for a strong – albeit maybe hidden in\ndrafts – algorithmic component in Riemann’s mathematics. Edwards showed how\nRiemann, while he may have been “primarily interested in grand general abstract\nconcepts,” on several occasions “did not venture into these higher realms\nwithout doing a lot of serious computation to lay the groundwork for his\nflights.” [",
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", p. 64].",
{
"type": "Note",
"id": "idm697",
"noteType": "Footnote",
"content": [
{
"type": "Paragraph",
"id": "footnote21",
"content": [
"Note that Arias-de-Reyna also\nclaims that Dedekind’s misinterpretation of Riemann’s fragments on modular\nelliptic functions is related to his overlooking the importance of computations\nfor Riemann."
]
}
]
},
" These observations are confirmed by Riemann’s archive, in\nparticular by the many parts that remain unpublished – which Carl Siegel, of\ncourse, knew very well."
]
},
{
"type": "Paragraph",
"id": "Sx3.p10",
"content": [
"This leads me to one last potential issue, or more exactly to a limitation any\neditor would face with the manuscripts of a mathematician such as Riemann:\nunderstanding their content. As Carl Siegel’s work on the Riemann–Siegel\nformula has shown, Riemann’s archive contained, and maybe still contains,\nimportant unpublished (even if not fully developed) results that escaped\nWeber’s and Dedekind’s attention. This shows the extent to which it can be\nuseful and fruitful to revisit mathematicians’ archives."
]
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"Acknowledgements"
]
},
"\n\nThis work was supported by a public grant as part of the Investissement\nd’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH."
]
},
{
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"id": "authorinfo",
"content": [
"Emmylou Haffner holds a PhD in history of mathematics from the Université Paris\nDiderot. She currently works at the Institut de Mathématique d’Orsay, where she\noversees the CollEx-Persée project Archives Mathématiques d’Orsay (AMOr). Her\nmain interests are in the history of mathematics and mathematical manuscripts\nin the 19th and 20th centuries.\n\n",
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