Revue d’Histoire des Mathématiques Tome 26 Fascicule 2 2 0 2 0 SOCIÉTÉ MATHÉMATIQUE DE FRANCE REVUE D’HISTOIRE DES MATHÉMATIQUES RÉDACTION Rédacteur en chef : Frédéric Brechenmacher Rédacteur en chef adjoint : Maarten Bullynck Membres du Comité de rédaction : Tom Archibald Andrea Bréard Veronica Gavagna Catherine Goldstein Tinne Hoff Kjeldsen Ralf Krömer Catherine Jami Marc Moyon Clara Silvia Roero Laurent Rollet Sabine Rommevaux Ivahn Smadja Directeur de la publication : Fabien Durand Secrétariat : Société Mathématique de France Institut Henri Poincaré 11, rue Pierre et Marie Curie, 75231 Paris Cedex 05 Tél. : (33) 01 44 27 67 99 / Fax : (33) 01 40 46 90 96 Mél : [email protected] / URL : http//smf.emath.fr/ Périodicité : La Revue publie deux fascicules par an, de 150 pages chacun environ. Tarifs : Prix public Europe : 94 e; prix public hors Europe : 105 e; prix au numéro : 43 e. Des conditions spéciales sont accordées aux membres de la SMF. Diffusion : SMF, Maison de la SMF, Case 916 - Luminy, 13288 Marseille Cedex 9 Hindustan Book Agency, O-131, The Shopping Mall, Arjun Marg, DLF Phase 1, Gurgaon 122002, Haryana, Inde © SMF No ISSN : 1262-022X, électronique : 1777-568X Maquette couverture : Armelle Stosskopf Revue d’Histoire des Mathématiques Journal for the History of Mathematics Tome 26 Fascicule 2 2 0 2 0 SOCIÉTÉ MATHÉMATIQUE DE FRANCE Revue d’histoire des mathématiques 26 (2020), p. 97–119 doi:10.24033/rhm.229 LA NOTION MÉDIÉVALE DE CONTRACTIO DANS L’ARITHMETICA INTEGRA DE MICHAEL STIFEL (1544) Sabine Rommevaux-Tani Résumé. — La notion de « contractio » apparaît au Moyen Âge dans différents contextes, en particulier dans le De ortu scientiarum de Robert Kilwardby (ca. 1215–1279), à propos de la question de la subordination des sciences. Ce dernier prend alors l’exemple du livre X des Éléments d’Euclide afin de montrer comment la classification des lignes irrationnelles est subordonnée à l’arithmétique au moyen de la contractio des nombres dans les grandeurs. Nous verrons que la lecture que fait Michael Stifel du livre X des Éléments d’Euclide dans son Arithmetica integra (1544) peut apporter un éclairage sur les propos de Kilwardby. Abstract (The medieval notion of contractio in Michael Stifel’s Arithmetica integra (1544)) The notion of “contractio” appears in the Middle Ages in various contexts, in particular in Robert Kilwardby’s De ortu scientiarum (ca. 1215–1279), on the question of the subordination of sciences. Kilwardby then takes the example of Book X of Euclid’s Elements to show how the classification of irrational lines is subordinated to arithmetic by means of the contraction of numbers into geometric quantities. We shall see that Michael Stifel’s interpretation of Book X of Euclid’s Elements in his Arithmetica integra (1544) may shed some light on Kilwardby’s remarks. Texte reçu le 18 juillet 2018, accepté le 30 août 2019, révisé le 18 février 2020, version finale reçue le 26 juin 2020. S. Rommevaux-Tani, CNRS, SPHere, UMR 7219, Univ. de Paris, Bâtiment Condorcet, Case 7093, 5 rue Thomas Mann, 75205 Paris cedex 13, France. Courrier électronique : [email protected] Classification mathématique par sujets (2000) : 00A30, 01A35, 01A40. Mots clefs : Euclide, Robert Kilwardby, Michael Stifel, Moyen Âge, xvie siècle, irrationalité, arithmétique, géométrie, livre X des Éléments d’Euclide. Key words and phrases. — Euclid, Robert Kilwardby, Michael Stifel, Middle Ages, sixteenth century, irrationality, arithmetic, geometry, Book X of Euclid’s Elements. © SOCIÉTÉ MATHÉMATIQUE DE FRANCE, 2020 98 S. ROMMEVAUX-TANI Michael Stifel fait paraı̂tre en 1544, à Nuremberg, une Arithmetica integra en trois livres, dont le premier livre contient une arithmétique des nombres entiers et fractionnaires, le deuxième livre une classification des nombres irrationnels et une lecture arithmétique du livre X des Éléments d’Euclide et le troisième livre une algèbre. Il s’agit donc fondamentalement d’un livre d’arithmétique dans lequel sont traités, de manière parallèle, les nombres entiers et fractionnaires, les nombres radicaux ou irrationnels et les nombres cossiques, soit les nombres de l’algèbre1. Dans cette Arithmetica integra, la géométrie n’est toutefois pas absente. Dans une précédente étude [Rommevaux-Tani 2014], j’avais déjà noté le lien particulier qu’établit Stifel entre arithmétique et géométrie, au livre II, grâce à la contractio de nombres rationnels ou irrationnels dans des carrés. Je voudrais revenir ici sur ce concept, en m’intéressant à son origine médiévale d’une part, et à l’usage qu’en fait Stifel d’autre part. 1. LA NOTION MÉDIÉVALE DE « CONTRACTIO » La notion de « contractio » apparaı̂t au Moyen-Âge, notamment à propos de la musique et plus généralement dans le contexte de réflexions sur les sciences dites intermédiaires, ou scientiæ mediæ. Ainsi, dans son ouvrage, Sinnlichkeit und Vernunft in der mittelalterlichen Musiktheorie, Frank Hentschel [2000, p. 144–145] dresse une liste de textes dans lesquels se trouve l’expression « numerus contractus » pour désigner l’objet de la musique, soit le nombre contracté dans les sons. La notion de contractio se trouve, par exemple, dans les Questiones mathematicæ de Raoul le Breton ou Radulphus Brito (ca. 1270–1320), maı̂tre ès art à l’université de Paris, puis théologien et proviseur de la Sorbonne. Dans la question 7 de ces Questiones mathematicæ, qui demande si le nombre des sciences mathématiques se limite à quatre, Raoul le Breton explique que le sujet de la mathématique est la quantité abstraite des qualités sensibles, qui se décline en la grandeur ou 1 Ainsi, on trouve dans chacun des livres un Algorithme (Algorismus) des nombres considérés, soit des entiers et des fractions (livre I, chap. i), des radicaux (livre II, chap. iv), des sommes et différences de nombres radicaux (livre II, chap. ix), des fractions entre sommes ou différences de radicaux (livre II, chap. xi), des des q racines p p pp p p binômes ou résidus, soit des nombres de la forme a b, a b et a b, où a et b sont des nombres rationnels, le premier terme de la somme ou de la différence étant toujours plus grand que le second (livre II, chap. xii) et enfin des nombres de l’algèbre que sont les nombres cossiques, soit l’équivalent de nos monômes (livre III, chap. iii). LA NOTION MÉDIÉVALE DE CONTRACTIO DANS L’ARITHMETICA INTEGRA 99 le nombre [Hentschel 2000, p. 282, 284]2. Quand le mathématicien considère le nombre absolument et par soi, il fait de l’arithmétique, quand il considère la grandeur absolument et par soi, il fait de la géométrie ; par contre, les objets de l’astrologie ou astronomie sont les grandeurs contractées dans les corps célestes et ceux de la musique sont les nombres contractés dans les sons ou plutôt dans les rapports qui sont dans les sons [Hentschel 2000, p. 284]3. Ainsi, un premier processus d’abstractio à partir des qualités sensibles produit la quantité, ou la grandeur et le nombre, puis un second processus de contractio dans le sensible, conduit de la grandeur aux corps célestes et du nombre aux sons4. Ce faisant, la géométrie et 2 « Utrum sint tantum quattuor scientiæ mathematicæ. [...] Ad istam questionem dico primo quod sunt quattuor scientiæ mathematicæ, scilicet arithmetica, musica etc. [...] Primum demonstratur quia : mathematica est de eo, quod est abstrahibile per naturam suam a qualitatibus sensibilibus et quantum ad suum esse non determinat sibi aliquam determinatam complexionem qualitatum sensibilium, modo omnis quantitas abstrahibilis a qualitatibus sensibilibus est vel magnitudo vel numerus, ergo omnis mathematica est de magnitudine vel numero. » (Est-ce qu’il existe seulement quatre sciences mathématiques ? [...] À cette question je réponds d’abord que les sciences mathématiques sont au nombre de quatre, soit l’arithmétique, la musique etc. [...] La première assertion est démontrée ainsi : la mathématique est au sujet de ce qui est susceptible d’être abstrait, selon sa nature, des qualités sensibles et qui, quant à son être, n’implique pas un mélange déterminé de qualités sensibles. Or toute quantité susceptible d’être abstraite des qualités sensibles est ou bien la grandeur, ou bien le nombre, donc toute mathématique est au sujet de la grandeur ou du nombre.) 3 « Modo magnitudo potest dupliciter considerari, uno modo secundum se et absolute, secundum quod est abstrahibile a qualitate sensibili, et sic de ipsa est geometria, vel potest considerari, ut est contracta ad corpus cæleste, et sic magnitudo, scilicet contracta ad corpus cæleste, est subiectum in astrologia. Et quomodo hoc sit verum, apparebit post. Si autem sit de numero, hoc est dupliciter, quia numerus potest considerari secundum se et absolute, ut est abstrahibilis a qualitatibus sensibilibus, et sic de ipso est arithmetica, vel ut est contractus ad sonos, et sic est musica de ipso, sive est contractus ad proportiones, quæ sunt in sonis. » (Or la grandeur peut être considérée de deux manières, d’une manière pour elle-même et absolument, selon qu’elle est susceptible d’être abstraite de la qualité sensible, et alors à son sujet on a la géométrie, ou elle peut être considérée comme contractée dans un corps céleste, et alors la grandeur, contractée dans un corps céleste, est le sujet de l’astrologie. Et pourquoi cela est vrai apparaîtra plus loin. Et pour ce qui est du nombre, c’est de deux manières, car le nombre peut être considéré pour lui-même et absolument, selon qu’il est susceptible d’être abstrait des qualités sensibles, et alors à son sujet on a l’arithmétique, ou bien selon qu’il est contracté dans les sons ou contracté dans les rapports qui sont dans les sons, et alors à son sujet on a la musique.) 4 On peut évoquer ici Aristote, Métaphysique, E, 1, 1026a [2008, p. 225] : « [...] certaines parties de la mathématique [traitent] d’objets immobiles, pourtant peut-être non séparables comme dans la matière [...]. » Par ailleurs, ce balancement entre abstractio et contractio peut être rapproché de celui entre l’« aphairesis » ou la soustraction, soit l’acte d’abstraction qui produit les objets mathématiques, et la « prosthesis », ou l’addition, par laquelle sont obtenus les objets physiques (je remercie ici Vincenzo De Risi qui m’en a fait la remarque). Voir à ce sujet [Cleary 1985]. 100 S. ROMMEVAUX-TANI l’arithmétique sont considérées comme proprement mathématiques, alors que la musique et l’astronomie sont en partie mathématiques et en partie naturelles [Hentschel 2000, p. 284–285]5 ; en d’autres termes, ce sont des scientiæ mediæ. Quelques années avant Raoul de Breton, Robert Kilwardby (ca. 1215– 1279), d’origine anglaise, fut lui aussi étudiant puis enseignant à l’université de Paris. Il rédige son ouvrage d’introduction à la philosophie, le De ortu scientiarum, probablement avant 1250, soit avant qu’il n’entre dans l’ordre des Dominicains et ne rejoigne l’université d’Oxford où il étudie la théologie et dont il deviendra régent [Sharp 1934, p. 1 ; Rodrı́guez Arias 1997, p. 467 ; Maierù 2013, p. 353]. Réfléchissant aux statuts des différentes branches des mathématiques, Kilwardby lie subordination des sciences et contractio [Kilwardby 1976, p. 45–46 ; Maierù 2013, p. 366–367] ; rappelons que pour les auteurs médiévaux, une science est subordonnée à une autre quand elle en tire ses principes6. Voici donc comment Kilwardby introduit la notion de contractio : Je ne vois pas d’inconvénient à ce que l’astronomie et la géométrie soient des sciences différentes et que pourtant l’une soit subordonnée à l’autre. En ef5 « Secundum demonstratur, scilicet quod duæ istarum sunt pure etc., et primo de arithmetica et geometria, quod si<n>t pure mathematicæ, quia illæ scientiæ sunt pure mathematicæ, quæ sunt de aliquo simpliciter et secundum se considerato, non contracto ad aliquam materiam sensibilem, modo arithmetica est de numero secundum se considerato, non contracto ad aliquam materiam sensibile, et etiam geometria est de magnitudine sic abstracta et considerata, quare istæ duæ pure sunt mathematicæ. Secundum demonstratur, scilicet quod musica et astrologia sint partim etc., quia istæ duæ considerant ens mathematicum contractum ad aliquod sensibile, sicut astrologia considerat magnitudinem contractam ad corpus cæleste. Musica autem sive musicus considerat numerum contractum ad sonos sive ad proportiones numerorum in sonis, ergo, cum istæ duæ considerent aliquid partim mathematicum et partim naturale, erunt partim mathematicæ et partim naturales. » (On démontre la seconde assertion, à savoir que deux d’entre elles sont pures etc. et d’abord au sujet de l’arithmétique et de la géométrie, que ce sont des mathématiques pures, car sont des sciences pures ces mathématiques qui traitent de choses considérées pour elles-mêmes et simplement et non pas contractées dans quelque matière sensible, or l’arithmétique traite du nombre considéré pour lui-même et non pas contracté dans quelque matière sensible et de même la géométrie est au sujet de la grandeur abstraite et considérée ainsi, donc ces deux sont des mathématiques pures. On démontre la seconde partie, à savoir que la musique et l’astrologie sont en partie etc., car les deux considèrent l’étant mathématique contracté dans quelque sensible, comme l’astrologie considère la grandeur contractée dans le corps céleste. Et la musique ou le musicien considère le nombre contracté dans les sons ou dans les rapports de nombres dans le sons, donc puisque les deux considèrent une chose en partie mathématique et en partie naturelle, elles seront en partie mathématiques et en partie naturelles.) 6 Pour une bonne étude de la notion médiévale de science subordonnée ou science subalterne, voir [Corbini 2009]. LA NOTION MÉDIÉVALE DE CONTRACTIO DANS L’ARITHMETICA INTEGRA 101 fet, la diversité d’une science vient de la diversité de son sujet ; la subordination vient de ce qu’un sujet est sous un sujet, en particulier lorsque la démonstration descend de la science supérieure vers l’inférieure, qui toutes deux s’épaulent. Note, comme cela est clair, que la diversité dans les sujets des sciences doit être considérée de trois manières : l’une est par séparation, comme la grandeur et le nombre harmonique, et celle-ci conduit à une véritable différence entre les sciences ; l’autre est par contractio et elle est double. En effet, ou bien la contractio est produite en raison de la différence dans un même genre et dans une même nature, comme sont les figures planes et le triangle, ou le triangle et l’isocèle, et une telle diversité ne produit pas une science différente, puisque par une même science sont considérés le sujet et les parties du sujet et les espèces. Ou bien la contractio est obtenue en raison de la différence avec un autre genre ou une autre nature, à partir de laquelle cependant ce qui est contracté est susceptible d’être fait un, comme sont la grandeur et le nombre l’une relativement à l’autre. En effet, ils sont de natures différentes et pourtant ils sont susceptibles d’être liés et d’être maintenus unis de sorte que le nombre est contracté en quelque sorte au moyen de la grandeur. En effet, les grandeurs peuvent être déterminées en nombre, de sorte qu’une même démonstration sera faite pour elles comme pour les nombres. En effet, une démonstration qui se déroule dans les nombres de manière absolue, se déroule dans les nombres des grandeurs ou, pour le dire mieux, dans les grandeurs déterminées de manière numérique7. Ainsi, les sciences diffèrent selon leurs sujets. Mais il y a plusieurs modes de différentiation de ces sujets. Soit les sujets sont de genres totalement différents, comme sont la grandeur géométrique et le nombre musical, de sorte que les sciences de ces sujets, la géométrie et la musique, sont des sciences réellement séparées. Soit le sujet d’une science est lié au sujet de l’autre science, ce lien étant effectué au moyen de la contractio. Deux 7 Kilwardby [1976, p. 45–46] : « [...] non videtur mihi inconveniens quod astronomia et geometria sint diversæ scientiæ et tamen una sit alteri subalternata. Diversitas enim scientiæ est ex diversitate subiecti ; subalternatio ex eo quod subiectum est sub subiecto, præcipue cum descendat demonstratio a superiori in inferiorem, quæ duo sese compatiuntur. Quod ut pateat nota quod est considerare diversitatem in subiectis scientiarum triplicem : unam, quæ est per disparationem, sicut magnitudo et numerus harmonicus, et talis facit scientiarum veram diversitatem ; aliam, quæ est per contractionem et haæc est duplex. Aut enim contractio fit per differentiam eiusdem generis et naturæ, ut sunt figura plana et triangulus, vel triangulus et isosceles, et talis diversitas non facit diversam scientiam quia eiusdem scientiæ est considerare subiectum et partes subiecti et species. Aut fit contractio per differentiam alterius generis et naturæ, ex qua tamen et eo quod contrahitur natum sit fieri unum, et sic se habent ad invicem magnitudo et numerus. Sunt enim diversarum naturarum, et tamen nata sunt coniungi et contineri ita ut quodammodo contrahatur numerus per magnitudinem. Possunt enim magnitudines disponi in numero quodam ita quod eadem demonstratio fiet de illis quæ de numeris. Quæ enim demonstratio tenet in numeris absolute, tenet in numeris magnitudinum vel si potius dicitur in magnitudinibus numeraliter dispositis. » 102 S. ROMMEVAUX-TANI cas se présentent alors : soit les sujets d’une science sont des sujets particuliers d’une autre science, comme sont les triangles relativement aux figures planes et dans ce cas la géométrie du triangle est une partie de la géométrie et n’est pas une science nouvelle ; soit un sujet d’un certain genre est contracté dans un sujet d’un autre genre ou d’une autre nature. Ainsi, en reprenant les exemples de la musique et de l’astronomie donnés par Raoul le Breton, celles-ci sont subordonnées respectivement à l’arithmétique et à la géométrie au moyen de la contractio des nombres dans les lignes et des grandeurs dans les corps célestes8. Kilwardby se propose quant à lui de contracter les nombres dans les grandeurs. Et il ajoute : Pour cette raison, une certaine partie de la géométrie, principalement le livre X d’Euclide dans lequel il est question de la ligne irrationnelle, est subordonnée à l’arithmétique, puisque la grandeur disposée numériquement se situe d’une certaine manière sous le nombre et la démonstration arithmétique descend en elle. Et c’est ce que dit Aristote au premier livre des Seconds analytiques, à savoir que la démonstration arithmétique ne descend pas dans la géométrie sauf si les grandeurs sont des nombres, c’est-à-dire disposées numériquement9. Ainsi, l’étude par Euclide des lignes irrationnelles, qui appartient bien pour Kilwardby au domaine de la géométrie, est toutefois subordonnée à l’arithmétique. Notons dans le passage cité la référence aux Seconds analytiques, qui peut sembler problématique. En effet, selon Aristote, la subordination est impossible entre des disciplines dont les objets sont de genres différents, de sorte que la géométrie, dont l’objet est la quantité continue, ne peut pas être subordonnée à l’arithmétique, dont l’objet est la quantité discrète. Ainsi, dans les Seconds analytiques (I, 7, 75a39), ?, p. 103 écrit : « Il n’est donc pas possible de prouver en venant d’un autre genre, par exemple, ce qui est géométrique par l’arithmétique ». Toutefois, quelques lignes plus loin (I, 7, 75b5), Aristote semble ouvrir une brèche : « il n’est pas possible d’appliquer la démonstration arithmétique aux propriétés des grandeurs, à moins que les grandeurs ne soient des nombres » (c’est moi qui souligne). Kilwardby 8 Cette subordination de la musique et de l’astronomie à l’arithmétique et la géométrie a été largement discutée au xiiie siècle, notamment par Robert Grosseteste et Thomas d’Aquin [Corbini 2009]. 9 Kilwardby [1976, p. 46] : « Et ideo aliqua pars geometriæ, præcipue X <liber Euclidis> ubi agitur de linea irrationali [il y a rationali dans le texte], subalternatur arithmeticæ quia magnitudo numeraliter disposita quodammodo est sub numero et descendit arithmetica demonstratio in illam. Et hoc dicit Aristoteles in I Posteriorum quod arithmetica demonstratio non descendit in geometriam nisi magnitudines numeri sint, id est numeraliter dispositæ. » LA NOTION MÉDIÉVALE DE CONTRACTIO DANS L’ARITHMETICA INTEGRA 103 s’appuie sur ces quelques mots pour justifier son argument. David Rabouin [2009, p. 89] note à propos de ce passage d’Aristote qu’il est généralement interprété par les commentateurs modernes comme « stigmatisant une erreur que ne devrait pas commettre celui qui a compris l’incommunicabilité des genres ». Mais, pour sa part, il souligne qu’« Aristote lui-même n’a pas hésité à traiter à l’occasion des grandeurs “comme des nombres” ». Il donne ainsi l’exemple du temps, considéré dans la Physique comme « nombre » du mouvement. Et il poursuit en disant [Rabouin 2009, p. 89–90] : De fait, l’idée de traiter des grandeurs “comme des nombres” ne constitue pas nécessairement une violation de la règle d’incommunicabilité sous deux conditions essentielles : la première est évidemment de disposer d’un langage opératoire commun, sans quoi aucune comparaison n’est possible ; l’autre condition est que l’équivalence ne porte pas directement sur les “sujets”, sans quoi on aura contrevenu au principe, mais sur leurs relations. David Rabouin voit alors dans le livre X des Éléments d’Euclide un exemple d’une telle approche, notamment dans les propositions X. 5 et X. 6 où les rapports entre grandeurs commensurables sont identifiés à des rapports entre nombres et il souligne alors le vocabulaire commun aux nombres et aux grandeurs utilisés dans les démonstrations de ces deux propositions. Dans le passage cité précédemment, Kilwardby en dit trop peu pour que l’on puisse savoir à quoi il pense en faisant référence au livre X. Kilwardby n’a pas la réputation d’être un mathématicien et il n’a pas écrit d’ouvrages de mathématiques de sorte qu’il est difficile de se faire une idée de ce qu’il savait sur le sujet. Notons qu’il aurait pu avoir eu connaissance de la lecture arithmétique que fait Anaritius ou al-Nayrı̄zı̄ (m. 922) du livre X, puisque son traité a été traduit en latin au xiie siècle par Gérard de Crémone [?] ; il a pu aussi avoir lu lui-même les premières définitions du livre et les premières propositions faisant le lien entre grandeurs commensurables et nombres. À la lecture du De ortu scientiarum, il apparaı̂t que Kilwardby est relativement familier avec le texte des Éléments, qu’il cite, dans la version de Robert de Chester ; il renvoie ainsi aux livres I, III, V, VII à IX, et XI, donnant parfois des énoncés exacts de définitions ou propositions [Kilwardby 1976, p. 30–31, 39, 72]10. Pour le livre X, toutefois, il n’y a dans de De ortu scientiarum, que la référence générale au traitement des lignes irrationnelles que nous avons donnée précédemment. 10 Les énoncés qu’il donne sont identiques à ceux qu’on trouve dans la version des Éléments d’Euclide que ses éditeurs, Hubert L. L. Busard et Menso Folkerts, attribuent selon toute vraisemblance à Robert de Chester [Busard & Folkerts 1992]. 104 S. ROMMEVAUX-TANI Signalons pour finir que, dans un autre passage de son texte, Kilwardby [1976, p. 39–40] se demande pourquoi et comment la géométrie considère les nombres, alors qu’elle traite de la grandeur : À la première question on doit répondre que la géométrie ne traite pas des nombres, si ce n’est par accident, puisque de fait les grandeurs sont susceptibles d’être nombrées. En effet, la grandeur est toujours divisible et diminuée et plus elle est diminuée en étant divisée, plus le nombre croı̂t. Pour cette raison, l’arithmétique descend dans la géométrie et par conséquent la géométrie, qui doit traiter entièrement de la disposition des grandeurs, ne peut suffisamment en traiter sans la science des nombres, c’est-à-dire l’arithmétique. Et par conséquent, quand elle doit traiter des grandeurs en tenant compte du fait qu’elles sont susceptibles d’être nombrées, elle traite des nombres, non pas en tant que nombres, mais en tant que grandeurs susceptibles d’être nombrées, et pour cela la géométrie est subordonnée à l’arithmétique11. Ainsi, le nombre apparaı̂t dans la grandeur, par accident, du fait que la grandeur est divisible à l’infini ; pour une division donnée de la grandeur, le nombre de ses parties peut être déterminé. Toutefois, cette explication ne permet pas de comprendre comment les nombres irrationnels sont aussi présents dans les grandeurs géométriques. La lecture arithmétique que fait Stifel du livre X des Éléments d’Euclide dans le livre II de son Arithmetica integra a pour but de le montrer clairement. 2. « CONTRACTIO » DE NOMBRES DANS DES CARRÉS AU LIVRE II DE L’ARITHMETICA INTEGRA DE STIFEL Si le livre X des Éléments d’Euclide contient une classification des lignes irrationnelles en treize espèces, Stifel considère qu’il y est question, de fait, des nombres irrationnels12. Stifel se propose donc de retrouver à partir 11 « Cap. XV. Quare geometria considerat de numeris et quomodo, cum sit de magnitudine. [...] Ad primum dicendum quod geometria non tractat de numeris nisi per accidens quia enim magnitudines numerabiles sunt. Dividitur enim magnitudo semper et diminuitur, et quantum ipsa dividendo minuitur, tantum crescit numerus. Propterea descendit arithmetica in geometriam et ideo geometria, quæ habet omnino dispositionem magnitudinum tractare, non potest sufficienter tractare de ipsis sine scientia de numeris, scilicet arithmetica. Et ideo quando debet de magnitudinis tractare in ratione qua numerabiles sunt, de numeris tractat non propter numeros sed propter magnitudines numerabiles, et in hoc subalternatur geometria arithmeticæ. » 12 « Ad integram tractationem numerorum irrationalium, pertinet liber decimus elementorum Geometriae Euclidis » (Le dixième livre des Éléments de géométrie d’Euclide a pour but de traiter complètement des nombres irrationnels) [Stifel 1544, fo 105r]. Sur le statut problématique du livre X, voir l’analyse qu’en fait Bernard Vi- LA NOTION MÉDIÉVALE DE CONTRACTIO DANS L’ARITHMETICA INTEGRA 105 des nombres irrationnels les différentes espèces de lignes présentées par Euclide grâce à un procédé particulier, la contractio de nombres dans des carrés, qui consiste, un nombre étant donné, à considérer le carré ayant pour surface ce nombre ; les lignes irrationnelles euclidiennes sont alors obtenues comme les côtés de ces carrés pour certains types de nombres irrationnels. Ainsi, Stifel explique que par la contractio des entiers dans des carrés sont obtenues les lignes qu’Euclide nomme exprimables, soit les lignes commensurables à une ligne exprimable donnée de référence, que Stifel appelle lignes rationnelles en longueur (provenant des entiers qui sont des carrés) et les lignes commensurable en puissance seulement à la ligne exprimable de référence, que Stifel nomme lignes rationnelles en puissance seulement (provenant des entiers qui ne sont pas des carrés)13. Stifel [1544, fo 105V] introduit alors la figure ci-dessous, qui permet de visualiser le procédé. Dans cette figure, les nombres 9 et 18 sont contractés (contrahuntur )14 respectivement dans le carré T (c’est le carré composé de neufs petits carrés) et dans le carré R (le plus grand des carrés) et donnent d’une part la ligne AM de mesure p 3, qui est rationnelle en longueur et d’autre part la ligne FD de mesure 18, qui est rationnelle en puissance seulement (c’est-à-dire que son carré est rationnel). Toujours dans cette figure, le trac dans le chapitre « La position du livre X par rapport à la classification des sciences mathématiques » [Euclide 1998, p. 14–18]. 13 Les lignes exprimables sont introduites ainsi par Euclide [1998, p. 34], dans la définition 3 du livre X : « Cela étant supposé, il est démontré que par rapport à une droite proposée, il existe des droites, infinies en multitude, commensurables ou incommensurables avec elle, les unes en longueur seulement, les autres aussi en puissance. D’une part donc, que la droite proposée soit appelée exprimable, et celles qui sont commensurables avec elle, soit en longueur et en puissance, soit en puissance seulement, exprimables ; d’autre part, que celles qui sont incommensurables avec elle soient appelées irrationnelles ». Dans la version de Campanus, qui est utilisée par Stifel (ce dernier le cite et en reprend les énoncés), on a la définition suivante [Busard 2005, p. 306] : « Omnis autem linea cum qua ratiocinamur posita vocetur rationalis. Lineeque ei communicantes dicuntur rationales. Eidem autem incommunicates dicuntur irrationales sive surde. » (Que toute ligne donnée avec laquelle nous raisonnons soit dite rationnelle. Et les lignes qui lui sont commensurables sont dites rationnelles. Et celles qui lui sont incommensurables sont dites irrationnelles ou sourdes). Et on trouve par ailleurs les notions de « lignes rationnelles en longueur » et de « lignes rationnelles en puissance seulement » chez Campanus [Rommevaux 2001, p. 99–100]. 14 Dans mon article [Rommevaux-Tani 2014, p. 178, n. 16], j’avais traduit « contrahuntur » par « sont concrétisés ». Il me semble aujourd’hui qu’il est préférable de garder ici le jeu de mots entre « abstrahere » et « contrahere », entre abstraire et contracter. Par ailleurs, on a aussi en latin le terme « concretus », participe du verbe « concresco », avec un autre sens que « contractus ». 106 S. ROMMEVAUX-TANI F p p D 18 pp C 18 p 18 162 R S B G Z 162 p N 162 K E Q T P A H pp 162 I O M Figure 1 p p nombre 162 est contracté dans le carré S , de côté CH , de mesure 4 162 et est ainsi obtenue la première espèce de ligne irrationnelle, la médiale15. Puis, en contractant ce que Stifel appelle les nombres binomiaux p p et les nombres résidus, c’est-à-dire les nombres de la forme a b, a b et p p a b (où a et b sont des nombres rationnels, le premier terme de la somme ou de la différence étant toujours plus grand que le second), il retrouve les douze autres espèces de lignes irrationnelles euclidiennes [Stifel 1544, fos 112r–113v ; Rommevaux-Tani 2014, p. 185–190]. 15 Cette ligne est définie ainsi par Euclide [1998, p. 151], à la proposition 21 du livre X : « Le rectangle contenu par des droites exprimables, commensurables seulement en puissance, est irrationnel et la droite qui peut le produire est irrationnelle ; qu’elle soit appelée médiale. » C’est la proposition X. 19 dans la version de Campanus [Busard 2005, p. 321] : « Omnis superficies, quam continent due linee potentialiter tantum rationales communicantes, est irrationalis, diciturque superficies medialis. Eiusque latus tetragonicum, scilicet quod in eam potest, est irrationale diciturque linea medialis. » (Toute surface que contiennent deux lignes rationnelles commensurables en puissance seulement est irrationnelle et la surface est dite médiale. Et son côté carré, c’est-à-dire ce qui peut produire celle-ci, est irrationnel et est appelé ligne médiale). 1 LA NOTION MÉDIÉVALE DE CONTRACTIO DANS L’ARITHMETICA INTEGRA Nombre contracté Racine du nombre contracté ou mesure du côté du carré p p 4 a a p p p a+ b, carré d’un bino- c + d avec c2 d mial com. ou incom.pà c ou p c + d avec c d2 p com. oupincom.pà c p ou c+ d avec c d p com. ou incom. à p c p p p p a+b, carré d’un bino- 4 c + 4 d avec 4 c 4 d mial p rationnel p p p p p a + b, carré d’un bi- 4 c + 4 d avec 4 c 4 d nomial irrationnel p p p a + b, non carré d’un a+ b binomial q p p p a + b, non carré d’un a+ b binomial q p p p a + b, non carré a+ b d’un binomial p p a p a 107 Lignes irrationnelles médiale (medialis) binomiale (binomialis) divisée en six sous-espèces bimédiale première (bimedialis prima) bimédiale seconde (bimedialis secunda) majeure (maior ) droite pouvant produire une aire composée d’une exprimable et d’une médiale (potens rationale & mediale) droite pouvant produire une aire composée de deux médiales (potens duo medialia) apotomé (residuum), divisées en six sousespèces p p b, carré de résidu c d avec c2 d com. ou incom.pà c ou p c d avec c d2 p com. oupincom.pà c p ou c d avec c d p com. ou incom. à p c p p p 4 b, carré de résidu 4 c d avec 4 c 4 d apotomé première rationnel d’une médiale (residualis bimedialis prima) 108 p S. ROMMEVAUX-TANI a résidu p b, carré de p p p 4 c d avec 4 c 4 d irrationnel p 4 p p p a b, non carré de a b résidu pp p a b, non carré de a b résidu q p p a b, non carré de a résidu p p apotomé deuxième d’une médiale (residualis bimedialis secunda) mineure (minor ) droite produisant, par adjonction d’une aire exprimable, un tout médial (componens mediale cum rationali) b droite produisant, par adjonction d’une aire médiale, un tout médial (componens mediale cum mediali) Cette construction est une bonne illustration de ce qu’affirmait Kilwardby : « une certaine partie de la géométrie, principalement le livre X d’Euclide dans lequel il est question de la ligne irrationnelle, est subordonnée à l’arithmétique, puisque la grandeur disposée numériquement se situe d’une certaine manière sous le nombre » (voir supra). 3. UN EXEMPLE D’UTILISATION DE LA « CONTRACTIO » DANS L’ARITHMETICA INTEGRA DE STIFEL : L’EXTRACTION DE LA RACINE D’UN BINÔME OU D’UN RÉSIDU Grâce à la contractio des nombres dans les carrés, une partie de la géométrie, l’étude des lignes irrationnelles, est donc subordonnée à l’arithmétique et se faisant, « la démonstration arithmétique descend en elle », selon l’expression de Kilwardby. Nous allons là encore en voir une illustration dans l’Arithmetica integra de Stifel. Nous venons de montrer comment, grâce à la contractio des binômes p et p des résidus, c’est-à-dire des nombres de la forme a b ou a b ou p p a b, dans des carrés, sont produites les lignes irrationnelles décrites par Euclide au livre X des Éléments. Et selon que ces nombres sont des carrés de binômes ou de résidus ou non, sont trouvées différentes espèces de lignes. Aussi, au chapitre x du livre II, Stifel se propose d’extraire la racine carrée d’un binôme ou d’un résidu. La règle est la suivante [Stifel 1544, fos 127v–128r ; Rommevaux-Tani 2014, p. 189] (notons que Stifel appelle LA NOTION MÉDIÉVALE DE CONTRACTIO DANS L’ARITHMETICA INTEGRA « parties » (particula) les nombres qui entourent le signe + ou binôme ou le résidu) : 109 dans le Premièrement. De ton binôme ou de ton résidu (dont la racine doit être extraite), pose la moitié. Deuxièmement. Prends les carrés des parties de cette moitié ainsi posée, soustrais ces carrés entre eux et conserve la racine de ce qui reste. Troisièmement. Prends la plus grande partie de la moitié (de ton binôme ou de ton résidu, posée précédemment à la place du tout) et ajoute-la à la racine conservée auparavant. Et la racine carrée de cette somme sera la première partie de la racine que tu cherches. Quatrièmement. Soustrais la même racine du reste, conservée auparavant, de cette partie plus grande précédente (de cette moitié que tu as d’abord posée à la place du tout, lorsque tu as divisé en deux ton binôme ou ton résidu). La racine carrée du nouveau reste sera la deuxième partie de la racine que tu cherches. Alors, entre les deux parties de ta racine trouvée, pose le signe des ajouts ou des soustraits. En effet, si on a la racine du binôme, le signe + doit être interposé. Et si c’est la racine du résidu, alors le signe doit être interposé16. En écriture moderne : sr r q p p a a + a b= 4 4 s r r q p a a ab= + 4 4 s r r q p a a2 a b= + 4 4 sr r a a b ; 4 4 4 s r r b2 a a b2 ; 4 4 4 4 sr r b a2 a b : 4 4 4 4 b 4 Stifel illustre la règle par plusieurs exemples, vérifiant que les calculs aboutissent au bon résultat en élevant au carré le nombre trouvé. Et il ajoute [Stifel 1544, fos 129r–v] : 16 « Radicum uero extractiones sic fiunt. Primo. Pro binomio aut residuo tuo (de quo radix est extrahenda) pone dimidium eius. Secundo. Recipe quadrata particularum dimidij illius sic positi, eaque quadrata subtrahe ab inuicem, & radicem relicti illius serua. Tertio. Recipe maiorem particulam dimidij (tui binomij, aut residui, primo positi loco sui integri) eamque primo adde ad radicem prius seruatam, & radix quadrata, illius aggregati, erit particula prima radicis quam quæris. Quarto. Eandem radicem relicti prius seruatam, subtrahe à priore illa particula maiore (illius dimidij, quod primo posueras, loco sui integri, dum dimidiares tuum binomium, aut residuum). Et radix quadrata illius relicti nouissimi, erit particula secunda radicis quam quæris. Duabus ergo particulis tuæ radicis inuentæ, interpone signum additorum uel subtractorum. Nam si radix binomij, interponendum est signum +. Si autem sit radix residui, tunc interponendum est signum . » 110 S. ROMMEVAUX-TANI Je tente maintenant de donner la raison de ce mode d’opération, les nombres (que j’ai pris pour exemples) étant contractés dans des surfaces carrées, p comme ce binôme 38 + 288 est contracté dans la surface carrée. 2 p 72 p p 36 6 72 p 2 + 2 6 Figure 2 p Tu vois de manière certaine que sa racine 6+ 2, par une multiplication par elle-même, forme cette surface partagée selon ses parties, dont deux (qui sont rationnels) s’associent dans un nombre abstrait et produisent sa plus grande partie et les deux autres s’associent aussi et produisent sa partie irrationnelle17. p Stifel lit sur la figure que le carré formé sur la ligne de mesure 6 + 2 se décompose en deux carrés dont les surfaces sont des nombres entiers etpdont la somme fait 38 et deux rectangles égaux qui, pris ensemble, font p 2 72, soit 288. Nous reconnaissons ici la figure associée à la proposition 4 du livre II des Éléments d’Euclide (fig. 3), qui, en termes arithmétiques, revient au développement du carré d’une somme, (a + b)2 = a2 + b2 + 2ab, mais qui dans les termes géométriques d’Euclide s’énonce ainsi [Euclide 1990, p. 331] : Si une ligne droite est coupée au hasard, le carré sur la droite entière est égal aux carrés sur les segments et deux fois le rectangle contenu par les segments18. 17 « Tentabo nunc huiusmodi operationum reddere rationem, numeris (quos p pro exemplis posui) contractis ad superficiem quadratas : ut binomium hoc 38 + ÿ288, p sic contrahitur ad superficiem quadratam. [figure] Certe uides, ut radix eius 6 + ÿ2, multiplicatione sua in se, constituat superficiem hanc, distributam per suas particulas, quarum duæ (quæ sunt rationales) in numero abstracto coeunt, faciuntque particulam eius maiorem : & reliquæ duæ etiam coeunt, faciuntque particulam eius irrationalem. » 18 Dans la version de Campanus, utilisée par Stifel, l’énoncé de la proposition 4 du livre II est formulé ainsi [Busard 2005, p. 97] : « Si fuerit linea in duas partes divisa, LA NOTION MÉDIÉVALE DE CONTRACTIO DANS L’ARITHMETICA INTEGRA A B 111 C H K D E F Figure 3 Stifel évoque cette proposition au livre I de son Arithmetica integra, dans une notabilia. Il y explique [Stifel 1544, fo 34r] : Si deux de ces termes [il vient d’évoquer les termes d’une progression géométrique commençant à l’unité] sont des carrés, alors, si on ajoute à ces carrés le médian doublé, la somme de tous fera nécessairement un carré, comme l’indique suffisamment cette figure19. Stifel explique que dans la progression géométrique de raison a et de premier terme 1, a2k + a2` + 2ak+` est un carré, pour k et l entiers donnés quelconques. Il ne précise pas que c’est le carré de ak + a` . Il ne donne pas de démonstration de ce résultat mais renvoie à la figure ci-dessous (fig. 4), dans laquelle des nombres sont contractés dans des surfaces (même si Stifel ne le dit pas ici) et qui montre avec suffisamment d’évidence (satis indicat) que la propriété est vraie dans le cas particulier des illud quod ex ductu totius in seipsam fit equum est hiis que est ductu utriusque partis in seipsam et alterius partis in alteram bis. » (Si une ligne est divisée en deux parties, ce qui est produit à partir de la multiplication de la totalité par elle-même est égal à ce qui est produit par la multiplication de chaque partie par elle-même et de l’une par l’autre deux fois). Le langage géométrique du carré est remplacé par l’expression « ex ductu » qui renvoie à l’opération de multiplication (pour cette signification du verbe « ducere », voir [Guillaumin 2020, p. 117–118]). Ainsi Stifel a accès à une version de cette proposition qui peut le conduire naturellement à faire le parallèle entre la figure géométrique et l’interprétation arithmétique. 19 « Et si duo illi termini fuerint quadrati, tunc si medium duplicatum superaddas illis terminis quadratis, tunc aggregatum ex omnibus illis necessario etiam fiet quadratum, ut hæc figura satis indicat. » 112 S. ROMMEVAUX-TANI carrés 4 et 16. Il fait alors le lien avec la proposition 4 du livre II des Éléments d’Euclide [Stifel 1544, fo 34r] : En plus de celle-ci la figure de la proposition 4 du livre II a de magnifiques usages en algèbre20. 4 8 8 16 Figure 4 À cette première propriété fait suite une seconde [Stifel 1544, fo 34v] : Si deux de ces termes [toujours d’une progression géométrique commençant à l’unité] sont des cubes, alors le regroupement de ces deux cubes et de chaque médian triplé, fait un cube, dont la racine cubique est la somme de deux racines cubiques des deux cubes plus petits susdits21. a3k + a3` + 3a2k+` + 3ak+2` est le cube de ak + a` . Là encore, la figure ci-dessous (fig. 5) vaut comme seule explication. 8 16 16 2 32 32 32 4 64 4 2 Figure 5 20 « Et præter hæc figura illa propositionis quartæ secundi Euclidis, usus habet magnificos in Algebra, ut suo loco ostendam. » C’est par exemple le cas pour la résolution de certains systèmes de deux équations à deux inconnues [Rommevaux-Tani 2016, p. 103–104]. 21 « Si duo illi termini extremi fuerint cubici, tunc coaceruatio ex illis duobus cubicis, & ex utroque medio triplicato, fit cubicus, cuius radix cubica sit aggregata ex duabus radicibus cubicis, prædictorum cubicorum minorum. » LA NOTION MÉDIÉVALE DE CONTRACTIO DANS L’ARITHMETICA INTEGRA 113 Une figure analogue est présente au chapitre x du livre II quand, après avoir donné la règle pour déterminer la racine carrée d’un binôme et d’un o résidu, Stifel [1544, de l’extraction de la racine p f 130r] donne l’exemple p cubique de 252+ 24 200, qui vaut 6+ 2. La figure p (fig. 6) fait apparaı̂tre les éléments du développement au cube de 6 + 2 et suffit à Stifel comme justification : Le binôme contracté dans le cube est vu nettement sous les parties qui le composent22. p 8 p 12 p 2 2592 p 216 6 2 12 p 2592 p 2 + 6 Figure 6 Ainsi, la figure géométrique dans laquelle sont contractés les nombres donnés dans la règle arithmétique considérée est un support qui fait voir le résultat. Notons que Stifel ne donne pas d’algorithme général pour l’extraction de la racine cubique d’un binôme ou d’un résidu comme il le fait pour l’extraction de la racine carrée. p À la suite de l’extraction de la racine carrée du binôme 38+ 288, Stifel [1544, fos 130v–131r] propose un deuxième exemple : l’extraction de la p racine carrée du résidu 18 4. Lapméthode consiste la racine p à chercher p carrée du binôme correspondant 18 + 4, soit 4 8 + 4 2. La racine de 22 « Binomium autem illud contractum ad cubum, cernitur sub istis particulis compositionis. » 114 S. ROMMEVAUX-TANI p p 4 18 4 est alors 4 8 2. Stifel le vérifie en faisant la multiplication, dont il illustre les étapes dans une figure (fig. 7) : p p 4 2 p 4 p 4 + 8 D 2 p 4 2 A B p 4 2 C D p 4 p 8 4 2 D p 4 p 8 4 2 Figure 7 p Son binôme est d’abord contracté dans une surface carrée. Et ce binôme est 18 + 4. Alors l’extraction de la racine du résidu ne diffère en rien de l’extraction de la racine du binôme, pour ce qui concerne cette opération qui est faite selon la règle, si ce n’est que le signe est posé dans les résidus, là où le signe + est posé dans les binômes. Et celui qui comprend la multiplication au carré comprend aussi l’extraction de la racine. Et on a la multiplication ainsi : p p 4 4 8 p 2 p 4 4 8 2 p p 8+ 2 2 2 p p Par la première multiplication (c’est-à-dire 4 8 par 4 8) est construite la surp p p 4 4 face ABCD, qui fait 8. Par la deuxième multiplication, soit 2 par 2 est p construite la surface carrée + 2. Par la troisième multiplication, c’est-à-dire p p 4 8 par 4 2 est faite la surface E 23 qui doit p être soustraite des deux surfaces construites précédemment, soit de ABCD et 2. C’est pourquoi la surface E retranche la surface AB . De même, la surface F , qui est faite à partir de la quatrième multiplication, retranche aussi une surface qui lui est égale, qui est 23 La surface E , à retrancher, est équivalente aux deux carrés A et B . Stifel la matérialise par des hachures verticales. Voir notre explication à la suite du texte. LA NOTION MÉDIÉVALE DE CONTRACTIO DANS L’ARITHMETICA INTEGRA 115 AC 24. Mais A avait p déjà été retranchée par la surface E .pQue l’on conçoive alors que la surface 2 soit mise à la place de p A, puisque 2 et A sont égales. C’est pourquoi la surface F soustrait C et 2. Et ainsi, il reste finalement la p surface D seulement, qui fait 18 4. Et c’est pla surface de laquelle est extraite la racine au moyen de la surface binomiale, 2EFABCD 25. p p L’algorithme d’élévation au carré de 4 8 4 2 se lit sur la figure, Stifel utilisant un jeu de hachures pour rendre visible ce qui est p ajouté p et ce qui retranché au cours p de l’opération. Il veut montrer que 8 + 2 2 2 p est bien p le carré de 4 8 4 2. Il commence donc par construire le carrépde p p 4 4 4 8, surface 18 + 4 et de côté 8 + 2. Il remarque que le carré de côté p formé des quatre carréspA, B , Cpet D, correspond au premier terme, 8, 4 obtenu enpmultipliant 8 par 4 8. Le deuxième terme,pobtenu en mulp 4 4 2 par 2, donne une surface de mesure 2, que Stifel ne tipliant nomme pas sur la figure, mais quip est le petit pcarré en haut à gauche. Au troisième terme, correspondant à 4 8 par 4 2, Stifel associe une surface E , qui équivaut aux deux carrés A et B . Il faut donc p retrancher ces deux carrés de A, B , C , D et de la surface de mesure 2 ; cette soustraction est signalée sur la figure par des hachures p verticales. p Au quatrième terme, correspondant à la multiplication de 4 8 par 4 2, Stifel associe une surface F , équivalente à A et C , qu’il faut là encore retrancher. p Toutefois, A a déjà été retranchée. Stifel remarque alors qu’elle vaut 2. La surface F 24 La surface F , à retrancher, est équivalente aux deux carrés A et C . Stifel la matérialise par des hachures horizontales. 25 « Contrahitur autem primo, eius binomium ad superficiem quadratam. Eius p autem binomium est ÿ18 + 4. Vnde nihil differt extractio radicis residui, ab extractione radicis binomij, quantum attinet ad operationem illam quæ fit iuxta regulam : nisi quod signum ponitur in residuis, ubi signum + ponitur in binomijs. Qui autem intelligit multiplicationem quadratam, ille etiam intelligit radicis extractionem. Sic autem habet multiplicatio. p p pÿÿ8 pÿÿ2 ÿÿ2 p ÿÿ8 p ÿ8 + ÿ2 2 2 p p Prima multiplicatione (id est ÿÿ8 in ÿÿ8)pconstituitur p p superficies ABCD , quæ facit ÿ8. Secunda multiplicatione, uidelicet ÿÿ2 in superficies p p ÿÿ2, constituitur p quadrata + ÿ2. Tertia multiplicatione, uidelicet + ÿÿ8 in ÿÿ2, fit superficies E p subtrahenda à superficiebus duabus prius constitutis, uidelicet ab ABCD & ÿ2. Tollit itaque superficies E , superficiem AB . Sic superficies F , quæ fit ex quarta multiplicatione, tollit etiam sibi æqualem, p ut est AC . Sed A prius sublata fuit p per superficiem E . Ideo intelligatur superficies ÿ2 inplocum A successisse, cum ÿ2 & A sint æquales. Itaque psuperficies F , subtrahit C & ÿ2. Et sic tandem manet superfiecies D sola, quæ facit ÿ18p 4. Et est illa superficies de qua extrahitur radix, mediante superficie binomiali, ÿ2EFABCD . » 116 S. ROMMEVAUX-TANI p est donc équivalente à C et au petit carré de surface 2. Stifel signale par des hachures horizontales qu’ils p doivent être retranchés. Ainsi, de A, B , C , D et du petit carré de surface 2, sont retranchés A, B , C et le petit carré, p p 4 4 de sorte qu’il reste D, soit le carré de 8 2. C’est bien ce que Stifel cherchait à montrer. Il est intéressant de noter ici comment Stifel identifie la surface avec le nombre p parle de « la surface ABCD qui p qui y est contracté, de sorte qu’il fait 8 » (superficies ABCD,pquæ facit ÿ8) et qu’il ne désignepmême p pas la surface correspondant à 2, mais la nomme simplement + 2(+ ÿ2) ; ici le signe +, qui est le signe des ajouts (signum additorum), indique que la surface doit être ajoutée. 4. CONCLUSION La notion de contractio est une notion couramment utilisée au Moyen Âge, dans le contexte universitaire, dans le cadre des discussions sur la subordination des disciplines. Les extraits des Questiones mathematicæ de Raoul de Breton que nous avons analysés à ce propos en montrent clairement les enjeux. Si les exemples de l’astronomie, et surtout de la musique, sont habituels, la référence au livre X des Éléments d’Euclide par Robert Kilwardby dans le De ortu scientiarum est plus étonnant. Ce dernier, s’il souligne que l’étude des lignes irrationnelles, objet du livre X, est subordonnée à l’arithmétique, n’explique pas comment s’effectue cette subordination. Le recours à l’Arithmetica integra de Michael Stifel permet d’en donner une illustration. Ainsi, nous avons vu comment la contractio de nombres irrationnels dans des carrés permet à Stifel de proposer une lecture arithmétique du livre X des Éléments d’Euclide qui le conduit à identifier la figure géométrique au nombre qu’il lui associe, comme dans le dernier exemp ple que nous avons p examiné, où Stifel utilise l’expression « carré 2 » (superficies quadrata ÿ2). Par ailleurs, il met en relation propriétés des nombres et propriétés des figures, la figure apportant un support visuel aux opérations sur les nombres. Ainsi, dans ce que propose Stifel, on a affaire non seulement à une descente des démonstrations arithmétiques dans la géométrie (selon l’expression de Kilwardby), mais aussi à une remontée des propriétés des figures vers les propriétés des nombres26. 26 Il s’agit sans doute pour Stifel de justifier ainsi l’usage des nombres irrationnels, dont le statut ne va pas de soi. Ainsi, dans le premier chapitre du livre II de l’Arithmetica integra, Stifel se demande si les nombres irrationnels sont de vrais nombres ou si ce LA NOTION MÉDIÉVALE DE CONTRACTIO DANS L’ARITHMETICA INTEGRA 117 Notons à ce propos que l’on retrouve chez Stifel le balancement entre abstractio et contractio que nous avons souligné dans les Questiones mathematicæ de Raoul le Breton. En effet, alors qu’il commente les deux premières définitions du livre X des Éléments d’Euclide, Stifel remarque que dans la première, Euclide définit les quantités commensurables et les quantités incommensurables27, alors que dans la deuxième il est question de lignes commensurables en puissance. Et il ajoute que, quoi qu’il en soit, Euclide ne traite que des quantités car, selon lui [Stifel 1544, fo 106r ; Rommevaux-Tani 2014, p. 177–178] : C’est comme s’il [Euclide] disait : nous parlons d’une manière de la commensurabilité et de l’incommensurabilité des quantités abstraites (c’est-à-dire des nombres irrationnels) et nous parlons d’une autre manière des mêmes quantités quand elles sont contractées dans les lignes28. Notons pour finir que si nous avons choisi d’illustrer les propos de Kilwardby par la construction de Stifel, nous ne prétendons pas que le texte de Kilwardby, ni même celui de Raoul le Breton, ont été une source des réflexions de Stifel. Il est en effet peu probable qu’il ait lu le De ortu scientiarum ou les Questiones mathematicæ, ces deux textes n’ayant pas été édités à la Renaissance. Mais les idées contenues dans ces introductions à la philosophie ont pu se diffuser dans les universités jusqu’au temps de Stifel. On retrouve par exemple la notion de contractio dans la Summa totius logicæ de ps-Thomas d’Aquin (Tractatus 6, caput 1) à propos de la définition des formes abstraites29 ou encore chez Jean Buridan (m. 1363) dans le cadre d’une discussion sur la subordination des sciences [Biard 2015, p. 158– 159]30. sont des nombres fictifs (numeri ficti) [Stifel 1544, fos 103r–104r] et dans le deuxième chapitre il justifie l’usage de ces nombres par leur intérêt pour le livre X des Éléments d’Euclide [Stifel 1544, fos 104r–104v ; Rommevaux-Tani 2014, p. 179–182]. 27 Dans l’énoncé de cette première définition, que Stifel reprend à Campanus, il est bien question de « quantitas » quand Euclide parle de « grandeur » [Rommevaux-Tani 2014, p. 176]. 28 « Quasi dicat : aliter loquimur de commensurabilitate & incommensurabilitate quantitatum (id est, numerorum irrationalium) abstractarum : atque aliter loquimur de eisdem, quando contrahuntur ad lineas. » 29 Voir http://www.corpusthomisticum.org/xpl.html 30 Notons que le balancement abstractio/contractio, que l’on trouve chez Stifel, conduit à chercher l’origine de la notion de contractio dans le cadre des commentaires à Aristote. Toutefois, l’usage du terme contractio se trouve aussi dans un cadre métaphysique et théologique, notamment chez Nicolas de Cues, qui explique la participation de Dieu dans le monde par le concept de contractio [Federici Vescovini 2002, p. 114]. Je remercie là encore Vincenzo De Risi qui m’en a fait la remarque. 118 S. ROMMEVAUX-TANI RÉFÉRENCES Aristote [2008] Métaphysique, présentation et traduction de Duminil (Marie-Paule) & Jaulin (Annick), Paris : GF Flammarion, 2008. Biard (Joël) [2015] La subalternation selon Jean Buridan, dans Biard (Joël), éd., Raison et démonstration : Les commentaires médiévaux sur les Seconds Analytiques, Turnhout : Brepols, 2015, p. 151–168. Busard (Hubert L. L.) [2005] Campanus of Novara and Euclid’s Elements, 2 vols., Stuttgart : Franz Steiner, 2005. Busard (Hubert L. L.) & Folkerts (Menso) [1992] Robert of Chester’s (?) Redaction of Euclid’s Elements, the so-called Adelard II Version, Basel, Boston, Berlin : Birkhäuser, 1992. Cleary (John J.) [1985] On the Terminology of ‘Abstraction’ in Aristotle, Phronesis, 30-1 (1985), p. 13–45. Corbini (Amos) [2009] Non ergo est ex alio genere descendentem demonstrare : The Prohibition of Descensus, Subalternation and the Relation between Sciences, Documenti e studi sulla tradizione filosofica medievale, 20 (2009), p. 292–328. Euclide [1990] [1998] Les Éléments, vol.1 : Livres I-IV, introduction générale de Caveing (Maurice) ; traduction et commentaires de Vitrac (Bernard), Paris : Presses universitaires de France, 1990. Les Éléments, vol.3 : Livre X, traduction et commentaires de Vitrac (Bernard), Paris : Presses universitaires de France, 1998. Federici Vescovini (Graziella) [2002] Nicolas de Cues et les transcendantaux, dans Federici Vescovini (Graziella), éd., Le problème des transcendantaux du xive au xviie siècle, Paris : Vrin, 2002, p. 103–120. Guillaumin (Jean-Yves) [2020] Dictionnaire de la terminologie latine ancienne de l’arithmétique et de la géométrie, Paris : Les Belles Lettres, 2020. Hentschel (Frank) [2000] Sinnlichkeit und Vernunft in der mittelalterlichen Musiktheorie, Stuttgart : Franz Steiner, 2000. Kilwardby (Robert) [1976] De ortu scientiarum, ed. Judy (Albert G.), Toronto : The British Academy and the Pontifical Institute of Mediaeval Studies, 1976. LA NOTION MÉDIÉVALE DE CONTRACTIO DANS L’ARITHMETICA INTEGRA 119 Maierù (Alfonso) [2013] Robert Kilwardby on the division of sciences, dans Lagerlund (Henrik) & Thom (Paul), éd., A companion to the Philosophy of Robert Kilwardby, Leiden, Boston : Brill, 2013, p. 353–389. Rabouin (David) [2009] Mathesis Universalis : l’idée de « mathématique universelle » d’Aristote à Descartes, Paris : Presses universitaires de France, 2009. Rashed (Roshdi) [2011] D’Al-Khwarı̄zmı̄ à Descartes : études sur l’histoire des mathématiques classiques, Paris : Hermann, 2011. Rodrı́guez Arias (Jesùs M.) [1997] La doctrina de Roberto Kilwardby sobre la abstracción cientifica, Ciencia Tomista, 124, no 404 (1997), p. 465–486. Rommevaux (Sabine) [2001] Rationalité, exprimabilité : une relecture médiévale du livre X des Éléments d’Euclide, Revue d’histoire des mathématiques, 7 (2001), p. 91–119. Rommevaux-Tani (Sabine) [2014] Irrationalité des nombres, irrationalité des lignes selon Michael Stifel et Simon Stevin, Revue d’histoire des mathématiques, 20-2 (2014), p. 171– 209. [2016] Michael Stifel, lecteur de la Practica arithmetice de Gerolamo Cardano, Bolletino di storia delle scienze matematiche, 36-1 (2016), p. 83–110. Sharp (Dorothea Elisabeth) [1934] The De ortu scientiarum of Robert Kilwardby (d. 1279), The New Scholasticism, 8-1 (1934), p. 1–30. Stifel (Michael) [1544] Arithmetica integra, Nürnberg : Johann Petreius, 1544. Revue d’histoire des mathématiques 26 (2020), p. 121–172 doi:10.24033/rhm.230 OBERWOLFACH IN THE FRENCH OCCUPATION ZONE: 1945 TO EARLY 1950s Volker R. Remmert Abstract. — This paper is part of a larger research project dealing with the history of the Oberwolfach Research Institute for Mathematics between its foundation in 1944 and the early 1960s. While the history of its foundation is relatively well understood, the development of the institute after 1945 has scarcely been touched on by historians (of mathematics). After World War II the challenge faced by the institute was twofold. On the one hand, it had to virtually reinvent itself, i.e. to strip itself of the agenda of war-related mathematical research and find a new identity suited for the post-war situation. On the other hand, the institute was without a budget as it had completely relied on funds from Berlin, which permanently stopped flowing with the end of the war. To understand how the institute and its director, Wilhelm Süss, dealt with this twofold challenge, the consequences of the institute coincidentally being situated in the French occupation zone, and the potential implications of this political happenstance for the history of mathematics in post-war Germany, will be centre-stage in what follows. The history of the Oberwolfach Institute in the late 1940s and early 1950s cannot be understood without embedding it into the political and cultural context of the French occupation zone, which had a long-term impact on its institutional identity. Co-operation with French mathematicians and with Texte reçu le 10 février 2020, accepté le 27 mai 2020, révisé le 24 juillet 2020. V. R. Remmert, Interdisziplinäres Zentrum für Wissenschafts- und Technikforschung (www.izwt.de), Bergische Universität Wuppertal, Gaussstrasse 20, D- 42097 Wuppertal. Courrier électronique : [email protected] 2000 Mathematics Subject Classification : 01A60, 01A74, 01A80. Key words and phrases : Oberwolfach Research Institute for Mathematics, FrancoGerman relations in mathematics after 1945, mathematics in Germany after 1945, Bourbaki. Mots clefs. — Oberwolfach, institut de recherche mathématique, relations francoallemandes en mathématiques après 1945, mathématiques en Allemagne après 1945, Bourbaki. © SOCIÉTÉ MATHÉMATIQUE DE FRANCE, 2020 122 V. R. REMMERT the French authorities became crucial for developing a new vision for the institute’s institutional identity. Résumé (L’institut Oberwolfach dans la zone française d’occupation : de 1945 aux années 1950) Cet article fait partie d’un projet de recherche de plus grande envergure sur l’histoire de l’Institut de recherche mathématique d’Oberwolfach de sa fondation en 1944 au début des années 1960. Tandis que le contexte de la fondation de cet institut est relativement bien connu, son développement après 1945 a peu été étudié par les historiens des mathématiques. La fin de la Seconde Guerre mondiale a placé l’institut face à un double défi. Contraint, d’une part, à se réinventer afin de se débarrasser du programme fixé par la guerre à la recherche mathématique, l’institut s’est cherché une nouvelle identité conforme au contexte d’après-guerre. Il s’est d’autre part trouvé dénué de tout budget suite à l’arrêt définitif des financements venus de Berlin et dont il avait été entièrement dépendant durant la guerre. L’objectif principal de cet article est d’étudier la manière dont cet institut, avec son directeur Wilhelm Süss, a relevé ce double défi en s’inscrivant dans la conjoncture, nouvelle, de la zone d’occupation française. Il s’agira ainsi de saisir les implications de cette conjoncture politique spécifique pour l’histoire des mathématiques dans l’Allemagne d’après-guerre. De la fin des années 1940 au début des années 1950, l’histoire de l’Institut d’Oberwolfach ne pourrait être comprise hors du contexte politique et culturel de la zone d’occupation française tant ce contexte a participé à une redéfinition de l’identité de l’institution sur le temps long. La coopération avec les mathématiciens français et les autorités françaises était désormais devenue essentielle au développement d’une nouvelle vision de l’Institut. This paper is dedicated to David E. Rowe on the occasion of his 70th birthday One of the first contacts the then National Institute for Mathematics (Reichsinstitut für Mathematik) in Oberwolfach had with French officials took place on Saturday, May 26, 1945. In his diary entry for that day, William Threlfall (1888–1949), deputy director of the Institute at the time, wrote: Süss is back from Freiburg. Second lieutenant Prudhomme of the Institut Pasteur comes to inspect the Institute and takes a calculating machine away with him. 1 John Todd (1911–2007) visited the Oberwolfach Institute in early July 1945 as a British Naval officer on behalf of the Admiralty Computing Service affiliated to the Combined Intelligence Objectives Sub-Committee (CIOS) while undertaking a survey of applied mathematical research in Germany [Todd 1 Diary of William Threlfall, May 26, 1945: “Süss aus Freiburg zurück. Vom Institut Pasteur kommt Prudhomme Souslieutenant, um das Institut zu besichtigen, nimmt eine Rechenmaschine mit.” I am grateful to Klaus Volkert for a copy from the Threlfall diary; cf. [Volkert 2018]. OBERWOLFACH IN THE FRENCH OCCUPATION ZONE: 1945 TO EARLY 1950S 123 1983, 19]. In 1946 he reported that “Süss had been interrogated on May 26 [1945] by Souslieutenant Prudhomme of Institut Pasteur” [Todd & al. 1946, 17]. While we know nothing about the interview or about the interviewer Prudhomme, these remarks indicate that the Oberwolfach Institute seemed to have somehow come to the attention of the French as well as the British in 1945. What did they find in Oberwolfach? 1. THE OBERWOLFACH INSTITUTE 1944/45 The National Institute for Mathematics in Oberwolfach had been founded in autumn 1944 as an institution geared towards organising and carrying out war-related mathematical research. The developments leading to the founding of the Oberwolfach Institute are well-known [Epple et al. 2005; Mehrtens 1996; Remmert 1999]. The Freiburg mathematician Wilhelm Süss (1895–1958), as president of the German Mathematicians Association (Deutsche Mathematiker-Vereinigung, DMV ), had been the driving force behind the founding. He became the first director of the institute and stayed in office until his death. Süss, while not at all a first-rate mathematician, was a first-rate organiser and had a golden diplomatic touch [Remmert 1999, 13f]. He had been president of the DMV since 1937 until the DMV petered out of existence after the war (to be newly founded without and against him by Erich Kamke in Tübingen in the French occupation zone in 1948) and rector of Freiburg University from 1940 to 1945. Thus, he was on rather good terms with the Ministry of Education and Research in Berlin as well as the Reich Research Council and the relevant Nazi officials in Berlin. I repeat my summary assessment of Süss’ political comportment, especially as president of the DMV, during the Nazi period as expressed in [Remmert 1999, 37]: [...] the DMV ’s professional policies had become closely entangled with issues at the very core of the Nazi state: its anti-Semitism, its anti-internationalism and its striving for autarky. The Ministry of Education and Research pursued the objective to transmit these issues to the sphere of the sciences. The collaboration of the DMV board and especially of Süss in this program, which was beyond their control, was the basis of their influence and their successful professional activities during the war. And, indeed, Süss’s efforts were rewarded with the foundation of the Oberwolfach Institute in the Black Forest in 1944, funded by the Reich Research Council, with a clear agenda to undertake and organize war-related 124 V. R. REMMERT mathematical research. To be fair, Süss was not exclusively interested in founding an institute for war-related mathematical research, but at the same time strove for “an institution that would dedicate itself, even beyond the (victorious) end of the war, to a wide spectrum of pure and applied mathematical research” [Epple et al. 2005, 151], in a model combination of the Italian institutes of Mauro Picone, the Istituto Nazionale per le Applicazioni del Calcolo (INAC), founded in 1933, and of Francesco Severi, the Istituto Nazionale di Alta Matematica (INDAM ), founded in 1939. 2 The official application for the Oberwolfach Institute, written by Süss in the summer and submitted in early August 1944, defined three essential tasks: (a) “Promotion of the mathematical sciences and their applications in the broadest possible sense,” (b) “expansion of departments into calculation institutes and mathematical production institutes with specific mathematical and technical equipment” and, finally, (c) “general tasks” including “a central office for mathematical reports,” “the drawing-up of a card-index on mathematicians for the tracing and optimum use of workers,” as well as “the establishment of a central information and inspection office for mathematical problems.” 3 These tasks were not aimed at actual mathematical work, but rather at the organization and consolidation of resources. The staff Süss envisioned consisted of a director (Süss), his deputy, three heads of department at the rank of professor, ten scientific and five technical assistants, and two draughtswomen, as well as librarians and further support staff. Such extensive plans could, of course, not be carried out in autumn 1944 as there was no way to secure the intended number of staff. However, the Oberwolfach Institute slowly started work under its director Süss. He appointed Emanuel Sperner, one of his colleagues on the board of the DMV, deputy director (to be replaced by William Threlfall, who knew French, in April 1945, as Sperner had been a member of the Nazi party since 1933). Sperner had done war-related mathematical research with the meteorological research group in Hamburg and brought his assistant Walter Stakowski to Oberwolfach. 4 The Dutch mathematician Gerrit Bol, who had taught in Greifswald from 1942 to 1945, and Herbert Seifert, who had worked at Adolf Busemann’s 2 On Picone and the INAC see [Epple et al. 2005, 141–148]; [Remmert 2017]; on Severi and the INDAM see [Goodstein & Babbitt 2012]; cf. [Guerraggio & Nastasi 2005, passim]; [Remmert 2017]). 3 Walter Gerlach’s application to the Reich Research Council, August 2, 1944 (University Archives Freiburg (UAF), C 89/4). On this and the following see [Epple et al. 2005, 152–154], paraphrased and quoted here. 4 Mentioned by Threlfall in his diary, Sept. 14, 1944. OBERWOLFACH IN THE FRENCH OCCUPATION ZONE: 1945 TO EARLY 1950S 125 Institute for the Dynamics of Gases in Braunschweig were appointed heads of department. Moreover, Hermann Boerner, who had formerly worked with the Reich Meteorological Service (Reichswetterdienst), and Wilhelm Maak, from the University of Hamburg, joined the institute as assistant scientists. All of them had experience in war-related mathematical research. These staff members were joined by mathematicians who were granted guest status having taken refuge in Oberwolfach from throughout the German Reich: Heinrich Behnke who had been bombed out in Münster; William Threlfall, like Seifert, had come to Oberwolfach from Braunschweig; and Henry Görtler, one of the leading mathematicians at Prandtl’s Institute for Fluid Dynamics in Göttingen who was designated for a professorship of applied mathematics in Freiburg, stayed in Oberwolfach from late 1944 and made a formal request to establish a “unit for mathematical fluid dynamics”; 5 George Lorentz, Wilhelm Magnus, Theodor Schneider and Leopold Vietoris spent time in Oberwolfach between October 1944 and May 1945. 6 Moreover, mathematicians from Freiburg, which had been massively bombed in late November 1944, came to Oberwolfach as well—not only Süss and his family, along with his assistants Hans Schubart and Hermann ter Hell, but also the Freiburgbased French mathematicians Frédéric Roger and Charles (Karl) Pisot. Hellmuth Kneser, Süss’s long-time friend and main mathematical advisor, visited from Tübingen frequently with his wife. Given that the war was nearing its end, the war research agenda of the Oberwolfach Institute could not be realized. Thus what the French and the British intelligence units found in Oberwolfach in May 1945 was the nucleus of an institute furnished with a library (mostly taken from Strasbourg and soon to be returned by Pisot) and a few calculating machines, but devoid of a mission and with its funding from Berlin cut off [Remmert 2019]. For Süss as director of the institute the challenge this posed was twofold. On the one hand, he had to virtually reinvent the institute, i.e. to strip it of its agenda of war-related mathematical research and find a new identity suited for the post-war situation. On the other hand, the institute was without a budget as it had completely relied on funds from Berlin, which per5 Görtler to Süss, April 9, 1945 (UAF, E6/1): “Anerkennung der Arbeitsgruppe für mathematische Strömungsforschung als eigene Abteilung des Mathematischen Reichsinstituts”. 6 Cf. the abstracts in the first abstract book (Vortragsbuch), starting in September 1944 with a talk by Pisot, accessible online via the Oberwolfach Digital Archive (https: //oda.mfo.de/). 126 V. R. REMMERT manently stopped flowing with the end of the war. To understand how Süss and his colleagues dealt with this twofold challenge, the consequences of the institute coincidentally being situated in the French occupation zone, and the potential implications of this political happenstance for the history of mathematics in post-war Germany, will be centre-stage in what follows. 2. OBERWOLFACH 1945/46: KEEPING THE INSTITUTE AFLOAT In order to adequately assess the situation of the Oberwolfach Institute in 1945 it is important to keep in mind that generally people in Germany could not have foreseen what we now know well, namely that between 1945 and 1949 no sovereign state would exist in Germany. For the Oberwolfach Institute this was particularly problematic as this implied that there was no central political actor who was competent in the field of science policy [Osietzki 1984; Stamm 1981; cf. Szöllösi-Janze 1996] and no institution that felt responsible for the institute. Money from Berlin ceased to flow and that was the end of it in 1945. After 1945, mathematics in Germany lay pretty much in ruins, as did Germany in general. 7 The sacking of Jewish mathematicians from German universities had been an immense loss for the discipline [Bergmann & Epple 2012; Siegmund-Schultze 2009]. International contacts had become increasingly fraught with difficulties given the political framework during the Nazi period [Remmert 2004, 228–234; Remmert 2017]. Opinions as voiced by first-rate mathematician (and close ally of Süss during the Nazi period) Helmut Hasse in a letter to Marshall Stone in 1939, “that there is a state of war between the Germans and the Jews”, did not really help to ease the situation. 8 During the war all universities suffered from a severe lack of academic staff, as many of the junior academic talents served in the war. Furthermore, many university mathematicians, such as Gustav Doetsch, Helmut Hasse, Wolfgang Krull, Herbert Seifert or Emanuel Sperner, just to name a few, were drafted into war-related mathematical research and thus withdrawn from research and teaching in universities. Often their research results were classified as secret and consequently, and because Jewish mathematicians were increasingly kept from publishing in German journals, publication activity was decreasing. On the other hand, publishing opportunities were severely restricted during the war by the rationing of paper. To 7 8 This passage draws on [Remmert & Schneider 2010, 265–267]. Hasse to Stone, March 15, 1939 (UAF, E4/44); cf. [Siegmund-Schultze 1993, 164]. OBERWOLFACH IN THE FRENCH OCCUPATION ZONE: 1945 TO EARLY 1950S 127 put it bluntly, mathematical Germany fell from a peer position to the second league between 1933 and 1945. Given the devastation in many German universities and the daily economic hardships, the rebuilding of mathematics and the mathematical publishing system progressed very slowly after 1945. Only after political and economic conditions had considerably improved in the 1950s did mathematical culture in Germany begin to develop dynamically. Before that mathematicians in Germany were in dire straits: the gaps that the firing of Jewish mathematicians had left in universities and specific subdisciplines, such as abstract algebra, could not be filled. International co-operation, while being resumed more speedily than after World War I, was still recovering slowly—not least due to travel restrictions in the early post-war years. The lack of paper persisted after the war (just as in France and other countries) and resulted in an enormous lack of teaching material, enhanced by the war losses in public and private libraries as well as in publishing houses, many of whose backlists had been destroyed (Teubner’s for instance). These developments were reflected in the fact that no mathematical journals were published in the immediate post-war years in Germany (see Section 9 below). At the same time mathematical literature and up-to-date knowledge from abroad did not easily flow into Germany. 3. GAINING OFFICIAL SUPPORT: JOHN TODD (MAY 1945), SZOLEM MANDELBROJT (OCTOBER 1945) AND FRENCH FIAT (MARCH 1946) Of the two visits mentioned above by Prudhomme and Todd, the latter turned out to be crucial for the Oberwolfach Institute. Not so much because Todd “saved” the Oberwolfach Institute, but because he recommended that Szolem Mandelbrojt visit it. The story of John Todd, “saviour” of the Oberwolfach Institute (“John Todd der Retter”), has often been told, in particular in the hagiographic brochure on the institute’s history that Süss’s widow compiled in 1967 [Süss 1967, 32] and also, more modestly, by Todd himself [Todd 1983, 21; cf. Todd 1997]: We were having a discussion on the patio when there arose a commotion among the servants. It was caused by a foraging party of Moroccan troops who wanted to occupy the building. I quickly got into proper dress with hat and in my best French persuaded them to leave the mathematicians and “même les poules” undisturbed. The very distinguished sergeant asked if he would be permitted to shake the hand of a British naval officer. Of course I said, “Yes”, and 128 V. R. REMMERT they left to try their luck elsewhere. However, they later appropriated Threlfall’s Mercedes-Benz. This incident kept “Lorenzenhof” [= the institute] intact until the local government was set up. Whether or not Todd’s calling the “foraging [...] Moroccan troops” to order by his authority as a (white) British naval officer and thus saving the Oberwolfach Institute seems a plausible story is not essential. However, what really mattered for the institute’s future was that Todd informed Szolem Mandelbrojt (1899–1983) of its existence and recommended that the Oberwolfach Institute be supported by the French. 9 Mandelbrojt, an early member of Bourbaki and the successor of Hadamard at the Collège de France in 1938, emigrated to the United States in 1940 where he taught at Rice University [Mandelbrot 1998; 2012]. In 1944 he, along with Jacques Hadamard, became a member of Louis Rapkine’s Mission Scientifique Française en Grande-Bretagne in London, founded to fill the research gap that had accumulated in France since the German invasion in 1940 and soon to be merged with the liberated CNRS [Dosso 1998, 353–356, 376–380]; [Guthleben 2009, 78–83, 95f]. Todd had been in close contact with Hadamard and Mandelbrojt in London and upon his return from Germany he wanted to “brief them about Oberwolfach, since it was in the French Zone” and not yet known to French mathematicians. The two had, however, already returned to Paris where Todd visited them in a CNRS office “when Mandelbrojt was being outfitted for a visit of inspection to Oberwolfach. He insisted on trying the revolver which had been issued to him—and there are presumably still two bullets in the floor of a CNRS office!” [Todd 1983, 21f] Indeed, Mandelbrojt visited Oberwolfach twice, on October 23, staying at least one night, and on October 31. In between he went to Tübingen where he met Kamke. 10 Mandelbrojt was not alone in going to the French occupation zone to check out German scientific resources, as by October 1945 the CNRS had already sent ca. 150 scientists on up to 400 such missions, including trips to the two universities, Freiburg and Tübingen, and the thirteen Kaiser-Wilhelm-Institutes in the French occupation zone [Defrance 2001, 8; Guthleben 2009, 99–104]. Mandelbrojt wrote to Todd in 9 Todd to Süss, October 16, 1945 (UAF, E6/11). Süss to Kamke, who had given Mandelbrojt a letter for Süss, October 31, 1945 (UAF, C 89/6). Cf. the list of guests Süss sent to the French administrator on January 29, 1947 (UAF, C89/108), mentioning that Mandelbrojt had arrived in Oberwolfach on October 23, 1945. Threlfall mentions Mandelbrojt’s second visit in his diary for October 31 and regretted having missed him at Tübingen the day before. 10 OBERWOLFACH IN THE FRENCH OCCUPATION ZONE: 1945 TO EARLY 1950S 129 November 1945 that his trip to Germany had been very interesting and that he would write a formal report on his journey, promising Todd a copy. He regretted that he had not been able to go to Göttingen, “due to flats (8!) and even more serious automobile accidents, but what I have seen, especially in Wolfach, in Tübingen, is very interesting”. 11 Mandelbrojt mentioned his trip to Germany twice in the ensuing correspondence with Todd. On January 1, 1946 he wrote to Todd, excusing himself for having been unable to go to London after the trip to Germany. He included two letters (now apparently lost) by Erich Kamke who had invited him to come back to Tübingen and give some lectures. However, Mandelbrojt could not bring himself to accept the invitation because, as he wrote, “Kamke is a fine man, I have all confidence in him, and if all the German mathematicians were like himself I would certainly accept his invitation, but you know, I know, and Kamke himself knows perfectly well that this is not the case”. 12 Mandelbrojt was in good company in his assessment of Kamke who had lost his professorship in 1937 because his wife was Jewish. Abraham Fraenkel in 1947 considered Kamke to be one of only four German mathematicians who had stood their ground during the Nazi period (the others being Erich Hecke, Oskar Perron, and Heinrich Scholz) [Remmert 2004, 245]. Thus, it is not surprising that Mandelbrojt in his letter to Todd made it clear, that he “really [was] not in favor of going to Germany otherwise than for military purposes”. And, to be true, the group of mathematicians he knew to be affiliated with Oberwolfach, such as Görtler, Kneser, Süss, Sperner, and Threlfall, definitely had not stood their ground during the Nazi period. Already in November 1945, after his return from Germany, he had made his stance on mathematicians in Germany quite clear, referring to two letters by Görtler and Kamke he had received: “I think I could arrange some advantages for those of the mathematicians who were sympathetic to my kind”, referring to his being Jewish. 13 In February 1946 Mandelbrojt wrote to Todd that the report on his trip to Germany was “being typewritten” and Todd would be sent a copy. As it was for the CNRS, he told Todd, he “did not speak at all of the ‘romantic’ side of the story” because the CNRS was “not interested in the political 11 Mandelbrojt to Todd, November 18, 1945 (Caltech Archives and Special Collections (CASC), Todd papers, folder 9.18, Mandelbrojt). 12 Mandelbrojt to Todd, January 1, 1946 (CASC, Todd papers, folder 9.18, Mandelbrojt). 13 Mandelbrojt to Todd, November 18, 1945 (CASC, Todd papers, folder 9.18, Mandelbrojt). 130 V. R. REMMERT opinions of German mathematicians, nor in their material (I mean financial) situation”. 14 We do not have a copy of Mandelbrojt’s report, but it can be surmised that Mandelbrojt spoke positively about the Oberwolfach Institute. In his letter he mentions that in Oberwolfach Süss had told him “that the financial situation of the Institute was bad”. However, Mandelbrojt had been able to arrange “a rendez-vous between him [Süss] and the French Directeur of the Universities at Baden-Baden”, Louis Sauzin, who had promised Mandelbrojt “to give the Institute the possibility to work”, probably by trying “to connect the Institute to the University of Freiburg.” In fact, Mandelbrojt’s intervention was successful, as we will see shortly. Süss, for his part, was immensely grateful to Mandelbrojt, as he frequently stressed in letters to colleagues in the next two years. Only a few days after Mandelbrojt’s visit to Oberwolfach, Süss wrote to a colleague in Heidelberg that due to Mandelbrojt’s intervention the Oberwolfach Institute “seemed to be safe as financial support had been promised”. 15 And in early December he wrote a letter to Mandelbrojt, the first in a series that Mandelbrojt did not respond to, thanking him heartily and reporting that he had in the meantime met with Sauzin, who had been very supportive of the Oberwolfach Institute and had even allotted the institute a modest budget. At the same time Süss suggested that some of the colleagues at the institute would like to further pursue their wartime work, namely Sperner in the field of meteorology, as well as Görtler and Seifert and Threlfall (in their cases he did not mention the context). 16 Indeed, in October Süss had given Mandelbrojt several manuscripts by Sperner and Görtler hoping that they might be of interest to the French or the British. It is not easy to assess whether or to what extent military-related mathematical work was done for the French in Oberwolfach or Freiburg (see Section 5 below). However, Mandelbrojt, while apparently and clairvoyantly keeping his personal distance from Süss, still used his influence to support the Oberwolfach Institute after his return to Paris. In particular, together with Joseph Pérès (1890–1962), who began to play a leading role in the CNRS at the time [Blay 2011; Charle 1989], Mandelbrojt apparently made some suggestions to the Section française d’information scientifique et 14 Mandelbrojt to Todd, February 24, 1946 (CASC, Todd papers, folder 9.18, Mandelbrojt). 15 Süss to Karl Freudenberg, November 3, 1945 (UAF, C 89/5): “Es hat uns jetzt ein in der mathematischen Kriegsforschung leitender französischer Fachkollege in den letzten zwei Wochen manche Erleichterung erwirkt und das Reichsinstitut [...] scheint gesichert, nachdem nun auch finanzielle Zuschüsse zugesagt sind.” 16 Süss to Mandelbrojt, December 6, 1945 ([Süss 1967, 58–60]; UAF, E6/11). OBERWOLFACH IN THE FRENCH OCCUPATION ZONE: 1945 TO EARLY 1950S 131 technique in Offenburg as to how to use the Oberwolfach Institute. The Section française d’information scientifique et technique in Offenburg had been founded in July 1945 on the model of the American and British Field Intelligence Agency, Technical (FIAT ), and also became known as French FIAT. While the original mission of FIAT had been to systematically exploit German research in science and technology [Gimbel 1990], French FIAT eventually took a different approach by shifting the emphasis to controlling German science [O’Reagan 2019]. The precursor of French FIAT, the Section T of the First Army, had already collaborated closely with the CNRS from the beginning of 1945. However, the CNRS mission in Germany only took off in the second half of 1945 when it was integrated with the French FIAT under the leadership of the geophysicist Louis Cagniard (1900– 1971) in Offenburg—just 50 kilometres, still a long way in 1945/46, from the Oberwolfach Institute [Defrance 2001]. While we do not know what exactly Mandelbrojt and Pérès suggested, Süss was very excited about it. He wrote to his old friend and closest advisor Hellmuth Kneser on April 1, 1946, telling him that he urgently needed to talk to him in person regarding several plans the French FIAT had proposed. The only specific plan he mentioned, counting on Kneser’s absolute discretion (“bitte absolut schweigen!”), was the idea to publish a new mathematical journal. He also stressed that Cagniard had discussed these ideas with Mandelbrojt and Pérès as well as with the people in charge of the American and British FIAT missions. 17 Given later developments it seems that these were the first steps towards two essential projects that would keep the Oberwolfach Institute afloat in the next few years: the publishing of a new journal, Archiv der Mathematik, starting in 1948 (see Section 9 below), and of the FIAT Review Reine Mathematik (see Section 6 below). In September 1946 Süss explicitly thanked Mandelbrojt for his “gracious intervention” that was “extremely valuable for the institute and gives us the opportunity to really work together” (with the French). 18 17 Süss to Kneser, April 1, 1946 (Niedersächsische Staats- und Universitätsbibliothek Göttingen (SUBG), Kneser papers, A 82: Süss). 18 Süss to Mandelbrojt, September 8, 1946 (UAF, C 89/7): “Wie Sie wohl gehört haben, arbeiten wir jetzt für die Section d’Information Scientifique (French FIAT, C.N.R.S.) an der FIAT-Review für Mathematik. Diese Verbindung mit Herrn Colonell [sic] Cagniard von der Section d’Information Scientifique verdanke ich Ihrer liebenswürdigen Vermittlung vom vorigen Sommer. Sie ist für das Institut ausserordentlich wertvoll und gibt uns Gelegenheit zur praktischen Zusammenarbeit.” Cf. Süss’ last, very similar letter to Mandelbrojt, November 26, 1947 (UAF, C89/332). 132 V. R. REMMERT 4. NEW MATHEMATICAL ORBITS: CHARLES EHRESMANN (APRIL 1946), BOURBAKI (AUGUST 1946) AND HENRI CARTAN (NOVEMBER 1946) After the war Franco-German relations in Oberwolfach clearly profited from the relative proximity of Oberwolfach to Strasbourg—roughly 70 kilometres—where Henri Cartan and Charles Ehresmann turned out to be highly supportive of the regional Franco-German rapprochement. The Oberwolfach Institute did not have too many options in 1946, if they wanted to meet and discuss mathematics with colleagues. Life in general was pretty much restricted to the French occupation zone and travelling beyond these confines, be it to Heidelberg in the American occupation zone or Basel in Switzerland, was nearly impossible for Germans. There were only three mathematical institutes in the French zone: Freiburg, Oberwolfach, and Tübingen. A fourth would be founded at nascent Mainz University in 1946. Thus Strasbourg was, in a way, a natural place to turn to, as was Basel. In order to attract mathematicians from abroad to visit Freiburg and Oberwolfach, Süss secured the support of the French authorities as early as 1946. The French were very interested in fostering academic and cultural life in their occupation zone, considering it to be a natural part in the process of rééducation [Defrance 1994; Högerle 2013; Zauner 1994]. Süss himself was very much aware of this when he wrote to his old friend, astronomer Paul ten Bruggencate (1901–1961) in Göttingen, that with respect to cultural affairs the French showed a lot of “sympathy as well as good will”. Süss thought “that the French would be the ones most likely to take into consideration the necessity to salvage the remains of European culture”. 19 This European perspective would reverberate in later policies of the Oberwolfach Institute (see Section 8 below). In these rather high spirits, Süss also wrote to Charles Ehresmann in Strasbourg in March 1946, inviting him to come to Oberwolfach. 20 He mentioned that Mandelbrojt had given his support to the Oberwolfach Institute and that Jacques Lacant (1915–2002), the officer responsible for controlling Freiburg University, had approved the invitation. While Lacant did not officially have a say in Oberwolfach affairs, Süss dealt with 19 Süss to ten Bruggencate, March 13, 1946 (UAF, C89/5): “Gerechterweise muss man anerkennen, dass gerade in unserem Bereich die Kulturangelegenheiten mit viel Verständnis und gutem Willen und auch individuell behandelt werden. Die Notwendigkeit, den Rest der europäischen Kultur zu retten, wird ja wohl auch am ehesten von den Franzosen berücksichtigt werden.” 20 Süss to Ehresmann, March 6, 1946 (UAF, C89/5). OBERWOLFACH IN THE FRENCH OCCUPATION ZONE: 1945 TO EARLY 1950S 133 him frequently as director of the Freiburg Mathematical Institute and former rector of Freiburg University. Mathematicians from abroad would often formally be invited to Freiburg (with Lacant’s consent) and then also travel to Oberwolfach. Ehresmann was very sympathetic to the invitation and visited Oberwolfach from April 25–27, 1946, his being the first entry in the institute’s guest book. He had early on been very interested in going to Oberwolfach in order to meet topologist William Threlfall. In December 1945 he wrote to Heinz Hopf in Zürich that he had hoped in vain to go to Oberwolfach to meet Threlfall during Christmas break. But, implicitly, he gave another reason for getting in touch with the Oberwolfach Institute, namely that mathematics in Strasbourg needed to be rebuilt in its international context. He invited Hopf to Strasbourg, stressing that “our Mathematical Institute would be very honoured to re-establish by your visit the relations with the mathematicians in neighbouring countries”. 21 In a way, Strasbourg and Oberwolfach as well as Freiburg shared the problem of a certain international isolation in the early post-war period. After his first visit in April 1946 Ehresmann went to Oberwolfach quite often and lent his support to the institute in various ways, for instance by putting Süss in touch with Cartan in 1946, introducing the institute to Bourbaki in 1946, joining the editorial board of the Archiv der Mathematik in 1948 (see Section 9 below), and taking his student Georges Reeb (1920–1993) with him to Oberwolfach in 1949 (see Section 8 below). It seems that Cartan and Süss had met before the war, but as can be seen from the correspondence between the two, Ehresmann acted as go between to re-establish contact in April 1946 by delivering a letter from Süss to Cartan in Strasbourg. 22 Süss started by offering his commiseration on the “sad fate” (“trauriges Schicksal”) of Cartan’s brother Louis, who had been executed as a member of the Résistance in December 1943. He went on to refer Cartan to Ehresmann for news about the Oberwolfach Institute and invited Cartan to Oberwolfach, pointing out that he had secured the consent of the French military government to extend such invitations, which had to be handled via Lacant in Freiburg. Cartan reacted kindly to Süss’ letter and stressed that he was glad to have received—orally via Ehresmann—news about his old friend Heinrich Behnke (1898–1979), whom he had first met in Münster in 1931. Behnke had travelled to 21 Ehresmann to Hopf, December 28, 1945 (ETH Zürich, library: Hopf papers, Hs 621: 465): “Notre Institut de Mathématique serait très honoré de renouer par votre visite les relations avec les mathématiciens des pays voisins.” 22 Süss to Cartan, April 26, 1946 (UAF, C89/5). 134 V. R. REMMERT Strasbourg in September 1941 to secure Cartan’s mathematical papers from Cartan’s apartment and deposited them in the Freiburg university archives. These papers included the notes of the first Bourbaki meeting in 1935. Cartan came to Oberwolfach in November 1946, having made sure that he would meet Behnke there [Cartan 1999, 783f; Remmert 2002]. The early visits by mathematicians from France (Cartan) and Switzerland (Hopf came from Zürich in August 1946) were very important for the international image and recognition of the institute. They were considerably facilitated through Behnke’s network. Behnke had done his best to keep in touch with mathematicians outside Germany under the critical eye of many a colleague in Germany as well as the Nazi authorities, not without difficulty. Behnke had at the same time given Süss advice when he started to seriously think about launching the Oberwolfach Institute [Remmert 2002]. After Cartan had left Strasbourg for Paris in 1947 Süss did not succeed in luring him to Oberwolfach again before 1950, but he did not quite give up on trying to secure Cartan’s goodwill for the benefit of the institute, and in 1948 Cartan agreed to jointly organise the Franco-German workshop that took place in Oberwolfach in August 1949 (see Section 7 below). In between the visits of Ehresmann and Cartan, Bourbaki arrived in Oberwolfach in August 1946. Of course, some knowledge about Bourbaki may have reached Freiburg and Oberwolfach earlier via Charles (Karl) Pisot, who had been a member of Bourbaki in the late 1930s. As an Alsatian, Pisot had found himself in a difficult position in 1940 and chose to work in Germany after his demobilization from the French army, starting at the Freiburg Mathematical Institute, spending a year in Greifswald in 1941/42 and then returning to Freiburg until the end of the war [Remmert 1999]. While Pisot had been out of touch with the Bourbaki group for five years, Ehresmann, a Bourbakiste since 1934, was well-aware of the group’s activities and publications. In August 1946 he sent the four Bourbaki volumes that had been published up to that point to Oberwolfach in exchange for books that Süss had given him in April (Set Theory 1939, four chapters of Topology 1940/42, first chapter of Algebra 1942). 23 Apparently, the reception of Bourbaki in Oberwolfach was rather enthusiastic, if we believe Süss’ report in letters to Cartan and Ehresmann in January 1947: 23 Cf. Süss to Ehresmann, August 23, 1946, acknowledging the receipt of the books, and Ehresmann to Süss, August 24, 1946 (UAF, C89/5). OBERWOLFACH IN THE FRENCH OCCUPATION ZONE: 1945 TO EARLY 1950S 135 It is with great interest that we at the institute have looked at the writings edited under the name of Bourbaki. From the inner circle of the colleagues it has been suggested to translate these writings into German and possibly publish them. We have not yet discussed this idea outside the institute and accordingly we do not know whether such a publication would be feasible at all. But we would be grateful to hear what you and your Bourbaki friends think about this. By way of such a translation the institute, which has set itself the task to foster international scientific contacts and in particular those with our neighbours, would see a possibility to pursue such a mediating role. Please consider this as a purely confidential request. 24 We know that Hellmuth Kneser definitely was quite intrigued with Bourbaki’s work and working methods (see Section 8 below). Süss, as it seems, was more interested in Bourbaki’s political potential for the Oberwolfach Institute, as he was not shy to say in the letter. Cartan quickly promised to convey the idea to his Bourbaki friends (“mes amis de Bourbaki”) and, if they agreed, to discuss it with their publisher. 25 Süss for his part jumped at this promise, invited Cartan to Oberwolfach to discuss the details, and suggested that Bourbaki’s publisher, Hermann, take “charge of publishing the translation”. 26 In a way this was a typical manoeuvre on Süss’ part, as he well knew that publishing mathematics in Germany was impossible for the time being (see Section 9 below). Cartan, too, was well aware of this and put it bluntly, when reporting Süss’s idea to André Weil (and Jean Dieudonné) in São Paulo in February: Süss proposes a possible translation into German. I don’t know whether this is a serious option for the near future given the current quasi-impossibility to publish in Germany. It is true that Süss proposes that Hermann takes charge of 24 Süss to Cartan, Ehresmann, January 10, 1947 (UAF, C89/5): “Mit grossem Interesse haben wir die unter dem Namen Bourbaki bisher herausgegebenen Schriften im Institut angesehen. Im engeren Kreis der hiesigen Kollegen ist nun der Gedanke aufgetaucht diese Schriften in die deutsche Sprache zu übersetzen und eventuell die Übersetzung zu veröffentlichen. Wir haben bisher mit niemand darüber gesprochen, wissen also auch nicht, ob eine derartige deutsche Veröffentlichung sich praktisch durchführen lässt. Vor allem anderen aber wären wir dankbar, hauptsächlich zunächst einmal Ihre und der französischen Freunde von Bourbaki Ansicht hierzu erfahren zu können. Das Institut, das sich ausdrücklich die Aufgabe stellt, fachlich die Verbindung mit dem Ausland und insbesondere mit den Nachbarn Deutschlands zu pflegen, würde in einer solchen Übersetzung eine derartige Vermittlertätigkeit gerne verwirklichen. Bitte betrachten Sie die Anfrage als zunächst ganz vertraulich gestellt.” 25 Cartan to Süss, January 23, 1947 (UAF, C89/5). 26 Süss to Cartan, January 31, 1947 (UAF, C89/5). 136 V. R. REMMERT the printing! But I think that you will agree with me that it is already quite a challenge for him to print the original [meaning the coming chapters]! 27 The reaction came quickly. Weil and Dieudonné didn’t see how this project could be achieved given the well-known difficulties and suggested to postpone it to the future, lamenting that “an English translation would be much more interesting, but nobody proposed it”. 28 Süss did not give up easily, and tried to pursue the idea further via Ehresmann, but nothing came of it. This did not, however, mean the end of Bourbaki’s impact on the Oberwolfach Institute (see Section 8 below). However, even if we take a certain amount of rhetoric into account, the assessment of Bourbaki Süss gives in a letter to Ehresmann from August 1947 is very interesting. In early August 1947 Christian Pauc, who was working with Otto Haupt on the revision of his textbook on the calculus and slipping some Bourbaki into it [Haupt et al. 1948], had visited the Oberwolfach Institute. On this occasion, as Süss enthused, Pauc had given three talks and, once again, we were left with a deep impression of the great importance to build mathematics on a modern fundament such as the Bourbaki project entails. Our continuous interest in Bourbaki has once more been reinforced. Thanks to your [= Ehresmann’s] visit last year we learned about the overall plan in so far as Pisot had not already made us aware of it. Pauc said that new volumes are about to be published. French science can only be complimented on this. 29 Süss went on to discuss a possible translation, which in his view could soon be published in Germany, but this topic then petered out in his correspondence with Cartan and Ehresmann. 27 Cartan to Weil, February 14, 1947 [Audin 2011, 174]: “Questions Bourbaki: tout d‘abord, Süss soulève la question d‘une éventuelle traduction en allemand. Je ne sais pas si c‘est bien sérieux pour l’avenir immédiat, étant donné la quasi-impossibilité actuelle des publications en Allemagne. Il est vrai que Süss suggère que Hermann se charge de l’impression! Mais je pense que vous estimerez comme moi que c’est déjà bien assez d’avoir à lui faire imprimer la version originale!” 28 Weil to Cartan, February 24, 1947 [Audin 2011, 182]: “Une traduction anglaise serait bien plus intéressante, mais personne ne nous la propose!” 29 Süss to Ehresmann, August 15, 1947 (UAF, C89/5): “dabei haben wir wieder einen tiefen Eindruck von der grossen Bedeutung des Aufbaus der Mathematik auf moderner Grundlage empfangen, wie das französische Bourbaki-Unternehmen sie darstellt. Unser stetes Interesse daran ist also nur noch mehr verstärkt worden. Ihnen verdanken wir dabei die Kenntnis des Gesamtplans durch Ihren Besuch im vorigen Jahr, soweit nicht Herr Pisot schon unsere Aufmerksamkeit darauf gelenkt hatte. Wie Herr Pauc erzählte, steht das Erscheinen einiger neuer Bände von Bourbaki bevor, wozu man die französische Wissenschaft beglückwünschen kann.” OBERWOLFACH IN THE FRENCH OCCUPATION ZONE: 1945 TO EARLY 1950S 137 All in all, even if nothing came of it, the idea to translate Bourbaki was very much in the mathematical spirit of the time (and in this respect the Oberwolfach Institute may have been ahead of quite a few contemporaries in Germany) as well as politically highly opportune in the French occupation zone. 5. A NEARBY MODEL FOR OBERWOLFACH? In June 1946 Süss attended a workshop on atmospheric physics at the Laboratoire de recherches de Saint-Louis in Alsace, a branch of the Laboratoire central d’armement in Paris. 30 He had attended the meeting by invitation of the institute’s (technical) director, Hubert Schardin (1902–1965). Schardin, one of the leading ballistics experts in Germany since the mid1930s, had been director of the Institute for Technical Physics and Ballistics of the Technical College of the German Air Force in Berlin-Gatow since 1935 and had been deeply involved in war-related research projects [Baumann 2007; 2008; Maier 2007, 261]. In 1945 he and a large part of his group relocated from Berlin to Biberach an der Riß in South West Germany, where they worked for the French military government from May 1945. For a while the French contemplated moving the institute to Paris. However, following the new French strategy to keep German scientific institutions within (or near) the French occupation zone [O’Reagan 2019, 80f and 90f], it was then decided that it was preferable as well as more to the benefit of the French to set it up in Saint-Louis in Alsace near the border to Germany, that is to integrate it into the French research system while the German researchers and employees could live nearby in Germany (in Weil am Rhein). Eventually it grew into the binational Institut franco-allemand de recherches de Saint-Louis inaugurated in 1959. Apparently, for a short while in 1946 Süss saw something like a model for the Oberwolfach Institute in Schardin’s institute and was very keen on co-operating with him. As he did not really have much to offer in terms of joint research interests, he instead helped Schardin receive an honorary professorship at Freiburg University. 31 The workshop in June had been the second of three that Schardin organized in 1946, the other two were on nuclear physics and gas dynamics [Schall 1988; Baumann 2007, 246.] 32 Süss was very impressed by what he experienced and learned at the insti30 Cf. the programme and the correspondence with Schardin (UAF, C89/363). Cf. his correspondence with Schardin in 1946/47 (UAF, C89/363). 32 For a short report on the third meeting see Zeitschrift für angewandte Mathematik und Mechanik 25/27(1947), 32. 31 138 V. R. REMMERT tute in Saint-Louis and at Schardin’s home in Weil. Shortly after the event he wrote to Helmut Hasse that Schardin had managed to set up an “almost international scientific meeting” of a “remarkable scientific level”. Meeting with “French and Swiss colleagues had been often friendly and always collegial”. 33 Süss had also told Schardin, who had a few mathematicians such as Robert Sauer (1898–1970) in his team, about Hasse’s war-related research and suggested that Hasse get in touch with Schardin, who thought that he might give a paid research grant to Hasse. All in all, Süss’ visit to Schardin’s group and workshop may have been important for the future shaping of the Oberwolfach Institute in two respects. On the one hand, it showed that the French might be willing to support meetings of French, German and Swiss mathematicians, and Oberwolfach might be the right place for that. On the other hand, it made clear that doing war-related or military research for the French, as Schardin’s group did, might be an interesting and realistic option for the Oberwolfach Institute. As has been mentioned earlier, the group Süss had brought to Oberwolfach had in principle the potential to engage in military-related mathematical work. However, based on the sources it is difficult to assess whether any was done or to what extent, even though some projects of Görtler and Sperner were closely related to their earlier war-related research (cf. table 1) and as such had to be approved by the French authorities. In general, research in Germany was closely monitored by the allies after the war [Osietzki 1984, 86f; Cassidy 1994; Heinemann 2001]. The Allied Control Council, in charge of the four occupation zones in Austria and Germany, dealt with the control of research in law no. 25 of April 29, 1946, and decreed in article 3 that “fundamental scientific research of a wholly or primarily military nature shall be prohibited” (Allied Control Authority Germany 1946, 103). Naturally this was a rather flexible definition, even though it was further specified in the law and in later implementary regulations by the Allied Control Council. 34 For research institutes such as the Oberwolfach Institute this meant that they had to be “authorized by the appropriate Zone Commander”, and technical reports in form of standardized questionnaires had to be handed in to the local military authorities every four months showing details of all its activities, with sufficient data to enable competent persons to verify the correctness of the results reported, together with all publications of the establishment and a complete report 33 Süss to Hasse, July 1, 1946 (UAF, C89/303). The French text was published in: Journal officiel du Commandement en chef français en Allemagne (= Amtsblatt des französischen Oberkommandos in Deutschland) 23(1946), 174–177; the implementary regulations followed in 54(1947), 553–557. 34 OBERWOLFACH IN THE FRENCH OCCUPATION ZONE: 1945 TO EARLY 1950S 139 listing the title of each problem studied, its scope, possible applied uses, sources of funds, amounts of funds expended, and the person in charge, and any other matter required from time to time by the Zone Commander”. Moreover, “all research and technical personnel employed in a research establishment” had to be “registered with the appropriate Zone Commander” (Allied Control Authority Germany 1946, 104f). While these reports were rather tedious to prepare, they now allow for a good overview of the official activities of institutes such as the Oberwolfach Institute. In this context two aspects were important for the Oberwolfach Institute. On the one hand the continued existence of the institute seems to have been approved by the French authorities by 1946. 35 On the other hand the obligation to regularly report on the institute’s activities offered an excellent opportunity for Süss to propagate and establish the relevance of the Oberwolfach Institute, as the recipients the reports targeted, the French authorities, were very specific (cf. table 1). To Süss the reports provided a platform and a forum for the “management of relevance” [Knorr-Cetina 1981, 110–112] of the Oberwolfach Institute as the central place for mathematics in Germany, just as he had planned it during the war. Naturally the rhetoric shifted a bit [Remmert 2019]. In the technical reports for 1947 and 1948 Süss gave three main reasons for the importance of the institute, namely (1) the editing of the FIAT reviews for French FIAT, (2) the project to translate and adapt “the reconstruction of mathematics through the French publications of Bourbaki”, 36 and, finally, (3) the selfgiven and rather expansive “mission to foster mathematics in every possible way”. 37 Süss gave a detailed agenda of the Oberwolfach Institute in a document for the French military government probably dated late 1946 (as it already mentioned the FIAT reviews). While drawing on the original application of August 1944 in describing the institute’s responsibilities he went beyond this text in assigning “general tasks” of a wide range for the future, namely mathematical research projects, a modest fellowship programme, workshops on specialized topics, promotion of research assignments, mathematical evaluation and information, procurement of 35 I have, however, not been able to find formal documentation of this. Süss, technical report (Tätigkeitsbericht) for the military government for 1948 (Staatsarchiv Freiburg (SAF), C37/1, Nr. 737): “Bearbeitung des Neuaufbaus der Mathematik durch die franz. Bourbaki-Veröffentlichungen”. 37 Süss, technical report (Tätigkeitsbericht) for the military government for 1947 (SAF, C37/1, Nr. 737): “Aufgabe, die mathematische Wissenschaft in jeder Weise zu fördern”. 36 140 V. R. REMMERT Table 1. Some of the projects of the Oberwolfach Institute mentioned in reports to the French responsible title origin date Süss (coordinator) Gerrit Bol FIAT 38 French FIAT Süss’ publishing programme (pre 1945) [Remmert 1999, 40] Süss’ publishing programme (pre 1945) war-related research 1946/47 1946/47 1946/47 self-given mission 1947 self-given mission 1947 Süss’ publishing programme ? 1947 Emanuel Sperner Emanuel Sperner Süss (coordinator) Süss (coordinator) Hermann Boerner Hermann Boerner Henry Görtler Henry Görtler Henry Görtler Süss (coordinator) Süss (coordinator) Monograph: Projective Differential Geometry 38 Textbook: Analytic Geometry 38 Monograph: Theoretical Meteorology 38 “foster mathematics in every possible way” 39 International contacts (Pauc, Stiefel, Hadwiger) 39 Representation Theory of Groups 39 Algebras of Dirac and Kemmer 39 Introduction to Mathematical Praxis 40 Research project: On the Theory of Laminar Boundary Layers 41 Research project: Oscillations in Fluids with Density Stratification and in Rotating Fluids 42 New journal: Archiv der Mathematik 43 “the reconstruction of mathematics through the French publications of Bourbaki” 44 1946/47 1947 Süss’ publishing programme war-related research 1947 1947 war-related research 1947 self-given mission, French FIAT self-given mission 1948 1949 specialized literature, promotion of mathematics teaching on all levels (including schools), organisation of vacation courses, and the editing of math38 Süss, technical report (Tätigkeitsbericht) for the military government for Sept.Dec. 1946 and Jan.-April 1947 (SAF, C37/1, Nr. 737). 39 Süss, technical report (Tätigkeitsbericht) for the military government for 1947 (SAF, C37/1, Nr. 737). 40 Ibid.: “Einführung in die mathematische Praxis”. 41 Ibid.: “Zur Theorie der laminaren Grenzschichten (Forschungsvorhaben)”. 42 Ibid.: “Schwingungen in Flüssigkeiten mit Dichteschichtung und in rotierenden Flüssigkeiten”. 43 Süss, technical report (Tätigkeitsbericht) for the military government for Jan.April and May-August 1948 (SAF, C37/1 Nr. 737). 44 Süss, technical report (Tätigkeitsbericht) for the military government for 1948 (SAF, C37/1 Nr. 737). OBERWOLFACH IN THE FRENCH OCCUPATION ZONE: 1945 TO EARLY 1950S 141 ematical journals. 45 Süss concluded by stating that the funding was to be provided by the Baden Ministry of Culture and Education in Freiburg. 46 On November 1, 1946 Jacques Lacant, the French cultural officer who was responsible for Freiburg University, visited the Oberwolfach Institute together with his deputy, Paul Falkenburger (1923–2010), apparently on the occasion of Cartan’s visit. Lacant’s report to his superiors stressed that the institute was geared towards “supporting mathematical research in all its domains” and thus testified to the fact that Süss had well succeeded in legitimizing the Oberwolfach Institute with the French. 47 Lacant gave a short description of the institute’s facilities (library, collection of offprints, 48 lecture room), which afforded the exchange and discussion of ideas in groups and workshops. He stressed that German mathematicians from other occupation zones as well as guests from abroad, especially from France, visited the Oberwolfach Institute to participate in the discussions (Cartan, Ehresmann, Joseph Pérès). Lacant also reported on the institute’s aspirations to publish a series of mathematical textbooks (see Section 9 below), much needed in view of the lack of mathematical literature in Germany as “everyone could easily see” (“une importance qui n’échappera à personne”), as well as on the work on the FIAT Reviews commissioned by French FIAT. After mentioning that the government of Baden in Freiburg had allotted the institute a modest budget of 10.000 Reichsmark, he summarized that the institute had left the favourable impression of a place of calm and serious work (“laisse une impression favorable de travail calme et sérieux”). The projects, Cartan had asserted, would also be of interest to French mathematicians and were in accordance with the regulations on the control of research. 45 Undated overview with French translation (Archives de l’occupation française en Allemagne et en Autriche (1945–1955), Paris: 1BAD1262): “a) laufende mathematische Forschungen, b) Förderung verdienter Fachleute, c) Arbeitsbesprechungen über Spezialgebiete, d) Förderung von Forschungsaufträgen, e) Mathem. Gutachten und Auskünfte, f) Beschaffung von Fachliteratur, g) Förderung des math. Fachunterrichts aller Stufen, h) Einrichtung von Ferienkursen und Herausgabe von Fachzeitschriften.” 46 Ibid.: “Mittel werden gestellt vom Bad. Ministerium des Kultus und Unterrichts”. 47 Lacant to Commissaire de la République, Délégué pour le G.M. de Bade, November 4, 1946 (Archives de l’occupation française en Allemagne et en Autriche (1945– 1955), Paris: 1BAD1262): “Le but de l’Institut est en premier lieu de permettre et d’aider la recherche mathématique dans tous ses domaines.” 48 Kurt Hensel’s collection of offprints (“Separatensammlung”) had been sent from Strasbourg to Oberwolfach in late 1944; cf. Karl Strubecker to Süss, October 7, 1944 (UAF, E6/15). 142 V. R. REMMERT Table 2. List of staff, August 31, 1947 49 name position salary paid by Süss, Wilhelm professor in Freiburg, director professor in Freiburg, deputy director professor in Freiburg, staff member adjunct professor (Munich), staff member lecturer (Dozent) in Freiburg, staff member lecturer (Dozent) in Freiburg, staff member staff member student assistant adjunct professor (Berlin), teaching assignment in Freiburg student assistant Freiburg Sperner, Emanuel Bol, Gerrit Boerner, Hermann Bilharz, Herbert Gericke, Helmuth Schwarzenberger, Rudolf Stakowski, Walter Hofmann, Joseph E. Bertling, Maria Freiburg Freiburg Oberwolfach Institute Freiburg Freiburg Freiburg Oberwolfach Institute no longer active no longer active Indeed, the lists of projects (table 1) and staff (table 2) that Süss regularly produced suggested significant activities at the Oberwolfach Institute, while in reality much of it relied on the staff of the bombed-out Freiburg Mathematical Institute. 6. THE FIAT REVIEWS OF GERMAN SCIENCE AND OBERWOLFACH After the intervention of Mandelbrojt and Pérès, the Oberwolfach Institute was commissioned with the publication of the volume on pure mathematics in the series FIAT Reviews of German Science. The goal of the series was to provide an overview of German findings in medicine, science and mathematics, covering the period between 1939 and 1946. The three FIAT branches, American, British and French, collaborated on the project, which eventually expanded to more than eighty edited volumes in the fields of medicine, pharmaceuticals, biology, chemistry, earth sciences, mathematics and physics published in English and German (as Naturforschung und Medizin in Deutschland, 1939–1946) [O’Reagan 2019, 138–141]. Seven of the volumes covered mathematics, Alwin Walther 49 Süss, technical report (Tätigkeitsbericht) for the military government for MayAugust 1948 (SAF, C37/1, Nr. 737). OBERWOLFACH IN THE FRENCH OCCUPATION ZONE: 1945 TO EARLY 1950S 143 (1898–1967) in Darmstadt editing five volumes on applied mathematics [Walther 1948] and Süss two on pure mathematics [Süss 1948]. Naturally, such a massive publishing project could hardly avoid drawing on co-operation with the old academic elites, irrespective of former Nazi affiliations. Thus, it is not surprising that Süss, well-informed about mathematics in Germany as (former) president of the DMV, seemed a good choice, when the French authorities were commissioned with the volumes on biology, geography and mathematics in the FIAT Reviews of German Science. 50 By early June 1946 Süss knew that he would be entrusted with the mathematics volumes, even though the official letter to that effect by the French FIAT was only sent on July 16, 1946. 51 Given his arrangement with French FIAT to cover both applied and pure mathematics he may have been surprised that on July 9 the American FIAT office had already delegated large part of the mathematics project to Alwin Walther, who eventually edited the five volumes on applied mathematics. 52 Already in June Süss started to write to prospective authors. The majority agreed to contribute to the project, including Max Deuring, Helmut Hasse, Hellmuth Kneser, Wilhelm Magnus, Herbert Seifert, William Threlfall, Helmut Wielandt, and Hans Zassenhaus, just to name a few. Clearly, the authors had many reasons to write for the FIAT Reviews, as historian of science Douglas O’Reagan put it, “in part to reconnect to the world’s scientific community, in part to receive a pay check in a brutal economy, and in part to rewrite their own collaboration with the Nazi government” [O’Reagan 2019, 140]. Indeed, the first aspect was very much stressed by the FIAT flyer coming with French FIAT ’s letter of invitation, which made the point, that “cooperation with German scientists must be obtained by pointing to the fact that the Reviews would help to re-establish contact with the international science community”. 53 Süss 50 The division of labour is mentioned in a letter by Cagniard to Lacant, July 26, 1946 (Archives de l’occupation française en Allemagne et en Autriche (1945–1955), Paris: 1BAD1265). 51 L’Ingénieur Général Gaston de Verbigier de Saint Paul to Süss, July 16, 1946 (Archives de l’occupation française en Allemagne et en Autriche (1945–1955), Paris: 1BAD1265). 52 American FIAT to Walther, July 9, 1946 (copy in UAF, C89/115); cf. Walther to Süss, June 18, 1946 (UAF, C89/115). 53 Flyer included in the letter of Verbigier de Saint Paul to Süss, July 16, 1946 (Archives de l’occupation française en Allemagne et en Autriche (1945–1955), Paris: 1BAD1265): “La coopération des savants allemands doit être obtenue du fait que les Reviews aideront à rétablir le contact avec la science internationale.” 144 V. R. REMMERT himself highlighted this aspect of the re-internationalisation of mathematics in Germany in his letters of invitation. In terms of remuneration nothing could be offered to Süss’ team of writers as French FIAT made clear in October. Because participation in the FIAT Reviews was such a good chance to catch up with international research, they argued, “contributors were expected to participate on a purely voluntary basis without dreaming of being paid in whatever way”. 54 The importance of the FIAT Reviews for the Oberwolfach Institute cannot be overestimated, because in 1946 and 1947 work on them was the institute’s only official task. Süss made this clear in a letter to the Ministry of Culture and Education in Freiburg in November 1946, demanding the financial support for the institute that had been promised earlier that year. Summarising the institute’s current situation, Süss, who was never shy to aggrandize himself and his endeavours, stressed that while it had “continued working undisturbed and expanded its agenda” its “main task in the near future lay in finishing and printing the great FIAT Review Mathematics for the United Nations”. 55 Apart from the overarching goal to keep the Oberwolfach Institute in existence, cooperation with French FIAT on the FIAT Reviews had several significant side effects: – to establish close and sustainable ties with the French Military Government, – to secure the financial support of the Ministry of Culture and Education in Freiburg, if only to a modest extent, – to safeguard Süss’ political influence within the discipline of mathematics, – to allow the staff at Oberwolfach/Freiburg to continue their work (including war-related research such as Görtler’s), – to further the re-internationalisation of mathematics in Germany with the Oberwolfach Institute as a node, a strategy in accord with the French cultural policies [O’Reagan 2019, 138], – to procure books and journals for the institute’s library, albeit on a modest scale. 54 French FIAT to Süss, October 4, 1946 (UAF, C89/9): “en demandant une participation purement bénévole à ses collaborateurs, sans vouloir songer à une retribution quelconque”. 55 Süss to the Ministry, November 11, 1946 (SAF C25/3, Nr. 243): “Das Institut hat seine Arbeiten ungestört und unter Ausdehnung der Aufgabengebiete [...] fortgesetzt. Seine Hauptaufgabe für die nächste Zeit besteht in der Fertigstellung und Herausgabe des großen FIAT-Berichts Mathematik für die Vereinten Nationen”. OBERWOLFACH IN THE FRENCH OCCUPATION ZONE: 1945 TO EARLY 1950S 145 The manuscript of the FIAT Reviews on pure mathematics was submitted on May 31, 1947. When the two volumes were published in 1948 they were an excellent piece of propaganda for the Oberwolfach Institute. On the one hand, they featured Süss and seven other authors (such as Hermann Boerner, Helmuth Gericke, Joseph Ehrenfried Hofmann, Emanuel Sperner and Georg Tautz) as members of the Oberwolfach Institute, thereby highly exaggerating the institute’s staff. On the other hand, Süss used the preface to highlight the role the Oberwolfach Institute had played in “planning and organising the work as well as seeing the review to the press” and, at the same time, refashioned the institute’s character by concluding: The Review will show that even in the times of this deplorable war the garden of true scientific research has been silently tended to by his friends. May it soon be in full blossom again! 56 Naturally, Süss envisaged a central role for Oberwolfach in achieving this goal, and indeed, soon the idea of Oberwolfach as a “paradise for mathematicians” began to spread within Germany (and beyond). 57 In 1952 Fritz Joachim Weyl (1915–1977) wrote a technical report for the British Office of Naval Research on a meeting on complex analysis he had attended in Oberwolfach in October 1951. He, too, picked up the thread of the “pleasant surroundings” of Oberwolfach inviting “peripatetic discussions, mathematical and otherwise” [Weyl 1952, 1]. But he also gave an interesting assessment, having obviously been briefed by Süss, of the influence that the production of the FIAT Reviews had had on the Oberwolfach Institute [Weyl 1952, 4]: The end of May 1947 saw the completion of the FIAT Review, and with it was terminated the presence of a permanent research group at Oberwolfach. The subsequent annual allocations of DM 10,000 by the Land Baden to the Institute covered little more than up-keep and rent of the premises. The year 1947/48 was a quiet one at the Lorenzenhof. In the meantime, however, the transient presence of refugees and FIAT-Review writers alike had transformed the shirtsleeve colloquia of the established group into shirt-sleeve symposia, held on topics of their selection by those who happened to be present. The log book, continued in compliance with the research control act, gives evidence that the char- 56 [Süss 1948, I, preface]: “Der Bericht wird zeigen, daß der Garten echter wissenschaftlicher Forschung auch in der Zeit dieses unseligen Krieges von seinen Freunden in der Stille gepflegt worden ist. Möge er doch bald wieder zu voller Blüte kommen!” 57 Horst Tietz used the phrase “Mathematiker-Paradies” in a letter to Süss, July 1, 1955 (UAF, C89/385). 146 V. R. REMMERT acteristic style of presentation, inviting group participation—well prepared yet showing clearly all loose ends—had on the whole been preserved. Out of this tradition has grown the activity of the Institute during the last three years, in the course of which it has been the meeting place for numerous gatherings, organized around one theme or another, either of a mathematical or regional character. What is hinted at here as being, in a way, a by-product of the work on the FIAT Reviews, namely the new format of “gatherings, organized around one theme or another, either of a mathematical or regional character”, is another testimony of the importance Süss attributed to the FIAT Reviews—irrespective of whether the link really existed, which is difficult to assess from the source material. However, from 1949 onwards thematic workshops gradually became the main characteristic of the Oberwolfach Institute, and their evolving into the typical Oberwolfach research tool was closely related to the specific situation in the French occupation zone. 7. THE BEGINNING OF THE WORKSHOPS IN OBERWOLFACH IN 1949 In July 1948 Süss wrote to Cartan, inviting him to return to Oberwolfach, and informing him that a delegation from the Sorbonne had visited Freiburg University, including Georges Darmois (1888–1960). Süss had used the opportunity to tell Darmois about the Oberwolfach Institute and proposed that it might be a good idea “to gather a small group of especially talented mature French and German students for one or two weeks in Oberwolfach in order that these young people could by working together come into intellectual and personal contact. Such contact often has a decisive lifelong influence and should be of priceless value for the future of our peoples”. 58 Süss asked Cartan for his support with the relevant authorities in Paris and Cartan made a note on the letter that he’d take care of it. The result of this was the Franco-German workshop in August 1949 [Remenyi 2011]. While this was not the first workshop held at the Oberwolfach Institute—pride of place goes to the topology workshop organized around Heinz Hopf’s visit to Oberwolfach in April 1949 -, it was 58 Süss to Cartan, July 24, 1948 (Cartan papers, I am grateful to Michèle Audin for a copy of this letter): “ob wir nicht in diesem Sommer einige wenige, besonders tüchtige, ältere französische und deutsche Studenten auf eine oder zwei Wochen in Oberwolfach zusammenbringen könnten, damit diese jungen Menschen in gemeinsamer Arbeit fachlichen und persönlichen Kontakt miteinander gewinnen könnten, der ja sehr oft für das ganze Leben entscheidenden Einfluß besitzt und für die Zukunft unserer Völker von unschätzbarem Wert sein dürfte.” OBERWOLFACH IN THE FRENCH OCCUPATION ZONE: 1945 TO EARLY 1950S 147 originally the first that had been planned. Moreover, it was substantially subsidised by the French authorities. In organising the Franco-German workshop Süss did not only rely on Cartan’s support, but also got in touch with Dieudonné, who promised to send an article for the new journal edited by the Oberwolfach Institute, Archiv der Mathematik [Bourbaki 1949], as well as to visit Oberwolfach in August 1949, independently of the proposed Franco-German workshop. 59 In January 1949 Süss reported to Cartan that the Institut Français in Freiburg had agreed to support the French visitors during the workshop while the Ministry of Culture and Education in Freiburg would cover the costs for the German visitors. 60 Süss suggested that Cartan should find six to eight French students while the Oberwolfach Institute would invite a similar number from Freiburg, Heidelberg, Mainz and Tübingen. In a follow up letter from February Süss asked whether Cartan and Dieudonné would like to take the occasion of the Franco-German workshop to come to Oberwolfach with members of the Bourbaki group. 61 Further details were arranged in spring in cooperation with Georges Deshusses, the director of the Institut Français in Freiburg, which was in close touch with Freiburg University. 62 At the same time Cartan was very actively trying to find volunteers to attend the Oberwolfach meeting. In April he wrote to Deshusses, that “despite of his (oral) propaganda among his colleagues he had only two commitments”. While one of the reasons he saw, was a lack of language skills among the younger generation, he conceded “that there also were without doubt other reasons” (“il y a aussi sans doute d’autres raisons”). 63 Obviously most of the young French mathematicians were not too keen on going to Germany. Jean-Pierre Serre later recalled that Cartan had “ordered us to go to Oberwolfach” [Remmert 2008, 1]. However, in late May Cartan came up with a preliminary list of ten possible participants: Jean Braconnier, Michel Cazin, Bernard Charles, Jean Colmez, Roger Decombes, Mercier (he did not give a first name), Jean Nordon, Georges Reeb, Jean Riss and René Thom. In July the list had been modified to still include Braconnier, Charles, Nordon, Reeb 59 Dieudonné to Süss, November 9 and 24, 1948 (UAF, C89/288). Süss to Cartan, January 21, 1949 (UAF, C89/286). 61 Süss to Cartan, February 14, 1949 (UAF, C89/286). 62 Cf. the correspondence in the Institut Français file (Centre des Archives Diplomatiques de Nantes: Fribourg 236PO/1/96). On the Institut Français in Freiburg see [Sid-Otmane 1992; Zauner 1994, 258ff; Högerle 2013, 105–142]. 63 Cartan to Deshusses, April 4, 1949 (Centre des Archives Diplomatiques de Nantes: Fribourg 236PO/1/96). 60 148 V. R. REMMERT and Thom, but Jean Arbault had been added as well as Jean-Pierre Serre and his wife. 64 With the exception of Nordon, who was supplemented by Alfredo Pereira Gomez, they all attended the meeting as did Dieudonné, while Cartan could not travel to Oberwolfach due to a traffic accident. As the workshop has been amply described by Maria Remenyi, I do not further discuss the details here [Remenyi 2011]. Suffice it to say that it marked a turning point for Oberwolfach in several respects: – Together with the topology meeting in April 1949 it set the stage for a series of conferences in the years to come (see table 3), with full moral and occasional financial support by the French authorities, and thus became the prototype of the new concept of conferences as the central research tool of the institute. – It was a deliberate and successful step towards a Franco-German rapprochement in mathematics. – It was a first small triumph of Süss’ ambitious and ultimately successful programme to not only keep the Oberwolfach Institute going, but to turn it into a focal point of mathematical research and communication in Germany as well as – an essential place for the re-integration of mathematics in Germany into the European (and international) community. – It further strengthened the interest at the Oberwolfach Institute in Bourbaki and their meeting model, the congrès Bourbaki (see Section 8 below). 8. FRANCO-GERMAN MATHEMATICAL RELATIONS IN OBERWOLFACH As we have seen, Cartan, Dieudonné, and Ehresmann were highly supportive of the Oberwolfach Institute in the late 1940s: they visited Oberwolfach, they made suggestions as to whom amongst the French mathematicians to invite to Oberwolfach (see table 4 for a list of visitors), they brought along Bourbaki, and they supported the idea of workshops in Oberwolfach in person as well as in spirit. From a practical point of view Ehresmann was crucial in this process as not only did he frequently visit Oberwolfach, but the mathematical energy he unfolded in Strasbourg in the 1940s and 1950s reverberated up to remote Oberwolfach. As mentioned above, Ehresmann was convinced that mathematics in Strasbourg needed to be rebuilt in its 64 Cartan to Deshusses, May 29 and July 7, 1949 (Centre des Archives Diplomatiques de Nantes: Fribourg 236PO/1/96). OBERWOLFACH IN THE FRENCH OCCUPATION ZONE: 1945 TO EARLY 1950S 149 Table 3. Workshops in Oberwolfach, 1949–1952 date topic 1949, April 2–4 1949, August 9–25 1949, October 27 to November 1 Topology Franco-German workshop Logic and foundations of mathematics Meeting of mathematicians from both sides of the Rhine Modern algebra and theory of numbers Complex analysis Geometry Logic and foundations of mathematics Modern algebra and theory of numbers 1950, November 24–26 1951, August 27–31 1951, October 20–26 1952, March 26–29 1952, June 1–6 1952, September 16–22 significant French funding international context. As it happened his own field of topology may have been particularly suited to achieve this goal. On the one hand some distinguished topologists, such as Beno Eckmann in Lausanne/Zürich, Heinz Hopf in Zürich, Georges de Rham in Lausanne were (relatively) nearby in the late 1940s (as well as Seifert and Threlfall who frequently went to Oberwolfach), and Ehresmann was in close touch with them. His seminar, colloque de topologie de Strasbourg, was a focal point of topology in the 1940s and 1950s [Audin 2008, 366f]. On the other hand topologists already had a certain, if young tradition to convene in specific, international conferences such as the meetings in Moscow and Geneva in 1935 [James 1999, 840ff; Apushkinskaya et al. 2019]. 65 The latter had been attended by de Rham, Ehresmann, Hopf, Seifert, and Threlfall [CISM 1935, 119f], who in April 1949 reconvened in Oberwolfach for the topology workshop, the first organized by the institute. In 1947 a meeting on algebraic topology was organized in Paris [James 1999, 844f], among whose participants Ehresmann, Hirsch and Hopf also came to the topology meeting in Oberwolfach in 1949. Thus Ehresmann was part of a closely interwoven international group and happy to share contacts and invitees with the Oberwolfach Institute (if the formalities could be arranged with the French authorities). To give just one example, Ehresmann wrote to Hellmuth Kneser in March 1949, informing him that he would come to Oberwolfach on April 3 in the 65 I am grateful to John McCleary for pointing this out to me. 150 year 1944 1945 1946 1947 1948 1949 1950 1951 1952 1953 1954 V. R. REMMERT Table 4. French visitors at the Oberwolfach Institute 67 name Pisot (cf. abstract book: first entry), Roger Mandelbrojt, Pisot, Roger Cartan, Cerf, Ehresmann, Pauc, Pérès Pauc (currency reform in Western Germany) Bouligand, Dieudonné, Ehresmann, Reeb, Vicensini—Franco-German workshop Bouligand, Braconnier, Cartan, Chabauty, Charles, Deny, Ehresmann, Koszul, Nordon, Reeb, Thom Ehresmann, Charles, Favard Brelot, Charles, Deny, Fourès (= Yvonne Choquet-Bruhat), Godeaux, Koszul, Lelong, Lichnerowicz, Pauc, Siddiqi, Thom Ehresmann, Gauthier, Lazard, Libermann, Lichnerowicz company of (“in Begleitung von”) Hopf, Eckmann, Hirsch and Reeb. 66 This kind of support, sanctioned and encouraged by the French authorities [O’Reagan 2019, 138], helped the Oberwolfach Institute to develop close relations with French and Swiss mathematicians and allowed Süss to pursue his programme to re-integrate mathematics in Germany into the European (and international) community. Ehresmann also introduced his student Georges Reeb (1920–1993) to Oberwolfach. Reeb later fondly recalled his first trip to Oberwolfach, together with Ehresmann, in April 1949, and returned quite often [Reeb 1994]. In particular, he stayed at the institute for six months from April to September 1950. Such visits usually had the full support of the French authorities, as it had become a French policy to control science in their occupation zone by placing French trainees (stagiaires) in German research facilities [O’Reagan 2019, 91]. While Reeb, who had taken his PhD in 1943, was not typical for such a “student spy” as O’Reagan calls them [O’Reagan 2019, 78], he still wrote a two-page report on his stay for the French authorities (Institut de Mathématiques du Lorenzenhof, see appendix). 68 Similar to the report Weyl wrote in 1952 Reeb had apparently been well-informed by Süss about the official view on the institute’s 66 Ehresmann to Kneser, March 18, 1949 (SUBG, Kneser papers, A 20: Ehresmann) The table is probably not complete. It draws on the abstract and guest books as well as the correspondence of Süss. 68 Copy of the 1950 report in: Archives de l’occupation française en Allemagne et en Autriche (1945–1955), Paris: 1BAD1262. 67 OBERWOLFACH IN THE FRENCH OCCUPATION ZONE: 1945 TO EARLY 1950S 151 history and objectives. Reeb started out by praising the “particularly pleasant atmosphere” in Oberwolfach, mostly due to Süss’ gift as a host, and pointed out that the institute had two main objectives, namely its publication activities and the “organisation of workshops and colloquia on specialised subjects”. Reeb went on to succinctly describe the “French influence” on the Oberwolfach Institute, highlighting the “important role of the influence of Bourbaki”, which extended beyond the institute into university teaching in Germany and had been welcomed with enthusiasm by Hellmuth Kneser. As examples he mentioned the spreading of the use of Zorn’s Lemma (Théorême de Lorx 69) and of the concept of filters. Inversely he pointed to the vitality of algebra in Germany as an instance of German influence on mathematics in France. Reeb’s eulogy was known to Süss, who in October 1950, ever the politician, used the opportunity to stress the international character and mission of the Oberwolfach Institute in a letter to Pierre Pène (1898–1972), French commissioner to Baden, while thanking him for a contribution of 3 000 DM to the institute’s budget: Indeed, we are convinced that each individual can contribute to the realization of a better future for Europe by practical work. As mathematicians we try to achieve this by professional co-operation with our colleagues beyond our borders and we are fortunate to have found a wide echo and have had good results in such a short time. 70 Naturally Süss made the point that in the wake of the Franco-German meeting a further step towards a Franco-German rapprochement in mathematics was imminent, namely the meeting of mathematicians from both sides of the Rhine in Oberwolfach in November 1950 (see table 3). A few years later, in the first extended report on the Oberwolfach Institute’s work and achievements (not specifically directed at French authorities), Süss again stressed the importance of the Franco-German workshop of August 1949 and of Bourbaki’s impact for the Oberwolfach Institute [Süss 1953, XIIf]: 69 I have to confess that I have no idea why it is called “Théorême de Lorx” by Reeb. Possibly LORX was just a misreading for ZORN while the report was being typed. 70 Süss to Pène, Commissaire pour le Land Bade, October 30, 1950 (Archives de l’occupation française en Allemagne et en Autriche (1945–1955), Paris: 1BAD1262): “Es ist in der Tat unsere Auffassung, daß jeder Einzelne von uns seinen Beitrag zum Zustandekommen einer besseren Zukunft Europas in der praktischen Arbeit leisten kann. Wir Mathematiker versuchen dies in der fachlichen Zusammenarbeit mit unseren Kollegen jenseits der Grenzen, und wir sind glücklich, dabei in wenigen Jahren ein so weites Echo gefunden und so gute Erfolge errungen zu haben.” 152 V. R. REMMERT For us Germans it was essential to not only be allowed an almost complete survey of the ambitions and works of the Bourbaki group, that has already published twelve books in the last few years, but also to get to know quite a number of their collaborators in their work on specific problems while employing the methods of Bourbaki. The impact of these methods can already be seen at the universities of Berlin, Mainz and Tübingen. For the most part this impact goes back to the [Franco-German] workshop in Oberwolfach. 71 This assessment reflects the deep appreciation for Bourbaki in Oberwolfach, which in turn left a distinct mark on the institute’s agenda. This development was not welcomed by all mathematicians in Germany. Wilhelm Blaschke (1885–1962), for instance, voiced criticism in a letter to Süss in September 1949, deprecatingly judging that “it would not be right to exclusively flirt with Bourbakistan” (“einseitig nur mit BOURBAKISTAN anzubändeln”). 72 However, Süss was determined to further forge the close alliance of Oberwolfach with Bourbaki. At the outset of the Franco-German meeting, on August 9, 1949, Dieudonné had presented the Bourbaki project in Oberwolfach (Exposé du but, de la méthode et du plan des “Eléments de Mathématique” de N. Bourbaki). 73 Apparently Dieudonné had also expressed his surprise that none of the Bourbaki volumes had yet been reviewed in the institute’s newly founded journal, Archiv der Mathematik. In January 1950 Süss wrote to his old friend Kneser, saying that he could understand Dieudonné’s surprise in view of “the close contacts we have to this French circle”, and asked him whether he would be willing to write a report on the Bourbaki project for the Archiv, the goal being to “objectively point to this after all rather important French project that is still rather unknown in Germany”. 74 71 “Für uns Deutsche war es von großer Bedeutung, nicht nur einen fast vollständigen Überblick über die Bestrebungen des BOURBAKI-Kreises zu erhalten, der ja in den letzten Jahren bereits 12 Bücher publiziert hat, sondern auch eine große Zahl der Mitarbeiter in ihrer Arbeit an speziellen Problemen nach den BOURBAKIMethoden genauer kennenzulernen. An den Universitäten Berlin, Mainz und Tübingen ist der direkte Einfluß dieser Methoden bereits auch stark zu erkennen. Er geht im wesentlichen auf jenes Kolloquium in Oberwolfach zurück.” 72 Blaschke to Süss, September 16, 1949 (UAF, C89/277). 73 Abstract book I, 17. Ralf Krömer is currently working on an assessment of Bourbaki’s activities in Oberwolfach and beyond in Germany in the 1940s and 1950s. 74 Süss to Kneser, January 5, 1950 (UAF, C89/316): “Dieudonné hat sich im Sommer darüber gewundert, daß wir im ARCHIV noch gar keine Besprechung von BourbakiVeröffentlichungen hatten. Die engen Beziehungen, die wir gerade zu diesem französischen Kreis besitzen, rechtfertigt [sic] diese Verwunderung. [...] Vollständigkeit in der Beurteilung des Stoffes ist ja nicht nötig, sondern mehr ein sachlich gut fundiertes OBERWOLFACH IN THE FRENCH OCCUPATION ZONE: 1945 TO EARLY 1950S 153 Kneser, in line with the enthusiasm for Bourbaki he had professed since 1946, wrote a glowing review [Kneser 1949]. He started by sketching out the overall design of the Bourbaki project—familiar to him from Dieudonné’s presentation in August 1949—and characterised the enterprise “to build the foundations of mathematics in this new order as an enormous and timely challenge” (“umfangreiche und zeitgemäße Aufgabe”). He emphasised that the books were suitable for talented mathematics students as experience in Germany had shown. Kneser stressed that he had “often particularly relished reading the historical notes, and even more so as no national barriers could be felt, that came so naturally and even in good presentations of this kind”. In concluding he expressed his high estimation for the Bourbaki group: “I believe that mathematicians of the decades to come will be grateful to Bourbaki that he has taken on this burden.” 75 Kneser was an excellent, widely read and (internationally) highly respected mathematician with a broad range of research interests [Wielandt 1974; Hofmann 2008, 132; cf. Kneser 2005] and Oberwolfach’s mathematical mastermind. Reeb put this nicely in his 1950 report, when he described Kneser as the “assiduous host of the Oberwolfach Institute, known for his almost universal knowledge of mathematics” (“hôte assidu du Lorenzenhof, connu pour sa connaissance à peu près universelle des mathématiques”; appendix). In 1946 he had shown immediate enthusiasm for Bourbaki (also mentioned by Reeb). In contrast to Süss, who as a mathematician with a certain lack of breadth and depth had, as it seems, mostly embraced Bourbaki for political considerations (see Section 4 above), Kneser’s work in the late 1940s shows clear traces of his mathematical engagement with Bourbaki. Both in his mathematical diary and in his correspondence with Süss the framework of the Topologie générale came up repeatedly, for instance, when following a hint of Ehresmann he requested that Süss look up the definition of absolute convergence of an infinite sum because he did not have a copy of the volume in Tübingen. 76 With Hinweisen auf diese immerhin recht bedeutungsvolle und in Deutschland noch wenig bekannte französische Unternehmung.” 75 “Diese Noten zu lesen ist öfters ein besonderer Genuß: insbesondere ist nichts von den nationalen Schranken zu verspüren, die ja sehr natürlich und auch in sonst guten Darstellungen dieser Art manchmal bemerkbar sind. [...] Ich glaube, die Mathematiker der nächsten Jahrzehnte werden Bourbaki dafür dankbar sein, daß er diese Mühe auf sich genommen hat” [Kneser 1949, 301f]. 76 Explicit references to this can be found in his diary (Tägliche Bemerkungen) on June 11, 1946 (SUBG, Kneser papers, D14: Absolut konvergente unendliche Summen und Produkte), and in his letter Süss of September 2, 1946 (UAF, C89/6). 154 V. R. REMMERT respect to Kneser’s own contacts with Bourbaki his interest in Bourbaki’s “Lemme fondamental” (Zorn’s Lemma) is more relevant. This had occupied him in August 1948 when he referred to the “Lemme fondamental” in the Éléments de Mathémathique in his mathematical diary. 77 The missing proof led to an exchange of letters with Jean Dieudonné between May and September 1949—before and after the Franco-German workshop— and eventually both of them published a proof [Bourbaki 1949; Kneser 1950]. 78 Dieudonné had sent his proof to Kneser who forwarded it to Süss suggesting that it might be published in the Archiv der Mathematik. Süss jumped on this idea and wrote to Dieudonné in September 1949 asking whether he would consent to this idea, arguing that if “we published such a fundamental theorem in our Archiv we would also once again formally show our [i.e. the Oberwolfach Institute’s] ties to the group of M. Bourbaki”. 79 Dieudonné discussed the idea at the Bourbaki meeting in Paris where no objections were raised and in October promised to submit “a small paper signed N. Bourbaki” (“un petit article signé N. Bourbaki”). 80 In his follow-up letter Süss inquired whether Bourbaki would also be willing to publish “a survey of the whole Bourbaki project as he had presented it in Oberwolfach”. Dieudonné did in fact send the survey for Süss’ personal use, pointing to the fact that it was not to be published as it still was “very imprecise and would surely have to be modified”. 81 In the wake of this exchange with Dieudonné, Süss asked Kneser to write a review of the Bourbaki volumes that had been published up to that point. Obviously Süss saw the political importance and the potential of cooperation with the Bourbaki group, namely Cartan, Dieudonné and Ehresmann who had all been extremely friendly towards the Oberwolfach Institute. In September 1949 he clearly stated this in a letter to Wilhelm Blaschke, praising the Bourbaki group for their “willingness to ignore 77 Entries in August 1948: “Beweis des “Lemme fondamental” von Bourbaki (El. de math. IR, p. 37)” (SUBG, Kneser papers, D14). 78 For his correspondence with Dieudonné see: SUBG, Kneser papers, A17: Dieudonné. On the history of Zorn’s Lemma see [Campbell 1978]. 79 Süss to Dieudonné, September 27, 1949 (UAF, C89/288): “Wir würden dadurch auch äußerlich erneut unsere Verbundenheit mit dem Kreis des Herrn Bourbaki dokumentieren, wenn wir ein so grundlegendes Theorem in unserem ARCHIV bringen.” 80 Dieudonné to Süss, October 13, 1949 (UAF, C89/288). 81 Süss to Dieudonne, October 26, 1949; Dieudonné to Süss, November 10, 1949 (UAF, C89/288): “Ce plan, qui comporte beaucoup d’imprécision et sera sans doute modifié ou complété par le suite, ne doit pas être publié, et je vous demande donc de le considerer comme confidential.” OBERWOLFACH IN THE FRENCH OCCUPATION ZONE: 1945 TO EARLY 1950S 155 the past in favour of a better future and to consider questions of prestige as well as personal or national vanities as secondary”. 82 As we have already seen, Blaschke quickly reacted, warning Süss not “to exclusively flirt with Bourbakistan”. Kneser, for his part, seems to have been deeply impressed by the Bourbaki method, including the congrès Bourbaki as a meeting model, and, crucially, by the consequences this might have for the future structure of mathematical research [Remmert 2021]. He became a fervent propagator of “teamwork” in mathematics in the 1950s against the opposition of the majority of elder mathematicians in Germany (Siegel being the most prominent, but, possibly, also the most old-fashioned one). When the future of the Oberwolfach Institute was at stake in the late 1950s Kneser did not tire to propagate the fundamental role of “teamwork” in modern mathematics and the importance of Oberwolfach as the place that had fostered this new way of doing mathematics in Germany via the specialised workshops the institute organised. His position was concisely put on record in 1960 when a committee of the Max Planck Society discussed the possible foundation of a Max Planck Institute for Mathematics and the Oberwolfach Institute’s relation to it: The critical status of current mathematical research is mainly due to the high pace of progress. Given the “structural change” (“Strukturwandel”) within mathematics new theories and results can so quickly come up and be further developed in the centres of current research that mathematicians who only learn about it in print cannot contribute to this work. It is a problem of communication. Frequent meetings between various teams are necessary to foster a healthy development. Because of the mobile character of mathematical work, it is possible in mathematics to a large extent that such results and methods cannot only be mutually communicated, but that this opens the way to real research. 83 82 Süss to Blaschke, September 12, 1949 (UAF, C89/5): “Überhaupt schien mir der Bourbakikreis gewillt, über alle Vergangenheit zu Gunsten einer besseren Zukunft hinwegsehen zu wollen und dabei Fragen des Prestiges ebenso wie persönliche oder nationale Eitelkeit als zweitrangig anzusehen.” 83 Kneser’s commentary, January 5, 1960: “Die kritische Lage in der heutigen mathematischen Forschung hat ihren Grund vornehmlich in dem Tempo der Fortschritte. Bei einem Strukturwandel“, wie er sich in der Mathematik vollzogen hat, können an den Zentren der aktuellen Forschung neue Theorien und Ergebnisse so schnell entstehen und weitergebildet werden, dass der Forscher, der erst durch den Druck davon erfährt, oft gar nicht in diese Arbeit eingreifen kann. Es ist also ein Problem der Kommunikation: Häufige Begegnungen zwischen den verschiedenen Arbeitsgruppen sind nötig, um eine gesunde Entwicklung zu fördern. Infolge des beweglichen Charakters der mathematischen Arbeit ist es in der Mathematik in besonderem Masse möglich, dass bei solchen Begegnungen nicht nur Ergebnisse und Methoden gegenseitig mitgeteilt werden, sondern dass dabei echte Forschung geschieht” (MPG com- 156 V. R. REMMERT Naturally, for Kneser, Oberwolfach was the perfect place in Germany to further pursue this new way of doing mathematics. 9. THE PUBLICATION PROGRAMME, ESPECIALLY THE ARCHIV DER MATHEMATIK AND THE STUDIA MATHEMATICA It has already become clear that publication strategies played a crucial role in the early years of the Oberwolfach Institute as they furnished a method to enhance the institute’s visibility and create a research agenda. From this perspective, the editing of the FIAT Reviews, the idea to translate Bourbaki, and the founding of the new journal Archiv der Mathematik are all in line with the general thrust of Süss’ endeavours to keep the institute afloat and with the drive to gain the support of the French authorities by pushing Franco-German co-operation. The history of the Archiv der Mathematik is a very good example to illustrate this process and its ramifications. As has been mentioned above (see Section 3 above) the idea to launch a new mathematical journal to be edited by the Oberwolfach Institute came up in March 1946, apparently following a suggestion of Mandelbrojt and Pérès and with full support of the French Military Government. Given the fact that no mathematical journals were published in Germany in the immediate post-war years (see table 5) and that it was totally unclear whether or when the major mathematical publishing houses, such as de Gruyter, Springer, or Teubner, would be able to start publishing again [Remmert & Schneider 2010, 265–268], it seemed worthwhile to contemplate the idea of establishing a new journal. While the British and US military authorities in Germany were often slow and sometimes reluctant to issue business or printing permits to established publishers, the French were very interested in fostering academic and scientific culture in their occupation zone [Mombert 1995]. For the Oberwolfach Institute a journal edited by the institute would naturally be a great asset, and for Süss, who had attempted in vain to get a foot into the mathematical publishing system during the Nazi period [Remmert 2000], [Remmert & Schneider 2010, Chapter 8], a new journal offered a chance to maintain his position and expand his influence as a major player in the German mathematical community after World War II. mittee “Institut für mathematische Forschung”: Hauptstaatsarchiv Stuttgart, EA 13– 201 Bue 333–2, page 3). OBERWOLFACH IN THE FRENCH OCCUPATION ZONE: 1945 TO EARLY 1950S 157 Abhandlungen aus dem Mathematischen Seminar der Hamburgischen Universität Archiv der Mathematik Archiv der Mathematik und Physik Deutsche Mathematik Jahresbericht der Deutschen Mathematiker-Vereinigung Journal für die reine und angewandte Mathematik Mathematische Annalen Mathematische Nachrichten Mathematische Zeitschrift Zeitschrift für angewandte Mathematik und Mechanik Zeitschrift für Mathematik und Physik 1950 1945 1938 1920 title 1900 Table 5. Mathematical Journals published in Germany, 1900–1950 1951 Süss formally submitted a “plan to edit a new mathematical journal” to French FIAT in June 1946. 84 He wrote: Since the collapse of Germany one of the main problems for German mathematicians has been the complete lack of publishing possibilities. The publishers of almost all mathematical journals who have been active in the past are based in the Russian zone and most of their publishing houses and their printing shops have been destroyed and are unable to work. Moreover, in the other zones the major printing shops capable of printing mathematics who had cooperated with these publishers seem to be in ruins as well. In order not to further paralyse German scientific work in the realm of mathematics because of the lack of a mathematical journal, we present the following plan to edit a new mathematical journal. 85 84 The following draws on [Remmert & Schneider 2010, 289–293]. Süss, June 1946 (UAF, E 6/13): “Eine der wesentlichsten Schwierigkeiten für die wissenschaftliche Arbeit deutscher Mathematiker besteht seit dem Zusammenbruch in dem Fehlen jeder Möglichkeit etwas zu veröffentlichen. Die Verleger fast aller mathematischen Zeitschriften der Vergangenheit haben ihren Sitz in der russischen Zone, wo ihre Verlagshäuser und die Druckereien meistens bis zur Arbeitsunfähigkeit zerstört sind; auch diejenigen größeren Druckereien mit mathematischem Satz in anderen Zonen Deutschlands, mit welchem jene Verleger zusammengearbeitet haben, scheinen durch Kriegsereignisse zerstört zu sein. Um die deutsche wissenschaftliche Arbeit auf dem Gebiet der Mathematik nicht noch länger durch das Fehlen einer mathematischen Zeitschrift lahmzulegen, wird folgender Plan zur Herausgabe einer neuen Zeitschrift vorgelegt: [...].” 85 158 V. R. REMMERT Süss suggested an editorial board representing the four centres of mathematical research in the French zone (himself for Oberwolfach, thus implicitly putting it on the mathematical map permanently, Gerrit Bol and Henry Görtler for Freiburg, Robert Furch for Mainz, where the university was just being founded, and Kneser for Tübingen). The journal would be open to papers written in German, English, French and Italian. Cagniard, as the relevant authority of French FIAT, quickly agreed to the proposal as it had “been created by our initiative” (“Avis très favourable, cette revue étant créée de notre initiative”) and gave French FIAT ’s full support. 86 Once the project had been authorised, it turned out to be quite difficult to find a publisher. Eventually, an agreement was signed with Braun in Karlsruhe in 1947 and the first issue of Archiv der Mathematik appeared in 1948. The Archiv was taken over by Birkhäuser in Basel in 1952 and became a well-respected journal over the next few years. In tune with the policies of the Oberwolfach Institute, namely to fashion itself into a place for international co-operation in mathematics, the journal had an international editorial board from the very beginning. Out of 18 members of the board 9 lived outside Germany: Enrico Bompiani (Rome), Charles Ehresmann (Strasbourg), Hugo Hadwiger (Bern), Heinz Hopf (Zürich), Trygve Nagell (Uppsala), Christian Pauc (Cape Town), Johann Radon (Vienna), Jan Arnoldus Schouten (Amsterdam) and Eduard Stiefel (Zürich). Naturally, Süss underlined the international mission of the journal in the foreword of the first issue. We are very grateful that many colleagues from Germany and abroad have agreed with our plans and are willing to support us in our efforts to re-establish contact between colleagues irrespective of national borders in unbiased cooperation. 87 When the news spread in 1946/47 that Süss was about to create a new journal the idea was not universally welcomed among mathematicians in Germany. William Threlfall and Herbert Seifert immediately wrote to Süss that they “would prefer the continuation of existing journals to founding 86 Note in handwriting, signed by Cagniard on the proposal, June 1946 (UAF, E 6/13). 87 Süss: Foreword, in: Archiv der Mathematik 1(1948), 2: “Wir sind glücklich, zu unseren Plänen schon die Zustimmung vieler deutscher und ausländischer Kollegen gefunden zu haben, die uns in dem Bestreben unterstützen wollen, in sachlicher Zusammenarbeit die Verbindung der Fachkollegen über die Grenzen hinweg wieder herzustellen.” OBERWOLFACH IN THE FRENCH OCCUPATION ZONE: 1945 TO EARLY 1950S 159 new ones”. 88 Erich Kamke in Tübingen, too, was not supportive as he was trying to get the Mathematische Zeitschrift licensed in the French zone. 89 The publisher Ferdinand Springer and his main mathematical advisor F. K. Schmidt also saw the new journal in clear competition to the inactive Mathematische Zeitschrift [Remmert 2000, 29]. However, for Süss the Archiv had at least two functions in the context of his legitimisation strategies. On the one hand, the Archiv played a crucial role in creating an international platform for the Oberwolfach Institute within the mathematical community. On the other hand, the Archiv as journal of the Oberwolfach Institute afforded the possibility of exchanging copies with other journals to help to build up a library at the institute in light of its modest budget. This had been among Blaschke’s motives in establishing the Abhandlungen aus dem Mathematischen Seminar der Hamburgischen Universität after World War I [Remmert & Schneider 2010, 159–163]. To this effect Süss had ensured that the publisher Braun guaranteed three free copies of the Archiv for the Oberwolfach Institute as well as the option to purchase as many copies as they needed at cost price. 90 The book series the Oberwolfach Institute started to edit under Süss’ auspice, Studia mathematica, basically followed the same rationale as the Archiv der Mathematik [Remmert & Schneider 2010, 274–281]. The scarcity of available and purchasable mathematical books in Germany was a crucial impediment to university teaching in mathematics. Süss clearly saw a chance here for himself and the Oberwolfach Institute and got in touch with the Göttingen publishing house Vandenhoeck & Ruprecht, being well aware that Hellmut Ruprecht (1903–1991) wished to expand Vandenhoeck & Ruprecht’s activities into mathematics and the sciences. Until 1945 Süss had co-operated with the Akademische Verlagsgesellschaft in Leipzig, which now lay in the Russian zone, and he had had a serious clash with Springer [Remmert 2000, 24–29]. Thus, if he wanted to get a foot into the market for mathematical books, Vandenhoeck & Ruprecht was about the only option he had. As it were, Ruprecht was quite interested in Süss’s idea to publish a series of monographs institutionally connected to the Oberwolfach Institute. The first volumes of the series Studia mathematica 88 Threlfall to Süss, June 18, 1946 (UAF, C89/114): “Wir würden die Fortsetzung bestehender Zeitschriften mehr begrüßen, als Neugründungen, die nach Mainzer Methoden schmecken.” They refer to the university of Mainz, which had been opened by the French in May 1946. 89 Knopp to Süss, August 23 and October 18, 1946 (UAF, C89/6). 90 Cf. the enclosure to Süss’ letter to the publisher, October 14, 1947 (UAF, E 6/13, p. 10–12). 160 V. R. REMMERT began to appear in 1948 and the series only ceased to exist in 1978 after 30 volumes had been published. Most of the early authors were members of Süss’s network: Vol. 1: Emanuel Sperner: Einführung in die analytische Geometrie und Algebra (1948) (cf. table 1), Vol. 2: Gerrit Bol: Elemente der analytischen Geometrie (first part 1948, second part 1949), Vol. 3: Walter Lietzmann: Elementare Kugelgeometrie (1949), Vol. 4: Gerrit Bol: Projektive Differentialgeometrie (1950) (cf. table 1). The series turned out to be a success for both the publisher and Süss. Ruprecht was pleased by the economic success of the series and by the fact that he had managed to get into a new market (mathematics). Süss had ensured that the series carried the subtitle “mathematical textbooks, edited by the Mathematical Research Institute Oberwolfach” (“Mathematische Lehrbücher, herausgegeben vom Mathematischen Forschungsinstitut in Oberwolfach unter der Leitung von Prof. Dr. W. Süss”), propagating and inflating the importance of Oberwolfach as a research institution. To sum up, the Archiv, following the publication of the FIAT Reviews, was an important tool to shape the institute’s identity as (1) a mathematical research institute, (2) a place of international co-operation in mathematics, (3) a nucleus of a Franco-German rapprochement in and beyond mathematics, and (4) a leading mathematical centre in Germany. This goal was not only pursued by the resumption of journal publication, but was flanked by further publication projects, that were partly connected to Süss’s extensive, but mostly ineffective publishing plans in World War II [Remmert 1999, 39–43], such as the book series Studia mathematica, the re-establishment of the Mathematisch-Physikalische Semesterberichte as well as a thwarted attempt to take part in the publication of a handbook of mathematics [Remmert & Schneider 2010, 274–281, 293–296]. 10. CONCLUDING REMARKS: SHAPING/FINDING A NEW INSTITUTIONAL IDENTITY This paper is part of a larger research project dealing with the history of the Oberwolfach Institute between 1944 and the early 1960s, when the Thyssen Foundation followed by the Volkswagen Foundation stepped in as major funding institutions, in the aftermath of which most of the OBERWOLFACH IN THE FRENCH OCCUPATION ZONE: 1945 TO EARLY 1950S 161 institute’s budgetary problems were solved [Gericke 1984, 37f]. 91 While the history of its foundation is relatively well understood (see Section 1 above), the development after 1945 has scarcely been touched on by historians (of mathematics). The project aims at filling this gap, namely, to analyse the history of the Oberwolfach Institute as it institutionally changed from a projected National Institute for Mathematics with a wide, but standard range of responsibilities into an international social infrastructure for research. This notion was completely new in the framework of German academia for years to come and has only been conceptually codified in 2011 when the term “social infrastructure for research” (“soziale Forschungsinfrastruktur”) as a specific category for research institutions was proposed—specifically with the Oberwolfach Institute in mind—as “meeting space for discursive exchange of current and the development of new research questions” (“in der Regel Begegnungsräume des diskursiven Austauschs von aktuellen und der Entwicklung von neuen Forschungsfragen”; [Wissenschaftsrat 2011, 20f]). Granted, the term may be awkward, but a look at the institutional identity of the Oberwolfach Institute shows that it clearly fills a conceptual gap. The Leibniz Association as institutional harbour of the Oberwolfach Institute adopted the concept, even though in 2014 only two of its 89 institutes fell in the category, the Oberwolfach Institute and the Leibniz Center for Informatics/Schloss Dagstuhl (founded in 1989; Leibniz Association 2014, 33–35). To historically understand the institutional evolution of the Oberwolfach Institute from National Institute for Mathematics to international social infrastructure for research means to focus on the evolvement of the institutional identity of the Oberwolfach Institute between 1944 and the early 1960s, namely the development and importance of the Oberwolfach Institute’s scientific programme (workshops, teamwork) and the research tools employed (library, workshops) as well as the corresponding strategies to safeguard the Oberwolfach Institute’s existence (for instance under the wings of the Max-Planck Society). This process cannot be understood without paying close attention to the French influence as it played out in Oberwolfach in the late 1940s. For an analysis of the institute’s history in the 1950s and 1960s, the concept of an institutional identity as developed and applied by Dania Achermann to the history of the Institute for Atmospheric Physics in Oberpfaffenhofen after World War II would be fruitful [Achermann 2016]. 91 The project is funded by the German Research Foundation: The Oberwolfach Research Institute for Mathematics, 1944–1963: From “National Institute for Mathematics” to an international “social infrastructure for research”. 162 V. R. REMMERT Achermann, following the work of Stuart Albert and David A. Whetten on organisational identities, conceives of an institutional identity as being shaped by three characteristics reflected in three questions: What was the core programme of the scientific institution (central)? What feature endured for a “long” time (enduring )? And what made this organisation unique and distinguishable from others (distinctive)? [Achermann 2016, 248]. For the Oberwolfach Institute the answer seems simple as, of course, seen from today’s perspective the workshop programme and the library guarantee these traits. However, further research must be done to better understand the evolution of the workshop programme and the library into the Oberwolfach Institute’s central, enduring and distinctive research tools during the 1950s and 1960s. The method Achermann proposes means to analyse the history of a scientific institution by way of understanding the historical transformation of its institutional identity. In 1945 the Oberwolfach Institute had lost its (young and fragile) institutional identity and was searching for a new one [Remmert 2019]. To study this process demands to closely intertwine the work of and at the institute with contemporary science policies as well as political contexts. The history of the Oberwolfach Institute in the late 1940s is a case in point, as we have seen that it cannot be understood without embedding it into the political and cultural context of the French occupation zone that had a long-term impact on the institute’s institutional identity. While control of research according to law no. 25 did not play a relevant role for the Oberwolfach Institute, co-operation with French mathematicians and with the French authorities became crucial for developing a new vision for the institutional identity. This new institutional identity focused on a publication programme and on turning Oberwolfach into a meeting place for mathematicians from Germany and abroad (workshops as well as individual visits), both with full support of the military government. Even though the publication programme did not endure as a research tool, it helped to build up the library as a research tool in times of a scarce budget—that is into the early 1960s. In 1956 Süss characterized the library as “the main tool for research” (“das hauptsächliche Werkzeug für die Forschungsarbeit”) 92. The third crucial influence on the institute’s institutional identity was the deep appreciation for Bourbaki in Oberwolfach, which in turn left a mark on the institute’s later agenda (Kneser and the structure of mathematical research, teamwork). Finally, the fact that the French authorities 92 Report on the Oberwolfach Institute by Süss for the Ministry of Culture and Education, April 26, 1956 (Hauptstaatsarchiv Stuttgart, EA 13–201 Bue 333–1). OBERWOLFACH IN THE FRENCH OCCUPATION ZONE: 1945 TO EARLY 1950S 163 modestly subsidised the institute entailed steady, if equally modest financial support from the Ministry of Culture and Education in Freiburg. All in all, this paper has shown the significant impact that its being situated in the French occupation zone in 1945 had on the further development of the Oberwolfach Institute. Beyond the realm of the history of mathematics, the institute is an excellent example of the French policy not to transplant research units, but leave them in place, which eventually was to the benefit of both sides. 11. ACKNOWLEDGEMENTS Research for this paper was kindly supported by the German Research Foundation in the framework of the project Das Mathematische Forschungsinstitut Oberwolfach, 1944–1963: Vom “Reichsinstitut für Mathematik” zur internationalen “sozialen Forschungsinfrastruktur” (The Oberwolfach Research Institute for Mathematics, 1944–1963: From “National Institute for Mathematics” to an international “social infrastructure for research”). Thanks for their help and comments go to Dania Achermann, Michèle Audin, Michael Barany, Corine Defrance, Christophe Eckes, Samantha Leong, John McCleary, Maria Remenyi, Norbert Schappacher, Antina Scholz, Reinhard Siegmund-Schultze, Klaus Volkert, the Revue’s referees and the Oberwolfach Research Institute for Mathematics. 12. APPENDIX Georges Reeb, 1950: Institut de Mathématiques du Lorenzenhof, typescript copy of the original report (Archives de l’occupation française en Allemagne et en Autriche (1945–1955), Paris: 1BAD1262): Ayant été souvent l'hôte du Lorenzenhof, et ayant séjourné durant cette année pendant six mois à cet établissement, je crois devoir résumer brièvement ici mes impressions toutes personnelles quant à l'activité et l'utilité d'un tel organisme. Avant de parler des questions scientifiques, je voudrais insister dès le départ sur l'atmosphère spécialement sympathique qui règne au Lorenzenhof. Cette ambiance est certainement due à la direction adroite et habile de Monsieur le Professeur W. SÜSS qui excelle dans l'art de réussir dans une atmosphère d'entente et de cordialité des gens de nationalité et de caractères très divers. Les possibilités de col- 164 V. R. REMMERT laboration scientifique offertes par le Lorenzenhof sont autant de possibilités d'entente sur un plan international. -Le Forschungsinstitut a été créé en juin 1944 par le Reichsforschungsrat ; le projet et les crédits étaient somptueux. Depuis l'institut vit sur des fonds très modestes mais il a conservé ses objectifs scientifiques : Édition et participation à l'édition d'ouvrages et publications scientifiques : (Rapport FIAT, Archiv der Mathematik, Math. Phys. Semesterberichte, Studia Mathematica, etc.) Organisation de congrès et de colloques sur des sujets spécialisés. -- A ces objectifs, le Lorenzenhof en a ajouté un autre : développer les relations et échanges de vue entre chercheurs sur un plan international. -Je voudrais indiquer maintenant, parmi tant d'autres, quelques exemples où cet objectif a été atteint. -LIAISON FRANCE - ALLEMAGNE : Les colloques organisés au Lorenzenhof ont souvent réuni Français et Allemands. Voici quelques résultats obtenus : a) Influence française : Il convient de réserver une place importante à l'influence de N. BOURBAKI dont de nombreux élèves (et collaborateurs) ont séjourné au Lorenzenhof. Dans le traité de mathématique qu'il a rédigé, N. BOURBAKI se propose de reprendre la mathématique à la base, et de construire un instrument de travail puissant et efficace. On peut dire que les mathématiciens allemands ont été très intéressés par ces idées neuves, qu'ils ont d'ailleurs très vite assimilées. Je pense en particulier à l'enthousiasme de M. KNESER (Tübingen) hôte assidu du Lorenzenhof, connu pour sa connaissance à peu près universelle des mathématiques. On peut relever les trois points suivants, où l'influence de N. BOURBAKI est très nette : Enseignement : dans certaines universités allemandes l'enseignement est directement influencé par N. BOURBAKI Théorême de LORX [Zorn's Lemma] : cet instrument était peu connu en Allemagne (où il était remplacé par un théorème moins commode) ; actuellement des mathématiciens allemands se servent du théorème de LORX. -Filtres : Enfin il n'est pas exagéré de dire que l'impression de méthode, d'ordre et de puissance n'a pas été sans surprendre les Allemands dans certains préjugés. -- OBERWOLFACH IN THE FRENCH OCCUPATION ZONE: 1945 TO EARLY 1950S 165 Comme d'autres facteurs de l'influence française on peut citer : La Géométrie Infinitésimale Directe (BOULIGAND, PAUC, CHOQUET ; HAUPT, AUMANN (Allemagne). -La Géométrie Différentielle et Algébrique Classique (VINCENSENI, d'ORGEVAL). -Influence des idées allemandes : Les visiteurs français ont peut-être été frappés par les faits suivants : Le soucis constant en Allemagne d'éveiller très tôt le goût de la recherche scientifique, et de créer de vocations de chercheurs. L'enseignement prépare très rapidement au travail personnel. -Le développement du calcul des variations, qui a abouti à la découverte de ``Chemin Royal'' de CARATHEODORY, et de son élève BOERNER La vitalité toujours considérable de l'Algèbre. Les idées de HILBERT E. NOETHER n'ont rien perdu de leur fécondité et de leur efficacité !Terminons par quelques mots sur : Les relations internationales : Pour ne citer que quelques faits saillants, on peut dire que le Lorenzenhof a été un terrain favorable pour la confrontation des idées sur les sujets suivants : Topologie : où l'imposante Ecole Suisse (HOPF, ECKMANN, STIEFEL) a repris contact avec l'Ecole Allemande (SEIFERT) Belge (HIRSCH) Française (EHRESMANN). Logique : 93 Géométrie différentielle : Les idées de W. SÜSS sur la géométrie différentielle relative, et les nouvelles conceptions de G. BOL sur la géométrie différentielle projective, ont trouvé des auditeurs intéressés. (RUND de Capetown, et d'autres). -Pour conclure je reviendrai encore une fois au caractère familier et sympathique des colloques du Lorenzenhof, qui favorise les échanges de vues et les conversations privées. Personne n'est pressé ; chacun se sent englobé dans la communauté. Ce climat est propice aux jeunes mathématiciens et étudiants (que M.SÜSS attire volontiers) et leur permet d'approcher leurs aînés dans les conditions qui ne leur sont offertes nulle part ailleurs. 93 It seems that in the process of copying the report a passage on the 1949 logic meeting in Oberwolfach (Paul Bernays) was skipped. 166 V. R. REMMERT REFERENCES Achermann (Dania) [2016] Institutionelle Identität im Wandel: Zur Geschichte des Instituts für Physik der Atmosphäre in Oberpfaffenhofen, Bielefeld: transcript, 2016. 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[1948] FIAT Review Angewandte Mathematik, 5 vols., Wiesbaden: Dietrich’sche Verlagsbuchhandlung, 1948. Weyl (Fritz Joachim) [1952] A Symposium at the Mathematical Research Institute Oberwolfach, Technical report ONRL-12-52, London: Office of Naval Research, 1952. 172 V. R. REMMERT Wielandt (Helmut) [1974] Hellmuth Kneser in Memoriam, Aequationes Mathematicae, 11 (1974), pp. 120a–120c. Wissenschaftsrat [2011] Empfehlungen zu Forschungsinfrastrukturen in den Geistes- und Sozialwissenschaften, Berlin: Wissenschaftsrat, 2011; https: //www.wissenschaftsrat.de/download/archiv/10465-11.pdf. Zauner (Stefan) [1994] Erziehung und Kulturmission: Frankreichs Bildungspolitik in Deutschland 1945–1949, Munich: Oldenbourg, 1994. Sommaire Sabine Rommevaux-Tani — La notion médiévale de contractio dans l’Arithmetica integra de Michael Stifel (1544) 97 Volker R. Remmert — L’institut Oberwolfach dans la zone française d’occupation : de 1945 aux années 1950 . . . . . . . . . 121 Contents Sabine Rommevaux-Tani — The medieval notion of contractio in Michael Stifel’s Arithmetica integra (1544) . . . . . . 97 Volker R. Remmert — Oberwolfach in the French Occupation Zone: 1945 to early 1950s . . . . . . . . . . . . . . . . . . . . 121 Revue d’histoire des mathématiques / Journal for the History of Mathematics Éditée par la Société Mathématique de France, la Revue d’histoire des mathématiques publie des articles originaux (en français ou en anglais) consacrés à l’histoire des mathématiques, de l’Antiquité à nos jours. Dans ces textes, les sciences mathématiques peuvent être considérées aussi bien dans leur développement propre que dans leurs rapports à d’autres disciplines ou dans leurs contextes (culturel, institutionnel, social). La Revue d’histoire des mathématiques a l’ambition de servir la communauté internationale des historiens des mathématiques en offrant un espace de débat critique ouvert à des bilans historiographiques et des notes prospectives ou programmatiques. Elle s’adresse, au-delà de cette communauté, aux mathématiciens, aux historiens et philosophes des sciences, aux sociologues, aux anthropologues, et à tous ceux qu’intéresse une réflexion sur les mathématiques et leur développement. Edited under the auspices of the French Mathematical Society (Société Mathématique de France), the Journal for the History of Mathematics publishes original papers (in French or in English) devoted to the history of mathematics, from Antiquity to the present. The Journal welcomes manuscripts dealing with the development of the mathematical sciences proper as well as papers bearing on relationships to other disciplines or on the institutional, cultural, and social contexts. The ambition of the Journal for the History of Mathematics is to serve the historians of mathematics’ international community by offering a forum for critical debate, open to historiographic essays and programmatic contributions. Beyond the professional community, the Journal is addressed to mathematicians, historians and philosophers of science, sociologists, anthropologists, and to all those interested in understanding mathematics and its development. À propos des indicateurs bibliographiques / Bibliographic indicators Le Comité de rédaction de la Revue d’histoire des mathématiques souhaite exprimer sa position sur les facteurs d’impact et autres indicateurs présents sur le marché des revues scientifiques, comme le taux d’acceptation des articles. En 2009, la Revue d’histoire des mathématiques avait déjà, comme la très grande majorité des revues d’histoire des sciences, signé l’appel « Journals under Threat : A Joint Response from HSTM Editors », contre le classement en A, B, C de ces revues. Tant les mathématiciens que les spécialistes de sciences humaines et sociales ont établi que les indicateurs bibliométriques usuels n’ont pas de pertinence individuelle — en particulier parce qu’ils varient d’une discipline à une autre, et même d’une sous-discipline à une autre —, qu’ils ne permettent pas d’évaluer la qualité scientifique d’articles ou d’auteurs, et que la plupart d’entre eux sont faciles à manipuler. Dans un domaine en plein développement théorique comme l’histoire des mathématiques, le Comité de rédaction estime aussi que la qualité d’une revue n’est pas mesurée par son refus d’une grande quantité d’articles (ce qu’il est toutefois amené à faire), mais par sa capacité à améliorer les articles par des rapports détaillés et par l’encadrement des auteurs jusqu’à la publication. Il ne rendra donc public aucun indicateur de ce type. The Editorial Board of the Revue d’histoire des mathématiques wishes to express its point of view concerning impact factors—and other indicators such as acceptance rates— that are currently being used in the scientific journal market. In 2009, the Revue d’histoire des mathématiques, like most history of science journals, signed the call “Journals under Threat: A Joint Response from HSTM Editors” against the ranking of these journals on an A, B, and C scale. Mathematicians as well as scholars in the social sciences and in the humanities have established that standard bibliometric indicators are meaningless for ranking individual papers; they vary from one discipline to another, and even from one sub-discipline to another, they also do not assess the scientific quality of articles and authors, and most are easy to tamper with. In a field in full conceptual development such as the history of mathematics, the Editorial Board also believes that the quality of a journal is not measured by its rejection of a large number of articles (which it is always obliged to do), but by its ability to improve articles through detailed referee reports and through working with authors at each step of the publication process. The Editorial Board of the Revue d’histoire des mathématiques will thus not make public any indicators of this kind. Sommaire Sabine Rommevaux-Tani — La notion médiévale de contractio dans l’Arithmetica integra de Michael Stifel (1544) 97 Volker R. Remmert — Oberwolfach in the French Occupation Zone : 1945 to early 1950s . . . . . . . . . . . . . . . . . . . . 121 Société Mathématique de France
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