Model And Mathematics: From The 19th To The 21st Century PDF
Preview Model And Mathematics: From The 19th To The 21st Century
Trends in the History of Science Michael Friedman Karin Krauthausen Editors Model and Mathematics: From the 19th to the 21st Century Trends in the History of Science Trends in the History of Scienceisaseriesdevotedtothepublicationofvolumes arising from workshops and conferences in all areas of current research in the history of science, primarily with a focus on the history of mathematics, physics, and their applications. Its aim is to make current developments available to the community as rapidly as possible without compromising quality, and to archive thosedevelopmentsforreferencepurposes.Proposalsforvolumescanbesubmitted using the online book project submission form at our website www.birkhauser-sci ence.com. More information about this series at https://link.springer.com/bookseries/11668 Michael Friedman · Karin Krauthausen Editors Model and Mathematics: From the 19th to the 21st Century Editors MichaelFriedman KarinKrauthausen TheCohnInstitutefortheHistory ClusterofExcellence“MattersofActivity. andPhilosophyofScienceandIdeas ImageSpaceMaterial” HumanitiesFaculty Humboldt-UniversitätzuBerlin TelAvivUniversity Berlin,Germany RamatAviv,TelAviv,Israel TheeditorsacknowledgethesupportoftheClusterofExcellence“MattersofActivity.Image SpaceMaterial”fundedbytheDeutscheForschungsgemeinschaft(DFG,GermanResearch Foundation)underGermany’sExcellenceStrategy–EXC2025–390648296. ISSN2297-2951 ISSN2297-296X (electronic) TrendsintheHistoryofScience ISBN978-3-030-97832-7 ISBN978-3-030-97833-4 (eBook) https://doi.org/10.1007/978-3-030-97833-4 ©TheEditor(s)(ifapplicable)andTheAuthor(s)2022.Thisbookisanopenaccesspublication. 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This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered companySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Contents HowtoGraspanAbstraction:MathematicalModelsandTheir VicissitudesBetween1850and1950.Introduction ....................... 1 Michael Friedman and Karin Krauthausen PartI:HistoricalPerspectivesandCaseStudies KnowingbyDrawing:GeometricMaterialModelsinNineteenth CenturyFrance ......................................................... 53 Frédéric Brechenmacher WilhelmFiedlerandHisModels—ThePolytechnicSide ................ 145 Klaus Volkert ModelsfromtheNineteenthCenturyUsedforVisualizingOptical PhenomenaandLineGeometry ......................................... 177 David E. Rowe ModelingParallelTransport ............................................ 203 Tilman Sauer The Great Yogurt Project: Models and Symmetry Principles inEarlyParticlePhysics ................................................. 221 Arianna Borrelli InterviewwithMyfanwyE.Evans:EntanglementsOnandModels ofPeriodicMinimalSurfaces ............................................ 255 Myfanwy E. Evans, Michael Friedman, and Karin Krauthausen The Dialectics Archetypes/Types (Universal Categorical Constructions/Concrete Models) in the Work of Alexander Grothendieck ............................................................ 267 Fernando Zalamea v vi Contents PartII:EpistemologicalandConceptualPerspectives ‘Analogies,’‘Interpretations,’‘Images,’‘Systems,’and‘Models’: Some Remarks on the History of Abstract Representation intheSciencesSincetheNineteenthCentury ............................ 279 Moritz Epple Mappings, Models, Abstraction, and Imaging: Mathematical ContributionstoModernThinkingCirca1900 .......................... 309 José Ferreirós Thinking with Notations: Epistemic Actions and Epistemic ActivitiesinMathematicalPractice ..................................... 333 Axel Gelfert Matrices—CompensatingtheLossofAnschauung ...................... 363 Gabriele Gramelsberger Part III: From Production Processes to Exhibition Practices Interview with Anja Sattelmacher: Between Viewing andTouching—ModelsandTheirMateriality ........................... 381 Michael Friedman, Karin Krauthausen, and Anja Sattelmacher Interview with Ulf Hashagen: Exhibitions and Mathematical ModelsintheNineteenthandTwentiethCenturies ...................... 403 Michael Friedman, Ulf Hashagen, and Karin Krauthausen InterviewwithAndreasDanielMatt:Real-TimeMathematics .......... 431 Michael Friedman, Karin Krauthausen, and Andreas Daniel Matt How to Grasp an Abstraction: Mathematical Models and Their Vicissitudes Between 1850 and 1950. Introduction Michael Friedman and Karin Krauthausen What is a model? Today, this question can only be answered either with a high degree of abstraction or generality, or with the most specific and precise con- textualization, since the concept of the model and the practice of modeling are ubiquitous—in all the sciences and arts, in engineering and design. To underline this fact, already in 2003, the model theorist Bernd Mahr suggested that mod- els could “become the semantic, combinatorial, and technical foundation of our culture, just as this was the case with numbers through mathematics and informa- tiontechnology[…].”1 Abrieflookatthehistoryofthemodelconceptremindsus thattheessentialambivalenceattributedtomodelsinthetwentiethandtwenty-first 1BerndMahr,“Modellieren:BeobachtungenundGedankenzurGeschichtedesModellbegriffs,” inBild,Schrift,Zahl,ed.SybilleKrämerandHorstBredekamp(Munich:WilhelmFink,2003),59– 86,here60:“inähnlicherWeisezumsemantischen,kombinatorischenundtechnischenFundament unserer Kultur werden [könnten], wie dies die Zahlen durch die Mathematik und die Informa- tionstechnik geworden sind […].” Whereas in the present volume the object area is limited to mathematicalmodels(mainly,ifnotentirely,between1850and1950),andtheauthorshavemostly chosenahistoricalapproach,theinvestigationscarriedoutbyMahr,amathematicianandcomputer scientist,aregearedtoanepistemologyofthemodel.Mahrtakesculturalhistoryasastartingpoint todevelopagenerallyvalidlogical-formaldescriptionof‘thenatureofthemodel.’Seealso:Bernd Mahr,“OntheEpistemologyofModels,”inRethinkingEpistemology,vol.1,ed.GünterAbeland JamesConant(BerlinandBoston:deGruyter,2011),301–52. B M.Friedman( ) TheCohnInstitutefortheHistoryandPhilosophyofScienceandIdeas,TheLesterandSally EntinFacultyofHumanities,TelAvivUniversity,RamatAviv,6997801TelAviv,Israel e-mail:[email protected] B K.Krauthausen( ) ClusterofExcellence“MattersofActivity.ImageSpaceMaterial,”Humboldt-Universitätzu Berlin,UnterdenLinden6,10099Berlin,Germany e-mail:[email protected] ©TheAuthor(s)2022 1 M.FriedmanandK.Krauthausen(eds.),ModelandMathematics: Fromthe19thtothe21stCentury,TrendsintheHistoryofScience, https://doi.org/10.1007/978-3-030-97833-4_1 2 M.FriedmanandK.Krauthausen centuries—theambivalencebetweenconcretionandabstraction(withtheemphasis moving increasingly in the direction of abstraction)—can be traced much further back. Etymologically, the word ‘model’ is derived from the Latin modulus, the diminutive of modus. Whereas modus generally stands for ‘measure’ (also tem- poral measure), ‘measuring stick,’ and ‘quantity,’ as well as for ‘aim,’ ‘rule,’ or ‘manner,’ modulus (in Vitruvius, but also in the early Middle Ages) is essentially determined via the practice of architecture, where it stands in a technical sense for the dimensions of columns or the relations of their parts.2 Both terms belong to the context of ‘form giving’ and design, but whereas modus “is a conceptual termdesignatingsomethingabstractthatispositedandnotgiven,”modulus refers to “something concrete.”3 In the Italian architecture of the fifteenth and sixteenth centuries,thepractical-concretecontextofthemodel(ormodello)becomesclearer with the growing importance of three-dimensional scale models of future (as well as finished) architectural projects, which, in the case of larger, more elaborate projects may have been used to win over a client—a famous example being the competition in 1418 for a model of the dome of Florence Cathedral (see Fig. 1).4 While not providing a direct blueprint of the construction to be built, these mod- els,whichweremostlymadefromwood,actedashaptic-concreteelementsinthe design process. Despite their only moderate accuracy, they provided a convincing description of the construction’s form—that is, they permitted a summary view of a future to be realized, but one that was sufficiently approximate in the detail to allow for adjustments and changes. Later, three-dimensional models of this kind were also found in the natural sciences—an example is the series of crystal models constructed by Jean-Baptiste LouisRomédel’Isleattheendoftheeighteenthcentury(Fig.2).Ratherthanrep- resenting a step in a design process for a building to be realized, however, these models are part of an epistemic process of ‘form giving,’ they visualize knowl- edge about crystals and allow both students and trained scientists to obtain (in combination with verbal explanations and visual representations in related trea- tises) an overview. In this way, they allow the student and scientist not only to acquire existing knowledge, but also to explore new lines of research. In the late 18th and early nineteenth centuries, crystallographers such as Jean-Baptiste LouisRomédel’IsleandRenéJustHaüymanufacturednumeroussuchmodelsof 2Mahr,“OntheEpistemologyofModels,”255–59. 3Mahr,“Modellieren,”61:“einBegriffswortist,dasetwasAbstraktesbezeichnet,dasgesetztist undnichtvorgefundenwird;”“etwasKonkretes.”FortheEnglishusageoftheword,seetheentry “Model” in The Oxford English Dictionary, vol. VI (Oxford: Clarendon Press, 1933), 568–69, aswellas“model,n.andadj.”inTheOxfordEnglishDictionary,OEDOnline(OxfordUniver- sity Press, December 2021): https://www.oed.com/view/Entry/120577?rskey=Hv9gMy&result= 1&isAdvanced=false(accessedDecember6,2021). 4See:AndresLepik,Das Architekturmodell in Italien 1335–1550,dissertation,seriesRömische Studien der Bibliotheca Hertziana, vol. 9 (Worms: Wernersche Verlagsgesellschaft, 1994)—on modelsintheprocessofdesigningandconstructingFlorenceCathedral,59–89. HowtoGraspanAbstraction:MathematicalModels… 3 Fig.1 GiorgioVasari,“FilippoBrunelleschiandLorenzoGhibertiPresentingtheModelofthe Church of San Lorenzo (Florence),” ca. 1556–1558. Fresco in the Palazzo Vecchio, Florence. Photo:PeterHorree,2017.©Alamy,allrightsreserved crystalsmadeofpaper,wood,orterra-cotta.5ForHaüy,thesemodels,whichrepre- sentedtheoretical,idealizedminerals,were“amenabletomathematicalabstraction and geometrical analysis.”6 Yet scientific models also acted as prestigious objects intended for public and private collections. Moreover, in the nineteenth century, they played an important role in the self-promotion of university departments. 5Onthehistoryofcrystallography,see:JohnG.Burke,OriginsoftheScienceofCrystals(Berke- ley:UniversityofCaliforniaPress,1966);HenkKubbinga,“CrystallographyfromHaüytoLaue: ControversiesontheMolecularandAtomisticNatureofSolids,”ActaCrystallographicaSection A: Foundations of Crystallography68,no.1(2012):3–29.Seealso:MarjorieSenechal,“Brief HistoryofGeometricalCrystallography,”inHistoricalAtlasofCrystallography,ed.JoséLima-de- Faria(Dordrecht,Boston,andLondon:KluwerAcademicPublishers,1990),43–59—onthepaper modelsofcrystalsmadebyNicolasStenoin1669,55. 6LydieTouret,“CrystalModels:MilestoneintheBirthofCrystallographyandMineralogyasSci- ences,”inDutchPioneersoftheEarthSciences,ed.JacquesL.R.TouretandRobertP.W.Visser (Amsterdam:KoninklijkeNederlandseAkademievanWetenschappen,2004),43–58,here57.
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