Table Of ContentTrends 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
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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
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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.
