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 contextualization, 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 models could “become the semantic, combinatorial, and technical foundation of our culture, just as this was the case with numbers through mathematics and information technology […].”Footnote 1 A brief look at the history of the model concept reminds us that the essential ambivalence attributed to models in the twentieth and twenty-first centuries—the ambivalence between concretion and abstraction (with the emphasis 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 temporal 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.Footnote 2 Both terms belong to the context of ‘form giving’ and design, but whereas modus “is a conceptual term designating something abstract that is posited and not given,” modulus refers to “something concrete.”Footnote 3 In the Italian architecture of the fifteenth and sixteenth centuries, the practical-concrete context of the model (or modello) becomes clearer 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).Footnote 4 While not providing a direct blueprint of the construction to be built, these models, which were mostly made from wood, acted as haptic-concrete elements in the 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.

Fig. 1
figure 1

© Alamy, all rights reserved

Giorgio Vasari, “Filippo Brunelleschi and Lorenzo Ghiberti Presenting the Model of the Church of San Lorenzo (Florence),” ca. 1556–1558. Fresco in the Palazzo Vecchio, Florence. Photo: Peter Horree, 2017.

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 Louis Romé de l’Isle at the end of the eighteenth century (Fig. 2). Rather than representing 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 knowledge about crystals and allow both students and trained scientists to obtain (in combination with verbal explanations and visual representations in related treatises) 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 Louis Romé de l’Isle and René Just Haüy manufactured numerous such models of crystals made of paper, wood, or terra-cotta.Footnote 5 For Haüy, these models, which represented theoretical, idealized minerals, were “amenable to mathematical abstraction and geometrical analysis.”Footnote 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. Hence, alongside their analytic-epistemic function, these models had a strategic and political function within the university system: scientific collections became a way for institutes and universities to call attention to themselves—a task assumed in the twentieth and twenty-first centuries principally by the complex technologies of instruments and laboratory machines (such as the various large and small electron microscopes).Footnote 7

Fig. 2
figure 2

a One of the 448 crystal models of unglazed porcelain made by Romé de L’Isle in Paris in ca. 1780. The complete collection was bought for Teylers Museum by Martinus van Marum in 1785 (Teylers Museum, Haarlem). CC BY-SA 3.0 NL (https://creativecommons.org/licenses/by-sa/3.0/nl/deed.en). b Planche VIII from Jean-Baptiste Louis Romé de L’Isle’s Essai de cristallographie, 1772. (ETH Library Zürich, Rar 2708, https://doi.org/10.3931/e-rara-16480) Public Domain Mark)

But what is meant when one speaks of mathematical models in the nineteenth and twentieth centuries? That is the subject of the present volume, and this question is answered by the contributors to this volume in detailed historical studies of mathematical model practices in the long nineteenth century in France and Germany, and, going beyond this initial focus, in a series of interviews on model practices in the sciences of the nineteenth and twentieth centuries as well as on twenty-first century digital visualization techniques.Footnote 8 However, there can and will be no single answer that encompasses all mathematical models: mathematical models have been and continue to be so productive because their definition, function, and appearance was and is capable of change. When the talk is of mathematical models, then, from a historical perspective, these include both the haptic-concrete model constructions of the nineteenth centuryFootnote 9 and the abstract, formal model concepts of the twentieth century. The central concern of the present volume is to show the historical diversity and capacity for change as well as the pedagogical and epistemic importance of mathematical models.

In the following introduction we concentrate on the ambivalent meaning of mathematical models between concretion and abstraction. This is examined by means of a few paradigmatic protagonists, concepts, and concrete examples from the period between 1860 and 1950.Footnote 10 Here, it will be necessary, for the nineteenth century, to refer to models situated at the interface between physics and mathematics. In this way, it can be shown whether and which ontological questions are raised by the mathematical objects in each respective context (see section I.) and what meaning is attached to the models, taken by themselves and in relation to James Clerk Maxwell’s notion of analogy, in research and theory (in section II.). From the question of what models represent, we then move on to considerations by prominent nineteenth century mathematicians and model builders (such as Felix Klein and Alexander Brill) about how models represent (section III.), before tracing the paradigm shift from ‘Anschauung’ (intuition) and ‘Bild’ (image) to ‘word’ or ‘text’ and ‘formal logic’ in the work of Felix Hausdorff and Alfred Tarski (section IV.(1)). In a final step we then briefly refer, with the example of mathematical biology, to the changed use of mathematical models in the natural sciences (section IV.(2)) and, with the example of structural anthropology (Claude Lévi-Strauss), to the deployment of mathematics and the model concept in the humanities of the first half of the twentieth century (section V.). The sequence of historical stations discussed in the introduction should show the development of mathematical models toward an increasing conceptual plurality. Whether and to what extent this plurality entailed an epistemic gain will be discussed at the end of the introduction (section VI.), before we close with an overview of the contributions to this volume.

I. Models at the End of the Nineteenth Century: Between Maxwell’s ‘Fictitious Substances’ and Boltzmann’s ‘Tangible Representation’

When talking about mathematical models at the end of the nineteenth century, it is important to be clear which objects are being referred to. This concerns not only particular examples of models but also the history of the term ‘model’ within the field of mathematics. In nineteenth-century mathematics, ‘model’ clearly referred to material, three-dimensional objects, that is, to objects that one could pick up in one’s hands—which is somewhat different from our current use of the term.

The nineteenth century understanding of the term ‘model’ is historically documented—for example, in Ludwig Boltzmann’s article “Models” published in 1902 in the Encyclopædia Britannica.Footnote 11 In his article, Boltzmann, an Austrian physicist known for his discoveries in thermodynamics and the founding of statistical mechanics, provides a paradigmatic summary of the term ‘model’ as it was understood in his time. He starts with a general definition that attaches particular importance to the concrete, material execution: “The term model denotes a tangible representation, whether the size be equal, or greater, or smaller, of an object which is either in actual existence, or has to be constructed in fact or in thought.”Footnote 12

For Boltzmann, a model is a physical entity with definite spatial relations. Following on from this definition, he first distinguishes generally between stationary and moving models, before looking more closely at speculative ‘kinematic’ models (James Clerk Maxwell’s models of the purely hypothetical particle motion of matter) and the more concrete categories of working models (showing the functioning of machines) and experimental models (for technical inventions). Finally, he looks briefly at the instrumental-mathematical and in no way trivial models found in the context of physics. These models facilitate complex calculations and (via a ‘physical analogy’Footnote 13) make such calculations possible in the first place.Footnote 14 In his examples and classifications, however, Boltzmann concentrates on the models in physics and mathematics, to which he attaches particular importance.Footnote 15 He describes mathematical models as follows: “In pure mathematics, especially geometry, models constructed of papier-mâché and plaster are chiefly employed to present to the senses the precise form of geometrical figures, surfaces, and curves.”Footnote 16

Here, it is important to note how restrictively Boltzmann describes the appearance of mathematical models: these are three-dimensional (i.e. haptic-concrete and material) models, which he understands as “precise” presentations of geometric figures. The function of these models was as an aid in teaching and research, whereby the modeling principally involved surfaces of second and third degree, and in the case of third order surfaces the model becomes a “complicated, not to say hazardous, construction” (see Fig. 3).Footnote 17

Fig. 3
figure 3

© Collection of Mathematical Models and Instruments at Georg-August-University Göttingen, all rights reserved

Thread model by Hermann Wiener of a cone of third order and genus 0 (Georg-August-Universität Göttingen, Collection of Mathematical Models and Instruments, model 81). See: Martin Schilling, ed., Catalog mathematischer Modelle, 7th ed. (Leipzig: Verlag von Martin Schilling, 1911), 122, no. 79.

In Germany beginning in the 1870s, three-dimensional mathematical models of this kind were produced in large series and sold through catalogues such as the Catalog mathematischer Modelle distributed from 1881 onward by the publishing house of Ludwig Brill.Footnote 18 These catalogues contained models designed by, among others, the mathematicians Alexander Brill (Ludwig Brill’s brother) and Felix Klein (see Fig. 4). Nevertheless, such models could (and can) also be seen in the scientific collections in the mathematical departments of universities. Mathematical models were also presented outside of the universities, however, particularly in exhibitions of scientific instruments, models, and apparatuses, such as the Special Loan Exhibition of Scientific Apparatus in South Kensington, London, in 1876, and an exhibition organized by Walther Dyck in Munich in 1893, accompanied by the publication Katalog mathematischer und mathematisch-physikalischer Modelle, Apparate und Instrumente.

Fig. 4
figure 4

Page from Ludwig Brill’s Catalog mathematischer Modelle (Ludwig Brill, ed., Catalog mathematischer Modelle für den höheren mathematischen Unterricht, 3rd ed. (Darmstadt: L. Brill, 1885), 44). Note that models were usually depicted next to their descriptions, and that these depictions were only rarely photos

Boltzmann, however, speaks not only about three-dimensional models of surfaces in mathematics but also about models of surfaces in physics, above all in thermodynamics, where they mathematically represented the behavior of gases and fluids:

In thermodynamics, […] models serve, among other purposes, for the representation of the surfaces which exhibits the relation between the three thermodynamic variables of a body, e.g., between its temperature, pressure and volume. A glance at the model of such a thermodynamic surface enables the behaviour of a particular substance under the most varied conditions to be immediately realized.Footnote 19

Here, while Boltzmann does not explicitly mention which models and which model builders he is referring to, one may well be reminded of Maxwell’s clay model of a thermodynamic surface from 1874.Footnote 20 Maxwell (1831–1879), one of the most important physicists of the nineteenth century, dedicated himself to the study of electromagnetism and the kinetic theory of gases. His clay model represents a substance with water-like properties by reproducing the latter’s energy-entropy-volume coordinates, thus allowing the different possible states (gaseous, liquid, solid) to be read. This modeling of thermodynamic phenomena is based on a diagrammatic and geometric precedent—the ‘graphical method’ of the American mathematical physicist Josiah Willard Gibbs, who proposed this in 1873 for the combined representation of the properties of volume, pressure, temperature, energy, and entropy of a given body in any state (see Fig. 5a, b).Footnote 21 On his own three-dimensional realization of Gibb’s geometric method of representation, Maxwell writes:

[Regarding] Prof. J. Willard Gibbs’s […] graphical methods in thermodynamics[:] […] I made several attempts to model the surface, which he suggests, in which the three coordinates are volume, entropy and energy. The numerical data about entropy can only be obtained by integration from data which are for most bodies very insufficient, and besides it would require a very unwieldy model to get all the features, say of CO2, well represented, so I made no attempt at accuracy, but modelled a fictitious substance […].Footnote 22

Fig. 5
figure 5

© National Museums Scotland, all rights reserved

a Josiah Willard Gibbs’s presentation of the relationship between the various variables of a given body. From Josiah Willard Gibbs, “A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces,” in Josiah Willard Gibbs, The Scientific Papers of J. Willard Gibbs, vol. 1 (New York and Bombay: Longmans, Green, And Co), 33–54, here 44, Fig. 2. b James Clerk Maxwell’s thermodynamic surface Collections of the National Museums Scotland, T.1999.385 / PF4433).

From Maxwell’s description one can infer the instrumental and productive character of his geometric model, which allows mathematical calculations, and thus also observations, about the development of the modeled object that are not possible with the real object. More important for our context, however, is Maxwell’s statement about what he has modeled here: in part this is “a fictitious substance.” The above quotation thus points to a fundamental question: how was the relation of a geometric-physical model of this kind to the reality investigated understood in the second half of the nineteenth century? Maxwell’s formulation makes clear that, in the question of the model, he distances himself from an ontological commitment. The geometric-physical model refers to empirical relations—that is, it is not an arbitrary representation and not a product of the imagination. But since the theoretical knowledge about the physical objects is unreliable or insufficient, or the available data is incomplete or the calculation too complex, the model stands in for a well-reasoned but speculative, and in this sense ‘fictitious,’ content. Notable is also the difference that can be observed at this point between the models of physics and those of pure mathematics. At the time, both disciplines were facing a problem of representation (physics with respect to electromagnetism, and mathematics with respect to, for example, continuous but nowhere differentiable curves or complex curves and surfaces of increasing complexity and, among other things, their singular points (see section III. below), but they reacted very differently: mathematics strove for exactness in the three-dimensional representation of curves and surfaces, whereas the physics of the 1870s suspended somewhat the faithfulness to reality of the representation in order to obtain the latitude needed to allow for further approaches to the object of study.

For Maxwell, the model (especially the kinematic model of the purely hypothetical particle motion of matter) is a continuation of thought by other means—this was Boltzmann’s conclusion in 1902. In the nineteenth century, physical theory could no longer be understood as the clear and final determination of the structure of matter, but was “merely a mental construction of mechanical models.”Footnote 23 And the functioning of these mechanical models must have just enough to do with the real phenomena to help the understanding of these phenomena, and thus have a heuristic effect on the formation of the physical theory. The radicality of Boltzmann’s and Maxwell’s position can be appreciated by recalling Pierre Duhem’s conception of physics published in 1906 under the title La théorie physique: son objet, sa structure. For Duhem, physical theory was an achievement of abstraction, and indeed as a system of well-founded hypotheses and logical deductions—he explicitly excludes the material models of thermodynamics from knowledge.Footnote 24

Already at the beginning of Boltzmann’s article, such a broad function is attributed to all scientific and technical models. The mode of representation of models is compared here with the mode of functioning of thought. Thus, in order to form a mental representation of the world, thought has to link the things of the real world with concepts. For Boltzmann, therefore, the relation of similarity between mental representations and the things of the world necessarily remains incomplete and unverifiable—but without such a link between concept and thing, knowledge of the world becomes impossible. Boltzmann describes the act of mental representation as follows: “The essence of the process is the attachment of one concept having a definite content to each thing, but without implying complete similarity between thing and thought.”Footnote 25 For Boltzmann, the absence of a “complete similarity” between mental representation and real thing belongs to what he calls “symbolization”Footnote 26—for which he cites language and writing as examples. If we transfer Boltzmann’s considerations to the scientific model, then we can conclude that the model expands the space of thought and action, and this, one imagines, is because it can (and must) be a well-founded, but not a complete, representation of a physical or a geometric object.Footnote 27

If Boltzmann ascribes such importance to the scientific model, then that is due not least to the model practice and theoretical considerations of a physicist whom Boltzmann mentions repeatedly in his article: Maxwell. In the next section, we must therefore turn our attention to Maxwell’s understanding of model and analogy.

II. 1850s/1870s: ‘Analogy’ and ‘Model’ in Maxwell

How is ‘analogy’ defined in Boltzmann’s article? ‘Analogy’ is first employed with respect to scientific quantities. If longitude, mileage, temperature, or other physical quantities are expressed by numbers, Boltzmann’s term for these numbers is “arithmetical analogies,” never ‘models.’Footnote 28 According to Boltzmann, however, also the “tangible models” of mathematics and physics belong to the category of analogy: they create a “concrete spatial analogy in three dimensions.”Footnote 29 The term ‘analogy,’ therefore, is broader and can include any representation of mathematical figures or scientific variables. The model, then, is a special class in the category of analogy insofar as it is extended in length, width, and height.

This power of analogy is founded in the changed claim of physics as represented by Maxwell. Maxwell is convinced that the structure of matter can be grasped neither by physical theory alone (i.e., a logical-deductive procedure) nor by a purely mathematical-analytic approach. And what is more, even the derivation of axioms from the experimental event—the procedure clearly favored by Maxwell—is not possible in the case of the structure of matter.Footnote 30 For the investigation of elastic solids, the motion of gas molecules, or electromagnetic fields, and thus for domains for which no satisfactory theory was yet available and for which the conclusions from experiments were not sufficient, Maxwell recommended another procedure: the method of physical analogies.Footnote 31 Boltzmann is referring to this implicitly when he states: “The question no longer being one of ascertaining the actual internal structure of matter, many mechanical analogies or dynamical illustrations became available […].”Footnote 32

Maxwell’s method of analogy rests on the fact that different physical phenomena can have surprisingly similar mathematical formulas describing them. For Maxwell, this formal mathematical similarity becomes a motor for further calculations, new hypotheses, and theoretical inferences. Thus, in his 1856 article “On Faraday’s Lines of Force,” Maxwell uses the mechanical analogy of the motion of a fluid to investigate the geometry of electromagnetic ‘lines of force’ (for an example of these lines of force as they were drawn by Maxwell, see Fig. 6 from Maxwell’s 1873 book A Treatise on Electricity and Magnetism).

Fig. 6
figure 6

One of James Clerk Maxwell’s examples for drawings of lines of force. From James Clerk Maxwell, A Treatise on Electricity and Magnetism, vol. 1 (Oxford: Clarendon Press, 1873), Plates, fig. 4

The lines of force are a conjecture on Michael Faraday’s part and not a certain, calculable finding. Maxwell assumes that these lines of force permeate space even when there are no objects on which such a force can act. Maxwell insists that electromagnetic phenomena are “not even a hypothetical fluid” and therefore appear in his considerations merely as a kind of imaginary substance.Footnote 33 The analogy to the motion of a fluid is nevertheless helpful and productive: it serves Maxwell to suggest to the understanding in a manageable form those mathematical ideas that are necessary for the study of the still largely unknown electricity.Footnote 34 In this case, according to Maxwell, neither a purely mathematical nor a purely hypothetical approach can help:

In the first case [of a purely mathematical approach] we entirely lose sight of the phenomena to be explained; [...]. [In the second case of a theoretical approach], we see the phenomena only through a medium, and are liable to that blindness to facts and rashness in assumption which a partial explanation encourages.Footnote 35

To compensate for the limitations of the mathematical formula and of the theoretical hypothesis, Maxwell employs the physical analogy (in this instance, the comparison between electromagnetic phenomena and the behavior of fluids) to allow mathematical descriptions and predictions even for the non-calculable and largely unknown domain. Maxwell defines this physical analogy as follows: “By a physical analogy I mean that partial similarity between the laws of one science and those of another which makes each of them illustrate the other.”Footnote 36

Ernst Mach, who in 1902 declared Maxwell’s method of analogy to be one of the most important methods of research, and considered physical analogy to be as central to scientific research as the experiment, aptly characterized physical analogy as an “abstract similarity.”Footnote 37 Hence, analogy here is explicitly an abstract representation. It might be objected, however, that this abstractness still allows a proximity to the empiricism of the physical phenomenon.Footnote 38 According to Maxwell, this is precisely the advantage of analogy over mathematical description and physical theory. In Maxwell’s work, analogy is thus distinguished by multiple and even partially contradictory properties: (1) it remains connected to empirical evidence and is therefore concrete; (2) it gives rise to a speculative but not at all arbitrary illustration, thus enabling the further examination of previously unknown domains; (3) it succeeds in this productive visualization by means of a mode of abstraction: precisely that abstract similarity of the actually different physical phenomena which is guaranteed in the identical mathematical formulas.

Yet Maxwell does not propagate analogy as the only method of research. In his research practice this method is combined with other representational practices, such as the diagram, the three-dimensional model, and mathematical calculation. The distinction found in Boltzmann’s article between models as tangible representations and other forms of representation or symbolization (such as arithmetical analogies, and thus with numbers) appears to be less rigorous in Maxwell’s case. When mentioning the drawings that help him to determine the lines of force of an electric field, Maxwell speaks explicitly of a “geometrical model of the physical phenomena.”Footnote 39 For Maxwell, these geometric representations are valid representations of the physical laws at work in the electrical phenomenon—and the transition between the geometric drawing of a space traversed by lines of force and the three-dimensional model resulting from this seems fluid. Two-dimensional drawing and three-dimensional model allow an approach to the physical object that the object itself may not permit. In Maxwell’s writings there is therefore a double extension of scientific representation: on the one hand, the model moves into the immediate vicinity of drawing (i.e., the difference between the two-dimensional and the three-dimensional visualization appears secondary); on the other, with the physical analogy, the methodological range of physics is expanded—this is done via a procedure that, as is expressly stated, is based on an incomplete, abstract similarity, and it is from this that it derives its heuristic force.Footnote 40 Nevertheless, for Maxwell as well as for Boltzmann, model and analogy do not coincide.Footnote 41 A partial overlapping (not a coincidence) of model and Maxwell’s method of physical analogy is found only at the end of Boltzmann’s encyclopedia article:

It often happens that a series of natural processes […] may be expressed by the same differential equations; and it is frequently possible to follow by means of measurements one of the processes in question […]. If then there be shown in a model a particular case of [the first process] in which the same conditions at the boundary hold as in [the second process], we are able by measuring […] in the model to determine at once the numerical data which [we may] obtain for the analogous case […].Footnote 42

As we have already seen, according to Maxwell, two different physical phenomena are formally analogous when both can be described mathematically by one and the same formula. Analogy, then, allows the transfer from the known (or better known) domain to the unknown domain. In Boltzmann’s description the model now assumes a decisive position insofar as one domain of the analogy is not constituted by the physical object but already appears as a model of this object. This model allows the developments of the physical object to be read, and these findings can then be transferred to the other side of the analogy, which is to say, they can lead to a further analysis of that domain which, taken by itself, would not be accessible to measurement. In this case, via the access to the model, the physical analogy is made fruitful for further research. The model can therefore become part of the physical analogy and also support the latter’s function as a motor of scientific knowledge.

At this point, however, the valorization of the model refers in Boltzmann’s case to a hybrid: the geometric-physical model whose similarity to the real physical phenomenon need not be complete to be scientifically productive. In the course of the nineteenth century, one would invoke other concepts and descriptions to determine the use of these purely mathematical models in pedagogical and epistemic contexts. These were above all the concepts of Anschauung (intuition) and Bild (image), which are the subject of the next section.

III. 1880–1900: ‘Anschauung’ and ‘Bild’ (Klein and Brill)

Boltzmann’s 1902 article “Models” gathers together the knowledge about models of the second half of the nineteenth century and in doing so discusses a variety of specific models: not only the traditional models of architecture and engineering, but also and in particular those of the natural sciences and mathematics. Here, not only the various functions of the models become visible, but also uncertainties about the definition and evaluation of in particular mathematical models. If one looks at the correspondence between Boltzmann and the man who requested the encyclopedia article from Boltzmann, Joseph Larmor (a professor of natural philosophy at Queen’s College in Galway until 1885, then first a lecturer at St John’s College, Cambridge, and from 1903 a professor of mathematics at Trinity College, Cambridge), then Boltzmann’s discomfort becomes palpable. On January 7, 1900, in response to Larmor’s request, Boltzmann writes:

I have now hurriedly looked around for information about models. […] I also went to the Vienna Polytechnicum in order to look at models of house, roof, and bridge construction as well as an endless variety of machines but came away persuaded that what we have on hand in Vienna is obsolete and miserable. How much better the article could be written were one exactly familiar with models in London and America!Footnote 43

Boltzmann’s first thought is about the use of models in architecture and engineering, which while being in use in technical universities were—this is the objection he raises—far more frequent in English-speaking countries than in Vienna, where Boltzmann was a professor of theoretical physics. These models had a clear instrumental or pedagogical function insofar as they were intended for the explanation of technical constructions—showing these constructions in a scaled-down form that often could be taken in the hands and occasionally even dismantled. Yet Boltzmann’s misgivings were not only related to the dearth of good examples, but concerned, above all, the broad, hardly summarizable variety of models, whereby it was particularly with the mathematical models that he felt the most uncertainty. In the same letter he writes:

I am also in the dark about how mathematical models should be handled. I don’t even understand particularly much about algebraic surfaces of pure geometrical models. […] If I dare to write about every kind of model, I am very much afraid that in spite of all of my efforts that I will fail to come up to the Encyclopaedia Britannica’s standards. I simply don’t know the material well enough. For this reason it would be my dearest preference if it would not cause any great difficulties to take the preparation of the article away from me […].Footnote 44

We should not be too quick to interpret Boltzmann’s doubts about his competence as a personal shortcoming; these doubts were rather the expression of an objective overload with respect to the diversity of models found in polytechnic university collections in the second half of the nineteenth century. With respect to the mathematical models, another factor plays a role: their construction and active distribution began only in the nineteenth century (first in France at the beginning of the nineteenth century; then, in the second half of the century, also in England and Germany), and their function was less straightforward than that of the technical models. The importance of mathematical models was undoubtedly in teaching, since they were employed in university education for the teaching of mathematics, particularly at the technical universities. Their function was thus the visualization of complex mathematical objects. Yet they did act as a blueprint for the calculated construction of technical constructions—as was the case, for instance, with models of bridges. Rather, they were purely mathematical objects that the haptic model should make graspable—and Boltzmann doubted precisely whether his knowledge of these abstract mathematical objects (he explicitly mentions algebraic surfaces) was sufficient. He had no doubts, however, regarding the function of the mathematical models: their importance lay in their Anschaulichkeit (their capacity to make certain phenomena available to the senses), and this is based on their no longer being merely abstract objects of reason but concrete objects of sensory perception. This position can be gathered from Boltzmann’s article “Über die Methoden der theoretischen Physik,” which Dyck, the editor of the Katalog mathematischer und mathematisch-physikalischer Modelle, Apparate und Instrumente, requested in 1892. In this article Boltzmann writes that the material mathematical models serve “to make the results of a calculation intuitable [anschaulich], and indeed not merely for the imagination, but also visible to the eye, graspable by the hand, with plaster and card.”Footnote 45 The material realization of the mathematical surfaces and curves translated into three dimensions addresses visual and haptic perception (eye and hand), and thus generates sensory evidence for the mathematical contents. According to Boltzmann, the great advantage of mathematical models lies in the quicker understanding of the contents. That is why these models are more than mere illustrations of the results of the ‘proper’—that is, the mathematical—procedure (the analysis); rather, they are equivalent to geometric construction:

In mathematics and geometry it was at first undoubtedly the need to save labor that led from purely analytic methods back to the constructive methods as well as to the visualization by models. […] What an abundance of shapes, singularities, forms developing from one another the geometer of today has to commit to memory […].Footnote 46

Mathematical models are suited to come to grips with the vastly increased set of objects of geometry. Their Anschaulichkeit aims at the economy of the acquisition of knowledge and the organization of knowledge. In this economy models represent, according to Boltzmann’s hope, a way of saving of labor. And while he does not formulate this explicitly, it can be supposed that via models and geometric construction, the economy of the mathematically trained engineer can also benefit from such a saving of labor. For in the second half of the nineteenth century, the relation between mathematics and empiricism—and that means also the application of mathematics in other sciences and in engineering—was no longer self-evident. In the discussions of mathematicians on the foundation of their discipline, the reference to empiricism was now either asserted as a necessity and a virtue or dismissed in favor of formalization and logicalization. Boltzmann’s formulation in his 1902 encyclopedia article that models “present to the senses” the form of geometric figures should therefore be taken seriously as a positioning.Footnote 47 Mathematical models stand for a mode of concretion that in the course of the development toward ‘mathematical modernity’ would be increasingly excluded. In this sense, these models were elements of a ‘counter-modernity,’ but this counter-modernity, as Herbert Mehrtens has pointed out, was an important accompaniment to the development toward modernity and not its reactionary adversary.Footnote 48 Mehrtens argument can perhaps be slightly reformulated: mathematical models played a key role insofar as they cushioned the development toward mathematical modernity. They could do this due to their dual nature, since while their contents was a purely mathematical object, the latter was lent Anschaulichkeit qua material modeling. Abstraction and concretion thus met in these models without this having to serve a purpose outside mathematics.Footnote 49

It was indeed the case that arguments of modernization and arguments of counter-modernity went hand in hand in the writings of the proponents of model building. This applies in particular to Alexander Brill and Felix Klein, both of whom were advanced mathematicians. In nineteenth century Germany, Brill and Klein were among the driving forces behind the construction and acquisition of models. They advanced a reform in the education of future engineers and mathematicians, where the use of models was necessary to exemplify the new concepts and the abstract objects being used in class.Footnote 50 Klein’s statements reflect the state of upheaval mathematics was undergoing at the time, and its search for its own foundations of knowledge. Thus, his influential Erlangen program, which he first set out in 1872, attempts a naming and unification of various geometries (parabolic, hyperbolic, and elliptic, as well as projective geometry). For this unification he suggested the use of an abstract, group-theoretical approach in order to investigate different spaces and manifolds, focusing on their groups of transformations. This new ‘motion geometry’ (Bewegungsgeometrie, as Hans Wussing termed it) positioned the concept of the group at the transition between geometry and analysis.Footnote 51 In this way, Klein initially distances himself from a sensory-empirical approach to space and spatial figures, as can be inferred from the beginning of the text in which he set out his Erlangen program:

We peel off the mathematically inessential sensory image [sinnliche Bild], and regard space only as a manifold of several dimensions, that is to say, if we hold to the usual idea of the point as spatial, three-dimensional element. By analogy with the transformations of space, we speak of transformations of the manifoldness; they also form groups.Footnote 52

In order to talk about groups as a mathematical structure that characterizes manifolds, Klein had to remark that the sensory image was inessential, and hence the concreteness of the sensory impression has to be ‘peeled off’ in favor of the mathematical abstraction. Here, Klein undertakes a hard confrontation between abstraction (understood as a concept or as formal mathematics) and concretion (understood as a sensory impression or as a physical fact), one that will appear repeatedly in the development toward mathematical modernity.Footnote 53 On the other hand, however—and in a way similar to Maxwell’s critique that one should avoid working only with a theoretical hypothesis or only with an abstract formula—Klein also distances himself from a complete, absolute abstraction of the research of space. Geometry aims at a “figurative reality” (“gestaltliche Wirklichkeit”) of the spatial figures, and that means that the transformation groups stand for ‘real’ movements, and the groups are therefore attributed Anschaulichkeit.Footnote 54 For this faithfulness to reality of geometry claimed by Klein, mathematical models play a particular role:

For geometry, a model—be it realized and observed or only vividly imagined—is not a means to an end but the thing itself [die Sache selbst].Footnote 55

Which is to say that Klein pointed out that three-dimensional models—despite being concrete, tangible objects, and therefore imposing concrete sensory images on the viewer—can serve not only as a means of visualization but also as “the thing itself.” The reality of the mathematical objects can be grasped in the model, and indeed without this mathematical object being identified with a physical fact or a sensory impression. The plaster model of an algebraic surface does not merely illustrate the abstract mathematical object but presents a conceptually correct representation and, in this sense, the Anschauung (intuition) of a geometric reality.Footnote 56

If one consults Klein’s 1873 lecture “Über den allgemeinen Functionsbegriff und dessen Darstellung durch eine willkürliche Curve,” with the distinction found there between a curve jotted arbitrarily on paper, a curve drawn on paper according to a specific law or formula, and a merely imagined curve as an aid, then three-dimensional mathematical models correspond to the curve drawn according to a specific law or formula on paper.Footnote 57 The latter is an empirical object of perception (unlike a mental image) and at the same time an idealization, since the conceptually conveyed law allows the representation to become mathematically precise in a way that is impossible for the merely approximate precision of the jotted curve. Even if the Anschauung of the built, drawn, or merely imagined models in Klein’s sense does not yield a “precision mathematics,” [“Präzisionsmathematik”] one can still infer that in a scientific respect the models operate in an orientating and even epistemic-heuristic way, and thus while not founding knowledge, are still able to guide it and give rise to new knowledge.Footnote 58 This is also suggested by Brill’s statements in his lecture from November 7, 1886:

The maker of a model was free to write a paper on this, the publication of which […] played no small part in encouraging one to carry out the often-arduous calculations and drawings at the basis of the practical execution. Conversely, the model often prompted subsequent investigations into the specific features of the represented structure.Footnote 59

Thus, according to Brill, mathematical models in the nineteenth century did not merely serve to visualize lengthy calculations; they were not simply a scientific tool. If one also considers the design stage, then they were also an object of research. The constructors of these models made use of solid theoretical knowledge and the methods connected with this (particularly calculation and geometric construction), but they needed to proceed innovatively and exploratively in order to understand a scarcely graspable—since abstract and possibly even speculative—object. The mathematical model is—at least in the design stage—an object of questioning, which qua construction should be transformed into an object that is both familiar and useful for the production of further knowledge. In this respect, the models served a process of intellectual appropriation and mathematical habitualization, however not as the passive learning of something already given, but as the active exploration of something at least partially unknown, and that means as the driving factor of present (linked to the object) and future scientific and application-oriented knowledge—what Klein calls “inventions and new mental connections.”Footnote 60 It was this that gave the mathematical models of the nineteenth century their experimental and thus epistemic function.

If one wants to classify the role played by these models in the process of Anschauung more precisely, then, on the one hand, one has to recall what the history of science and the history of mathematics of the last decades has already brought to light. Particularly in German-speaking countries, the power of Anschauung has been acknowledged as the prerequisite and achievement of knowledge since Immanuel Kant, but this ennoblement of the concept-led, so-called pure Anschauung (the nonempirical but sensory manifestation of a truth given qua reason) lost its persuasiveness in the course of the nineteenth century. Particularly the attempt to secure a pure Anschauung in an immutable, single ‘a priori’ of space and time was discredited with the rise of non-Euclidean geometries and could no longer be considered as the self-evident foundation of mathematical knowledge.Footnote 61 Moreover, the potential of sensory perception for deception was thoroughly explored by nineteenth century experimental physiology and psychology, for instance in relation to the blind spot of the eye or the misperception of continuous movement when looking at moving images (e.g., through a phenakistiscope). Visual perception was converted into an image production with its own dynamics.Footnote 62 Hence, the understanding of Anschauung underwent significant changes in the nineteenth century: on the one hand, the concept could no longer refer to a reliable a priori endowment; on the other, it was no longer related to visual perception and mental images alone, but was interpreted particularly by Charles Sander Peirce as an intuition of signs, and thus linked with the field of semiotics. For the mathematicians of the nineteenth century, however, this latter expansion was evidently not a well-known or convincing argument.Footnote 63

In Klein’s work, alongside Anschauung, one frequently finds also the concept of Bild (image), both in an empirical-concrete sense (visible drawing or haptic model) and, above all, in a sense related qua Anschauung to the imagination (mental image). Exemplary for this is an article from 1874, in which Klein discusses the Bilder (images) of mathematical functions and asks about their completeness. While explaining how to visualize a complex curve \(y = f(x)\), he notes that the sketch of the real part of the function is incomplete. A second method concerns the representation of a surface whose coordinates are \((Re(x), Im(x), Re(f(x)))\), but although, according to Klein, this yields “a complete image,”Footnote 64 it does not allow a sufficient visualization of complex singularities. He formulates the problem as follows:

In the investigation of the algebraic functions y of a variable x, one is accustomed to use two different intuition-related aids [anschauungsmäßiger Hilfsmittel]. One represents, namely, either y und x consistently as coordinates of a point of the plane—whereby the real values of these alone come into evidence and the image of the algebraic function becomes the algebraic curve—or one spreads the complex values of one variable x over a plane and designates the functional relation between y and x by the Riemann surface constructed over the plane. In many relations, it must be desirable to possess a transition between these two intuition-based images [Anschauungsbildern].Footnote 65

Here, the meaning of the term ‘image’ (Bild) goes beyond an empirical-concrete visualization on paper (a drawn algebraic curve as an image of the algebraic function) and designates in addition a three-dimensional model (Riemann surface). Moreover, it should be noted that Klein invokes several “intuition-based images” (“Anschauungsbilder”) (algebraic curve and Riemann surface) and searches for a “transition” between them. If Anschauung should serve the “investigation” of mathematical objects and productively advance this, then it must draw on several images. And to search for the transition between these images would then be—one must assume—the task of the understanding and the imagination. Klein’s solution to the problem formulated in the above quotation is in any case to investigate, together with the given curve, a further curve: the dual curve in the projective complex plane.Footnote 66

Most of Klein’s drawings do not have the visualization of singularities for a theme but follow Julius Plücker’s investigation on the relations between the invariants of algebraic curves and the corresponding invariants of their dual curves; specifically, Klein aims at a “Veranschaulichung” (“visualization”)Footnote 67 of these relations. At the end of his contribution from 1874 there is a sketch of what a branch point of a singular curve of degree three looks like (see Fig. 7): “one [confers] to the surface an […] outgoing branching […], as it is visualized, for instance, in a symmetrical way by the included drawing.”Footnote 68

Fig. 7
figure 7

Felix Klein’s visualization of a second order branch point. From Felix Klein, “Ueber eine neue Art der Riemannschen Flächen (Erste Mitteilung),” Mathematische Annalen 7 (1874): 558–566, here 566

If one considers this example, and above all how Klein refers to the different ‘images’ of a complex curve, then one notices in the first quotation that while the first ‘image’ is a two-dimensional drawing (the algebraic curve as an image of the algebraic function), the second is a three-dimensional model (or even two models; see Fig. 8). Consequently, the term ‘image’ (Bild) functions in Klein as an umbrella term that, first, allows one to think together sensory-concrete three-dimensional models and two-dimensional representations, and, second, starting from this empirical level, points to a transition to be sought between the representations that has not yet been drawn or built, and in this respect should probably be conceived as an action of understanding and imagination. Klein’s remarks suggest that he understands the two-dimensional drawing and the three-dimensional model as different but equally valid interpretations of the same mathematical object, whereby only their interaction does justice to the fact that there are several ways of exploring the mathematical properties of the object in question. That these two empirical methods of visualization—three-dimensional models and two-dimensional drawings—are thought together in order to revive and advance mathematical research is also clear in Ludwig Brill’s introduction to the third edition of his catalogue of mathematical models in 1885: “[…] it will continue to be the publishing house’s aim to serve those scientific circles that see the use of models and drawings as an aid and a strong support for the promotion and stimulation of mathematical studies.”Footnote 69

Fig. 8
figure 8

© Collection of Mathematical Models and Instruments at Georg-August-Universität Göttingen, all rights reserved

Model by Adolf Wildbrett of the real part of the (complex) function \(w = 1/z\). Similar models were made for the imaginary part of this function, as well as for other functions. (Georg-August-Universität Göttingen, Collection of Mathematical Models and Instruments, model 253).

According to Klein, the ideal approach to mathematical objects lies in a ‘transition’ between different ‘images.’ This transition, however, appears to go beyond the visualization on paper and/or in plaster or with threads and is probably reserved for mental Anschauung, which qua combination of images can drive the investigation further. This combinatorics of the representations that aims, in the sum, to arrive at a better—since more precise and, in an epistemic respect, productive—image is stated more explicitly by Dyck in 1892. In connection with his construction of models for the real and imaginary part of complex functions, he notes:

The present series of models was made following an introductory lecture on function theory. […] In order to visualize the course of a function of a complex variable in the vicinity of certain singular points […] by a spatial representation, both the real and the imaginary part of the function values are plotted in the familiar way over the plane of the complex argument as ordinates. Thus, each function of a complex argument is made sensible [versinnlicht] through two surfaces designated R and I, whose simultaneous observation provides an image of the function’s course.Footnote 70

Only the simultaneity—and that means the combination of the sensory-concrete representations of mathematical objects—can deliver the ‘image’ (Bild) as such, which is the image of a process and basically a ‘movement image’ that arises in the imagination. As a result, the image concept is distanced in mathematics from a simple representational function (Abbild) and is to some extent abstracted as well as, and above all, made mental. The change in meaning of the image extends beyond mathematics and can be seen particularly clearly in the arts of the outgoing nineteenth and early twentieth centuries, for instance when in Impressionism the focus was on the (physiological) reality of the eye, or, in the programs of abstract painting, the autonomy of art became a theme, as in Wassily Kandinsky’s book Über das Geistige in der Kunst (Concerning the Spiritual in Art), first published in 1912.Footnote 71 Yet while the fine arts programmatically expanded their understanding of the image, the image in mathematics was forced to take a back seat behind the word and language. When in a lecture given in 1886 Brill summarized the model movement as a successful one, he also noted a growing tendency toward or limitation to what he called ‘the word,’ that is, to a more formal language or to a more language-oriented conception of mathematics, embodied in synthetic geometry. This tendency led to an “underestimation” of the model in particular and of the image in general: “[…] this limitation to the word, which the synthetic geometers fondly favored for a time, could not fail to lead to an underestimation of the image and the technical skills required to produce it.”Footnote 72 If one follows Brill here, one may think that it is the growing abstraction in mathematics at the turn of the nineteenth to the twentieth century that slowly brought the model tradition to its decline.

Indeed, for Brill and Klein, the word—that is, the abstract approach (represented here by synthetic geometry)—should and could not function as that which unifies two phenomena or two images as if to reconcile between them, since, to quote from Klein’s 1893 lecture in Chicago, “mathematical models and courses in drawing are calculated to disarm […] the hostility directed against the excessive abstractness of the university instruction [of mathematics].”Footnote 73 Less than ten years later, Boltzmann, in his article from 1902, suggested a rather different approach, pointing to a shift in how the term ‘model’ was considered: models in mathematics had been reduced to material models of surfaces and curves, only “elucidating […] singularities,”Footnote 74 and the goal of mathematical modeling had shifted toward another scientific activity in which this modeling could be understood as ‘reconciling’ between two physical phenomena via a completely symbolic, abstract practice: the finding of an equation.Footnote 75

IV. 1900s–1930s: From Material Analogies and ‘Geometric Models’ to Formal Analogies and Language-Oriented Models

Boltzmann’s view that the aim of mathematical modeling was the finding of a unifying equation exemplifies how the understanding of the term ‘model’ changed in the first decades of the twentieth century from a material to a symbolic, language-based concept. We would like to examine two examples of this shift: to sketch the rise of the term ‘model’ as an instantiation of a mathematical system of axioms; and to examine the biological ‘paper and pencil’ models.

(1) 1891/1899/1936: Mathematics and the New Definition of ‘Model’

One of the challenges for mathematicians in the second half of the nineteenth century consisted in finding a visualization for hyperbolic geometries. In the 1860s the Italian mathematician Eugenio Beltrami built material models that should do just that. He described this realization in the concrete model as an “interpretation” (“interpretazione”) of hyperbolic geometry.Footnote 76 Later, one constructed also for other surfaces (or parts of surfaces) of non-Euclidean geometries equivalents in three-dimensional Euclidean space that can be understood as interpretations. In his paper “Les Géométries non Euclidiennes” from 1891, the French mathematician Henri Poincaré discusses possible ‘interpretations’ of non-Euclidean geometries, and uses for this also the concept of the dictionary:Footnote 77

[…] let us construct a kind of dictionary by making a double series of terms written in two columns, and corresponding each to each, just as in ordinary dictionaries, the words in two languages which have the same signification correspond to one another […]. [We can obtain theorems from one interpretation within the second one] as we would translate a German text with the aid of a German–French dictionary.Footnote 78

The metaphor of the dictionary can be read as a sign that, already at the turn of the nineteenth to the twentieth century, mathematicians understood Beltrami’s concept of interpretation with respect to mathematics as a language, although Beltrami himself probably hardly meant it that way. Poincaré’s use of a metaphor borrowed from the realm of language did not remain without echo, however. In his book Non-Euclidean Geometry, published in 1906, Roberto Bonola points out that the model for non-Euclidean geometry was a material model, and he includes a photo showing one of Beltrami’s paper models. Bonola emphasizes this analogical relation and describes it as a translation: “There is an analogy between the geometry on a surface of constant curvature […] and that of a portion of a plane, both taken within suitable boundaries. We can make this analogy clear by translating the fundamental definitions and properties of the one into those of the other.”Footnote 79 A few years later Hermann Weyl writes about “euclidian model” (“euklidisches Modell”) of non-Euclidean geometry and understands this again as a translation with the help of a dictionary: “We now take up a dictionary with which the concepts of Euclidean geometry are translated into a foreign language, a ‘non-Euclidean’ one.”Footnote 80 These examples already indicate that in the early twentieth century there was a shift in meaning that detached the mathematical model from its characterization qua visual Anschaulichkeit and concrete, material appearance and initiated a language-based understanding of the model. This shift becomes explicit in the formal-logical concept of the model that became widespread following Alfred Tarski’s writings of the 1930s. For Tarski, a Polish-American logician, the model should be defined as a correspondence between certain logical formulas and a certain mathematical structure. More precisely, the model was an instantiation—that is, a “realization”—of a formal axiom system, as in the following formulation from 1936:

Let L be an arbitrary class of propositions. We replace all extra-logical constants occurring in the propositions of the class L by corresponding variables. […] An arbitrary sequence of objects satisfying each propositional function of the class L’ will be called a model or realization of the propositional class L (in just this sense one usually speaks about the model of the axiom system of a deductive theory).Footnote 81

Tarski’s approach does not come from nowhere, but is prepared for by Weyl, among others, who had already undertaken a dematerialization of the concept of model and understood this as the instantiation of an axiom system. Another influence was probably the writings of the mathematician David Hilbert. In his book Grundlagen der Geometrie, published in 1899, Hilbert discusses various axiom systems and presents a number of different geometries that satisfy these axioms. Here, Hilbert does not use the term ‘model,’ however, but speaks of a “system.”Footnote 82 He is also familiar with Maxwell’s research. In a letter to Gottlob Frege from December 29, 1899, shortly after the publication of Grundlagen der Geometrie, Hilbert mentions not only the mathematical theory of duality but also Maxwell’s theory of electricity, and understands both as examples of a kind of abstract-analogical “transformation”: “Any theory can always be applied to an infinite number of systems of basic elements. One only needs to apply […] a transformation.”Footnote 83 And this statement comes immediately after the famous sentence: “If among my points I think of some system of things, for example, the system love, law, chimney sweep…, and then assume all my axioms as relations between these things, then my propositions, for example, the Pythagoras, are also valid for these things.”Footnote 84

At this point it is essential to mention another protagonist who is important for the development of an abstract model concept, between Hilbert and Tarski: Hausdorff, in whose writings one can observe such a change in the use of the model concept already in 1903/4—thus even earlier than in Weyl. Hausdorff was among the early readers of Grundlagen der Geometrie.Footnote 85 Hilbert’s axiomatic method attracted Hausdorff’s attention and inspired him to undertake his own studies, that is, to find his own examples and counterexamples as well as single axioms and axiom groups of geometry. At first, however, he did not call these examples models of the systems of axioms. This happened only in 1903 in the context of his discussion on the consistency of non-Euclidean geometries—here, he already anticipates the later formal-abstract meaning of the term.

In a manuscript titled “Nichteuklidische Geometrie,” probably written in 1901 or 1902,Footnote 86 Hausdorff declares Abbilden (the image of a mathematical mapping) to be a general strategy of a step-by-step proof of the consistency of an axiomatic system. Also the “Beltrami-Cayley’sche Bild” (“Beltrami–Cayley image”) or generally a “euklidisches Bild” (“Euclidean image”),Footnote 87 as Hausdorff now formulates it, is given this new role as an instrument of a proof of consistency. Yet in Hausdorff’s manuscript the term ‘model’ does not appear. This is no longer the case, however, in Hausdorff’s later texts on the problem of space.Footnote 88 Thus, in “Das Raumproblem,” which was his inaugural lecture at Leipzig University, he presents an axiomatic method that introduces a variety of different geometric systems that are limited solely by the criterion of consistency. Here, a terminological shift can be observed when Hausdorff writes: “The absence of a contradiction has been directly demonstrated by appropriate mappings [Abbildungen] of non-Euclidean geometries onto Euclidean models and of Euclidean geometries onto pure arithmetic.”Footnote 89 What Hausdorff had previously still designated as a “Euclidean image” now becomes a “Euclidean model,” by which he does not mean material models but Euclidean geometry. As Moritz Epple has pointed out, this is one of the earliest occurrences of an abstract model concept.Footnote 90 At the same time, however, Hausdorff continues to use the term ‘model’ in its earlier mathematical sense (as a material model). This is the case, for example, in his inaugural lecture.Footnote 91

One finds the new use of the term ‘model’ in an even more explicit form in the lecture “Zeit und Raum,” which he gave during the winter semester 1903/04. Here, Hausdorff argues that, between two different “systems” of geometric objects that satisfy the same axioms, one can find a “translation” (thus drawing on the metaphor that had already been used by Poincaré in 1891): “One can visualize this principle of mapping [Abbildungsprincip] as a kind of translation from one language of geometry into another on the basis of a dictionary.”Footnote 92 Here, it is a matter of the validity of Euclidean geometry, which is demonstrated by the mapping of geometric concepts onto their arithmetical counterparts.Footnote 93 For these images of mathematical mapping ((Ab-)Bilder), Hausdorff uses—as in his inaugural lecture—the term ‘model,’ but only with the image of a geometric system of objects of Euclidean geometry. For instance, he selects spherical geometry as the first example for the discussion of the consistency of a non-Euclidean geometry,

because a Euclidean model can be immediately located for it. Instead of constructing a non-Euclidean geometry directly from the corresponding axioms (as Bolyai and Lobachevsky […] have done), we want to search for Euclidean images of non-Euclidean relations, that is, to use our transformation principle or to conceive of a dictionary by means of which Euclidean propositions can be translated into non-Euclidean ones.Footnote 94

In his introduction to the abovementioned texts by Hausdorff, Epple argues convincingly that the above quotation as well as the corresponding brief passages in “Das Raumproblem” are among the earliest occurrences of the abstract model concept in Hausdorff’s work, and in this respect they open up a path that leads beyond the author: “they simultaneously provide very early proof of a usage that points in the direction of later mathematical model theory.”Footnote 95

Thus, already at the beginning of the twentieth century, in Hausdorff’s work, the model concept appears in an abstract usage. This does not occur consistently, however, but parallel to the earlier material meaning of the term. Hausdorff’s significance for the development of abstract discourse is therefore difficult to determine. Rudolf Carnap and Kurt Gödel use the term ‘model’ explicitly in the 1920s in the sense of Hilbert’s ‘system.’Footnote 96 Carnap speaks of models of an axiomatic system in 1928,Footnote 97 and Gödel uses a similar terminologyFootnote 98—despite both coming from different logic traditions. These mathematicians are representative of the environment from which mathematical model theory emerged. This environment can be seen with Tarski’s statement that a model is a realization of a class of propositions if one can verify that the theorems have been satisfied. The investigation of the differences between Carnap, Tarski, and Gödel, and the answer to the question of how the modern conception of the model emerged from these different approaches is beyond the scope of this introduction. The previous remarks will have to suffice to make clear that it was irrelevant for the mathematicians mentioned above whether a tangible model existed on which or with which the theorems could be located or demonstrated. What was relevant was now only the language-based reality of mathematics.

(2) 1931/1925–6: The ‘Pencil and Paper Models’ of Biology and the Precursors of Modeling

When Tarski speaks of the realization of a propositional class, this underlines an aspect that was already implicit in Brill’s lecture from 1886—the realization in language.Footnote 99 At roughly the same time as Tarski’s statement about realization, Nicolas Rashevsky, one of the pioneers of mathematical biology, also introduced the term ‘model’ into his work. For Rashevsky, the mathematical formalization of biological phenomena was a central concern.

Rashevsky initially studied theoretical physics at the University of Kyiv in Ukraine, before fleeing to the USA in 1924 and eventually developing the idea of biomathematics at the University of Pittsburgh in the 1930s. His attempt to shed light on the complexity of biological phenomena led him to the study of the cell,Footnote 100 from whose actual appearance he nevertheless abstracted by equating its structure with a simpler geometric entity—a sphere or an ellipsoid. As Maya Shmailov has shown in her 2016 book Intellectual Pursuits of Nicolas Rashevsky,Footnote 101 Rashevsky was able to use this theoretical cell to develop mathematical equations that described the cell’s growth and division. He referred to these equations as “pencil and paper models” and attached a higher value to them than to the “actual ‘experimental’ models” of physics. Already in 1931 he noted:

The use of models is not unfamiliar to the physicist. However the physicist uses models in a somewhat different way. […] [N]o one of them [the physicists] did actually ‘build’ any such models, nor experiment with them. These models were, if we may call them so, ‘paper and pencil models.’ […] [A physicist] may satisfy himself by investigating mathematically, whether such a model is possible or not. The value of such ‘paper and pencil’ models is not only as great as that of actual ‘experimental’ models, but in certain respects it is even greater. […] It is through the study of models that Maxwell finally arrived at his equations, which are and probably will forever remain one of the cornerstones of physics.Footnote 102

In order to show whether and to what extent Rashevsky adopted and changed the understanding of the model in physics and mathematics, it will be necessary to refer to another example from the field of biology in which it is also a matter of mathematical representation, namely, the Lotka–Volterra model, as it is called today, or the predator–prey model.

Alfred James Lotka was an American mathematician, physicist, and statistician. Vito Volterra was an Italian mathematician and physicist known for his contributions to mathematical biology and integral equations. Both worked out the predator–prey equations independently and at approximately the same time, in 1925–26. Their equations describe the interaction of predator and prey populations, whereby the primary variable is the size of the predator or prey population. Because predator and prey populations interact with each other, their population dynamics are interconnected: whereas the predators reduce the population of the prey by eating them, the prey increase the population of the predators by providing them with food. In the current literature these equations are mostly designated as the Lotka–Volterra model—for example, in Michael Weisberg’s 2013 book Simulation and Similarity.Footnote 103 Nevertheless, it is important to recall the actual terms used by Lotka and Volterra themselves. In Lotka’s book Elements of Physical Biology, published 1925, for example, the word ‘model’ (and the corresponding verb ‘to model’) do not appear. In his preface, however, Lotka expressly notes that the spirit of his work is mathematical systematization:

It is hoped that the mathematical character of certain pages will not deter biologists and others […] from acquiring an interest in other portions of the book. […] I may perhaps confess that I have striven to infuse the mathematical spirit also into those pages on which symbols do not present themselves to the eye.Footnote 104

Similar formulations are found in Volterra’s writings from 1926—although Volterra too does not describe his work as modeling or use the term ‘model.’ He does, however, set out the reasons for constructing a simplified representation, that is, a representation that reduces biological complexity and only provides an “approximate image” (“immagine approssimata”) of this:

In order to deal with the question [of predation] mathematically, it is better to start from hypotheses that, even if they distance themselves from reality, give an approximate image of it. Even if the representation will be, at least at first, coarse, […] it will be possible to apply the calculation […]. [Hence it is] advisable […] to schematize the phenomenon by isolating the actions to be examined, assuming that they work alone and neglecting the others.Footnote 105

Volterra addresses the problem of complexity by abstracting from it: the various elements have to be isolated in order to be represented mathematically. This is in keeping with Lotka’s appeal to systematization and with Rashevsky’s approach—although Rashevsky explicitly defines his activity as modeling.

Whereas, in the 1920s, the term ‘model’ was only rarely used explicitly for the procedures of the abstraction and mathematical representation of certain well-selected (biological) processes, this situation changed decisively in the 1950s and 1960s. The Proceedings of the Fourteenth Symposium in Applied Mathematics of the American Mathematical Society (1961) is devoted entirely to “Mathematical Problems in the Biological Sciences,” and almost all the contributors describe their activity as ‘modeling’ or explicitly use the term ‘model.’ A few passages from the Proceedings should make this clear:

  1. (1)

    “[…] [one can] show how a particular model is formulated mathematically.”Footnote 106

  2. (2)

    “In all these respects our model idealizes reality […]. [The drawn] conclusions […] with respect to the model are of physical interest only insofar as they do not depend too grossly on these specific features of the model.”Footnote 107

  3. (3)

    “One of the great advantages of setting up model biochemical systems is that they can be tested in new situations with little difficulty.”Footnote 108

Against the background of this frequent use of model vocabulary, it becomes apparent that the sense of the term ‘model’ differs considerably depending on whether it appears in a purely mathematical context or in the context of applied mathematics (in physics, biology, or in other disciplines). In the mathematics of the 1930s, the term ‘model’ was given a particular molding: here, it is defined as the ‘theoretical realization’ of a formal axiomatic system. Also in physics and biology the term underwent a clarification in the first half of the twentieth century; however, this differs considerably from the mathematical convention: in the natural sciences the mathematical model now designated a local, temporary mathematical abstraction of complex processes. Moreover, the term ‘model’ no longer appeared alone but as part of a terminological network: it was associated with other terms that already referred to the process of abstraction, such as ‘theory,’ ‘analogy,’ ‘idealization,’ and ‘representation.’

V. 1940s: Lévi-Strauss and Mathematical Models in Anthropology

In the twentieth century, this new conception of mathematical modeling—as a ‘translation’ of carefully selected ‘specific features’—was not unique or limited to the discourse of the natural sciences. This is to be seen in the writings of Claude Lévi-Strauss (1908–2009), whose work was essential for the development of structural anthropology. His book Les structures élémentaires de la parenté (The Elementary Structures of Kinship), published in 1949, is considered one of the most important anthropological works on kinship of the period.Footnote 109 In this work Lévi-Strauss draws directly on the help of the mathematician André Weil to grasp the marriage customs of the Murngin of northern Australia as a coherent system of the exchange of women, and thus to decipher the various structures of kinship.Footnote 110 The empiricism of possible and impossible marriage ties seems too complex to be reduced to general laws and thus rules of combination. The ethnologist therefore had to limit himself to writing lists of commandments and prohibitions as well as descriptions of the observed relations, and could at best compile statistics. The mathematician Weil, however, kept himself at a distance as much from the qualitative descriptions of the ethnographer as from the quantitative assessment of social behavior. Instead, he enlisted the concept of the mathematical groupFootnote 111: first, he selected the characteristic permutations of the phenomenon at hand to then classify these in terms of their algebraic properties and deduce the group that enclosed them. Weil remarks:

The most difficult thing for the mathematician, when it comes to applied mathematics, is often to understand what is at issue and to translate the data of the question into his own language. Not without difficulty, I finally saw that it all came down to studying two permutations and the group they generate.Footnote 112

Concretely, he begins with the observation that the Murngin are divided into marriage classes and that each man or each woman can seek his or her marriage partner only in certain classes that correspond to his or her own class. Membership of a class is determined by one’s ancestry, so that the alliance of the parents dictates the possible alliances of the son and daughter. Weil assumes four classes and identifies first the four corresponding marriage types of the parents, then the marriage types of the sons and of the daughters derived from these. To these three series, in which the marriage type of the sons and of the daughters is already in a relation of permutation to the marriage type of the parents, he then adds specific conditions, for instance that each man may marry the daughter of his mother’s brother. The types and the conditions, or the resulting possible constellations for son and daughter, are translated by Weil into an algebraic description, which in turn is subjected to certain hypotheses and in this way developed further. In the end he obtains two function formulas (one each for son and daughter), which together define the structures of the marriage system, and which should provide an answer with regard to both possible and impossible alliances.Footnote 113 The (utopian) long-term goal of this mathematically processed ethnology is to describe the marriage behavior and the resulting kinship systems of a large number of societies by means of algebraic formulas, and thus to guarantee a formal comparability. The precondition remains, however, that a manageable number of permutations can be identified, and that the added conditions have a broad validity (–neither could be guaranteed for the investigation of modern Western societies, for example).

When Weil describes applying mathematical structures to structural anthropology as a translation into one’s own language, he is clearly borrowing or drawing on the common understanding of ‘model’ in the sciences of the time. This becomes evident in an explicit way when Lévi-Strauss, in the chapter “Social Structure” of his book Structural Anthropology,Footnote 114 complements the concept of structure with the concept of model, whose definition he adopts in the form of a quotation from John von Neumann and Oskar Morgenstern’s book Theory of Games and Economic Behaviour (1944):

Such models [as games] are theoretical constructs with a precise, exhaustive and not too complicated definition; and they must be similar to reality in those respects which are essential in the investigation at hand. To recapitulate in detail: The definition must be precise and exhaustive in order to make a mathematical treatment possible. The construct must not be unduly complicated so that the mathematical treatment can be brought beyond the mere formalism to the point where it yields complete numerical results. Similarity to reality is needed to make the operation significant. And this similarity must usually be restricted to a few traits deemed ‘essential’ pro tempore—since otherwise the above requirements would conflict with each other.Footnote 115

By taking up this model concept from the publication of the mathematician and economist,Footnote 116 Lévi-Strauss succeeds in strengthening his approach in three respects: First he reconciles the formalism and universalism of mathematics with the ongoing ethnographic observation of the richly detailed and changeable empirical phenomena by promoting the selective choice of significant aspects and thereby the transition to abstraction, while nevertheless insisting, with von Neumann and Morgenstern, on a “[s]imilarity to reality.”

Second, he is able to do justice, via the legitimation of the game as a model, to the connection between necessity and arbitrariness, for example in his study of kinship systems. That is, he need relinquish neither the rational claim nor the contingency of the cultural phenomena. With the help of the model, the kinship relations become recognizable as a generalizable system made up of independent parts, precisely because this kind of modeling can describe the diverse and processual “form” of reality as an “action of laws which are general but implicit.”Footnote 117

Third and finally, the structuralist approach is conceived and legitimized as a strategic intervention of the scientist in the empirical data. In order to obtain an instructive modeling, both the ethnologist and the scientist must select wisely from the diversity of possible aspects. Hence, the object of ethnology is not a concrete family (or a concrete community of families), but the abstract relations between elements of what in the different cultures is recognized as a family.Footnote 118 The social structure is not defined inductively but must be read from the constructed model. While the hypothetical prototype in ethnology aims at an “empirical knowledge of the social phenomena,” these are not objectively given but are only inferred with the help of modeling.Footnote 119 To achieve this it is important to limit the case examples and the aspects to be considered in order to obtain the striven for meaningful simplification and to transpose the observations into a systemic context—that is, to a model in which developments can be experimentally played through and universal structures deduced.

VI. Conclusion: The Model in the Twentieth Century: Fictitious, Fragmentary, Temporary

Lévi-Strauss’s use of the term ‘model’ is remarkably similar to the way it was used at the same time in the natural sciences. Moreover, in the quotation from von Neumann and Morgenstern’s book cited above, one can observe a terminological plurality that has already been noted: modeling demands “[s]imilarity to reality” and at the same time “formalism,” although this similarity is usually “restricted” when choosing the traits to formalize.

To sum up, one can say that the range of meanings associated with the model concept had already been prepared by Maxwell’s research and his use of the terms ‘analogy’ and ‘model.’ Maxwell emphasized the ‘fictitiousness’ of the object of his mathematical-physical model and the incomplete (or, for Mach, abstract) similarity that accompanied the physical analogy. In this way, scientific representation is provided with a crucial degree of latitude, which, rather than obstruct its heuristic function, strengthens it. Boltzmann, for his part, remarks that the mathematical models of the nineteenth century are ‘tangible representations’ that visualize abstract geometric objects—the ontological question is not present here.

For the period between 1850 and 1950, it can be tentatively stated that scientific representation moves away from the ideal of complete similarity in two respects: On the one hand, mathematical discourse distances itself from material mathematical models, and the term ‘model’ acquires a purely logical meaning that no longer depends on the similarity to a geometric or physical object. One can suppose that this specialization of the meaning was also prepared by Maxwell’s understanding and use of the terms ‘analogy’ and ‘model,’ inasmuch as it was Maxwell’s expansion that led Hilbert to discuss how, qua transformation, different systems could be derived from a theory. Nevertheless, starting in the 1930s, the concept of the model in mathematics becomes increasingly precise, since it was now grasped significantly more restrictively and independently of physical empiricism.

On the other hand, one can note that while in the discourse of the exact sciences the model operated as a shifting mediator between theory and reality, the model itself could be entirely temporary, as Maxwell observed. In the twentieth century this temporary character can be seen in the way the relation of the model to reality was to be understood as fragmentary and local (indeed, as a reduction of complexity). The scientific actors were well aware of this fragmentary, local relation of the model to reality. Here, one can detect a twist in the idea of the mathematical representation of ‘reality’: the models move away from the obligation to a complete representation of reality to become a productive tool of knowledge that can be readjusted according to the context. It is with this productivity of the model, with its local and fragmented character in mind that the contributions and interviews in this volume should be read.

* * *

The various contributions in this volume explore, from different perspectives, this productivity of the mathematical model and of modeling in mathematics. We start with a historical perspective, presenting both a long durée history and detailed case studies. We then continue in the second part with an examination of the epistemological and conceptual perspectives. Finally, the third part of the volume investigates not only how material mathematical models were produced, but also how they were exhibited.

Part 1: Historical Perspectives and Case Studies starts with the contribution by Frédéric Brechenmacher, which looks back at the early and parallel history of material mathematical models. At the center of Brechenmacher’s investigation is the practice of the drawing and construction of mathematical models, a practice that was widespread in France starting in the late eighteenth century and still visible in the 1870s. Many studies on the golden age of material models have focused on the impressive models of higher mathematics from the period starting in 1860; however, in France the drawing of mathematical models of all kinds (i.e., beyond higher mathematics) goes back much further and became a well-established practice for the acquisition of geometric knowledge, particularly at the École polytechnique (which became so important at the end of the eighteenth century), and therefore in the teaching of technicians and engineers. The models of higher mathematics designed by prominent mathematicians partially break with this ideal of a general model-drawing practice, forcing it into the background. Brechenmacher’s contribution corrects this by presenting model drawing as part of an important non-textual and practical tradition of (French) mathematics, while also identifying the central protagonists, model builders, and the communities in which the construction and the drawing of models took place. Finally, the author links the tradition of model drawing to the advance of the ‘graphical method’ and the related successful ‘modelization’ of empiricism at the end of the nineteenth century.

The following three contributions (by Klaus Volkert, David Rowe, and Tilman Sauer) provide detailed case studies of material mathematical models from the late nineteenth and early twentieth centuries. Klaus Volkert’s contribution examines the spectrum and function of mathematical models in polytechnic schools using the example of the ETH Zürich in the second half of the nineteenth century. At the center of this study is the model collection that Wilhelm Fiedler assembled there in the 1860s as a professor of descriptive geometry and geometry of position (Geometrie der Lage). This collection was so highly esteemed by Walther Dyck that he invited Fiedler to describe it for the catalogue of an exhibition planned to accompany the conference of the German Mathematical Society in Nuremberg in 1892 (an invitation that Fiedler turned down). The author examines Fiedler’s importance for the teaching with and the research on models. Models were traditionally found in polytechnic schools, where they played an important role in the education of technicians and engineers, and in this case also of teachers. The teaching of descriptive geometry was not based on theorems and demonstrations, but presented problems that were solved by means of drawing or by graphic approximation. Fiedler, however, decisively expanded the existing model collection by acquiring additional models (e.g., a series of teaching models by Jakob Schröder) and by working on the production of models himself. As the author is able to show, this was the case also with regard to models for objects of higher mathematics. Already in 1865 Fiedler designed a model of the Clebsch surface (a cubic surface with 27 straight lines)—hence a few years before Christian Wiener’s model was made, which today is considered the first. In doing so, Fiedler made use of relief perspectives: a geometric practice that was closer to the practice of artists and architects than to that of mathematicians. His rod model is only an approximation of the geometric Clebsch surface; nevertheless, it demonstrates the importance of polytechnic schools with respect not only to their use of simple models in teaching but also to their handling of models of higher mathematics.

David E. Rowe’s contribution addresses the models of complex surfaces (specifically quartic surfaces) designed by Julius Plücker in the 1860s. At that time these models of higher mathematics represented “the more exotic geometrical knowledge” (Rowe), and were of some interest to mathematicians in England, but also to Plücker’s student Felix Klein, as well as to Klein’s students—before being forgotten at the beginning of the twentieth century. The author reconstructs the early history of these models and their nameable epistemic function in the mathematics of the late nineteenth century. To this end, he initially examines the interaction between mathematics and optics research using the example of the geometric representation and analysis of rays with the help of line congruences in three-dimensional space. Of central importance in this field is Fresnel’s wave surface, a model that arose as a derivation from a quartic equation for a wave front of light passing through biaxial crystals, which led to further research in the 1820s, for example in the work of William R. Hamilton, who developed a general theory of ray systems (as infinitely thin pencils of rays), and, later, in the work of Plücker. The path to Plücker’s models passes via Ernst Kummer, who built on Hamilton’s theory and developed it in a purely mathematical way for a large class of quartics, also designing thread models for these. The actual goal, however, and not a trivial one, was the general classification of quartic surfaces. This generalization was achieved by, among others, Sophus Liu, and as a result of the rethinking of models by Albert Wenker, Klein, and Klein’s student Karl Rohn. Plücker’s line geometry and his models of complex surfaces represent an important bridge for this progress in mathematical knowledge: they take up special quartics (that is, surfaces of degree four), which are enveloped by subsets of lines in a quadratic line complex. These central objects of line geometry are a degenerate type of the Kummer surface; but Plücker begins with canonical cases from which he systematically derives all other types. Klein and his student completed Plücker’s models, and in this way were able to show the transition (by deformation) from Kummer surfaces to a number of central complex surfaces.

Tilman Sauer’s contribution concentrates on a very specific type of model: the plaster models of curved surfaces by the Dutch geometer Jan Arnoldus Schouten. These were used by Schouten to illustrate the novel geometric concept of parallel transport, a concept developed in the framework of differential geometry for manifolds of an arbitrary number of dimensions. Sauer points out that, while this concept arose in a somewhat nonvisual mathematical setting (geometry of higher dimensions, n-dimensional manifolds and their associated curvature), Schouten, in order to assist the geometric Anschauung (intuition), turned to a material model to visualize geodesic transport. This material model was illustrative of a conceptual problem that was still being explored at the time; in that sense, the model’s physical properties function epistemically—they not only help understand the abstract concept but may have played an important role for the conceptual development of Schouten’s work.

The last three contributions in the first part deal with the transformation undergone by modeling during the later decades of the twentieth century in physics (Arianna Borrelli), ‘applied’ mathematics (Myfanwy E. Evans), and pure mathematics (Fernando Zalamea). Arianna Borrelli’s contribution deals with models and symmetry principles in early particle physics. Borrelli discusses the development of theoretical practices between the 1950s and the early 1960s and presents examples of the complex relationship between mathematics and the conceptualization of physical phenomena. Indeed, a closer look at theoretical practices in this period reveals a tension between the employment of advanced mathematical tools and the ‘modeling’ of observation, when ‘model’ is understood as a construction enabling the fitting and predicting of phenomena. Due to this tension, the question arises whether it is even possible to make a general claim about the relationship between mathematics and models. This contribution, then, examines a general tension expressed at the time: an opposition between local mathematical constructs that fit phenomena (such as models) and those expressing the more general (hidden) principles of the constructs’ coming-to-be (such as theories or structures).

The interview with Myfanwy E. Evans explores the connections between pure and applied mathematics and the research on materials. More concretely, Evans shows how these fields intersect with material models and animation software. Through a consideration of the tradition of constructing material models of periodic minimal surfaces—a tradition initiated by Hermann Amandus Schwarz and his student Edvard Rudolf Neovius in the last third of the nineteenth century and continued by Alan H. Schoen during the 1960s—Evans shows how these models help to understand the structure of human skin, and how this applied research, in turn, prompts a mathematical theory of entanglement. Accordingly, these models and their twenty-first century digital visualizations can be considered as epistemic objects, both for applied and pure mathematics.

Last but certainly not least, Fernando Zalamea presents in his short but thorough contribution the work of Alexander Grothendieck, who, according to Zalamea, explored the pendulum movement between the abstract and the concrete, the universal and the particular in mathematics. Zalamea stresses two basic directions that Grothendieck examines in the space of transition between archetypes (universal categorical constructions) and types (concrete models): on the one hand, projecting archetypes to types in the 1950s (in his work on the Tôhoku paper and the Riemann–Roch theorem) and, on the other, embedding types into archetypes in the 1980s (in the works Pursuing Stacks and Les Dérivateurs). This pendulum movement between the abstract and the concrete is also seen in Grothendieck’s general remarks about models in Récoltes et semailles. Zalamea points out, however, that it is both the abstract and the concrete, the universal and the particular (not the choice between one or the other) that are necessary to nurture the mathematical imagination.

The four contributions in Part 2: Epistemological and Conceptual Perspectives present epistemological perspectives on the model and modeling, while also expanding the discussion to include new terms and conceptions. The first contribution, by Moritz Epple, presents a panoramic view of the history of abstract representation in the sciences of the nineteenth and early twentieth centuries. Here, Epple concentrates on a number of terms that were employed during this period, such as ‘analogy,’ ‘interpretation,’ ‘image,’ ‘system,’ and, last but not least, ‘model.’ By focusing on actors’ categories and reviewing various protagonists (physicists, mathematicians, and philosophers such as Ludwig Wittgenstein, Heinrich Hertz, James Clerk Maxwell, Hermann von Helmholtz, Eugenio Beltrami, Felix Klein, and Felix Hausdorff), the author delineates the epistemic ruptures, continuities, and symmetries between these various concepts. Hertz, for example, believed in a correspondence between causally structured reality, the activity of the mind, and scientific theory formation; however, this correspondence was based on an epistemic symmetry of the relations in his ‘images’ and ‘models.’ On the other hand, at the end of the nineteenth century and, more strongly, at the beginning of the twentieth century, geometry underwent an epistemological rupture, whereby its ‘systems’ and ‘models’ no longer necessarily reflected reality as such. As Epple points out, with the rise and development of the various interpretations, systems, and images of non-Euclidean geometry, one was forced to reshape and restructure the relations between inequivalent mathematical representations of physical space.

José Ferreirós’s contribution explores a similar theme but from a different perspective by concentrating specifically on mappings and models and their affinity to acts of abstraction and imagination, and on how mathematics contributed to modern thinking at the turn of the nineteenth to the twentieth century. What was the role of the term ‘Abbildung’ (mapping, representation) in the thought of Bernhard Riemann, Hermann von Helmholtz, or Richard Dedekind? What were the historical vicissitudes of the pictorial terms ‘image’ and ‘representation,’ ‘Bild’ and ‘Abbildung’ in the mathematics of this period? And how is this history reflected in theoretical physics—in Heinrich Hertz’s reflections, for example, or in the conceptual and philosophical difficulties being encountered in physics and mathematics?

The contribution by Axel Gelfert deals with models from a more general perspective. For Gelfert, mathematical actions are mediated by symbol systems, notations, and formalisms, which actively shape mathematical practice. By stressing the role of mathematical practice and by reviewing the philosophies of this practice, this contribution examines the interweaving of notations, formalisms, and models in mathematics. This leads Gelfert to examine epistemic mathematical actions—that is, actions that can be considered as constitutive of mathematical practice. To answer in a concrete way what these epistemic actions in mathematics might be, Gelfert gives three main examples: the use of gestures and symbolic operations, the construction and use of material mathematical models, and the re-proving of mathematical theorems. Returning at the end of his paper to the issue of notation, the question arises whether notation can be considered as a long-term epistemic action. Moreover, while notations and formalisms, in order to function properly, may need to be aided by physical actions and material models, one can certainly ask how material models function in the long run with respect to formalism.

In the last contribution to the second part, Gabriele Gramelsberger describes the decline of Anschauung (intuition) in the nineteenth century and the subsequent declaration of Anschaulichkeit (palpable visuality) as a model in geometry. Gramelsberger surveys the debates on Immanuel Kant’s concept of Anschauung, and follows these debates particularly in nineteenth century mathematics and twentieth century physics. Gramelsberger notes that the debate between Werner Heisenberg and Erwin Schrödinger, for example, can be reconstructed as a mismatching debate between Heisenberg’s conception of Anschauung and Schrödinger’s conception of Anschaulichkeit. While Heisenberg emphasized the loss of spatiotemporal Anschauung in the Kantian notion, Schrödinger ignored this loss in favor of the traditional spatiotemporal concept of (Newtonian) Anschaulichkeit. Moreover, while for a certain period—mainly in the mathematics of the nineteenth century—material models succeeded in replacing Anschauung by Anschaulichkeit, the loss of Anschauung in physics, especially in early quantum theory, was subsequently compensated for by formalism and—as in Gramelsberger’s example—matrices. The author hence argues that in early quantum theory, in order to understand all possibilities of reality, one first needed to overcome certain notions inspired by geometric Anschauung.

The last part of this volume, Part 3: From Production Processes to Exhibition Practices, examines with three interviews the before and after of models—that is, how these models were produced (before they became finished models) and how they were presented (once they were ready). We start with an interview with Anja Sattelmacher, which partly continues the themes of the second part of the volume, dealing also with conceptions of Anschauung (intuition). The interview with Sattelmacher concentrates on the materiality of material mathematical models and on their processes of production. The interview has two interconnected foci: first, a consideration of how, during the nineteenth century, conceptions of Anschauung were to be seen as one of the goals of these models, that is, the visualizing of a mathematical formula; second, a detailed discussion of the models’ materials and material techniques. At the center of this discussion is the question of how the choice of a material (such as paper, wire, or wax) influenced not only the production processes of the model, but also the epistemic values transmitted through them.

The interview with Ulf Hashagen examines the history of the numerous exhibitions of mathematical models, mainly in the second half of the nineteenth century and the first decades of the twentieth century. ‘Exhibition’ here refers not only to two exemplary exhibitions—the 1876 exhibition at the South Kensington Museum, London, and the 1893 exhibition at the Technische Hochschule München—but also to collections of models at various universities, such as the collection in Göttingen. How was the act of exhibiting mathematical models understood at the time? How did the manner of exhibiting mathematical material change during these decades? And how were the various agendas of the different disciplines at that time (engineering, mathematics, physics) reflected in and promoted by the use and presentation of models?

The last interview in this volume concerns not only future modes of presenting models, but also the future production of these material models. The interview with Andreas Daniel Matt examines the history of digital mathematical models at the end of the twentieth century. The IMAGINARY project developed by Matt and his team enables, one may claim, real-time mathematics—namely, digital models of surfaces presented within seconds in an interactive manner. The history of this computer program is more convoluted than one might think, however, and it is entangled with the history of the various exhibitions in which these digital models have been presented. If one considers the various exhibitions of material mathematical models during the nineteenth century—as discussed in the interview with Ulf Hashagen—the question arises whether IMAGINARY breaks with this former tradition? Or can one also detect certain continuities and common characteristics?

Translated by Benjamin Carter