Interview with Anja Sattelmacher: Between Viewing and Touching—Models and Their Materiality

Abstract

XXX.

KK: You recently published your book Anschauen, Anfassen, Auffassen: Eine Wissensgeschichte mathematischer Modelle,¹ where you discuss material mathematical models from the nineteenth and early twentieth century, concentrating mainly on Germany and particularly on the processes of manufacturing, collecting, and distributing these models. Before discussing these issues in more detail, we would like to begin with your understanding of ‘model.’ Against the background of your research in the history of science and cultural studies, what would you say a mathematical model is?

AS: In my research the term ‘model’ is closely tied to the concept of Anschauung,² with an emphasis on the material context. Both of these tend to be overlooked when considering the conceptual superstructure of mathematics. If one focuses on materiality, then the mathematical model is something that is subjected to a process of production, which then leads to a three-dimensional, haptic object. A proper account of the cultural and epistemic techniques and practices at work in this production process takes us beyond a history of the discipline of mathematics. These practices extend into a number of very different fields of study, ranging from the applied arts to industrial engineering to the fine arts.

KK: And mathematical education is another of these fields in which the model is firmly rooted.

AS: Exactly: educational practices, mathematical knowledge, and, of course, artisanal know-how. In my book I show that many model builders have backgrounds with close ties to the fields of craft or engineering or architecture. It is not by chance that it was precisely these mathematicians who so strongly embraced and promoted the making of models.³

MF: The beginnings of the tradition of making mathematical models lie in the nineteenth century in France, Germany, and England (see Fig. 1a for models made by the French mathematician Théodore Olivier and Fig. 1b for models made in Germany by Ludwig and Alexander Brill). In what way do these traditions differ, and how are they anchored in the respective cultures of science and knowledge?

Fig. 1

a A string model made by Théodore Olivier of an intersection of an oblique cylinder and an oblique conoid (yellow and blue), ca. 1830–1845, fruitwood, brass, thread, lead weights (Union College Permanent Collection, 1868.28 UCPC). Courtesy of the Union College Permanent Collection, all rights reserved. b Various mathematical models from plaster at the inner cover of Ludwig Brill, ed., Catalog mathematischer Modelle für den höheren mathematischen Unterricht (Darmstadt: L. Brill, 1881)

AS: The way science was studied and taught in France or Germany was of course very different, particularly since France had a centralist and Germany a more ‘federalist’ system. Furthermore, the culture of model building in France began earlier and initially arose in the military sector (that is, in the École Polytechnique, which had its roots in the military) before being used for civilian purposes.⁴ That then also gave rise to the concept of the civil engineer.⁵ This development started around 1830 with models by the mathematician Olivier (see Fig. 1a), a graduate of the École polytechnique. That is the period in which the modern understanding of the engineer emerged, and it was then that these model-building practices started.

This situation cannot be directly transferred to Germany, although at that time there was of course academic research and teaching on descriptive geometry in Germany too. But in Germany model building remained a local matter—that is, it only took place in universities where there were professors of mathematics who were interested in model building and actively promoted it. Of course, German mathematicians around 1870 also travelled to France and looked at the mathematical models at the Musée des Arts et Métiers. There is not a continuous line, but certain techniques and conceptions were adopted from France. That can be seen, for example, in the founding of a Polytechnikum in Karlsruhe, one of the oldest technical colleges in Germany, which was based on the idea of the École polytechnique. The first professors taught in Karlsruhe on the basis of drawings by the mathematician Gaspard Monge, who in 1794, along with the engineer Lazare Nicolas Marguerite Carnot, founded the École polytechnique. Christian Wiener, a professor of descriptive geometry at Karlsruhe Polytechnikum, was also aware of Monge’s works and probably used them as a basis for his own work. Already in the 1860s he constructed models for the teaching of higher mathematics. It is with Wiener that the German model-building tradition begins—a tradition continued by his son, Hermann Wiener, who was very active in model building.

MF: Did French and German model building have the same aim: roughly speaking, to visualize an abstract mathematical formula?

AS: The basic idea was certainly the same. The objective of descriptive geometry—or, at first, of geometric drawing—was to visualize the abstract formula.⁶ But in France, at the end of the eighteenth and the beginning of the nineteenth century, there was a greater focus on architectural design,⁷ that is, on the construction of buildings. In descriptive geometry there was also an interest in how the line is materialized—that is, the course the line takes when, for example, it rotates around a pole, the forms this gives rise to (for instance, a cylinder), and what an engineer can do with this in his or her practice.

In Germany, on the other hand, the focus was on Anschauung, and that gave rise to the idea of using descriptive geometry and the mathematical model for pedagogical purposes, as a teaching aid—for example, in schools. Which is to say that in Germany the epistemic objectives were broader. It should be borne in mind, however, that there is a gap of more than seventy years between the beginning of the model tradition in France and the dissemination of model building in Germany, and that much changed in this time. This relates to the different ways in which the models were constructed, but, of course, also to the development of mathematics itself—from an epistemic as well as a cultural and political point of view.

But ultimately you are right: model building is concerned with visualizing a mathematical formula. For me, however, the important question is another one: what is actually happening in such a process of model building? What do the different steps that are necessary look like concretely? How is the drawing made, and how on the basis of the drawing does one decide which materials should be used? What tools and what practical knowledge is linked to the processing of the materials? So there is a chain of multiple translations, from the drawing to the model. In my book I did not only want to look at the beginning and the end of the process, I wanted to consider the whole procedure of model building—that is, to describe the practices that generate knowledge. These are epistemic practices.

KK: This leads to two far-reaching questions: First, what role does ‘Anschauung’ play in the discourses and practices of mathematical model building? And, second, what exactly do the material and practical steps in the translation of an abstraction (the mathematical formula) into a concrete model look like? Let us perhaps first proceed with the meaning of Anschauung. The concept does not exist in the same way in other languages, and particularly in English it has either to be reformulated or to be reduced by speaking, for instance, of ‘intuition.’ Could you describe what the concept of Anschauung reveals in the German discussion about mathematics and model building?

AS: Without the concept of Anschauung, the mathematical model cannot be thought at all, at least not in the sense that the mathematical model had in the nineteenth century. Anschauung does not only mean that I look at something (etwas anschauen), or that I observe something. It goes beyond visual perception, because, first, it sets a thought process in motion, and, second, this process can be learned, practiced, taught, and, if you like, conditioned. Anschauung generates knowledge, and Anschauung instigates a learning process. Therefore, in nineteenth-century sources in the field of education, the concept of Anschauung is ubiquitous. It is not only used by the leading educational reformers of the early nineteenth century—such as Friedrich Fröbel and Johann Heinrich Pestalozzi—but basically by anyone involved with students or with formal education, that is, with pedagogical work.

In this context, one needs to refer to the discussions about formal education taking place in Germany at the time. In the nineteenth century there were two conflicting ideologies: on the one hand, the new humanism, which put its hopes in classical education with Greek and Latin; and, on the other, an educational reform movement that had a more practical orientation. The latter espoused the concept of Anschauung in order to argue for the acquisition of newer languages and for the teaching of an application-oriented mathematics. The mathematics professor Felix Klein was one of the most prominent figures to embrace the concept of Anschauung and to opt for model building among his students. He was well acquainted with these discourses and was also aware of the political situation of education in Germany.

The German-speaking mathematicians working with models in the nineteenth century had internalized what Anschauung meant—that is, that it also aims at a learning process and at a process of ‘making graspable.’ Indeed, Anschauung does not only mean that I show the object to my students, and that in this way they grasp something; rather, Anschauung also means that the students construct the model themselves, and this constitutes the actual learning process. Hence, Anschauung is also related to the touching of materials—that is why my book is called Anschauen, Anfassen, Auffassen (Viewing, touching, grasping). The model is not only observed with the eyes; the other senses are addressed too, in particular the sense of touch, since it is with the hands that one builds the model.

KK: Over the last thirty to forty years the history of science has been interested in the concrete practices and procedures that accompany, promote, or even simply make possible the production of knowledge.⁸ Here, it is no longer a matter of insight or truth but of ‘knowledge in the making’ or of artisanal knowledge. Could one say that in the mathematics of the nineteenth century there was a greater awareness of the pedagogical and, further, of the epistemological meaning of the work with material and of the concrete process of production—the “motorische Empfindung” (motor sensation) as Klein calls it?⁹

AS: I would certainly agree with that, and it is not only limited to mathematics. If we look at the models made by the Blaschkas (Fig. 2a),¹⁰ with these works it was certainly a matter of acquiring knowledge through model building. Or if one thinks of Adolf Ziegler’s wax models (Fig. 2b),¹¹ in this case too the intention was to generate and pass on knowledge, whereby knowledge here arose both in the making of the object itself and through the interaction with the model. I would say that that was a particular characteristic of the time.

Fig. 2

© Stadtmuseum Tübingen, all rights reserved. b Wax model of the vascular system of a human liver (1912–1914), Adolf Ziegler workshop, Freiburg im Breisgau (Institut für Anatomie und Zellbiologie, Universität Heidelberg). © Universität Heidelberg, all rights reserved

a Glass model of a heliozoan (1885), Blaschka workshop, Dresden, now in the zoological collection of Universität Tübingen. Photo: Peter Neumann.

MF: I would like to return to the role of Anschauung at the end of the nineteenth century, since it was then that the ‘crisis of Anschauung’ erupted, in the sense that mathematicians discovered several mathematical objects and domains that were sometimes regarded as ‘monsters’ and considered as objects which cannot be visualized, one of the famous examples of these objects being the Weierstrass function, a continuous but nowhere differentiable function.¹² Another example is mentioned in an observation by Christian Wiener concerning a surface described by ‘moving’ one Weierstrass function along another one. The described surface, which exactly like the Weierstrass curve is nowhere differentiable, “cannot be represented by means of drawing or a model.”¹³ How is this crisis of Anschauung to be reconciled with the tradition of model building? How did the mathematicians that championed model building—such as Klein, Alexander Brill, or Hermann Wiener—react to this crisis and the problem of the mathematical objects that elude Anschauung?

AS: One could say that for Klein this did not present a problem because in his opinion not every conceivable mathematical phenomenon should be expressed visually. Models were a means to understand one aspect of mathematics, not the whole of mathematics. Apart from that, however, the crisis of Anschauung was fundamentally important. This crisis did not only last one or two years but from the end of the nineteenth century to the 1920s. And yet this crisis and mathematical model building did not have so much to do with one another, because the mathematicians working in a purely theoretical way were concerned with entirely different questions and issues. Of course, the model builders were aware that non-Euclidean hyperbolic geometry reduced the possibility of visualization and also of Anschauung.¹⁴ However, there were certainly attempts to build models for objects of non-Euclidean geometry. There was no clear opposition in the sense of their being, on the one hand, a Euclidean mathematics that was an object of Anschauung and could be visualized and, on the other, a non-Euclidean mathematics that was purely abstract and could not be visualized. Another example of the attempt to go beyond the possible is the modeling of four-dimensional space (see Fig. 3 for Victor Schlegel’s model of a convex regular four-dimensional polytope).¹⁵

Fig. 3

© Mathematical Institute, Georg-August-University Göttingen, all rights reserved. See also: Victor Schlegel, Ueber den sogenannten vierdimensionalen Raum (Berlin: H. Riemann, 1888)

Victor Schlegel’s model of a projection of the regular four-dimensional 24-cell into three-dimensional space (Göttingen Collection of Mathematical Models and Instruments, model 352).

However, one has to admit that after the peak of the crisis of Anschauung models became less important within the class curriculum at technical colleges. In the 1930s they gradually disappeared from the classroom and were relegated to university collections. The production of models completely stopped. That had already begun of course when set theory became more important, and, for this, other modes of academic teaching were used. At that point the mathematical models migrated to the glass cabinets of the collections and disappeared from production.

MF: The loss in the importance of models can be gathered from the famous lecture “Die Krise der Anschauung” given by Hans Hahn in 1932.¹⁶ Although Hahn, as he states himself, used drawings to illustrate the mathematical objects that triggered this crisis, he does not mention any models. On the other hand, also in 1932, the book Anschauliche Geometrie by David Hilbert and Stefan Cohn-Vossen was published,¹⁷ and this work contains numerous photos of models (see, for example, Fig. 4). So, to put the question concretely, what role do models play in the 1930s?

Fig. 4

Photos of plaster models of Dupin cyclide. From David Hilbert and Stefan Cohn-Vossen, Anschauliche Geometrie (Berlin: Springer, 1932), 193. © Springer, all rights reserved

AS: Here, it is important to be precise and to remember that Hilbert’s book did not appear suddenly; it was based on a series of lectures titled “Anschauliche Geometrie” that Hilbert gave between 1920 and 1921 in Göttingen, and at that time he was involved in an intense study of the mathematical models found there.¹⁸ These lectures were then published in 1932, which means that another ten years lie between the propositions and their publication. Moreover, these lectures were intended for teaching, and the book too should be understood as a textbook—that is not the same thing as a research finding that is addressed to other researchers. It is important to differentiate. How do mathematicians work at their desks when they want to solve mathematical problems? And how do they then teach what they have discovered at their desks during this research? One can gather from Hilbert’s book that he valued the pedagogical and epistemic function of models. From a historical perspective, I would not want to make such a clear distinction between mathematical model builders or representatives of Anschauung on one side and formalists on the other,¹⁹ at least not for the 1920s.

KK: The conflict between descriptive geometry and abstract or formal mathematics cannot be decided on the basis of the significance of the built models. Could one say, however, that another tension becomes noticeable at this point, one that would continue to grow in the twentieth century: the difference between the applied sciences and a ‘pure’ mathematics?

AS: Yes, I think that it is precisely here that one finds the division. That can clearly be seen if we look at the history of the mathematical institute in Göttingen.²⁰ Basically, descriptive geometry in Germany around 1900 became increasingly disconnected from mathematics and migrated to the architecture faculty. In Göttingen Klein wanted to build a science campus with a wind tunnel for applied physics etc. This was a place for the applied sciences so to speak, and it was precisely there that mathematics should also be located. The building for mathematics was opened in 1929—hence, only after Klein’s death—but with this location mathematics was indeed brought much closer to the applied sciences. In purely institutional terms, however, mathematics in Göttingen still remained tied to the faculty of philosophy—and these contradictory alliances brings to light a conflict in which Klein also found himself. Klein had a humanist education and explicitly stated in his biography: “mathematics is essentially a pure Geisteswissenschaft [humanities discipline].”²¹ But he also wanted it to serve the applied sciences. Of course, the ambivalent position of mathematics also related to mathematical models. And this ambivalence was partly a result of university politics: the chemists had their laboratory, the biologists had their formaldehyde specimens, and because the mathematicians did not only have chalk and board, pencil and paper, but also models, they now also had something to show, and that entitled them to demand more space. If one looks at Klein’s correspondence,²² then what one sees is indeed this question: how big do the spaces of the mathematical institute need to be in order to house all the models? Hence, the mathematical models in Göttingen played an important role in the founding of the institute.

The history of this Göttingen building gives us a good picture of the turf war that had a formative role in the mathematics of the 1920s. On the one hand, this building was built in the heyday of mathematical formalism and, on the other, the building was literally built around the mathematical models. Every mathematics student that enters the Auditorium Maximum in Göttingen had to pass these models. And that was at a time when these models actually no longer played a role at all.

MF: I would like to return to the second topic, which was mentioned before, namely the material techniques. The models were made from different materials: plaster, threads, card, etc. Can one assign different mathematical meanings, or rather interpretations, to the various materials the models are made from? Or do the materials refer in each case to different epistemic practices in or beyond mathematics?

AS: That is an interesting question, and one that was actually at the beginning of my work on mathematical models. I travelled throughout Germany visiting the collections of mathematical models, and in doing so—at least within descriptive geometry—I frequently came across the same forms (for example, a cylinder), but often in different materials, that is, in plaster or wood or brass and thread. I asked myself why these various materials were used and what kind of epistemic meaning was linked with this. I think that there were entirely pragmatic reasons for this, because the materials are of course also expressions of their times. So the models that work with brass and thread were made at a completely different time from the plaster models, and these in turn were made at a different time from the wire models. There is a chronological order of the materials that points beyond mathematics. The use of brass and silk thread is derived from the electromagnetism of the eighteenth century, and that was ultimately borrowed by the model-building mathematicians from the physicists of the École polytechnique. The early French models are reminiscent of certain instruments in the experiments of Michael Faraday or André-Marie Ampère.²³

If we look at the model of a rotating cylinder by Olivier (Fig. 5a), then the mathematics on this cylinder is limited—the ‘pure’ model as such is of course only this cylinder, but this in turn is attached to a wooden box and a brass frame—a little like in the physics instruments or the physics models of the time. The wooden box has a function, but this function seems to have more of an aesthetic origin, as with the physics models too. In the more recent mathematical models, these additional constructions mostly disappear or are reduced. When one looks at a later example—for instance, by Hermann Wiener from 1912 (see Fig. 5b)—then one sees that the cylinder now stands alone. One can rotate the cylinder without having to turn some screw on the side of the frame.

Fig. 5

a One of Théodore Olivier’s models from the 1830s: a rotating cylinder with a wooden box (Collection of Historical Scientific Instruments, Harvard University). © Collection of Historical Scientific Instruments at Harvard University, all rights reserved. b A string model of a rotating cyliner. From Hermann Wiener and Peter Treutlein, Verzeichnis von H. Wieners und P. Treutleins Sammlungen Mathematischer Modelle. Für Hochschulen, Höhere Lehranstalten und technische Fachschulen (Leipzig: Teubner, 1912), 12

KK: From a mathematical point of view it is to some extent the same model, but from an aesthetic point of view it has a different material finish, and above all it does not have the massive ‘frame’ represented by the wooden box.

AS: Exactly. In 1910 it is still the same model of a cylinder that can transform, but now without the frame.

One has to remember that the mathematicians did not actually have a tradition of experiment, and therefore for their modeling practice they drew inspiration from the natural sciences. The company Pixii père et fils, which built the early models in France and in particular the models for Olivier, also constructed experimental assemblages—which are basically also models—for Ampère and Faraday, so it is not surprising, therefore, that these models resemble each other.

Of course, there are also biographical contingencies. Olivier was from Lyon and therefore from a center of the silk manufacturing industry, which partially explains the use of silk thread. However, it was not uncommon in France to use silk thread in scientific models, especially in the physics of magnetism. In Germany, on the other hand, initially other materials were used. Alexander Brill produced his mathematical models around 1870/80 using plaster. Brill at that time was a professor at the Technische Universität in Munich, and plaster was also the material used by the art students to make casts. The plaster cast collection in Munich was certainly another influence. Joseph Kreittmayr, who made the plaster models for Brill also cast models for the Bayerisches Nationalmuseum.²⁴ In order to produce the mathematical models, Brill first had to find the craftspeople and organize the production, and he therefore drew on techniques and infrastructures that were already established. Moreover, plaster was cheap, and once the negative form had been made it was simple to reproduce the mathematical model.

Hermann Wiener’s models belong to yet another tradition. Although Wiener had studied with Brill and both were also related (they were cousins), Wiener made his models at a time when plaster no longer appeared adequate to the requirements of mathematical model making. Instead he used wire, which was flexible and could be easily shaped. Wiener criticized plaster models for not being able to show the essential: the movement of the line in space.²⁵ He wanted to show this movement in the model, but could not do this with plaster. Although one could score the lines and then use the model to examine their course, one could not manipulate the model. With Olivier’s models, that was still possible: there were screws that could transform the cylinder into a paraboloid. In Wiener’s case it is the hinge that enables the transformation from the planar to the spatial (and vice versa). This can be seen clearly in his model of an ellipsoid (see Fig. 6b).

Fig. 6

© Anja Sattelmacher, all rights reserved. c The same model of the ellipsoid (as in (b)), now unfolded. Photo: © Anja Sattelmacher, all rights reserved

a Construction drawing by Hermann Wiener—in this case, for an elliptic paraboloid. Wiener draws the wire sizes at a ratio of 1:1 to the model so that the points at which the wire should be cut and soldered (or connected by a hinge) can be transferred directly from the drawing to the model. Private collection Friedhelm Kürpig, undated. © Friedhelm Kürpig, all rights reserved. b Model of an ellipsoid in folded form, built following the construction drawing by Hermann Wiener. Model by Friedhelm Kürpig. Photo:

The model has to be flexible so that one can show the generation and the course of a curve. But models should also be robust and capable of being produced in large numbers. For this reason they need to be standardized. Wiener’s model designs meet these different requirements. In Fig. 6a one can see his construction drawing for the ellipsoid. With such a drawing, any precision mechanic schooled in model building could and can reproduce this model. That was not so simple with the plaster models because they were less standardized.²⁶

KK: Can you describe in more detail the work process necessary for the design and the making of the plaster models?

AS: To produce a prototype of a model—for instance, of an ellipsoid or a hyperbolic paraboloid—the individual sections for the solid were first calculated and then cut out from a zinc sheet. This procedure was identical to the production of the flexible card models (Figs. 7 and 8), only this time the sections slotted together are made from zinc sheet. Once assembled as a model, the structure has to be held together with card, bits of wood, modeling clay, or scraps of metal in such a way that the form—for instance, a sphere—was preserved and no longer moved. It was subsequently filled with modeling clay.

Fig. 7

© Anja Sattelmacher, all rights reserved

Markings on sections of a sphere. The numbers refer to the points at which the sections should be slotted together (Göttingen Collection of Mathematical Models and Instruments, models 13 and 19). Photo:

Fig. 8

© Anja Sattelmacher, all rights reserved

Two models of an ellipsoid: on the right a version made of cut steel plate following the method used in Brill’s card models; on the left a version made of bent wire following Hermann Wiener. Models by Friedhelm Kürpig. Photo:

The negative of the surface produced in this way was hollow on the inside and could now be filled with plaster—assuming the models were produced as hollow casts. That means that an approximately three-centimeter layer of plaster was poured into the body.

KK: So the work process involves a chain of translations and probably also a number of different actors. It begins with the mathematical formula, which is translated into a drawing, which is then in turn translated into card or steel and slotted together to form a three-dimensional structure. From this one obtains a casting mold, if I have understood you correctly. But alternatively a wooden mold can be lathed by a professional carpenter, which can likewise be used for the creation of a mold for the plaster.

AS: The intermediate stage with the wooden mold does not always take place. When the sections are made of a tougher material—for example, thicker card—one can pour the modeling clay directly into the form, and then one already basically has the master mold for the plaster models.

MF: The people in the model-building workshops were not necessarily mathematicians; there were also craftspeople—carpenters, for example. How much mathematical knowledge did the craftspeople need to have? Mathematicians of course were not present at all stages of the work.

AS: To answer this question, one has to take a more detailed look at when which people were involved in the production process of the models. If one considers the process from the formula to the finished model, then at the beginning, for the calculation, rather a lot of mathematicians and mathematics students tended to be involved, and at the end fewer. If we take the card model already mentioned as an example, this consists of cut sections, and both the calculation and the drawing of the curves would have been carried out by mathematicians, because that has to be very precise. The prototypes of the sections were therefore prepared and also cut out by mathematicians. But, for the plaster models, the actual work of reproduction, such as the production of the master mold and the pouring of the plaster, was outsourced to the workshops. Once the prototype was ready, this was followed by purely artisanal techniques, and these were always the same regardless of whether one was casting a Greek Minerva or a model based on a mathematical formula. This being said, right at the end of the modeling process, the mathematicians were brought back to score the curves (compare the photos of Hilbert’s plaster models in Fig. 4, on which the scored curves are clearly visible). Although, of course, it is difficult to say in retrospect who actually did the scoring.

MF: But the scoring would have had to have been guided in some way, and such templates could only have been created by a mathematician, right?

AS: Exactly, such templates were made by mathematicians. Who actually did the scoring? I can imagine that it took place in the workshops—hence, one can assume the scoring was carried out, for instance, by those doing the casting, on the basis of templates, since these models were produced and distributed serially. One could not score the curves in the master mold, because plaster expands during the casting and would cover over the lines.

A few sources, such as the recorded recollections of a female descendant of the model publisher Martin Schilling, suggest that this manual, often very precise and delicate work was carried out for the most part in a domestic setting by the wives of the mathematicians. However, this is not mentioned or documented by the mathematicians themselves.²⁷

MF: If one looks at the process of manufacturing the models (as you have just described it) in the framework of seminars, then this may have been suitable when one taught a class of ten to fifteen students, since there probably would have been a need for assistance and close contact between the persons involved. But, if I understand correctly, the number of students at the end of the nineteenth century in Germany did not remain fixed over the years. Did the change in student numbers affect the way models were used?

AS: Here, we come to an important point, since one can in fact ascertain that in the 1910s projective media became more important for teaching purposes. That was certainly linked to the growing number of students at the technical colleges. Wiener in Darmstadt, for example, had considerably more students than Brill and Klein in Munich. In their modeling seminar, Klein and Brill had six to fifteen students, whereas Wiener had around fifty. So naturally he needed to consider how it would still be possible to work with models with such a large number of students—that is, what these models would need to look like to be useable in a large lecture hall; he was not satisfied by the idea of simply passing the models around. Therefore, he constructed the models in such a way that they could also be projected, quite simply with the help of a light source.²⁸ An example of these new projection techniques can be seen in Fig. 9, showing a demonstration of Erwin Papperitz’s kinodiaphragmatic (kinodiaphragmatische) projection apparatus.

Fig. 9

Quelle und Meyer, 1916), Plate 2

Four photos of a demonstration of Erwin Papperitz’s kinodiaphragmatic projection apparatus showing various light and shadow projections. The first photo (top left) shows the conic sections of a paraboloid resulting from the shadow projection. The second shows an intersection curve on a sphere likewise produced by means of shadow. From Erwin Papperitz, Methodik des Mathematischen Unterrichts (Leipzig:

MF: Wiener projected the models in his lectures?

AS: Yes. That was also his argument—if one can move the model and demonstrate how the form changes, then one can show the course of a curve, how the sections change. And wire for him was the ideal medium because, with the help of light, the wire models allowed spatial projections that still functioned as a material of Anschauung even in front of a larger group of students. But what this also shows is that Wiener lived at a time in which film and projective media were beginning to appear. At the beginning of the twentieth century, Wiener was aiming at effects that were clearly different from those of Brill in the 1870s—despite the fact that they were both mathematicians, came from a similar school, and that their models visualized the same mathematical objects.

MF: With these new media techniques, it is also a matter of another kind of knowledge transfer. In the German model seminars of the 1870s and 1880s, one of the tasks of more advanced students was to produce models themselves. So the models were not always intended to be looked at. Could one say that, with the new projection techniques, the haptic character of the models gets lost?

AS: One can certainly see it that way, since the production of models by the students themselves is either reduced or no longer takes place. That is also related to the fact that descriptive geometry had become less important in mathematics and likewise in teaching. Also the concept of Anschauung changes at that time.

MF: In what way?

AS: Between 1870 and 1910 one sees a gradual shift in the way one thought about mathematical models, and that also had consequences for the understanding of Anschauung. That can be seen clearly in Wiener’s teaching practice, which is based on the idea that projection allows the eye to see the model and that the rest arises via the process of thought. Hence, the eye is the organ that senses the model…

MF: …and not the hands.

AS: Exactly. A transfer from the hand to the eye takes place.

MF: With this disappearance of the haptic, the era of three-dimensional mathematical models essentially comes to an end. From the 1930s onwards basically no more model series were produced.²⁹ To finish we would like to talk about what seems to be a revival of mathematical models at the end of the twentieth century. How do you view the new digital models that can be produced and shown with the help of virtual reality software, 3D printers, touch screens, and other technical tools? Can one understand these new visualization techniques as a revival of mathematical models? Especially since these new techniques can now also visualize mathematical objects that in the nineteenth century were thought to be impossible to visualize. What in your opinion is the new epistemic and cultural meaning of these techniques?

KK: To give a concrete and very popular example for Michael’s question, IMAGINARY is a platform that works rigorously on the visualization of mathematical formulae (e.g., of curves and surfaces), and it does this among other things by using ‘swarm intelligence,’ since the visualization program is put at the disposal of the website’s users.³⁰ The visual modeling of the mathematical objects happens interactively and in real time: by changing the mathematical parameters, the user simultaneously changes the visualized object. This gives rise to a ‘haptic moment,’ simply because the objects change with the interventions of the user—as when Olivier turns the screws or when Wiener pulls on the hinges.

AS: The question is: what possibilities do these technologies offer and what limits? What types of models could I not yet realize ten years ago, and what can I realize today because the technology has changed in a certain respect? That does not have to be tied to a progress-oriented way of thinking but can be based on the simple fact that the technology has changed. If one wants to think the model process further, then one must consider, for example, the computer and also the 3D printer as an object that already contains various technologies itself, and not as something into which one enters something—as into a black box—and in the end something comes out. We historians of science will not get anywhere if we only consider the beginning and the end. We have to look at what kinds of 3D printers and touch screens existed ten years ago. What could they do, what kinds of technologies did they operate with? Why are we doing things differently now, and what haptic qualities does this give rise to? What does that generate in the model itself? As with the models of the nineteenth century, with today’s visualizations too, one has to look very carefully at the practices and technologies that are involved.

Translated by Benjamin Carter.

Anja Sattelmacher is a historian working at the intersection of the history of science and media. She studied Media and Cultural Studies in Weimar and Lyon and holds a master’s degree in Museum Studies from Macquarie University in Sydney. For her PhD on the history of knowledge of mathematical models at the history department of Humboldt-Universität zu Berlin, she studied numerous collections of material mathematical models throughout Germany, France, and beyond. What most struck her was the willingness to experiment with different materials for making the models. Sattelmacher is now an academic adjunct at Humboldt-Universität zu Berlin in the media studies department. For her current project, she examines collections of historic film documents to write about the history of political education in 1950s and 1960s (West) Germany. Together with Sarine Waltenspül and Mario Schulze, she recently published a Focus section in the journal Isis on research film.