‘Analogies,’ ‘Interpretations,’ ‘Images,’ ‘Systems,’ and ‘Models’: Some Remarks on the History of Abstract Representation in the Sciences Since the Nineteenth Century

Abstract

Despite the rapidly growing interest in the history of ‘thinking in models’, a detailed historical understanding of the different modes and forms of such thought-forms is still some way off.

A German version of this paper has been published under the title “‘Analogien,’ ‘Interpretationen,’ ‘Bilder,’ ‘Systeme’ und ‘Modelle’: Bemerkungen zur Geschichte abstrakter Repräsentationen in den Naturwissenschaften seit dem 19. Jahrhundert,” Forum interdisziplinäre Begriffsgeschichte 5, no. 1 (2016): 11–30. This English version has been slightly revised and corrected. Many thanks are due to the translator Benjamin Carter and the editors of this volume for their great help and patience.

Despite the rapidly growing interest in the history of ‘thinking in models’, a detailed historical understanding of the different modes and forms of such thought-forms is still some way off. This is mainly due to the fact that, when considered retrospectively, not enough attention is paid to the at times surprisingly diverse epistemological language used by the actors in this history. Accordingly it is not uncommon that not only the language use but also the thought-forms themselves are characterised with some vagueness in historical accounts.

This can be illustrated by a frequently quoted passage from the introduction to Heinrich Hertz’s Prinzipien der Mechanik, published in 1894. Here Hertz remarks:

We form for ourselves images [innere Scheinbilder] or symbols of external objects; and the form which we give them is such that the necessary consequents of the images [Bilder] in thought are always the images of the necessary consequents in nature of the things pictured [abgebildeten Gegenstände]. In order that this requirement may be satisfied, there must be a certain conformity between nature and our thought. Experience teaches us that the requirement can be satisfied, and hence that such a conformity does in fact exist. When from our accumulated previous experience we have once succeeded in deducing images of the desired nature, we can then in a short time develop by means of them, as by means of models [wie an Modellen], the consequences which in the external world only arise in a comparatively long time, or as a result of our own intervention. We are thus enabled to be in advance of the facts, and to decide as to present affairs in accordance with the insight so obtained. The images [...].¹

Around two decades later Ludwig Wittgenstein, on whom Hertz’s reflections on images of external reality had made a strong impression, would—with a barely perceptible alteration—form from this the famous proposition 2.12 of his “Tractatus logico-philosophicus”: “The image is a model of reality.”²

The combination of these two texts and motifs has occasionally been made the centrepiece of a genealogy of thinking in abstract and, particularly, mathematical models in scientific modernity. In Wittgenstein’s formulation, however, a difference is lost that, if read carefully, is still found in Hertz’s text: namely that between images that are ‘like models’ and the models themselves. After Wittgenstein it was possible—although initially still uncommon—to speak also in the sciences of abstract, immaterial models of real relations. Until Hertz—and still in the language of his introduction—the models that the images resembled without being identical with them were not abstract but concrete, such as the models that had been used for centuries in a range of artisanal, artistic and academic contexts, from architecture to painting, and had recently acquired a certain prominence in the sciences in the form of the material structures representing scientific objects that can still be found in great quantities in our collections and museums. Some examples selected at random from an abundance of similar ones are the once commercially distributed plaster or string models of mathematical objects (Figs. 1 and 2) and the material models of biological and chemical phenomena (Fig. 3).³

Fig. 1

© Research group Differential Geometry & Geometric Structures, TU Wien, all rights reserved

Model of a space curve with singular points (Sammlung mathematischer Modelle, Institute of Discrete Mathematics and Geometry, TU Wien).

Fig. 2

Plaster model of a third-order surface (Sammlung historischer mathematischer Modelle, Martin-Luther-University Halle-Wittenberg. All rights reserved

Fig. 3

Photo: David Ludwig. CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/)

Model of a spore of the fungus Puccinia graminis (grain rust) (Botanisches Museum Greifswald).

The historiography of thought-forms and epistemological motifs that are related to what scientific modernity—following Wittgenstein and many others—first designated as (theoretical as well as mathematical) ‘models,’ stands before a fork in its potential path at this point. A history of models in the strict sense would have to carry out a detailed study of the aforementioned material objects and their uses. While some contributions to this first way of approaching a detailed history of models are beginning to emerge, it, too, remains largely unwritten. However, the sciences of the nineteenth century and of earlier periods were of course also familiar with different forms of more or less abstract, more of less non-material representations. For this second strand in the history of models in science, Heinrich Hertz’s ‘images’ are only one late example.

Besides ‘images,’ nineteenth-century scientific discourses also spoke of a broad range of other ways in which to represent objects of knowledge, even without being (material) models as these were understood at the time. In particular the notions of ‘analogies,’ ‘interpretations’ or ‘systems’ of real or mental objects were widespread in this function. The examples associated with these notions have frequently been of substantial importance for the development of science. However, as I hope to suggest in the following, they often stand for quite different forms and functions of abstract representation. Here I use the expression ‘abstract representation’ somewhat vaguely and naïvely as a generic term for diverse ways of describing a set of scientifically interesting objects or facts by means of something else, and to thematise this for scientific practice without thereby referring to material, tangible objects—as was the case with the notion of ‘model’ in the language of the nineteenth century. At the same time (pace Wittgenstein and notwithstanding the inflationary use of the term ‘model’ beginning in the mid-twentieth century) it is important to avoid premature talk of ‘thinking in models’. As we shall see, the abstract representations in question occasionally had very concrete epistemic functions. Therefore the word ‘abstract’ should not be overstated. In the following I set out to characterise in particular the respective epistemic situation—i.e. the specific circumstances of knowledge and knowledge practices—in which recourse to a form of abstract representation took place and seemed promising to those involved.

I understand these remarks as prolegomena in two respects: first, historically, as preliminary remarks on a history of thinking in abstract representations that is oriented to actor’s categories and that sees in the variation of these contemporary categories a field of important historical and epistemological differences that should not be levelled out too quickly; second, epistemologically, as preliminary remarks contributing to a differentiated analysis of the forms and functions of abstract representations in the sciences of modernity. On both levels I aim in particular to show that a reduction of the discussion of the historical function of abstract representations to the problem of perspectivity or relativity (a set of facts or phenomena can have many—and even inequivalent—abstract representations/models, which for a thinking in models ultimately calls into question the category of the real) would fall short. The recognition of this problem has rightly been regarded as an essential signum of scientific modernity, and it received its first thorough treatment in Gaston Bachelard’s Le nouvel esprit scientifique, in which Bachelard set out to develop his ‘non-Cartesian’ epistemology.⁴ In the historical material, however—as I hope to make plausible—the issue of perspectivity or of the potential multitude of abstract representations of the same segment of ‘reality’ is by no means the only noteworthy aspect, and sometimes one finds almost the opposite epistemological tendencies to this development. This does not speak against Bachelard’s thesis; rather it speaks against an underestimation of the complexity of the epistemological history of the modern sciences.

Dynamical Analogies, Physical/Mechanical Analogies, Mathematical Analogies

I shall begin with a form of abstract representation that was frequently referred to in the mid-nineteenth century and that was of great importance particularly in mathematically-based areas of physics and still played a terminologically fixed role in Hertz’s mechanics: namely ‘analogy.’ More accurately, this form of abstract representation should be divided into a number of sub-varieties, since it was referred to sometimes as ‘physical analogy,’ sometimes as ‘mathematical analogy’ and sometimes (in the case of Hertz) as ‘dynamical analogy.’⁵ An early explicit epistemological treatment is found in James Clerk Maxwell’s famous 1856 text “On Faraday’s Lines of Force.”⁶

The text begins with a description of the problematic epistemic situation in the area of physics that Maxwell was engaged in: “The present state of electrical science seems peculiarly unfavourable to speculation.”⁷ For Maxwell there were several reasons for this: While the laws of static electricity and some parts of the mathematical theory of magnetism were known, in other parts there was still insufficient experimental data. There were mathematical formulae for the flow of currents in conducting materials as well as for their mutual attraction, but the relationship of these formulae to other areas of the theory of electricity and magnetism remained unclear, and so on. In this situation the development of a theory of electricity required that the various known parts be correlated with mathematical precision and that proposals be made for unknown areas without, however, making “physical hypotheses,”⁸ that is to say without making uncertain assumptions about the precise nature of the facts underlying electrical and magnetic phenomena. Given the partial knowledge of experimental data at the time, such hypotheses were always subject to error. Hence a ‘physical theory’—i.e. a realistic, causal explanation of electricity and magnetism—was still unattainable, not in principle but owing to the particular conditions of knowledge at the time.

Maxwell concluded that in this situation of highly incomplete and partial knowledge it was requisite “to obtain physical ideas without adopting a physical theory.”⁹ And in order to do that, he continued,

we must make ourselves familiar with the existence of physical analogies. 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.¹⁰

The examples that Maxwell went on to provide—particularly one taken from the earlier work of his colleague William Thomson (later Lord Kelvin)—clearly show that with the “partial similarity between the laws” of two sciences he meant the agreement of their mathematical form. Of note in the example taken from Thomson was that certain mathematical formulae that regulated the flow of heat in bodies were consistent with those describing the attraction of bodies under the influence of a reciprocal force that was inversely proportional to the square of their distance. Both cases could be described by means of so-called potentials—i.e. by solutions of Laplace’s differential equation given certain boundary conditions:

We have only to substitute source of heat for centre of attraction, flow of heat for accelerating effect of attraction at any point, and temperature for potential, and the solution of a problem in attractions is transformed into that of a problem in heat.¹¹

Maxwell pointed out that the physical and causal circumstances of both cases were very different and not analogous. In the case of heat, a phenomenon in a continuous material medium was concerned; in the case of attraction, discrete material bodies interacted with each other through forces across an empty space. It was merely the partially correlating mathematical form that conveyed the analogy. Hence the analogy between the two domains was a matter of translation, of symmetry and of a common formal structure. Unlike the modern model, analogy here was not an asymmetrical relationship between representation and represented but a relation of reciprocal representation. Moreover, it was not the mathematical form itself that was declared to be the representation of a domain of physical phenomena; rather Maxwell’s ‘physical analogy’ was a symmetrical relation between different domains of physical phenomena placed on the same level, each representing and illustrating the other. When, as in the example taken from Thomson, one of the domains belonged to mechanics, Maxwell spoke of a ‘mechanical analogy.’ This was a particularly attractive class of analogies for the explanation of nature, since mechanics had to be regarded provisionally as the best-known area of physics. Once, through Thomson and others, ‘dynamical theory’ became the catchphrase of a more abstract theoretical and mathematized version of mechanical theory, the expression ‘dynamical analogy’ was also frequently found in the texts of English-language natural philosophers.¹²

The physics of the second half of the nineteenth century discussed a broad range of additional examples of such analogies. In the following years Maxwell himself endeavoured to find analogies for electromagnetism. Among other things, he developed a ‘mechanical analogy’ for the dynamical interaction between electrical and magnetic fields, as illustrated in Fig. 4. (If the secondary literature has sometimes described this analogy as a ‘mechanical model,’ this is strictly speaking yet another anachronism).

Fig. 4

James Clerk Maxwell’s mechanical analogy. From James Clerk Maxwell, “On Physical Lines of Force,” in The Scientific Papers of James Clerk Maxwell, vol. 1, ed. William Davidson Niven (Cambridge: Cambridge University Press, 1890), 451–513, here 489, Fig. 2

In the epistemic situation in which Maxwell made use of this analogy it was difficult to interpret it as anything other than an attempt, with the help of mechanical analogy and controlled by mathematical equations, to find out something about the microstructure of reality itself, even if this should turn out not to be mechanically structured. Moreover Maxwell presented his attempt not least as an offer of dialogue to those who considered magnetism a micromechanical phenomenon.¹³

A particularly interesting example of physical analogy was introduced into the literature by Hermann Helmholtz in 1858. This was the analogy between hydrodynamics—i.e. the continuum mechanics of fluids—and the static theory of magnetic fields generated by electric currents:

This [hydrodynamic] problem leads to a peculiar analogy between the vortex motions of water and the electro-magnetic effects of electric currents. Thus, if in a simply connected space filled with a moving fluid there is a velocity potential, then the velocities of the water particles are equal and in the same direction as the forces exerted on a magnetic particle in the interior of the space by a certain distribution of magnetic masses on its surface. If, on the other hand, vortex filaments exist in such a space, then the velocities of the water particles are to be set equal to the forces on a magnetic particle by closed electric currents that in part flow through the vortex filaments in the interior of the mass, in part in its surface, and whose intensity is proportional to the product of the cross section of the vortex filaments and the velocity of rotation.

In the following I shall therefore frequently avail myself of the fiction of the presence of magnetic masses or of electric currents, simply in order to obtain a briefer and more vivid representation of the nature of functions that are the same kind of functions of the coordinates as the potential functions or attractive forces that attach to those masses or currents with respect to a magnetic particle.¹⁴

Here too the aim was to demonstrate analogical formal-mathematical relationships between different subject areas. It should be borne in mind, however, that Helmholtz brought epistemic asymmetry into play in two respects: on the one hand electromagnetism was used here to explain hydrodynamics, not mechanics to explain electromagnetism; on the other—and this asymmetry is even more striking—the electromagnetic side of the analogy also explains the abstract mathematical objects involved: namely the (in certain situations multivalued) potential functions that Helmholtz used to describe vortex motions.¹⁵ Felix Klein would later make this (or a closely related) epistemic asymmetry the starting point for a more comprehensive ‘physical’ representation of certain epistemic objects in mathematics—a representation which he attributed not to Helmholtz, however, but to Bernhard Riemann.¹⁶

To sum up, the epistemic functions of physical analogies provided, firstly, a very fruitful heuristic and, secondly, a way out of the research-practice dilemma of uncertain “physical hypotheses,” as Maxwell wrote. In this respect one can also see in them an initial distancing from a strong (and at least temporarily unattainable) version of realism in the physical sciences. Of course the recourse to physical analogies also involved a clear tendency to emphasize formal similarities in the physical world and to identify uniform mathematical forms in the diversity of the physical world. This tendency was clearly directed against a relativistic epistemology. Indeed it may even have worked directly towards a realism of physics at a more abstract level. Where, then, are ‘physical analogies’ to be classified on a scale between concrete and abstract thought-forms? Here the evidence is clear. Analogies were principally a matter of constructing bridges between different concrete imaginations, possibly even of the provision of concrete imaginations that could represent abstract mathematical objects (such as multivalued potential functions). Such concrete imaginations as those developed in the analogies could and should be used for the development of new ‘physical ideas,’ as can be traced in particular in the physics of ether in the second half of the nineteenth century.¹⁷ The explanatory side of analogies thus gave rise—even against Maxwell’s intention and assertion—to tentative imaginations about the unknown physical nature of the phenomena at hand. Used in this way, analogy also helped, in spite of everything, an early form of perspectivist explanation of physical reality.

Interpretations of Non-Euclidean Geometry

I shall now discuss another central chapter in the development of the exact sciences in the nineteenth century, an episode that retrospectively and anachronistically has often been described as a shift to a thinking in mathematical models: namely the emergence and propagation of so-called non-Euclidean geometry. For Bachelard it constituted the paradigmatic development in relation to which the formation of the new scientific spirit of modernity could be most clearly understood, at least in some of its important respects, and which also represented the earliest episode of this transformation.¹⁸

On closer inspection, however, it turns out once again that the actors for good reason did not speak of models (and if they did, then again in the traditional sense; see below)—and nor did they refer to the physical analogies discussed above. Instead the Italian mathematician Eugenio Beltrami, in an important step in this development, drew on the notion of ‘interpretation’—and more precisely on the interpretation of a system of basic propositions or principles of geometry. Here a certain form of pluralism came into play: such a system of basic propositions—this was the thrust of Beltrami’s intervention—could admit multiple different interpretations. Already before Beltrami the complementary question had been raised: How many possible systems of basic propositions (principles) of geometry are there? This indicates that another epistemological concept was in play: namely that of ‘systems’ (of basic propositions of geometry). Moreover it is in this episode that we reencounter the notion of ‘image’, invoked at the beginning of this article, in the historical material. Again I shall briefly sketch the epistemic situation in which Beltrami made his attempt to provide an interpretation of non-Euclidean geometry (as the Italian title of his essay suggests).¹⁹

Even if well known in its general outline, the gradual, dispersed beginnings of the development that would eventually be summarised under the name non-Euclidean geometry still continue to occupy historians. Here I shall limit myself to a few cursory remarks that are essential for understanding the principal point, and ask the more initiated reader to be patient.²⁰

The first thing to note is that during the 1820s and early 1830s a number of mathematicians had come to the conclusion that, in addition to the traditional system of geometry that could look back on a continuous development since antiquity and was considered the perhaps clearest example of a science that was both demonstrative and descriptive, there was still (at least) one other system of geometry that could (and should) be developed mathematically. Both mathematical outsiders, such as János Bolyai, and established scientists, such as Nikolai Lobachevsky in Kazan and Carl Friedrich Gauss in Göttingen, agreed that a geometry could be developed in which Euclid’s much-discussed postulate on three intersecting lines (usually, but somewhat misleadingly called ‘parallel postulate’) was not valid (Fig. 5).

Fig. 5

Euclid’s ‘parallel postulate.’ If two straight lines intersect a third in such a way that the sum of the inner angles between them on one side is less than two right angles, then the two lines on this side will meet if extended indefinitely

In the eighteenth century several geometers (in particular Girolamo Saccheri in a book published in 1733, shortly before his death, with the telling title Euclides ab omni naevo vindicatus) discussed logical alternatives to this postulate or axiom²¹ with the intention and hope of showing that these alternatives were absurd, if the other axioms of Euclid’s geometry were retained. This discussion yielded a tripartition of possible cases, which can be illustrated for example as in Fig. 6: Suppose two line segments of equal length AB and CD are perpendicular to a third segment AC, while the free end points are connected by a fourth segment BD.

Fig. 6

Girolamo Saccheri’s quadrangle

Three cases are then possible:

• 1. The angles $\sphericalangle ABD$ and $\sphericalangle CDB$ are equal and smaller than a right angle

  (acute angle hypothesis).

• 2. The angles $\sphericalangle ABD$ and $\sphericalangle CDB$ are both equal to a right angle

  (right angle hypothesis; this hypothesis is equivalent to the parallel axiom).

• 3. The angles $\sphericalangle ABD$ and $\sphericalangle CDB$ are equal and greater than a right angle

  (obtuse angle hypothesis).

After a few intermediate steps the third case led to a contradiction with the assumption that lines could be extended to infinity, which while not being an explicit axiom of traditional geometry was nonetheless taken for granted by most participants in the discussion. The second case was that of traditional geometry. For the first case—as the eighteenth century geometers who were occupied with this question believed—a contradiction could also be found. Remarkably enough, however, this came to light only after a lengthy and rather subtle argumentation, the weaknesses of which were obvious to experts.

It was precisely here that the aforementioned authors announced their disagreement. Bolyai, Lobachevsky, Gauss and a few others believed that the acute angle hypothesis could not be refuted but, on the contrary, could be developed into a valid system of geometry. This system—and this is the decisive epistemic point—was different in content from the traditional system of geometry. Thus in the new geometry, given a line and a point not on this line, there were several lines in the plane defined by the line and the point, passing through this point that do not intersect the given line (and not just a single line, as in traditional [Euclidean] geometry); also in this new geometry the sum of the angles of a triangle was always less than 180 degrees, and so on. Both systems of geometry could not be regarded as simultaneously correct descriptions of the structure of physical space and be true in this sense—an unprecedented epistemic situation and a completely new problematic for the venerable science of geometry.

Lobachevsky and Gauss regarded this problematic as an empirical question to be decided for instance on the basis of astronomical observations, a position that a few decades later would strongly promote empiricism in the natural sciences. However, for the time being it remained impossible (for a number of reasons) to confidently make such an empirical decision. The ‘young radical’ Bolyai, in turn, believed he possessed speculative proof that the new deviant form of geometry was in fact the true one—unfortunately this ‘proof’ has not come down to us.

Important for the historical situation in which Beltrami wrote was certainly also the fact that the overwhelming majority of the mathematically educated were still firmly convinced of the truth of the traditional geometry of Euclid and his successors.²² The epistemological irritation caused by the alternative geometry, later so celebrated (not least by Bachelard), initially had very little effect. Indeed, who could be sure that the acute angle hypothesis did not simply conceal a more remote error that had not yet been detected? Gauss remained cautious, expressing himself on the subject only in letters to fellow scientists, never in print. Bolyai was virtually unknown. While Lobachevsky was a professor and rector at the prestigious University of Kazan, this was far from the centres of European learning.

Beltrami’s intervention in 1868 occurred precisely in this epistemic situation in geometry. In the years immediately before, a number of Gauss’s statements in his correspondence had been posthumously edited, and interest among mathematicans in the obscure texts of Lobachevsky and Bolyai had grown somewhat. The main question, however, was what to do with them. Beltrami decided to try to interpret them. Precisely what he had in mind is the point that now needs to be addressed.

Beltrami’s principal interest was in mathematical objects in ordinary, traditional geometry that had internal relations corresponding to, or at least approximating, those of Bolyai-Lobachevskian geometry (henceforth BL geometry). Beltrami found several such objects. In the first step of his argumentation he referred to curved surfaces in the three-dimensional space of ordinary Euclidean geometry, in particular to surfaces that could be obtained through the revolution of a suitable curve around a spatial axis that had surfaces with constant negative curvature.²³ Figure 7 shows a slightly later model (in the nineteenth century sense of the term) of such a surface. If the points on a surface of constant negative curvature were interpreted as points on a plane of BL geometry (henceforth BL plane) and the ‘shortest lines’ (geodesic lines, lines of least curvature) on such a surface as lines on the BL plane, the BL geometry used for these was the usual length and angle measurement in certain limited areas. However, there was still a difficulty: all known concrete surfaces of constant negative curvature in three-dimensional Euclidean space were unable to represent the entire BL plane (this led, among other things, to multiple self-intersections of geodesic lines, and generally the regions of a BL plane outside these limited areas could no longer be represented).²⁴

Fig. 7

© Mathematical Institute, Georg-August-University Göttingenn, all rights reserved

Surface of revolution of a tractrix with constant negative curvature, designed by student of mathematics Isaak Bacharach, Munich 1877 (Göttingen Collection of Mathematical Models and Instruments, model 188).

Hence Beltrami went a step further in his argumentation and considered the interior of a circle with a length and angle measurement that deviates from the (Euclidean) norm. Such an ‘auxiliary circle,’ with the correct specification of its metric determinations, could also represent a surface of constant negative curvature that was no longer thought of as being in three-dimensional Euclidean space, and that in this sense could be imagined as an abstract surface of constant negative curvature. And this new object (or imagination) in turn could also be interpreted as a BL plane. Thus Beltrami’s interpretation combined two stages of representation: the auxiliary circle (to which appropriate determinations of length and angle measurement were assigned) represented an (unlimited) surface of constant negative curvature, which in turn represented a (complete) plane of BL geometry. As he went on to summarise:

It follows from the above that the geodesic lines [of the BL plane regarded as a surface of constant negative curvature, ME] are represented in their total (real) development by the chords of the limit circle, whereas the extensions of these chords outside this circle have no (real) representation. On the other hand two real points of the [BL] plane are represented by two likewise real points inside the limit circle which determine one chord of this circle. It can thus be seen that two arbitrarily chosen real points of the [BL] plane always determine one geodesic line, which is represented on the auxiliary plane by the chord passing through their corresponding points.

[...] What is more, the theorems [of non-Euclidean planimetry] are only accessible to concrete interpretation when one relates them not to the [Euclidean] plane but precisely to these surfaces [of constant negative curvature], as we shall demonstrate in detail below.²⁵

By means of this two-stage representation Beltrami now also established an epistemic symmetry for his interpretation. All elements of BL geometry (both figures and relations between them) now found a counterpart in both stages of the interpretation: in the (abstract) surface of constant negative curvature as well as in the interior of the auxiliary circle with its metric determinations. Just as in the physical analogies, now also in the interpretation of non-Euclidean geometry two imaginations mutually explained each other. In order to achieve this epistemic symmetry Beltrami was prepared to forgo a degree of imaginative concretion: His unlimited surface of constant negative curvature was an imagination that (unlike the surfaces of revolution with constant negative curvature with which he had begun his considerations) no longer found a place in the world of traditional geometric imaginations. It is an example of one of the early epistemic objects of mathematics beyond the sphere of ordinary spatial intuition that would become so characteristic of mathematical modernity.²⁶ Nevertheless Beltrami insisted that even this imagination was still concrete, as is shown by the final sentence of the quote above.

Indeed Beltrami’s interest in the concretisation of the problematic epistemic object ‘non-Euclidean plane’ is documented in yet another way. He made several models (in the literal sense of the time) of the non-Euclidean plane from paper, which he frequently described in his correspondence—at least one version of which has survived and can today be found at the Istituto Matematico of the University of Pavia (Fig. 8).²⁷

Fig. 8

© Dipartimento di Matematica, Università di Pavia, all rights reserved. The photograph was kindly provided by Professor Maurizio Cornalba at the Dipartimento di Matematica, Università di Pavia

A (paper) model of non-Euclidean geometry by Eugenio Beltrami, ca. 1869–1872.

About twenty years later, in 1889, it was incidentally also Beltrami who brought Saccheri’s aforementioned book, Euclides ab omni naevo vindicatus, back to the attention of a wider audience of mathematicians and historians of science.

Beltrami’s successful attempt at an interpretation of non-Euclidean geometry—along with two other texts published in 1868: Riemann’s previously unpublished inaugural lecture “Über die Hypothesen, welche der Geometrie zu Grunde liegen” and Helmholtz’s “Ueber die Thatsachen, welche der Geometrie zum Grunde liegen”—substantially contributed to the reception of this obscure object of growing mathematical desire in wider mathematical circles. In contemporary mathematical literature in other languages, his notion of interpretazione was often rendered with the help of the term ‘image’.

Thus Felix Klein in his 1871 paper “Ueber die sogenannte Nicht-Euklidische Geometrie,” in which he turned his attention to the new subject, initially introduced Beltrami’s motif of interpretation to then consistently use the terminology of the ‘image’ for the representation of non-Euclidean geometries, as can be seen in the following quotation that links Beltrami’s construction with a technique of projective geometry developed by the English geometer Arthur Cayley:

For the ideas of hyperbolic geometry, in the light of the above, we immediately obtain an image if we draw an arbitrary real conic section basing it on a projective determination of measure.²⁸

The expressions Abbild (roughly: image of a mathematical mapping) and anschauliches Bild (intuitive image) are also frequently found in the autographed 1893 lectures on non-Euclidean geometry, which introduced Klein’s ideas on non-Euclidean geometry to a new generation of mathematicians.²⁹ In these lectures, and in line with his general mathematical orientation, Klein was explicitly concerned with the Versinnlichung (the making available to the senses) of the new geometries—another frequently used catchword for the epistemic function of the images of non-Euclidean geometries. As with Beltrami here too the choice of language reflects the main epistemological interest in the material models of this period.

When Klein’s 1893 lectures appeared in print for the first time posthumously in 1928 in a completely revised version by Walter Rosemann, one found in addition to the still widely used terminology of the ‘image,’ ‘representation’ (Abbild) etc. the word ‘model’ used in the modern sense. Thus in an anachronistic semantic shift one reads for example about the “model of planar hyperbolic geometry already known to Beltrami,”³⁰ where “model” no longer refers to Beltrami’s paper model, which was presumably unknown to Rosemann, but to his ‘interpretation’ BL geometry. The precise source from which Rosemann adopted the modern usage of the term ‘model’ must remain an open question here. Felix Hausdorff’s inaugural lecture ‘Das Raumproblem,” discussed below, and Hermann Weyl’s lectures on space, time and matter (first published in 1918), both of which adopt the new usage at least to some extent, are two possibilities, however.³¹

Thus the epistemic functions of Beltrami’s ‘interpretation’ and the ‘images’ of non-Euclidean geometry before the semantic shift of the notion of ‘model’—just like the ‘physical analogies’ of the time—mainly comprised a heuristic of new research objects with unclear status. And as with analogies, it is also very clear that the representations called an ‘interpretation’ added concrete imaginations of more familiar mathematical objects to a still problematic system of geometric principles. This step could well be interpreted as a means of safeguarding the plausibility or ‘existence’ of non-Euclidean geometry (or geometries); this function would, however, shift historically. The images and/or interpretations of non-Euclidean geometry led to a relativisation of the real to the extent that in the course of a long debate involving many authors it gradually became clear that the correspondence of the epistemically symmetrical, distinct images of geometries with the faits accomplis of experience was not fully determining such images, and thus that even mathematically inequivalent ‘images’ of geometric relations—or different systems of basic geometric propositions, in Beltrami’s terms—could be made to correspond to the same empirical facts (see below).³²

As we have seen, in the construction of the interpretations and images of non-Euclidean geometry, too, the main aim of the researchers involved was the production of epistemic symmetries. What was valid and existed in an interpretation or an image also existed and was valid in the domain of thought that was interpreted or pictured (abgebildet). The geometers of the last third of the nineteenth century (besides Klein one should at least mention Wilhelm Killing and Henri Poincaré) outdid each other in creating more and more ‘images’ of non-Euclidean geometries and in studying the ‘mappings’ (Abbildungen) between them. This gave rise to an increasingly broad range of different possible representations of the ‘same’ geometric relations by means of different images. Under the title “Interprétation des géométries non euclidiennes” Poincaré for instance explicitly invoked the obvious idea of the translation between such images with a matching lexicon, again underlining epistemic symmetry.³³

As a consequence of these developments, the plurality of geometric systems possessed (at least) two dimensions at the end of the nineteenth century. Not only were there now different, mathematically inequivalent geometries, but also each of these geometries in turn admitted several interpretations or images between which there was epistemic symmetry. In this way, in particular the (symmetrical and asymmetrical) relations between various possible representations of different possible geometric systems themselves came to light. Bachelard would later say that it was not one representation of spatial relations that constituted the subject of geometry but the entire collection of interrelated, not completely equivalent representations of such relations; geometry was not the science of one unequivocally determined mathematical form of spatiality but the science of the multiplicity of possible mathematical forms of spatiality and their complex interrelationships.³⁴

Systems, Spielräume, Euklidische Modelle: Some Remarks by Felix Hausdorff, Ca. 1900

In order to take a closer look at how far the discussion of the representations of epistemic objects in geometry at the turn of the century could go beyond the limits of traditional intuition and imagination, and in order to discuss one of the earliest uses of the term ‘model’ in a non-material (albeit still limited) sense in mathematics, I shall now look briefly at how the mathematician and epistemologist Felix Hausdorff dealt with the subject of non-Euclidean geometry. At this point in time Hausdorff, who was born in 1868, was a außerplanmäßiger Extraordinarius, i.e. a professor by title but without a regular salary, at the University of Leipzig. It was only later when he was made a planmäßiger Extraordinarius (a salaried professor) at the University of Bonn in 1910 and through his groundbreaking monograph Grundzüge der Mengenlehre, published in 1914, that he eventually received professional recognition as a mathematician. During his Leipzig years, alongside his mathematical career, he also tried his hand as a Nietzschean writer—under the pseudonym Paul Mongré—and radical advocate of scientific and, in particular, mathematical modernism.³⁵ In his epistemological monograph Das Chaos in kosmischer Auslese, published in 1898, Mongré introduced the still novel language of Georg Cantor’s theory of infinite sets into the world of literary metaphors, among other things, by using the idea of an arbitrary map between sets as a tool to describe variations of temporal and spatial relations. In the monograph, his main emphasis was a radical and comprehensive critique of any and all kinds of metaphysical ideas about time, and an illustration of the consequences of this critique in other fields from the natural sciences to morals.³⁶

In 1903 Hausdorff gave his inaugural lecture as associate professor in Leipzig on the problem of space, published in the same year and followed in the winter semester of 1903–1904 by a lecture course on time and space, carefully drafted in a manuscript which remained unpublished until very recently.³⁷ In these papers Hausdorff first identified himself publicly as Mongré, and the mathematician now engaged in an open exchange with the epistemologist. Around the same time he also wrote an undated paper, “Nichteuklidische Geometrie,” aimed at a non-specialist audience; it was intended for publication in a natural science journal but never made it to press.³⁸ In this paper Hausdorff—not least in reference to the publication a few years earlier of David Hilbert’s Grundlagen der Geometrie—drew on the notion of ‘system’ already used by Beltrami in order to describe the still new, deviant geometries:

By a single non-Euclidean geometry we understand any system of geometric propositions that deviates in any more or less important respect from a particular system of Euclidean geometry. Non-Euclidean geometry as a mathematical discipline sets itself the task of testing and comparing all these single systems.³⁹

It was precisely this shift that prompted Bachelard to speak in relation to this scientific field of a new scientific spirit. Hausdorff now also pushed ahead with the epistemological as well as ontological problem associated with this shift:

But that particular single system, the underlying normal system as it were, compared with which all the rest are already in the naming mere negations, abnormalities and varieties, is logically speaking no more just, natural or necessary than the others. In this respect mathematics cannot object strongly enough to the prejudice popular among many philosophers that wishes to deduce Euclidean space with all its peculiarities as the a priori ‘product of logical positing.’⁴⁰

If traditional [Euclidean] geometry ceased to appear natural or necessary, then the problem of the empirical validity of geometric systems became just as pressing as that of understanding the unavoidable Spielräume—the spaces of play or ranges of freedom—in the construction of such systems. In his inaugural lecture of 1903, Hausdorff distinguished the kinds of such Spielräume— there was a Spielraum of experience, of intuition, and of thought.⁴¹ Given these ranges of play in the construction of geometrical systems, almost every specific, concrete determination of geometric space lost its binding character, as Hausdorff continued in his popular article on non-Euclidean geometry:

As strangely reactionary as it may sound, even propositions such as those that speak of the infinity and limitlessness of the world in space and time are strictly speaking very premature conclusions from the centre of the universe to its fringes and have about the same value as the opinions of a jellyfish in the Atlantic Ocean about the shape of the American coast.⁴²

Like other representatives of mathematical modernism Hausdorff experienced the rejection of such strictures as a liberation. This was primarily due to the liberation of language, which had been practised not least by the extensive use (and translation) of the various analogies, interpretations and images of new mathematical objects that had accumulated over the preceding decades:

Modern mathematics owes its essential and enlightening insights to the radicalism with which it has proceeded [in the uses mathematical language], but which for the uninitiated observer has something arbitrary and harmful to his or her dearest habits. Here one calculates with ‘numbers’, where 2a can be equal to a [...], the shortest lines on a [curved] surface are designated ‘straight’ according to need, a straight line of ordinary space is interpreted as a ‘point’ of a four-dimensional space, and so on. Why does mathematics do that? [...] [B]ecause an appropriate name can reveal the most important connections between seemingly distant domains, while an inappropriate one can obscure them. The transfer of a language use to unusual, deviant cases serves in particular to make visible one by one the numerous presuppositions that float unresolved in the ordinary sphere of concepts. It is almost the ‘logical experiment’ […], the chemical analysis of concepts.⁴³

At the same time the free use of language in mathematics was able to call forth intuitive ideas insofar as mathematics—as in the interpretations and images of non-Euclidean geometries—“seeks to intuitionally imagine its pure thought-figures and herein proceeds with the free use of the elements of reality.”⁴⁴ Of course Hausdorff did not intend to suggest a solid fundament for mathematical language here through the back door of intuition. Unlike Klein, for example, Hausdorff emphasised that “the word ‘intuitive’ means too many things, and indeed to each person something different.”⁴⁵ Nevertheless the use of this intuition, imagination or fantasy, while always an individual use, was a legitimate means of gaining access to the thought-objects of mathematics. On Beltrami-Cayley’s image (das Beltrami-Cayley’sche Bild) of pseudospherical geometry Hausdorff commented: “For these relations [of the formal system of pseudospherical geometry, ME] we can obtain an intuitive analogy which at present will only prove a helpful symbol, but later will prove to be the true equivalent of the matter.”⁴⁶ The image ‘versinnlichte’ (made available to the senses)⁴⁷ while the object itself was the formal system of pseudospherical (or BL) geometry. Moreover, Beltrami’s image provided an additional epistemic function for non-Euclidean geometry: Assuming that traditional, Euclidean geometry was free of internal contradictions, Beltrami’s image allowed to conclude that pseudospherical (or BL) geometry was equally free of internal contradictions, since any such contradiction would have to manifest itself as a contradiction in Beltrami’s image:

From the consistency of Euclidean geometry one will be able to conclude the consistency of a non-Euclidean [geometry] if, to speak somewhat vaguely and generally, one is able to produce a Euclidean image of this geometry, i.e. if one can set up a correspondence between non-Euclidean elements (points, lines, surfaces, etc.) and Euclidean elements in such a way that the non-Euclidean relations between the former elements are represented by the Euclidean relations between the latter.⁴⁸

Despite this emphatic plea for the freedom of mathematical expression, imagination and thought, and despite the epistemological twist just mentioned, the term ‘model’ was not used by Hausdorff in this manuscript, written probably between 1900 and 1903. The term appeared just once in it, and not as part of Hausdorff’s own text, but in a longer passage quoted from an earlier article by Helmholtz in which the latter had pointed out that even in traditional, Euclidean geometry there were topics that challenged the abilities of spatial intuition, such as complicated knotted threads in space, many-surfaced ‘crystal models,’ or complex architectural drawings.⁴⁹ Such ‘models’ were still the traditional, tangible models of the nineteenth century, understood as material aids to overcome the weaknesses of spatial intuition (which, as Hausdorff insisted, could never serve as a foundation for mathematical truth anyway).

In his inaugural lecture “Das Raumproblem” of 1903, and in the lecture course on “Zeit und Raum” (given in the winter term 1903/1904), Hausdorff revised his terminology in a slight but telling fashion. Now, in a very specific sense, he spoke of “Euclidean models” of certain geometrical systems where he had used the term “images” before. Again, the immediate context was the proof of consistency of geometrical systems:

The absence of an internal contradiction has been proved directly by suitable mappings of non-Euclidean geometries to Euclidean models, and of Euclidean geometry to pure arithmetics.⁵⁰

Similarly, and explicitly, Hausdorff introduced the term ‘Euclidean model’ (euklidisches Modell) in some passages of his lecture course on time and space, given in 1903/1904. Once again, the immediate context was a proof of the consistency of non-Euclidean geometries. We can see, therefore, that the epistemic role of what now was an immaterial ‘model’ was still to provide a familiar, concrete imagination to an unfamiliar system of geometrical axioms, and in this sense epistemologically similar to the material models of earlier decades, and to Beltrami’s ‘interpretation.’ As far as I can see Hausdorff never used the term ‘model’ in these years except in this limited sense of ‘Euclidean models’ or ‘images’ for non-Euclidean geometries.⁵¹

Underlying Hausdorff’s remarks on non-Euclidean geometries was not merely a plea for a liberated construction of mathematical thought-objects. The purpose of the “chemical analysis of concepts” based on a free use of language and on the exploration of the (conceptual, intuitional and experiential) Spielräume for providing mathematical descriptions of reality was ultimately also a “self-criticism of science.”⁵² In order to do science—understood as an attempt of the human intellect to find rational order in experienced reality—, Hausdorff insisted, one had to be aware of the irreducible indeterminacy of any given attempt to describe reality in mathematical terms, be it in geometry or in any other field of mathematized science. Mongré’s Das Chaos in kosmischer Auslese thus ended with the following passage—which is obviously in need of further explanation⁵³:

In the passage to the transcendent the whole wonderful and richly articulated structure of our cosmos collapsed into chaotic indeterminacy. Thus on returning to the empirical even the attempt to establish the simplest forms of consciousness as the necessary incarnations of appearance fails. As a result the bridges that in the fantasy of every metaphysician pass between chaos and cosmos are destroyed and the end of metaphysics is declared—of the acknowledged one no less than the hidden one. And the natural science of the next century will not be spared the task of extracting the latter from its current structure.⁵⁴

Images and Dynamical Models: Heinrich Hertz Once Again

In the last step of my considerations I shall briefly look again at Hertz’s use of the terms ‘image’ and ‘model’.⁵⁵ Remember that for Hertz always several ‘images’ of the same segment of the real were possible, with the significant restriction that each of these images had to stand in an ‘inference equivalence’ (Folgenäquivalenz) to the causality of the real. This epistemological structure was closely related to that of the images of non-Euclidean geometry. In the domain of geometry, the equivalence between on the one hand the inference relationships (Folgerungsbeziehungen) within a geometric system and on the other those in its images (or interpretations) was a logico-mathematical one. Hertz, however, was concerned with the causal relationships between physical causes and effects. In both cases the images assisted in the comprehension of something that was unfamiliar and difficult for the human mind to access.

In this context Hertz also developed a very idiosyncratic, technical sense of the notion of ‘model’ that broke away from the still common understanding of a model as a concrete, material representation of a scientific object (as we saw at the beginning Hertz was not unfamiliar with this sense). Once more the notion of ‘system’ or of a variety of systems was essential. In this new, technical sense Hertz stated:

Definition. A material system is said to be a dynamical model of a second system when the connections of the first can be expressed by such coordinates as to satisfy the following conditions:

• 1. That the number of coordinates of the first system is equal to the number of the second.

• 2. That with a suitable correspondence of the coordinates in both systems the same defining equations for the systems hold.

• 3. That by this correspondence of the coordinates the expression for the magnitude of a displacement agrees in both systems.⁵⁶

And he concluded:

Corollary 1. If one system is a model of a second, then, conversely, the second is also a model of the first. If two systems are models of a third system, then each of these systems is also a model of the other. The model of a model of a system is also a model of the original system.⁵⁷

Just as in the earlier contexts of physical analogies the point here was also a—definitionally enforced, complete—reflexivity, symmetry and transitivity of the model relationship. Being models of each other was a relation of equivalence between “material systems.” Even Hertz’s technically defined “models” were, therefore not models in the modern, reductive and relativizing sense. And, as in the case of the proliferation of images of non-Euclidean geometries, Hertz was also aware that there could always be, or indeed were, many models of the same system:

Corollary 3. A system is not completely determined by the fact that it is a model of a given system. An infinite number of systems, quite different physically, can be models of one and the same system. Any given system is a model of an infinite number of totally different systems.⁵⁸

Limited by this requirement of equivalence there was, however, a certain measure of perspectivity in Hertz’ models. This was desirable and useful principally from an epistemological point of view, since the central ‘proposition’ was that the dynamical conditions of systems that were models of each other were in mutual correspondence⁵⁹ so that insights into the dynamics of a system could be obtained by examining an (equivalent) model system. In the middle of these definitional considerations Hertz once again brought the concept of mental images of material processes used in the introduction to his book into play in an observation that can hardly be described as anything other than cryptic:

Observation 2. The relation of a dynamical model to the system, of which it is regarded as the model, is precisely the same as the relation of the images which our mind forms of the things to these things themselves. For if we regard the state of the model as the representation of the state of the system, then the consequents of this representation, which according to the laws of this representation must appear, are also the representation of the consequents which must proceed from the original object according to the laws of this original object. The agreement between mind and nature may therefore be likened to the agreement between two systems which are models of one another, and we can even account for this agreement by assuming that the mind is capable of making actual dynamical models of things, and of working with them.⁶⁰

If we take Hertz’s consideration seriously, then it is an ambitious, even if empirically entirely unsupported hypothesis of cognitive science—a sort of mechanics of the mind.

Despite its avowed perspectivity, Hertz’s discussion of images and models therefore remains bound to a rigorous idea that is ultimately deeply rooted in the presented epistemological traditions of the nineteenth century—the idea of a correspondence between causally structured reality, the activity of the mind and scientific theory formation. Hertz believed in this correspondence on the basis of his consistent emphasis of the epistemic symmetry of the relationships in ‘images’ or ‘models’ (just as Maxwell had believed in the epistemic symmetry of his ‘analogies’). The epistemological rupture that geometry had already undergone or was in the process of undergoing, as we have seen particularly in the case of Hausdorff, was not embraced by Hertz.

Epilogue: The Rise of (Modern) Mathematical Models

The rise (and inflation) of the modern concept of the mathematical model as an actor’s category is a mid-twentieth century affair that I cannot go into here. It was only conceivable after the break noted by Bachelard that brought about the nouvel esprit scientifique of the twentieth century. To tell the history of this rise in detail it would be necessary not only to examine further how the concept of models in geometry began to acquire its modern, abstract meaning, the beginnings of which we have traced in Hausdorff and, among other places, in Rosemann’s pseudo-Klein from 1928, but also in particular to trace the epistemological shifts in mathematical economics of the same period. There too one finds in numerous early-twentieth-century texts mechanical analogies in the style of the nineteenth century⁶¹ before the modern sense of the term ‘model’ came into general use. Several historians of mathematical economics refer to Jan Tinbergen’s 1935 paper “Quantitative Fragen der Konjunkturpolitik” (Quantitative Issues of Economic Policy)⁶² as the place where the concept of the mathematical model was first introduced into economic theory, a concept that then spread rapidly in this field.⁶³

For as long as the epistemological history of the shifts in the understanding and use of abstract representations in the natural sciences remains unwritten—one that ultimately led to the repression of the earlier and the rise of the modern concept of models—the historian can only cautiously recollect the epistemic functions of analogies, interpretations, images and, not least, tangible models of the natural sciences and mathematics of the nineteenth century.

What for us today has become an irreversible insight into the perspectivity and relativity of scientific representations as well as into the unavoidable and epistemically not indifferent reduction of the complexity of the real in any theoretical model—and even in any biological model system—did not obtrude on the thought of the physical sciences of the century before last. At that time the focus was on other epistemic functions of analogies, images and material models: namely their collective or individual heuristic value, the relief or replacement that they could offer in epistemic situations where causal hypotheses were not (yet) available, the epistemic symmetry or equivalence between different concrete representations of the same supposedly more abstract structures, the development of uniform mathematical structure of different segments of nature.

However, something else came into play besides such functions in the interpretations, systems and images of non-Euclidean geometry: namely the task of dealing with inequivalent mathematical representations of something real (physical space) that, for that very reason, had to be interpreted, versinnlicht and evaluated anew as free imaginations. Hence wherever the question of the relationship between such imaginations and a scientifically explored segment of reality was posed, a persistent epistemological problematic was raised that ultimately forced a “self-criticism of science”⁶⁴—one that only the “natural science of the next century” would carry through.⁶⁵

Translated by Benjamin Carter and Nathaniel Boyd.