Boehm Iconic Knowledge

September 6, 2017 | Author: Anonymous Ynn1VrHWs1 | Category: Analogy, Conceptual Model, Concept, Representation (Arts), Geometry
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Gottfried Boehm

Iconic Knowledge The Image as a Model

Knowing or Doing? Working with models has always been a widespread practice in artists’ ateliers and in the workshops of technicians and scientists. Related finds date back to ancient Egypt and one can assume the use of model sketches starting in the early stages of arithmetic and geometry in Babylon. The role of the model has intensified markedly since the Early Modern Period and has since experienced a rich blossoming. In the context of the evaluation of the model from the perspective of the discourse, it has been heavily imposed upon since the 18th century by an antagonism that, without a semblance of connection, pits cognitive knowledge against aesthetic experience. Kant attempted to confirm that this was the case as well as to overcome the situation in his Critiques. All the same, he was not able to prevent models (especially of artists and scientists) from being relegated to the vestibules and ghettos of applied arts or the history of science by an appraisal that oriented itself on taste and style. This was only to fundamentally change in the 20th century. The model shifted from the periphery into the center of art; test experiments appeared as the artworks themselves. The dynamic of scientific knowledge and its rapid paradigm shifts increased the need for constructing models. Some models became popular icons of knowledge, for example the atom model or the DNA spiral. It became clear that in doing so, they tied in to an older tradition of models that had brought forth revered representations of the night sky with the astrolabe or the armillary sphere. Classrooms and living rooms alike are still furnished with globes. All of these are examples in which representing with models

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Iconic Knowledge. The Image as a Model


is connected with instrumental practical application. This is also a quality of geographical maps or other forms of mappings.

This complex history makes the question of the model seem boundless and urgent at the same time. Two main pathways would seem useful for discussion here: the historical case study, or a reflexive procedure that keeps an eye out for the conditions of iconic models. We have decided to go with the first option, well aware that it always leads back to the first.

To begin, we will relate the great variety of model types, model concepts and practices back to a single differentiation criterion, namely the manner in which models relate to their objects, and how their individual reference is constituted. A spectrum emerges, on the one end of which the full-scale simulation of an (in principle) accessible original can be found. The evocativeness of such a simulation, for example that of the model, consists in the minimizing of difference. It looks like a real train and moves like one too. As many of the visible details as possible are recreated faithfully and to scale. If the users were to “liliputanize” to the same scale, they would return to an original world and the “real” train would appear to them as a monumentally enlarged model. Here we have an inkling of the dialectic that can generally be observed in the translation of originals into secondary representations: the model can itself function again as the original.

In any case, the simultaneous model enjoys widespread usefulness in design, industry, architecture, and art and should therefore not be confused with simple bricolage. What characterizes it is always the availability of an original, a precise concept of the real that would still be accessible without a model. Nonetheless we are dealing with real models that display some of these characteristics: -

Adherence to scale, and the shifting of scale


Selection of characteristics, and thereby:


the directing of attention, as well as:


room for interpretation.

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Iconic Knowledge. The Image as a Model


Simulative models capture our interest first and foremost through the recognition that they make possible.

At the other end of the spectrum, we encounter models with a decidedly open relationship of reference; one could thus also call them heuristic. These stem from a practice that deals with realities that can be inaccessible, invisible, or visible but unknown. There is no definite concept that captures them, not even the concept’s substitute: palpability. This applies especially to models of the world and the heavens. The night sky is certainly one of the strongest visual sensations, a cinema open since the dawn of time, and whose iconic substrates have likely been explored since the cultures of the Megalithic – measuring, representing, and ritualizing. Pictorial figurations were ascribed to the twinkling chaos of the heavens, the paths of the planets were identified, the cyclical movements of the constellations were represented in the zodiac, and the phases of the moon and the equinoxes were calculated. Since the earliest times, model images have been helpful in the search for organizing patterns. Their characteristic reduction of complexity allowed for inscrutable relationships to be opened up in a lucid manner.

The deictic power of models always relies on the correct simplification. This is, after all, what activates the image-specific overview, and a channeling of the gaze allows for visual evidence, despite how provisional, historically speaking, models may have proven to be. The history of cosmos models remains to this day an unfinished process of revisions, accelerated once again by the Hubble photographs. The gaze into the heavens and the model practice that conveys it – both always reflect the limits of knowledge and of reason. But hardly a more potent, more epistemic source of impulses has emerged than our bottomless fascination with the starry sky, the residuum of imagination. And even the event of the so-called Copernican turn, whose model change also deeply influences our own time, can be traced back to a dry visual constellation, to a few spheres on prospective orbits. What a

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Iconic Knowledge. The Image as a Model


concentration of insight! It is the reconstruction of a model framework with the subsequent costs of an unforeseen revolution in “mentality”.

In the following, we will be dealing with heuristic models. They also allow for a plausible transition from the sphere of artifacts to that of knowledge. The close interaction between knowing and doing, between connaître and construire characterizes a guideline of our argument. The following quote from Immanuel Kant might serve as the motto for what is to come: “We cannot conceive of a line without drawing it in our thoughts; nor a circle without first delineating it.” (Kant KrV B154) We add to this: the line, guided by the eye and hand, also wants to be drawn in actuality.

Schema – Image The model constructors’ career success was made possible by their inherent ability for cognitive revelation (deixis, demonstratio). They show something that can only be seen in this way; they have specific evidences at their disposal. Which ones?

This does not contradict the fact that, as is the case for all scientific construction of hypotheses, a selection of external data is constantly flowing into the practice of models. As prosaic as models might appear, they have a surplus of the imaginary at their command, they maintain a difference vis-à-vis the real. It is precisely this difference that opens up the space for the free play of experiment that so often accompanies models.

Let’s return to the armillary sphere: the framework of wires, the swinging of bodies into one another, is not the sky, and yet it is. A deictic power inherent in the model attracts us; however, we gladly allow that the model is part of the quotidian of the scientist in disciplinary discourses, a means of scientific explication among others. We are always dealing with an internal and an external functional relationship in models. This doesn’t just prompt the question “What?” but also the others, carrying a

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distinctly historical index, namely “When?” “For what purpose?” and “For whom?”

Interest in the iconic conditions of knowledge brings with it, vis-à-vis the self-image of the scientist, a clear shifting of attention – one on which we must insist, since it is what allows us to occupy ourselves with the demonstratio of the model itself. How does this shift come about?

Concealed inside this question, like a matryoshka doll, is a further elementary layout that deserves discussing before we return to the model type in a more specific sense in the next segment.

The intended questions read: How can knowledge be generated by images in the first place? Does something like iconic knowledge exist? How does it expand the horizon of experience? How does it come about?

Since scientific knowledge must insist on a verifiable solidity, our problem can be further honed. Now, the question is: How do image and concept meet each other in the image itself?

What kind of question is this? How can we proceed with it? Who could help us here? After all, categories like “visual thought” and the older “parler peinture” are not questioned, and a discipline like art history presupposes that in the space of its phenomena, concepts with images come to the rescue. To really understand this discursivity of the pictoral and thus to explain it or make it known through scientific means is one of the biggest challenges before us.

We will employ in this matter an elementary tool of the trade, but one that seems to be appropriate for images, namely the simple means of precise observation and of description, which will then prove capable of being implemented in further arguments and in particular theoretical

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didactic plays. In any case, our premise is simple and hopefully stable: when concepts combine with images, this interlacing should also be detectable in the images themselves in one way or another.

The elementary problem before us is that concepts are by their nature abstract, i.e. completely invisible. If there is a way, then it must overcome the difficult passage, the notable transition between the two separate worlds – back and forth and back –, that is, it must interweave invisibility with visibility.

But first, we are dealing with two separate evidences.

On the one hand, with the reality of the concept: We can come up with any number of concepts, and we use them constantly, for example right this very minute.

On the other hand, there are concretions in images that involve the conceptual at least to the extent that they can be named. We ceaselessly identify sky, clouds, mountain, tree, house, table, and chair in images and call them by names.

What we are missing is a joint that flexibly interweaves the abstractinvisible with the concrete-sensory, a missing link. How can it be tracked down? Is the thing that combines the invisible with the visible itself visible? In what way?

In order to shed light on this matter, we will take a look at a drawing that Paul Klee published for didactic purposes in his “pedagogical sketchbook” from 1925. We will detach it from its context and transfer it to that of our argument. We see a strange concretion of the concept house, or more precisely, the facade of a house. Contrary to our expectation, this house lies flat on the ground, but yet does not show its floor plan. Put plainly: it is lying in the horizontal and attempts with all its might to raise itself from it. This effort is supported by a perspectival

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shortening, but one that disregards its own laws at a critical juncture. Klee himself highlights this section in his monosyllabic commentary: “the animal” – and by that he most likely means the observer – “(wants), for the benefit of his balance, to see all of the verticals of reality also projected as verticals...” He says it, but he doesn’t follow through. Technically speaking, he undermines perspectival convention by not combining the floor plan and vertical section together at a right angle.

Klee’s offensive exposes first and foremost simply the fact that representations follow rules that guide them. Apparently, there are quite different rules than just the traditional ones of perspective. A glance at Klee’s work in the twenties, which shows him occupied with, among other things, using the house to question a classic paradigm of perspective, also teaches one to discover other rules of possible houses. “The L-Platz under Construction” (1923), the “Houses-Projection” (1923), or “House-Inside” (1919), all of them water-colors, confirm this activity. Klee’s very “different” houses contort and stretch, play out their ground lines to all sides, overlay inner and outer, allow the focal point to wander, operate with paradoxes, and so forth.

Thus, Klee shows that a plethora of rules can be devised with the means of the image: rules that represent the same thing, that allow themselves to shape “ways of worldmaking” (Goodman), and that lead to quite unseen houses. These rules, like maps, complete a projection onto a plane – it is not just by coincidence that we find the word PLAN written into the picture field of the L-Platz –; but frontal projections also appear abruptly, one encounters the overlay of color onto a squared ground with a linear frame, and so forth.

The concept of the rule, which encompasses the most varied modi of representation, has involuntarily crept into our description. Could it perhaps be that it is trying to alert us to that ominous joint that conveys the invisible concept (“house”) with its manifold manifestations?

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In order to pursue this suspicion, we will now take up the drawing pencil ourselves, keeping in mind our motto and the expansion that KIee granted it. He doesn’t want to think the line drawn, but rather to let it promenade, to activate it as an “agent”, as a “point that shifts itself” (6).

We will now draw a house, adhering to a series of the simplest steps with the intent of better understanding the modus operandi of the rule. We will even abstain from using a ground line, i.e. the thought of any projection, and head off, quite mundanely, to drawing-kindergarten. We will take what appears: a straight, vertical line on a bright surface as literally as possible.

First, the line itself appears, along with the distances to the edges. The numerical comes into play, which would allow itself to be checked, but which as it stands remains under the threshold of attention.

Then, a second vertical line, parallel at a distance. What we have just observed becomes more complicated. What remains manifest is the sparse structure – lines of lead on paper in an undecided order. One can already recognize here that two lines were placed onto a field, and begin to constitute themselves vis-à-vis an Inner and an Outer.

The third line branches off of the first one diagonally to the upper right. Since we are on our way to a concept of the house, it now begins to ring a bell. It is at this point that we can finally say something more: This is not simply a third line, but at the same time also half of a roof. The factualness of the lines changes into something meaning-bearing. We are close to our goal.

And voilà: the fourth line, which turns off to the left from the second line, meets up with the first slant: and the house is complete! An elementary viewness built itself up step by step and became optically discernible in the fourth move. One can imagine that children explore the possibilities of drawing in a similar way once they have left the phase of spontaneous

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Iconic Knowledge. The Image as a Model


actions behind them. We are dealing here with forms of what one could call the arché, the beginning, the ur-scene of the iconic: the emergence of meaning from out of material substrates.

The other drawings demonstrate how arriving at the sight of a house through drawing is by no means a matter of course. Just one wrong line, just one that is out of control, and one exits the perimeter of possible houses.

Our kindergarten visit has allowed us an insight into how the rule functions. It is, one can now say, itself invisible, but guides the formation of the visible, of a view in a fundamental manner. It is also for this reason that Kant, falling back on rhetorical conceptuality, called it a schema and assigned it to the image. In doing so, he also employs the rhetorical concept of the hypotypose, which literally means: draft, outline, essential trait, also: what lies beneath. In short: the schema rule is not itself a picture, but it serves, “steering”, as its basis. The verb “to steer” is not unimportant here, since it implies that we are dealing with a process. Indeed: without this transfer, without the transition from the factual to the effect, without this act, “viewness”, that is, the image, would never appear. In its structure it is not reality, but a condensate of the real. The joint of the schema that opens the way from the factual to representation is itself of a temporal nature. Images must “build themselves up” if they are to appear. They imply an act of showing, through which they – still and mute as they are – visibly come before us as an event, as their coming-to-be. But the schema doesn’t just sort out what belongs to the perimeter of possible houses – a boundary that may not be overstepped – but also the How, the form of possible appearance that only unveils itself when a real hand draws a real pencil over a real page.

Let’s keep in mind that the schema rule articulates iconic difference and determines its moments reciprocally. Itself invisible, it makes decisions that become visible. It allows an infinite amount to disappear into the abyss of the simply possible, thereby precisely making visible. When this

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house appears, millions of others have disappeared into the nirvana of potential.

The simplicity of our examples should not lead astray. We find ourselves in an area that structures the entire appearance of representation, as complex as it may be. A schema already exists when we bring a sign into a material field and activate a boundary against the Outer, thereby making it into representation. A schema exists when – as in this case – the lines appear in front of, that is, on the ground. The in- and on-, the upper- and lower-schema and so forth, fall into place as a motoric schema of an overview, that – as our drawing showed – activates the visual forces of the field. Let us now move the lower left line to the upper right – it then appears optically lighter, it appears elevated, making something like gravity indirectly noticeable, and brings our bodily experience into play. Even the simplest pictorial relationships thus implicate the eye, the force of seeing. The approaching observer is a methodical fictional character who has always come too late.

At the same time, it becomes clear that the iconic has been governed by the number from the start; that a mapping of the gaze arises, especially when it’s a matter of simultaneously honoring the field. Incidentally, cognitive science has begun using strong arguments to derive the inherence of mathematics as an episteme from the visual field. “Viewness” or “aspectation” thus proves itself to be a key to understanding number, as does language (because in linguistics, aspect also refers to temporal changes of the same verb: it transports it into very different “viewnesses” of reality, for example: to go, I go, I had gone, I will go, etc...). We have now arrived at the point at which images emerge. Their origin is puzzling, since it consists of the process of an immemorial transfer through which the perceptual can be experienced as something conceptual, the material as a spatial or logical determination. The schematism is thereby only circumscribed in passing, and we can only point out here that the ability to pre-conceive the perimeter of possible images corresponds with an inner pictorial ability

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that functions as the governor of the imaginary. Drawing even the simplest house requires recourse to imagination. It’s no coincidence that Kant granted the imagination a decisive role in his philosophical project of integrating the concepts of the intellect with outlook. A theory of images, however it may look in its details, will confront this theme.

To conclude this line of thought, let’s take a look once again the “L-Platz” with its houses. How do things really stand with the relationship between “the” concept and “a” house; in other words, one of the houses in the variety of the possible?

What do we see?


We see how this house looks


We see how houses can look in general


We see how Klee represents houses


We see the difference between houses in art and those on the real LPlatz


We see the logic of the moved focal point and patterns of perspectival deviations


We see the difference of line and color


We see, We see... much, much more and nothing that is arbitrary.

In other words, this image opens up a room for play of the possible that is just as distinct as it is broad, and within which certain decisions have been made. So in the image, what is revealed is not “the” concept of a house in the sense of a repraesentatio singularis, but rather: something like a house. Something like – this is a linguistic turn of phrase with quite a bit of freedom as it were, it states what opens the perimeter of the schema rule. The invisible unity of the concept reveals itself in different sensory views. The room for play must not be damaged in the process. There is also a tower on the L-Platz. A tower is not a house. Here, one schema is transgressed in order to make room for another one.

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The Specifics of the Model-Image Can our house drawings be called models? Only in a very unspecific sense. Because what houses are “in general”, what they are “in principle”, is so well known that no models are required for them, even though they would undoubtedly do justice to this role in a pinch. However, an architectural model in a strict sense, displaying a very specific construction, looks completely different. It is structured on the basis of specific data, and reaches a high level of precision – opening at the same time a view onto something that exists, does not yet exist, isn’t conceivable or doesn’t exist anymore. It creates a paradigm or an exemplum – which we will talk about later.

The generation of iconic knowledge, as we said, is a necessary, but not yet sufficient component of model construction. Further specifics enter into the picture, a few of which we will now discuss. In doing so, we will move along two paths: -

On the first one, we are dealing with a quantitative exactitude that avails itself of mathematical means in order to narrow the margins of the indeterminate, thereby creating the clearest evidences possible.


On the second path, what is of interest are the inherent determinations visible in the model itself and their referencibility, which can be reflected in a simultaneous overview. A sort of standing observation, a thoughtful lingering sets in that we are certainly familiar with for example in old models of the heaven. It’s no coincidence that they also play over to modalities of aesthetic perception again and again.

But this inner rule-boundness and its play of reflection isn’t a means in itself, but rather it establishes a relationship of analogy to the represented circumstances.

Both paths, the quantitative as well as the reflexive, cross one another, and differentiating the two does not necessarily imply completely

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separating them from one another. On both paths it is a matter of coherence and evidence in the face of a thing.

We have already encountered a subliminal presence of number relationships in the simplest positings of an image. Models raise them – as a rule – up to a thematic level, without this necessarily leading to a unified standard of representation, as is the case for example in modern maps with their accompanying legends. There are still very different kinds of iconic adherence to scale, and we have determined that they all are also closely related to one another in terms of their conditions with the intention of an overview, of an evident view. Some models emphatically carry a simultanizing framing characteristic, for example a strong, partially repeated circle or the sphere. Others achieve a directing of size, whereby the circumstances are very rigidly structured in the anticipation of the whole. It dominates over the parts, the details.

There are any number of examples of this, such as the medieval T-Omaps, which divest the structure of the three continents known then, Europe, Africa und Asia, of their individual geographic characteristics, in order to insert it into the form of a cross, which functions as the scale of the terrestrial world, within the circle.

Other premodern models of the world are based on a mare nostrum, i.e. on a relatively common living world, around whose central basin the neighboring and the distant lands, peninsulas, islands, and subcontinents accumulate – far removed from a continuous or metric adherence to scale. It was only the practice of complete surveying, initiated by military strategists of the 19th and 20th centuries, which make exact maps possible – despite the projection, introduced by Mercator in the 16th century, of the spherical surface of the earth onto a plane. These maps, to which one can ascribe model quality, distinguish themselves precisely due to very selective choosing from the total data. And vice versa: any observer of detailed travel maps knows that he or she could easily miss the forest for the trees. Those attempting to travel,

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for example, with the aid of the Ebstorfer map of the world, would most certainly not have arrived at their destination. The marching maps of the Romans or those of coastal shipping or of the Periplous offer a flow of significant characteristics, but practically no model-like valence.

The premise of the number is no less applicable to models of the heavens. The zodiac divides the whole of the circle into exactly twelve segments that allow for the exact angular calculation of aspectations, as well as for the creation of an order of relationships among the “houses”, the foundations of astrological knowledge, which operates on the premise of calculating the constellations of the stars for a particular individual. In this case, the adherence to scale combines exact geometry with the idea of the human being as a central focus of the cosmos model. If the suspicion should prove true that the zodiac developed from out of the model of the tree of life – with six fields each on the left and right –, then it would be an indication of the need to endow the model with more mathematical precision and calculability. The shift from an organic founding metaphor to a construed one emphasizes the role of the mental component of model determination.

One final example involves models of the cosmos under the augury of the harmony of the spheres. The Pythagorean – one could also say the historical – discovery that the fleetingness and affectivity of the tones could be traced back to a rational order of number relationships also left behind deep traces in certain model designs. We see the orbits of the planets, whose relationships constitute the harmonia mundi, represented in a precisely calculated proportional order.

The role of the number in the iconic model, along with its increasing precision, become clear when we observe their allocation to empirical reality. We again follow an argument that was already contained in our line motto. But it’s not enough that the line has to be drawn; if one wishes to imagine it, the act of drawing itself can be subjected to a continuous rationality. This thought also plays a fundamental role in what

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is probably the oldest mathematically-oriented image-theory model, the point-line-plane postulate. From late antiquity through Leon Battista Alberti and up until Kandinsky, it returns in different variations. It evolves the bold idea that the abundance of phenomena in the visible world - or the world worthy of representation - can be pictorially verified via a rational calculation. The rationale is that the invisible ideality of the geometrically defined point is explicated to the empiricism of a concrete point and from there, following the logic of geometry, further to line, surface, and complex bodies.

Here, the one recognizes the notion of descriptio, which gains considerable importance with increases in the precision of models. Descriptio doesn’t simply mean description in the sense of ekphrasis, which creates the image of a reality with rhetorical or at least linguistic means. Instead, descriptio is much more the construction of a primarily geometricizable circumstance, from a mathematical arché. This meaning of the word “description” has been pushed back into the realm of technical terminology in German, where one hears people speak, for example, about the description of a geometrical curve.

It is apparent that this valence of constructive descriptio stands to gain a great deal of importance for models. Its “showing” becomes substantially more precise, due to the possibility of substituting geometric guidelines for the visible and of “describing” it in this way. Thus, the inner consistency of the model increases as does its chance of accurately “hitting” a circumstance, of creating the suggestion of an evidence – for the reason alone that within the theoretical economy of European thought, the axiom “verum et factum convertuntur” (the true and the created converge), in various manifestations, has played a role that can hardly be overestimated.

And yet: the guiding of the gaze in the model along a mathematically conceived order of images does not mean that the result can be seen or

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understood to be the simple explication of calculations. Quite the contrary.

Despite constructive descriptio, what is decisive in the end are the demonstrative arguments in the model and their power to bring before the eyes, to convince. This can be quickly explained with models of the heavens. The different radii of the planetary orbits – which incidentally have been recognized since Kepler as ellipses – are not represented in the models. On the contrary: they are consistently “purged”, just when one should have known better, for the reason that their extreme differences would have exploded any demonstrative model conception. If one were to conceive of the earth as the head of a pin, then the moon would circle it at a distance of three centimeters as a piece of a grain of sand, while the sun, a sphere with a diameter of 11 centimeters, would be positioned at a distance of twelve meters. It was only the model theater of the planetariums that was better equipped to handle this image-negating disproportion. It engenders a combination of knowledge, teaching, amazement, and grandeur by means of a most technical artifact whose importance for the broader history of knowledge deserves a more thorough analysis.

The second way to incorporate the specific possibilities of the modelimage is that of an indirect representation based on reflection. As a reminder: Classical architectural models offer structure and scale at the same time, as well as the direct view of what they stand for. Heuristic models are much more open; they generally don’t achieve a direct reproduction of their circumstances – structure and dimensional accuracy provided for the sake of visualizing, as shown by the models of the heavens, are able to cope with substantial deviations. Nevertheless they also achieve the purposes for which they were created, namely setting up analogies between the construction and reality. Of course, the concept of analogy circumscribes references of a very particular kind, namely correspondences without similarity. This concept of representation can easily be explained with the help of an example. One

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thinks of a balance – the epitome of precise equivalents – such as those used at markets. An object on the one side, for example a piece of fruit, can be balanced by a completely dissimilar, different object on the other side, generally using standard weights. Of course, a radical selection of all characteristics also takes place here that tends towards the sole determination of the weight. Be that as it may: the notion of analog representation, of accurate representation functions under a complete relinquishing of similar depictions. Kant took up this notion and continued it when he introduced the concept of symbolic representation. His example is often cited. It draws a comparison between political despotism and the hand mill, whereby two methods, two inner forms of reflection enter into an evident relationship with one another.

It will come as no surprise that modern art - since Cézanne and up to and including Abstraction - also offers many examples of dissimilar analogies, in which an inner coherence, created in the image, explicates an unknown reality and functions as its model of interpretation. This can involve formal structures of image organization, of movement, energy, or of constellation, in short, anything that lends itself to being judged through reflection.

We will take this opportunity to briefly return to the previously mentioned antagonism between an autonomous concept of art and scientific images and to suggest a conceivable way out of this dead-end.

Our example involves Mondrian, who between 1915 and 1919, the years during which he approached and crossed the threshold of abstraction, developed strikingly model-oriented picture forms. The first work that comes to mind is “Pier and Ocean” (also more cautiously referred to by the technical title of “composition”), in which perspectival projection fuses together with the horizontal of a map. The perception changes accordingly between a frontal view and the overview or glance of an unfixed eye. The artist, incidentally, has done much not to place the simple pattern of lines coordinated at right angles on the surface, but

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rather to allow them to emerge out of the chromatic white – which also fundamentally changes the meaning of the representation. On a symbolic level, the picture resembles a visually eventful world-egg, and a cursory glance at the chessboard-images that follow it or the “worldsquares” of abstraction that came afterwards clarify Mondrian’s adoption of the model into his artistic representational practice. We add to these few hints the fact that Mondrian also annuls the rationality of calculation by using it. Constructs are so saturated with visual agitation that the observer is confronted with paradoxical experiences of suspension that take possession of the apparent explicitness.

In the case of other artists of his time, the drawing-out of the figureground-relationship emerges even more clearly, including the emphasis of the pictorial underside or reverse. A plethora of alternative world models developed in the context of modernity, which plumb beyond mimesis, the unfixed-complex and energetic character of reality, and avail themselves of an aesthetic heuristic. For Mondrian, it is not a matter of pictorially constituting a specific object driven by scale, but rather, with recourse to elementary conditions of the image or of painting, of discovering reality, not reconstructing it. Art, especially that of the 20th century, thus offers in its own way models for interpreting the world, for understanding it. The key used here is, as already mentioned, not mimetic, but formed according to the pattern of symbolic reflexivity, i.e. of analogy. It should be possible to overcome in this way the widespread opinion that the practices of art, science or mathematics mutually and essentially exclude one another. Not because their individual specificity – which could be referred to as an indispensable increase in the differentiation of cultural development – should be leveled; the level of the relationship between the various areas of knowing (including mathematics) and of doing seems a given in the context of their exemplarity. The “mixed” image-form of the model is the prime example of the crossing of epistemic with artistic methods – a crossing that, as we have shown, is enacted on iconic terrain.

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Iconic Knowledge. The Image as a Model

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The catchwords “exemplarity” or “paradigmatic” lead into a few concluding remarks that will attempt to situate the role of the very different models in the household of thought. When it is a matter of teaching and proving, in other words of epistemes – Aristotle used the verb DEIKNYNAI (which corresponds with deixis or demonstratio) in his rhetoric – the model is granted the character of a beginning, of a first reason, of an arché. Models are necessary and irreplaceable because they can bestow a demonstrative plausibility upon work with concepts, laws, formulas, and hypotheses, which these would never be able to generate on their own. What reveals itself implies a conclusiveness that cannot be trumped by concepts, only by more powerful image designs. Aristotle referred to this as a proteron.

The power of model-images lies in the fact that most of their visual roots, for example the tree, the body, the wheel, the spiral, proportional relationships, the circle or sphere, etc. were pre-shaped in the sphere of the living world and the everyday, and have been saturated with evidences. The vehicle of the imagination, of which the model-images avail themselves, thus come from metaphors that have deeply engrained themselves in the observers and their worlds of experience.

The discussion of the model-image, like that of images in general, learns from its historical variety, from the precise demonstrative analysis, but also from didactic plays of theory, especially those that endeavor to implant an eye into thought, to discuss the mutual reliance of the empty concept and blind visualizing. Coming to terms with this transition, however, relies, as we have seen, on representation.

Model-images belong therefore to the category of paradigms, i.e. of the example. The exemplary opens up a view - in every excluded and optimized segment; in the most visually intense one that it opens - onto something that is otherwise inaccessible or uninterpreted. It offers criteria and arguments, makes orientation possible, and additionally allows those at a distance from the sciences comprehensibility by

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Iconic Knowledge. The Image as a Model

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providing connections to otherwise too-difficult and hermetic knowledge. Seen from this angle, the model resembles windows – which can be both looked into or out of.

Put to the test, used thoughtfully and in reflection, the model mobilizes the most potent resource humans possess and the only one that holds up against reality in the long run: their imagination.

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