Diagrammatic Thinking Alexander Gerner
Introduction
“Come on, my Reader, and let us construct a diagram to illustrate the general course of thought [ … ]”1
“[ … ] we have no knowledge a priori of how to inquire—there can never be a time when we will know, for sure, that we are proceeding in the right way or even that there is a right way to proceed. We can only go by the evidence we have so far acquired, in faith that there is an impersonal truth, that is, a final opinion toward which an ideal inquiry would tend. The evidence that supports that faith is extensive and compelling and yet conceivably erroneous. It is shot through with uncertainty, unanswered questions, unresolved problems, and vague formulations.”2
“The diagram is no longer an auditory or visual archive but a map, a cartography [ … ] It is an abstract machine. It is defined by its informal functions and matter and in terms of form makes no distinction between content and expression, a discursive formation and a nondiscursive formation [ … ] If there are many diagrammatic functions and even matters, it is because every diagram is a spatio-temporal multiplicity.”3
The use of a diagram enables us to create a new way of relating to the unknown, of unfolding the dynamics of orientation in the world.
The diagrammatic is a method or a tool which enables us to know more, and to differentiate what is initially vague in a new way. In this way the structural parts of any-body, any-force, any-thing or any-world appear and show themselves more clearly. The power of diagram-making could be described as the skill of change and transformation by reason, and may be “cartographic,”4 “topologic,”5 or diagrammatic. Diagrammatic thinking is, however, not so much about the concrete shapes and forms of the geometrical configuration of knowledge represented as about the dynamic of how the structures of connectivity and separation—together with attentive abstractions and the relation of points of connectivity (territorialization) and disconnection (deterritorialization) and reconnection (reterritorialization)—are performed, evolve, and show forces of change.
In Erik Beltrán’s concept map (2008), rectangular patterns of lines and curved lines partially intersect but also open up hodologically closed spaces with conceptual names and attributed or imagined significations. The white-inked middle square on the black plane has at its very center (thus orienting the dynamic open constellation) the concept word diagrams (although it also presents the words “ethics,” “category,” and “map” as well as the disciplines “physics,” “chemistry,” and others in the same-sized lettering). The square outlined in white on the left enables the observer to focus on the tension in language between the material techniques of such scriptural notations of “linguistics” as “calligraphy” and “typography” and the oral tradition of knowledge and authority, “voice.” The lower left starts with an ellipse including concept “signs.” An ellipse-like line enters the top of this dynamic diagrammatic map machine suggesting the concepts “Intuition,” “Statistics,” and “Mathematics” and, on the curved part already inside the line that shows a middle square, “topology,” and “geometry” are written. All these concepts are important when reflecting on how “the” diagram can show a structure or a scheme of thought that is perceived and observed as both discursive and iconic at the same time; a structure that is in movement or serves as a dynamic navigation tool of complex relations in the organization of knowledge and experience when using diagrammatic thinking and experimentation.
We can never find our way in diagrammatic experimentation (thinking, reasoning [perception and cognition], observing and art practices such as map-making and affective becoming) without the orientation of our experiential habits and our oriented body (left/right hand etc.). Visual material artifacts (alias visual diagrams as in the following section) are the outcomes of constructed knowledge tools, which by being laid out or noted spatially will show more about the world than we already knew.
I
Visual diagrammatics: operating with diagrams as spatially extended and perceivable notations between visibility and readability
For Mersch, the diagrammatic builds its own category6; not a hybrid between discursivity and iconicity but—by putting discursivity and grammaticality into tension with what shows up, for instance in the perceptive principle of the visual—a proper category that creates spatial differentiations via contrasts, lacunae, nonfilled places or distance-relations. It therefore properly constitutes the visual operational space which shows how the structure of visual arguments is principally topological. This first notion of the diagram could be constituted by the necessity of a “lineature” that gives way to an epistemology of the line7 in all its different forms (arrows, linking structures, Jordan curves, folds, knots, etc.) in the disegno tradition8 of the Renaissance for a diagrammatic theory of diagrams as a creative tension between the visible and the knowledgeable, or the tension between the visually perceptible that exceeds the discursively readable notation. One could also speak of thinking, discovering, and creating with a hand that notes by drawing iconic scriptural and spatial localizations and positional extensions of unfolding lines and their folded points or knots on a plane perceived by the cognitively instructed eye, describable as the relation of eye, hand, and thinking.9 This widens the famous notion of Kleist that posits the creation of thought by/while speaking and the generation of knowledge with creating thought by/while drawing and observing, and thus unfolds lines or knots to form perceivable and notable spatial orders that not only accompany thought with illustrative graphics, but that actually show thought in the making, in which the universe itself is seen as a multiplicity of knotted and folded spaces that thought unfolds or explains on a perceivable plane.10 In this narrow account of the diagram as a visual diagrammatic, the diagram category is put into perspective by spatially given visual orders of notations on a plane or a on a sheet of existence onto which the diagram is projected and scribbled on, thus acquiring a history. In this way, by aligning inscriptions, transformations of information are transferred from one medium to another or from one level of a medium to another. The artifactual diagram-objects thus produced perform and constitute a structure and ordering of space in order to represent and show something. In this account of the visually perceptible diagrammatic, diagrams can be defined as visual-graphic schemata that form and perform arguments in a visual medium. Through them, arguments can be inferred, proved, refuted, and hypothesized.11 Of first importance in the visual diagrammatic is the “leading principle” of spatiality or spatial orders that rationalize space, in which topological relations are the most basic, and the lines and points interact with a plane. Secondly, the operationality of diagrammatic performance dislocates the importance of logical relations from discursivity and syntacticity toward relations of spatiality, in which iconic elements and scriptural language are superimposed on a plane and copresented by diagrammatic notations. For Krämer12 the focus lies on diagrams as hybrids between images and scriptural notations that make discursivity visible through interspatiality.13 Diagrams in this account have a double linear and nonlinear “readability” in which the directedness of the eidetic element of the diagram (seeing something in something else) points beyond a mere discreteness of texture and its referentiality toward a fixed symbolic system and its conventionalized objects.
On the other hand, diagrams do not just include pictorial representations that can be read in a nonlinear way, but (as in infographics) add to the mere pictorial presentations the operationality to think and reason further with the diagram, and to distinguish and differentiate spatially syntactical or other formalized structures with it.
Fundamental, however, is the realization that even writing itself is a hybrid construct in which the linguistic and the iconic, the double act of telling and showing, intersect, and can with Krämer be called an example of “notational iconicity” (Schriftbildlichkeit).14 The diagrammatic category is here interpreted as making thought visible through perceivable marks that contain both scriptural and iconic elements. Such a diagrammatic category could not exist without the capacity of the nonautomatic, nonprogrammable constitutive creature attention15 that lies at the basis of the capacity of constructive observation and the perceptively influenced creation of the new.
II
Charles Saunders Peirce’s account of the iconic diagram sign and the process of diagrammatic reasoning as the quest of invention and discovery
“Remember it is by icons only that we really reason, and abstract statements are valueless in reasoning except so far they aid us to construct diagrams.”16
In Peirce’s sense, diagrams are first of all eidetic operations or icons of relations. According to Peirce and Stjernfelt diagrams are icons, signs that work operationally in the moment they are used, and that therefore enable us to learn something new about the world according to its objects and scenes. Exploring unknown territory, real or imaginary, calls for the drawing of lines, paths, and trajectories. I assume here that the drawing of lines and diagrammatic relations is to be found at the very foundation of spatial orders that later lead to scientific activity (geometry, invention of numbers, scriptural fixation of notations as writing/alphabets or pictograms). Most probably the oldest material diagram artifacts are the orientation map-like artifacts that made it possible to orient oneself in a given life-world territory over 25,000 years ago. These were carved into mammoth bones and stones as primary human knowledge tools that made it possible to orient and discover something as well as to project and memorize places and food supplies.
Diagrams as “skeletonized icons” represent their object analyzed into parts, among which rational relations hold, be they implicit or explicit. They are also responsible for the proper genesis of objects. These “objectiving” relations can be spatial relations (topological) as well as any other kind that the diagram allows us discover and think with. Thus diagrammatic experimentation not only enables us to find and discover new knowledge about an object, but may enable us to reinvent a whole system of how to draw diagrams (see the debate on hypostatic reasoning/abstraction); for example, in the transformation of Euclidian to non-Euclidian geometry,17 and in introducing new objects or ens rationis, unknown before the diagrammatic thinking process started, by construction—that is, “the principal theoric step” of a demonstration.18 Peirce also underlines the fact that the observation of diagrams is essential to all reasoning—even if no auxiliary constructions are performed; in deductive reasoning there is always a step from the general statement to the singular. In 1903 Peirce distinguished between the terms “precisive abstraction,” which separates one element from another, and “hypostatic abstraction,” which by skeletonizing and abstracting at the same time introduces a new object and designates a transformation e.g. from good > goodness. Diagrams are thus pilots of complex relations that reveal new knowledge about the world through which we pilot our way by introducing or making explicit hidden or new elements of thought which transform the diagram or the complete representation system by “abduction” which is “parallel” (introducing hypostatizing abstractions from other areas), “creative” (creating new hypostatizing abstractions) or “theoretical”(creating a new perspective on the problem diagrammatized).19 Diagrammatic thinking therefore first makes the principle of diagrammatic reasoning accessible by introducing new elements through iconicity and through the operations of “hypostatic abstraction” and theoretical abstraction.
One has to be careful not to adopt a trivial similarity definition when speaking about diagrams as part of iconicity. From a non-trivial standpoint similarity cannot be equated with “identity” with the object, nor can similarity be psychologized to refer to merely subjective judgments or feelings of resemblance.20 For Peirce, it is through the icon and the operationality of similarity, and not through indexes or symbols, that knowledge of an object grows:
“For a great distinguishing property of the icon is that by the direct observation of it other truths concerning its objects can be discovered than those which suffice to determine its constructions.”21
The operationality criteria22 of iconic signs means that icons are signs from which more new information can be derived by observation and manipulation than that which sufficed for their construction.23 There is no new information, no learning possible, according to this theory, other than through the participation of iconic signs that need to be interpreted. However, the degree of image iconicity and its virtuous vagueness as the vague possibility of an introduction of the new is reciprocal to the preciseness and degree of determination toward its object. In Peirce’s taxonomy of the trios of signs (icon, index, symbol) in the “Syllabus” of 1903, the Peircean diagram forms a second subtype of the three hypoicons that can never show up in a pure form but are always in use: first, images (not necessarily in pictures!); second, diagrams (not necessarily visually represented); and third, metaphors (not necessarily linguistic metaphors!).24 When considered to be an icon in use, the diagram is defined by its similarity to the object it represents and performs. Peirce’s account of the iconic image sign presents its object by simple qualities (color, form, etc.) and Peirce defines metaphor by its similarity parallel with a third object (e.g., a symbolic system). The diagram (d), however, presents its object with a “skeleton-like sketch of relations,”25 so that we a) construct a diagram for the sake of prototypicality while attentively prescinding and eidetically abstracting from accidental characters26 and creatively synthesizing (eidetically abstracting for a schema) as in the subsequent conceptual map (d1, d2, etc.). Then in a second step b) we are enabled, through diagram observation, to read new information directly off the diagram; and c) through the technique and praxis of the third step we can drop (abstract) unnecessary elements or creatively insert new elements—e.g., by introducing auxiliary lines in geometry—and then from the transformed diagram (d1´) deductively conclude new knowledge.
We may come to the conclusion that we have to redraw the diagram completely (d2) or rethink a complete way of how to draw and think diagrams (d*) according to the presumptions or a priori knowledge we held, but may not hold any more after the first diagrammatic thinking process. Thus the sign process is seen not as a fixed moment, but in a succession of elements in time and space, and can be “[ … ] characterized with great truth as presenting before our eyes a moving picture of thought.”27
III
Mappings as artistic creation practices of thought: from orienting thought by diagrammatic reasoning (Peirce) to cartographic orientation (Deleuze) before representing thought diagrams
Concrete artistic diagrammatic/cartographic praxe and techniques may include
“[ … ] fragmenting known maps and rearranging them in novel ways; juxtaposing far with near; distorting space into a relative or egocentric form; changing orientation; manipulating projection, scale, and generalization to infringe accepted mapping standards; drawing on standard cartographic tropes such as the border, or naming to question social norms; abstracting and overcoding a known form; employing recognizable country shapes in new ways; shifting novel conceptual frames onto familiar icons such as the globe or tube map; and mapping onto different media so as to ask questions about the world or our identities.”28
“But we do not make a diagram simply to represent the relation of killer to killed [ … ] and the reason why we do not, is that there is little or nothing in that relation that is rationally comprehensible.”29
One could say that Deleuze’s account of the map category30 located before any rational representative order of the diagram, takes seriously what Peirce calls “the relation of killer to killed.” For Peirce, this contains little or no rationality to be read off a possible diagram and thus presents a complementary concept that investigates the ontologies that constitute diagrams. However, for Deleuze, cartographic ontology precedes the diagram category, while for Peirce the map is seen as a subtype of the diagram.
The “causality” of the “event” (of killing, alias the creation of a first diagram) and the rationality of a diagram as a logic of relations toward the event of the genesis of a diagram is put into question in the artistic practices and creations of real maps that for Deleuze are located on a subpersonal and sublogical level before being represented in any kind of medium. For Deleuze “art attains this celestial state that no longer retains anything of the personal or rational. In its own way it says what children say. It is made up of trajectories and becomings, and it too makes maps, both extensive and intensive.” Maps are constructive, and develop new knowledge; they also exercise power31 and can promote a change of orders (political, epistemological, social, and representational diagram); or they can transform or project “identities.”32
However, before having a thought, a history or a representation, we, as embodied and embedded creatures, navigate and create a spatio-temporal cartographic ontology that an artwork shows, according to Deleuze, as a “[ … ] map of virtualities, superimposed onto a real map, whose distances [parcours] it transforms [ … ] Every work is made up of a plurality of trajectories that coexist and are readable only on a map, and that change direction depending on the trajectories that are retained.”33
It is therefore not a matter of deciding whether the map is indexical or related with a material object (the territory), or whether it is not a territory34 in the sense of representing a possible object to which we attribute meaning. In Deleuze’s sense, it is very important that maps are above all dynamic, and evaluate displacements:
“Maps [ … ] are superimposed in such a way that each map finds itself modified in the following map, rather than finding its origin in the preceding one: from one map to the next it is not a matter of searching an origin, but of evaluating displacements. Every map is a redistribution of impasses and breakthroughs, of thresholds and enclosures, which necessarily go from bottom to top [ … ] Maps should not be understood only in extension, in relation to space constituted by trajectories. There are also maps of intensity, that are concerned with what fills space, what subtends the trajectories [ … ]”35
It is also not just a question of how artistic artifacts of maps “kill” and deconstruct “identities” or conventionalized and pre-existing logic of representational diagram relations, their symbolic practices of representation, or offer “clandestine” counterproposals to a theoretically possible (but actually impossible) “all knowing map,”36 but of underlining how artistic maps and mappings propose new possibilities of orientations and create new projection types concretizing first spatial orders and the proper understanding of the genesis of the diagrammatic, besides their material historical representations and expositions of representing the already-existing force and power relations of the diagram.
It is therefore no coincidence that geographers today show an interest in reflecting artistic maps and mapping processes in order to clarify by contrast what maps in scientific orders can do, should do, or should not do: “I studied maps,” said Fromentin “not in geography, but in painting.”37 In this way the diagram is not so much a functional machine as a factory of meaning production, closer to a theater in which scenes and diagrammatic relations change permanently on the map-stage.
“This is a different kind of diagram, a different machine, closer to theater than to the factory; it involves a different relation between forces. [ … ] This is because the diagram is highly unstable or liquid, continually churning up matter and functions in a way likely to create change. Lastly, every diagram is [ … ] constantly evolving. It never functions in order to represent a persisting world but produces a new kind of reality, a new model of truth. It is neither the subject of history, nor does it survey history. It makes history by unmaking preceding realities and significations, constituting hundreds of points of emergence or creativity, unexpected conjunctions or improbable continuums. It doubles history with a sense of continual evolution.”38
Thus the picture can become clearer; the traditional field of the conscious logic of relations (and its Noeta) will have to be seen as superimposed on the map-plane of the aesthetic (and its Aistheta) and the subtending map-points of intensities as affective events (and its Affects) and its movement in spatial trajectories, where hidden topological ontological assumptions or subtending forces must deal with the cartographic a priori notion. The focus should be on the appearing or individuations of artistic processes of map-making, as well as on the level of subconscious “cartographic activity.”39 This means that we have to start to investigate the dynamic aspect of the genetic cartographic a priori of orientation itself,40 in which a posteriori diagrams or structures of relations are produced as in the diagrammatic art of map making in hodological spaces, and not so much in the science of fixed geographies. Maps and their diagrammatic structures of iconic relations to create knowledge are therefore dynamic, and have something in common with “milk,”41 as they are fluid and have a “sell-by” date. As proposed in the title of a brilliant book by Karl Schlögel, Im Raume lesen wir die Zeit (Reading Time in Space):
“Always when a world comes to an end and a new is initiated, is the time of the map. Map-times stand for the transformation from one order (of space) toward another.”42
Illustration
I / Map 1 In “L’image du monde au Sudan,”43 Marcel Griaule (1949) interprets this diagrammatic graphic, representing and codifying the spatial knowledge of a culture painted in caves on the Bandiagara escarpments in Mali, as a map of the “Dogon” cosmos, known as aduno kine (the life of the world). According to Griaule, the oval “head” signifies “the celestrial placenta” and the “legs” show the “terrestrial placenta.” Here “[ … ] the torso and the arms form a cross, representing cardinal directions [ … ] This orientation toward the aduno kine is replicated in other patterns of everyday life such as in the layout of fields, weaving designs where men and women sleep.”44
II / Map 2 Erick Beltrán (2008) “The personal social order,” from the “Calculum Series,” Poster b/w.45 A meta-diagrammatic machinic map experimenting with orders of thinking, representations, and operations in different orders of knowledge, the subconcepts and lines of which can be related, folded/unfolded in multiple representative and non-representational “clandestine” ways.
III / Map 3 Terry Atkinson & John Baldwin, “Map of Itself” (1967), Bookprint 14.5 x 18 inches, from the series “Art & Language.”46 The introduction of the grid as a spatial order—“the map of itself”—or the standardized construction of an isotropic, homogenous space in “geometrism” 47 within the model of mathesis universalis reduces and abstracts a real or qualitative and perspectivated (embodied and embedded) experiential place to a quantifiable, measurable algebraic spatio-temporal extension. Thus the introduction of the grid is shown here as an important topic of map-making explored by map artists.
IV / Map 4 Recently a team led by the Spanish archaeologist Pilar Utrilla48 of Zaragoza University deciphered the oldest prehistoric map-like cartographic engraving to date in Western Europe, found in 1993 on a hand-sized stone (1kg) in a Grote in Abauntz, Navarra. In the above map-like stone the complex etchings engraved around 13,660 years ago, probably by Late Magdalenian hunter-gatherers at various times and using various diagrammatic design styles, superimpose the discovery’s reference point of the “mountain” San Gregorio with animal layers and other geographic layers. Just as the spoken language existed long before written languages, mental maps preceded the physical evidence of materialized map-making artifacts. The cartographic, in a wide notion of orientation, path-finding, etc., precedes the modern notion of maps and may include spatial planning ahead as a cognitive tool for orientation. This ability of prospection to navigate mentally as a possibility for the planning of future actions seems essential for adaptation and human brain evolution, as well as for preserving, relinking (religio), and reexperiencing past events real or imagined by the mnemonic functions of cartographic structures49 linked to present or future actions. This all shows how the diagrammatic imagination and the external epistemic diagrammatic tools of the cartographic are complementary in creating orientation on all imaginable levels. For the complementarity account of a) mental maps and b) material map artifacts (or map-like artifacts) for orientation in two anthropological models based on ethnographic research to account for the “way-finding” ability of early humans.50 Defining the map as a representation of a part of the Earth’s surface naturalizes and thus universalizes the map: “it also obscures its origins in the rise of the state; and it ignores its role in the establishment and maintenance of social relations in those societies where it exists.”51 It is not our purpose here to conflate maps with specific projection types and the modern science of mapmaking, nor with fundamental human abilities like orientation, way-finding, and other features of spatial intelligence.
V / Map 5 For Peirce, the operational definition of diagrammatic thinking is the following:
“We form in the imagination some sort of diagrammatic, that is, iconic, representation of the facts, as skeletonized as possible. The impression of the present writer is that with ordinary persons this is always a visual image, or mixed visual and muscular [ … ] If visual, it will either be geometrical, that is, such that familiar spatial relations stand for the relations asserted in the premises, or it will be algebraical, where the relations are expressed by objects which are imagined to be subject to certain rules are, whether conventional or experiential.”52
“By diagrammatic reasoning, I mean reasoning which constructs a diagram according to a precept expressed in general terms, performs experiments upon this diagram, notes their results, assures itself that similar experiments performed upon any diagram constructed according to the same precept would have same results, and expresses this in general terms. This was a discovery of no little importance, showing, as it does, that all knowledge without exception comes from observation.”53
Schematically, the diagrammatic thinking process shown above in this metadiagrammatic map would be better exemplified in a moving image in which the arrows and the schematic fields of gray (results) and blue (operations) and the precepts and preconditions (red), the immaterial operations (white notations) and the material results (black notations) diagrammatic parts 1,2,1´2´are seen as momentary fixations and directions of a developing and changing process of diagram experimentation
VI / Map 6 Druksland—Physical and Social—January 15, 1974, 11:30 am 1974 Offset print (unnumbered edition) 18 x 15 inches.54 The superposition and not the “causal” relation of personal and political orders in the map by the Israeli artist Michael Druk that marked out a cartographic depiction with different pictorial color elements, an explanatory legend, a scale of distance, a title, and such symbolic fixations of topographic areas as, for example the “occupied territories,” show in the first place the attribution of a spatial order that can be intentionally indicated as related to the map projections of the occupied territories in Israel with what can be called a “psychogeography”55 in his series of self-portraits, metamorphosing geographical into psychographic maps and vice versa. Spatio-temporal boundaries are traversed in which the territory and the face of the artist are becoming an immanent plane of the other (the map), showing the folded relation between the topography of the territories in its occupied and unoccupied parts and body orientation (left Duks, right Duks etc.), as rethinkable parts and embodied parts that are not rational but subconscious parts. There is no Cartesian ego of self-affirmation here, but the form of a dated map, a “fractured I of a dissolved ego.”56 The map’s title is dated very precisely, which makes it a proper time-diagram of meaning attribution “Druksland—Physical and Social January 15, 1974, 11:30 am.” This fixation of a momentary map that proposes that there could be changing maps each measurable instant of time shows how maps are located in between a mirror or copresentation of a reality and a diagrammatic structure of relations. In Duksland the presented spatial order of a physical territory superimposed on the pictorial element of a persona-mask on a flat plane representing a human skin-face shows in the lines that are named as “occupied,” the points and areas of intensity (located at the forehead) and might propose that the reality of the map could be a momentary thought map already outdated when visualized on a material fixed plane or image-medium.
VII / Map 7 Louisa Bufadeci’s diagrammatic “Ground Plan” (2003), digital Print 330 x 900 cm.57
Note
1 / Charles S. Peirce, Manuscripts on Existential Graphs. Reprinted in Peirce (1906), CP 4.530.
2 / T.L. Short, Peirce’s Theory of Signs, Cambridge University Press, Cambridge 2007, p. 347.
3 / Gilles Deleuze, Foucault, Continuum, London 2006.
4 / Franco Farinelli, “Von der Natur der Moderne: eine Kritik der kartographischen Vernunft,” Räumliches Denken, ed. Dagmar Reichert, ETH, Zurich 1996, p. 267–302. Franco Farinelli, “Did Anaximander Ever Say (or Write) Any Words? The Nature of Cartographical Reason,” Philosophy and Geography 1(2) (1998), p. 135–144.
5 / Sigrid Weigel, “Zum ‘topographical turn.’ Kartographie, Topographie und Raumkonzepte in den Kulturwissenschaften,” KulturPoetik 2 (2002), p. 151–165.
6 / Dieter Mersch, “Visuelle Argumente. Zur Rolle der Bilder in den Naturwissenschaften” in Bilder als Diskurse—Bilddiskurse, ed. S. Maasen, T. Mayerhauser, C. Renggli Velbrück Wissenschaft, Weilerswist 2006, p. 95–116.
7 / Sybille Krämer, “‘Epistemology of the Line’: Reflections on the Diagrammatical Mind,” Diagrammatology and Diagram Praxis, ed. A. Gerner, O. Pombo, CFCUL, Lisbon 2010 (forthcoming).
8 / Wolfgang Kemp, “Disegno. Beiträge zur Geschichte des Begriffs zwischen 1547 und 1607,” Marburger Jahrbuch für Kunstwissenschaften 19 (1974), p. 218–240.
9 / Elke Bippus, “Skizzen und Gekritzel. Relationen zwischen Denken und Handeln in Kunst und Wissenschaft,” Logik des Bildlichen. Zur Kritik der ikonischen Vernunft, ed. Martina Hessler, Dieter Mersch, Transcript, Bielefeld 2009, p. 76–93. Horst Bredekamp, Die zeichnende Denkkraft. Überlegungen zur Bildkunst der Naturwissenschaften, Einbildungen (=Interventionen 14), Zurich 2005), p. 155–171. Horst Bredekamp, Galilei der Künstler: der Mond, die Sonne, die Hand, Akademie Verlag, Berlin 2007.
10 / Gilles Deleuze, The Fold: Leibnitz and the Baroque, Continuum, London 2006. Horst Bredekamp, Die Fenster der Monade. Gottfried Wihelm Leibniz´ Theater der Natur und Kunst, Akademie Verlag, Berlin 2004.
11 / Martina Hessler, Dieter Mersch, “Bildlogik oder Was heißt visuelles Denken?” Logik des Bildlichen. Zur Kritik der ikonischen Vernunft, ed. M. Hessler, D. Mersch, Transcript, Bielefeld 2009, p. 8–62.
12 / Sybille Krämer, “Operative Bildlichkeit. Von der ‘Grammatologie’ zu einer‚ ‘Diagrammatologie’? Reflexionen über erkennendes Sehen,” Logik des Bildlichen. Zur Kritik der ikonischen Vernunft, ed. M. Hessler, D. Mersch, Transcript, Bielefeld 2009, p. 94–123.
13 / Krämer, “Operative Bildlichkeit,” and Sybille Krämer, “Writing, Notational Iconicity, Calculus: On Writing As a Cultural Technique,” Modern Language Notes—German Issue 118/3 (2003): p. 518–537. Martina Hessler, Dieter Mersch, “Bildlogik oder Was heißt visuelles Denken?” Logik des Bildlichen. Zur Kritik der ikonischen Vernunft, ed. M. Heßler, D. Mersch, Transcript, Bielefeld 2009, p. 8–62.
14 / Krämer, “Writing, Notational Iconicity, Calculus,“ p. 518–537.
15 / Bernhard Waldenfels, Phänomenologie der Aufmerksamkeit, Suhrkamp, Frankfurt 2004.
16 / Charles S. Peirce, The Logic of Quantity (1893), CP 4.127.
17 / Michael Hoffmann, Erkenntnisentwicklung, Klostermann, Frankfurt am Main 2005.
18 / Charles S. Peirce, Manuscripts on Existential Graphs, CP 4.616.
19 / Michael Hoffmann, “Seeing Problems, Seeing Solutions: Abduction and Diagrammatic Reasoning in a Theory of Scientific Discovery,” Abduction and the Process of Scientific Discovery, ed. O. Pombo, A. Gerner, CFCUL, Lisbon 2007, p. 213–236.
20 / Frederik Stjernfelt, Diagrammatology: An Investigation on the Borderline of Phenomenology, Ontology and Semiotics, Springer, Dordrecht 2007.
21 / Charles S. Peirce, Manuscripts on Existential Graphs, CP 2.279, 1895.
22 / Krämer, “Operative Bildlichkeit.”
23 / Frederik Stjernfelt, “Diagrams as Centerpiece of a Peircian Epistemology,” Transactions of the Charles S. Peirce Society 36/3 (2000), p. 357–384. Frederik Stjernfelt, “Two Iconicity Notions in Peirce’s Diagrammatology,” Conceptual Structures: Inspiration and Application. Lecture Notes in Artificial Intelligence 4068, Springer Verlag, Berlin 2006, p. 70–86. Frederik Stjernfelt, Diagrammatology.
24 / Charles S. Peirce, Manuscripts on Existential Graphs, CP 2.277.
25 / Stjernfelt, Diagrammatology, p. 90.
26 / Charles S. Peirce, New Elements of Mathematics IV, The Hague, Mouton 1976, p. 317.
27 / Richard Robin, Annotated Catalogue of the Papers of Charles S. Peirce, University of Massachusetts Press, Worcester 1967.
28 / Chris Perkins, “Cultures of Map Use,” The Cartographic Journal 45/2 (2008), p. 156.
29 / Peirce, New Elements of Mathematics IV, 316.
30 / Gilles Deleuze, “What Children Say,” Essays Critical and Clinical, trans. Daniel W. Smith and Michael A. Greco, Verso, London 1995, p. 61–67.
31 / Yves Lacoste, “An Illustration of Geographical Warfare,” Antipode 5 (1973), p. 1–13.
32 / John Pickles, A History of Spaces: Cartographic Reason, Mapping, and the Geo-Coded World, Routledge, New York 2004, p. 12.
33 / Deleuze, “What Children Say,” p. 67.
34 / Jane England, “The Map Is Not the Territory,” Essays by Jane England, England and Co., London 2001.
35 / Deleuze, “What Children Say,” p. 61.
36 / Perkins, “Cultures of Map Use,” p. 156.
37 / Deleuze, “What Children Say,” p. 66.
38 / Deleuze, Foucault, 30/1.
39 / Deleuze, “What Children Say,” p. 64.
40 / Werner Stegmaier, Philosophie der Orientierung, Walter de Gruyter, Berlin and New York 2008. Farinelli, “Von der Natur der Moderne,” p. 267–302. Weigel, “Zum ‘topographical turn,‘“ p. 151–165.
41 / See: Mark Monmonier cit. in Karl Schlögel, Im Raume lesen wir die Zeit. Über Zivilisationsgeschichte und Geopolitik, Carl Hanser Verlag, Munich 2003, p. 86.
42 / Schlögel, Im Raume lesen wir die Zeit, p. 87.
43 / Thomas J. Bassett, “Indigenous Mapmaking in Intertropical Africa,” Cartography in the Traditional African, American, Arctic, Australian, and Pacific Societies, ed. David Woodward, Malcolm Lewis, University of Chicago Press, Chicago 1998, p. 24–48.
44 / Bassett, “Indigenous Mapmaking in Intertropical Africa,” p. 26.
45 / Edition of the Barcelona Exhibition at Galeria Joan Pratts December–January 2008/2009.
46 / Katharine Harmon, The Map As Art: Contemporary Artists Explore Cartography, Princeton Architectural Press, New York 2009, p. 13.
47 / Gaston Bachelard, The Poetics of Space, Beacon, Boston 1994.
48 / Pilar Utrilla et al., “A Paleolithic Map from 13,660 calBP: Engraved Stone Blocks from the Late Magdalenian in Abauntz Cave (Navarra, Spain),” Journal of Human Evolution 57 (2009), p. 99–111.
49 / Bassett, “Indigenous Mapmaking in Intertropical Africa,” p. 30–33.
50 / Kirill Istomin and Mark Dwyer, “Finding the Way: A Critical Discussion of Anthropological Theories of Human Spatial Orientation with Reference to Reindeer Herders of Northeastern Europe and Western Siberia,” Current Anthropology 50 (2009), p. 29–49.
51 / Denis Wood and John Krygier, “Maps,” http://makingmaps.owu.edu/elsevier_geog_maps.pdf (accessed 2009).
52 / Charles S. Peirce, Manuscripts on Existential Graphs, CP 2.778.
53 / From “Peirce’s application to the Carnegie Institution,” July 15, 1902 (from Draft C), p. 90–102.
54 / Online: http://itp.nyu.edu/isco/1/img/IMG_1426.JPG.
55 / Guy Debord, “Essay: Introduction to a Critique of Urban Geography,” (1955), http://library.nothingness.org/articles/SI/en/display/2. Kanarinka, “Art-Machines, Body-Ovens and Map-Recipes: Entries for a Psychogeographic Dictionary,” Art and Mapping: Special Issue of Cartographic Perspectives, ed. Denis Wood, John Krygier (2006a), p. 24–30, http://www.nacis.org/documents_upload/cp53winter2006.pdf. Kanarinka … (2006b), (see above). “Designing for the Totally Inconceivable: Mods, Hacks and Other Unexpected Uses of Maps by Artists (and Other Regular People),” paper presented at the Association of American Geographers Annual Conference.
56 / Gilles Deleuze, Difference and Repetition, Columbia University Press, New York 1994, p. 194.
57 / Online: http://www.louisabufardeci.net/site/pages/gp.html.
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