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E2 European long distance path

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The European walking route E2
The European walking route E2

The E2 European long distance path or E2 path is a 4850 km (3010-mile) series of long-distance footpaths that is intended to run from Galway in Ireland to France's Mediterranean coast and currently runs through Northern Ireland, Scotland, England, Belgium, Luxembourg and France, with an alternative midsection equally designated via the Netherlands and east coast of England. It is one of the network of European long-distance paths.

The paths are aimed at walkers; alternative routes exist in some parts for horse riders and cyclists.

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  • ✪ Jeffrey Hamburger on "Mindmapping: Diagrams in the Middle Ages – and Beyond"


My name is Daniel Donoghue. I'm a Professor of English, and the Chair of the Committee on Medieval Studies. Diagrams are ubiquitous. There is the mandala, for example, used in Buddhist meditation. Or the mandorla, used in Byzantine and Western Christian iconography. But we don't have to go so far afield. I can recall, for example, the seductive symmetry of diagrams in textbooks from my earlier school days in fields like geometry and biology. In more recent decades, computers have allowed us to turn many kinds of data into visual representations of various kinds of data, including 3-D imaging. If we think of diagrams as techniques of visualization that give order to knowledge and perception, then the Middle Ages have a special claim on our attention, because much of its art is diagrammatic. We might be tempted to dismiss diagrams as a convenience, perhaps visually interesting, but still a tool that recedes into the background behind more pressing questions. But as our talk this evening will demonstrate, the topic opens into important philosophical and aesthetic questions such as, what is an image? Jeffrey Hamburger is the Kuno Franke Professor of German Art and Culture at the Department of the History of Art and Architecture, here at Harvard. The recipient of grants from numerous organizations, including the John S. Guggenheim Foundation, the American Philosophical Society, the NEH, and the Humboldt-Stiftung, Professor Hamburger was elected a Fellow of the Medieval Academy of America in 2001, and a member of the American Academy of Arts and Sciences in 2009. He is the author, the co-author, the editor and the co-editor, of too many books to list in an introduction like this. So I will mention only a few from recent years. I hope all of you had the opportunity last year to visit the stunning exhibition of medieval illuminated manuscripts, held in various institutions in the Boston area. They were assembled and displayed at the Isabella Stewart Gardner Museum, at the McMullen Museum of Boston College, and here at the Houghton Library. Professor Hamburger was one of the prime movers of the effort, and a co-editor of the beautiful volume to accompany the exhibition-- Beyond Words-- illuminated manuscripts in Boston Collections. Also in 2016, with three collaborators, he produced Liturgical Life and Latin Learning at Paradies Bei Soest, 1300-1425-- Inscription and Illumination and the Choir Books of a North German Dominican Convent. In 2015, with Nigel Palmer, he edited The Prayer Book of Ursula Begerin. And between 2015 and 2016, he produced a 12-volume survey of masterpieces of book illumination in central Europe. In 2016, with Harvard University Press, he co-edited Sign and Design-- Script as Image in Cross-cultural Perspective, 300 to 1600 CE. Next April, Professor Hamburger is organizing a symposium at Dumbarton Oaks, on the Diagram Paradigm-- Byzantium, the Latin West, and the Islamic World. As the Chair of the Committee on Medieval Studies, I am constantly reminded that to be a Medievalist means to be interdisciplinary. And that interdisciplinarity is even more true, if that's possible, in the field of Medieval Art History. Professor Hamburger's scholarship moves with disarming ease among various specialized fields, such as Late Medieval Theology, including Mysticism, liturgy, Latin and vernacular literature, and history-- whether the micro-history of a particular convent in Bavaria, or the particular interest in-- or, I should say, the broader movements of political and cultural history. His work has shown a particular interest in women's spirituality and mysticism, especially in convents throughout the German-speaking world, with special attention to book illuminations. But his work is always reaching toward big questions, so that when he examines a particular instance of mystical art produced in a Bavarian convent, for example, he crosses a threshold into a much broader discussion of the paradoxical relationship between art and mysticism. This evening, we are going to be treated to a similar move, that opens onto important questions facing us today. Finally, let me say, it is a special pleasure for me to make this introduction, because Jeffrey and I have been friends since our graduate school days. He will be speaking to us on mind mapping, diagrams in the Middle Ages, and beyond. [APPLAUSE] Thank you very much, Dan, for that more than generous introduction. And thank you all. I'm overwhelmed by the turnout. It's great to see so many people here. I begin with a diagram. The caption at the top, written in a child's hand, identifies the circle as the Earth. Das is die Erde. The four quadrants are arranged in a chiastic fashion along the opposing diagonals. The diagonal running from lower left to upper right marks the distance between Germany-- Deutschland-- and England. The opposing diagonal contains vectors in the form of arrows. In addition to attaching text to image, these pointers set the orb in motion. The arrow with the lower right, which points to Germany, lends motion and meaning to the dyadic declaration, da wohne ich. I live there. That at the upper left, which points to England, to the equally emphatic statement, da wohn stu. You live there. In its elementary abstraction, the simple drawing resembles a medieval T-O map, in which the simplest of geometrical schemata exemplifies the basic structure of God's creation. A typical map of this kind is far from being eurocentric in orientation. The upper semicircle represents Asia, the lower two quadrants Europe and Africa. The continents converge on the world's true center, the Holy Land, with Jerusalem sometimes, although not always, at the middle point. Such maps indicate that medieval cartography was diagrammatic to its core. To what extent, however, are diagrams cartographic, in the sense of modeling reality in spatial terms? No less than a medieval map, a medieval diagram embodies a Weltanschauung, a world view. If not as literally, then in its structuring of a reality thought to govern the world and mankind's place within it. The letter from which this child's diagram is taken explains the circumstances of its making. It begins, in translation, "Dear Mummy, Papa made a face like this when you were here, and cried like this." It continues. "This is the earth, and on the earth I see how close we are living." All these points are underscored by indexical arrows. A trio of additional pointers draws the reader's attention from the text to the diagram in the upper left hand corner, lending added force to its plaintive affirmation of proximity, as if its declaration of inconsequential distance could, in fact, effect a reunion. Adjacent to the diagram stands a crudely drawn face, if not quite round, then in most respects no less schematic than the cartographic schema with which it's juxtaposed. The tears to which the letter refers stream not from the large black eyes, but rather from a second pair above them, perhaps doubling as eyebrows. And, as if one pair of eyes couldn't weep enough, their duplication adds weight to the letter's words. Below the face, a trapezoid, representing both neck and torso, extends to embrace the word, Gesicht, a face. The overlap assures the linking of text and image, as does the long loop representing a tie, which knots the two types of graphic traces together. The other side of the letter includes a simple exercise in English, conducted, it seems, in anticipation of the desired reunion. The letter in question was written by my mother in 1934, when she was 10. She had been left behind in Leipzig by her parents, who had already fled Nazi Germany for London, where they were seeking to establish themselves before bringing over the rest of the family. The arrows that punctuate the letter lended a hope for instrumentality. The paintings of Paul Klee, whose notebooks are crammed with diagrams, deliberately and deceptively ape the naivete of children's drawings. They incorporate arrows and pointers to similar ends, transforming his playful yet haunting images into diagrams of desire. Like a medieval diagram of the micro- and the macrocosm, Klee's schematic drawing, Ich, Du, Erde, Welt-- I, you, earth, world-- also mixes iconic and indexical elements to diagram the relationship between the self and the world. In this case, his eye and his I-- his self. The horizontal arrow at the center pointing from the teardrop-shaped Du on the right, labeled, "you-- visible interiorization, object," to the reciprocal Ich on the left, labeled, "eye and I," punning, "artist," traces the path of optical appearances, "optical, physical-- uh-- opticalischa Weg?" I can't quite read it from here. The orb of the earth placed, as in a medieval diagram of the four elements, at the bottom, marks one half of an elliptical pathway which converges from above and below, on the eyeball at the left, the focal point of all the arrows. The lower half of the path represents the earth-- non-optical path of terrestrial or material rootedness, or stasis. In contrast, that at the top, which traverses the upper region of the all-embracing circle, labeled, Welt, represents the world-- non-optical path of cosmic communality, or dynamism. Klee's drawing encapsulates the argument presented in his essay, Paths for the Study of Nature, from 1923. By the standards of common parlance, clays poetic diagram could not be farther from a naturalistic representation of the world. Rather than reflecting reality, however, Clay's poetic diagrams participate in his credo of making visible. In his essay, clay describes the diagram in the following terms. Quote, "On the lower path which gravitates towards the center of the Earth lie the problems of static balance, which can be defined with the words, stand despite all the possibilities of falling. The upper path leads desire to break away from earthly attachment through swimming and flying to free impetus, to free movement." Klee explicitly identifies his diagram, which provides a pattern for his paintings, as a means of charting desire. The arrows in my mother's less ambitious cosmic diagram are no less indexical. They not only convey information, or express an affective state-- in this case, an overwhelming case of separation anxiety-- they represent an attempt to impose order on the world-- in this case, a world in which order was quickly slipping away, and in which nothing would soon make sense. More an expression of distraught desire than crude cartography, the diagram, despite its simplicity, raises fundamental questions. In the words of Nelson Goodman, "how is worldmaking related to knowing?" To paraphrase Goodman's reply, diagrams do not map the world. They make a world. Of course, medieval authors would not have accepted Goodman's account of multiple actual worlds, although in the context of discussions of divine omnipotence, possible worlds represented a different matter. His question, however, would nonetheless have made sense to them. I wish to argue that, no less than my mother's diagram, those made in the Middle Ages should be viewed, not simply as tools for thinking, or representations of knowledge, but also as vehicles of deeply held desires. Rather than simply representing the world, they make and model worlds. Diagrams and desire might seem far removed from one another. No less a logician than Charles Sanders Peirce, however, saw fit to connect them. In his essay, "Logic as Semiotic," written around 1897, Peirce spoke of the habit of "abstractive observation," that formed the foundation of learning from experience. Peirce defines the process in terms of desire. And this is the first of a few longer quotations, so I put them up for you to follow. "The faculty which I call abstractive observation is one which ordinary people perfectly recognize, but for which the theories of philosophers sometimes hardly leave room. It is a familiar experience to every human being to wish for something quite beyond his present means, and to follow that wish by the question, should I wish for that thing just the same if I had ample means to gratify it? To answer that question, he searches his heart, and in doing so, makes what I term an abstractive observation. He makes in his imagination a sort of skeleton diagram, or outlying sketch, of himself, considers what modifications the hypothetical state of things would require to be made in that picture, and then examines it, that is, observes what he has imagined, to see whether the same ardent desire is there to be discerned. By such a process, which is at bottom very much like mathematical reasoning, we can reach conclusions as to what would be true of signs in all cases, so long as the intelligence using them was scientific. On Peirce's account, which remains indebted to Kant's notion of the schema of imagination, namely the rule or procedure by which a pure non-empirical concept or category is associated with a sense impression, the process of perception itself involves a form of diagrammatic projection. In this model, diagrams are not simply something seen, they are also an inherent part of constitutive cognitive patterns, which observers bring to their experience of the world. And just as diagrams were central to, even constituentive of Peirce's logic, so too I hope to demonstrate logic is critical to understanding medieval diagrams in ways that have not always been fully appreciated. All of you, no doubt, have an instinctive sense as to what constitutes a diagram. Diagrams constitute a staple of our textbooks, as Dan pointed out. Especially in the German philosophical tradition, they are thought to appeal to intuition-- Anschauung. All the way back to Plato, diagrams have been associated with transcendental truth. Diagrams, however, at least as materially produced and transmitted, are very much a product of their time and place. Despite their association with laws, logic, and truth, diagrams have contingent dimensions. A diagram, some might say, is a two-dimensional schematic representation-- usually linear or geometric-- of relationships between various interrelated objects or concepts, that is designed to show how something works. If only it were that simple. Even this rudimentary definition is riddled with equivocation, and makes all kinds of assumptions. For starter, with what one might call the representational regime of the diagram. Does a diagram in some way reflect or embody reality? Or does it possess its own reality, whether ideational or material? Are diagrams platonic or pragmatic? Moreover, if one is to speak of the diagrammatic as well as the diagram, per se, why need one limit oneself to two dimensions? In the age of virtual reality, diagrams readily assume three-dimensional form. Also in logic, in which tautologies have given way to typologies. With the rise of computer-generated design, which encourages less the representation of preexisting forms than the playful generation of perception-bending spaces, architectural historians and even some architects refer to their solid structures as diagrammatic, not simply in character, but also in conception. Modern design practice serves as a reminder that a diagram is always spatial and relational in character. Some diagrams, such as the Feynman, Penrose, and Minkowski diagrams employed by physicists to graph spacetime-- and please don't ask me too many questions about those after the talk-- and the interactions among particles, explicitly integrate the component of time. As in the Compendium Historiae in Genealogia Christi of Peter of Poitiers, the shift from the third into the fourth dimension can unfold or unfurl on paper or parchment, not to mention, on a computer screen. Diagrams chart. They map. They interrelate. But they also unfold, sometimes literally. Diagrams deal with process, both in the world and no less importantly, in the mind. They plot rationality in spatial terms, and map out cognitive as well as mechanical practices and procedures. Cognitive psychologists ask whether spatial schemas offer a metaphor or a mechanism for cognitive processes. Medieval diagrams suggest that the answer is both. Take, for example, a single diagram found in a 14th century historical compendium from the Diocese of Konstanz, consisting of biblical paraphrases and summaries. The diagram takes the form of a seven arm candelabrum, and in this particular copy, can be considered either the last part of Peter of Potiers' work, with which it often circulated, or the first in a series of nine diagrams at the end of the manuscript, the remainder of which belong to the set known as the Speculum Theologiae. Its placement alone permits it to serve a mediating role between the historical works, which constitute the bulk of the book, and the moralizing diagrams at the end, whose emphasis is tropological, that is, moral, rather than typological, that is, historical. Whereas in an early 13th century English example here at Harvard-- it's one of the earliest surviving copies-- the candelabrum, accompanied by a Tree of Consanguinity, comes at the very beginning of Beiter's Timeline of Scriptural History. In the 14th century version, in which, moreover, the work has been adapted from roll to codex, the candelabrum comes, rather, at the end, thereby endowing it with an eschatological aura, embedded within the text, which ostensibly explains it. The diagram represents the church, by means of a seven-branched menorah from the temple. Transplanted to a Christian context, the seven-armed candelabrum, similar to monumental bronze examples found in churches throughout the empire, indicates, as does the genealogy that precedes it, that Judaism, synagoga, has been superseded by Christianity, ecclesia. The candelabrum represents part of the symbolic as well as the literal beauty of the church triumphant. The upright central arm represents Christ. The three on one side, the left, represent the three orders, or grades, of the faithful during the period before, the three on the other side, that is the right, after the Incarnation. The text, in turn, defines these three orders within the church in terms of the prophecy in Ezekiel regarding Noah, Daniel, and Job. Whereas Noah represents the prelates, Daniel represents the virgins, and Job, those who are married. All await the day of judgment. Above each of the seven arms an upturned, chalice-like cup pours out oil into one of the bowls held aloft by the arms, which in turn consist of pieces which are linked by small spheres, from which spring decorative lilies, terms which are repeated obsessively across the surface of the page, as if to match the number of parts in real examples. And for example, the candelabrum in Braunschweig, which I just showed you, has 74 different parts. The redundant denomination of the depicted object's parts provides a plethora of places-- loci-- not unlike the segments of the Guidonian hand, a common mnemonic device. How does such a diagram work? The accompanying text barely makes any mention of it. Some terms are allegorized. The majority, however, in particular the components of the candelabrum, are not. In contrast to what one finds in an authentic work by Peter, the Allegoriae Supertabernaculum Moysi. It is left to the viewer to make sense of the image. More than a line diagram, but less than a freestanding image, it hovers between two modes of representation. The object diagram figures both time-- sacred history-- and space-- heaven and hell, as well as the ranks within the ecclesiastical hierarchy. Like the diagram, all of history and all people are divided in two by the coming of Christ. The image's meaning, moreover, is conditioned by its context. Whereas the preceding schematic rendering of salvation history, in the form of Peter's chronology, identifies the diagram as an image connecting past and future, even as it distinguishes between the blessed and the damned, and by implication, Christians and Jews, the subsequent image, a Tree of Life based on Bonaventure's Lignum vitae, provides a visionary vehicle for spiritual ascent, by way of the imitation of Christ. In the words of the inscription at the bottom, a paraphrase of Apocalypse 22, "I saw the tree of life bearing twelve fruits, yielding its fruits every month. And the leaves of the tree were for the healing of nations." Just as the viewer of the candelabrum is instructed that the gifts which Christ confers on mankind surpass human understanding, so too the viewer of the Tree is invited to ascend with Christ and aspire to those heights. Both diagrams serve a mnemonic function, permitting the reader to hang various concepts on the branches of both candelabrum and tree, just as does the artist. The metaphors, however, merge-- a complementarity encouraged and enhanced by the fact that the two images face one another across an opening. Alone, by virtue of the two images' juxtaposition, the candelabrum becomes a type of Christ on the cross. The branches of both menorah and tree become intertwined, generating new associations. Metaphor becomes mechanism, a mechanism whose parts the viewer is expected to manipulate. The images both represent a Christian conception of truth, and provide the viewer with vehicles for striving towards it. The use of diagrams as demonstrations, whether mathematical or philosophical, or as vehicles with which to attain those truths, is as old as Plato, Euclid, and Archimedes. Each of the 42 diagrams developed by Lothar of Segni, later Pope Innocent III, to accompany his Treatise on the Mass, is paired with an explanation beginning, haec figura demonstrat-- this figure demonstrates. The repeated rubric was hardly necessary. The diagram itself serve to make the point. Throughout the Middle Ages, diagrams made particular types of claims on their viewers and demanded specific forms of cognitive response. Philosophical tracts regularly contain diagrams, whether to represent their contents, or to provide the reader with tools for conceptualizing problems. The diagrammatic tradition's roots in science-- in particular, astronomy and cosmology-- made it natural to associate medieval diagrams with reason rather than with feeling. Their abstraction, in turn, made it easier to associate them with the realm of the intellect, rather than with that of experience. Of course, the truth claims of diagrams do not need to be accepted at face value. In particular, religious devotional and theological diagrams, which deal with such topics as life and death, prophecy, and salvation, or the underlying order of creation-- you're looking here at a Genesis initial from a 12th century Bible-- necessarily grapple with powerful emotions, which, no matter how hard they try to impose it, escape the establishment of order. Not without reason, postmodern philosophers have subjected diagrams to scathing suspicion, if for different reasons than did late 19th century mathematicians and logicians, such as Frege, Pasch, and Hilbert. Writing on figures such as Gilles Deleuze, Francois Laruelle, and Alain Badiou, a neoplatonist who elevates set theory to the role of geometry in Plato's corpus, the historian of philosophy, John Mularkey, has declared, quote, "There is no Truth in diagrams, nothing sacred in geometry. The fluctuous line is not an intimation of the divine. Its immanence, its materiality, keeps it at some distance from the infinite lines, pure circles, and perfect triangles of Nicholas of Cusa. Such infinities, purities, and perfections smack of the virtual, the transcendental, the vision of God." This iconoclastic critique of the diagram, which casts it not as an instrument of reason, but rather as a rhetorical device, opens its seemingly cogent demonstrations of irrefutable logic to accusations, if not necessarily of mystical irrationality, than at least of fiction and ideological persuasion. Following David Freedberg, art historians now regularly speak of the power of images. Within anthropological perspectives, less often is it asked, to what end such power, whatever form it might take? The power that derives from persuasion, let alone animation, is very different from a demonstration of irrefutable truth. To what extent do diagrams lend one the appearance of the other? Part of diagram's persuasive power lies in their uncanny ability to both mimic and to structure the process of ratiocination itself. To these functions the diagrammatic method, underlying some forms of artificial intelligence, bear witness. Peirce invented a diagrammatic language called, by him, existential graphs, with which he thought he could address any philosophical problem, in particular those in logic. And this is a set of those so-called existential graphs from his notebooks across the street, in the Houghton Library. In a celebrated passage written shortly before his death in 1913, Peirce declared, quote, "reasoning is dependent on graphical signs." Peirce, who said of Aristotle, and again I quote, "have read and thought more about Aristotle than about any other man," echoes the philosopher's treatise on memory, in which he observes that, quote, "it is not possible to think without an image." That's the famous passage from Aristotle. Neither Peirce nor Aristotle, however, refers to representational images, whether actual or mental. And this passage is often misquoted. In fact, Peirce insists that neither on paper, nor in his dreams, do his diagrams engage with the imagination. This is a wonderful passage. "There is nothing fanciful about my diagrams. I do not, for example, see numbers with colors attached to them and placed upon some curve, and it perfectly astounds me to find how useful some persons are able to make such strange constructions. When I am in health I am not aware of having any dreams, unless perhaps of a problem in algebra where no real significations are attached to the letters, or something equally abstract. My 'Existential Graphs' have a remarkable likeness to my thoughts about any topic in philosophy." In contrast to his approximate contemporary, the philosopher, logician, and mathematician, Gottlob Frege, whose concept schrift-- Begriffsschrift-- a formal language for pure thought modeled on that of arithmetic represented a revolution in logic, but also had the effect of banishing pictures, even diagrams. Peirce insists not simply on the utility, but also the necessity of diagrams. The stakes here are high. Moreover, they can be framed in terms that a medieval theologian would have understood. Namely, what purchase do the imaginative faculties, which operate with images, have on truth? Just as medieval thinkers disagreed when it came to this issue, with iconoclasts representing the radical skeptics, so too modern philosophers and students of psychology have swung from one pole to another. In the words of one commentator, "we live at the end of a period, which perhaps more than any other, has hidden in the pictorial life of the mind from intellectual view." Philosophy in the mid-century regarded sense data as fictions, arrived at by bad inference, and suitable for disposal as an undergraduate exercise. It was seriously maintained that all inner representation was propositional." That's the end of the quote. Peirce's view could not have been more different. In his essay on improving reasoning, Peirce explains, "By 'graphical' I mean capable of being written or drawn, so as to be spatially arranged. It is true that one can argue viva voce, but I do not believe one can go very deeply into any important and considerably large subject of discussion with calling up in the minds of one's hearers mental images of objects arranged in ways in which time, without space, is incapable of serving as the field of representation, since in time of two quite distinct objects one must be antecedent and the other subsequent." In insisting on this spatial dimension of thought, Peirce again echoes Aristotle, who speaks not of mental pictures, but rather of diagrams. Having affirmed that, quote-- this is Aristotle speaking-- "it is not possible to think without an image," the philosopher observes, for the same effect occurs in thinking as in drawing a diagram. In comparing cogitation to the process of drawing, Aristotle goes beyond asserting that a diagram represents the content of thought. Rather, it is the procedure of producing the diagram which resembles, and even enables, the process of thought. Both, in his view, involve a step by step method of defining and drawing relationships that point towards particular conclusions. If drawing a diagram and the process of thinking represent two sides of the same equation, then it is not simply a matter of thought generating diagrams, but also of diagrams generating thought. Rather than a representation, the diagram structures the patterns according to which one thinks. The diagram is more than a mere representation. It is an active, or operative. This understanding of the diagram, not as an illustration but as a tool for thinking, and in religious contexts, a vehicle for transporting the soul to God, was of major importance in the Middle Ages. When thinking about medieval diagrams, Peirce provides a useful point of entry, and not only because he assigns them such importance. Peirce employed diagrams not simply to map out a pre-cogitated philosophy of signs, but to generate it in the first place. Peirce famously discusses the diagram in the context of his complex differentiation of signs into three overarching categories-- icons, indices, and symbols. In discussing specific instances of icons, Peirce defines the diagram in scholastic terms, which portray an obsession with triads worthy of the medieval theologian struggling to understand the Trinity. Just as theologians spoke of hypostases to refer to the persons of the Trinity as individuated instantiations of the Godhead, so Peirce-- and I love this detail-- he converted from Unitarianism to Trinitarianism-- called his three classes of individuated icons, hypoicons. "Hypoicons may roughly be divided according to the mode of Firstness-- that is, the pre-reflexive immediacy of which they partake. Those which partake of simple qualities or First Firstnesses--" he sounds like Duns Scotus-- "--are images-- those which represent the relations, mainly dyadic, or so regarded, of the parts of one thing by analogous relations in their own parts, are diagrams. Those which represent the representative character of a representamen by representing a parallelism in something else, are metaphors." Peirce can be heavy going. Peirce outlines a hierarchy of connectivity. Whereas images involve, in his view, one-to-one relationships predicated on shared qualities-- a kind of phenomenology-- and metaphors, a higher order set of parallel relationships that are mediated without any actual correspondence among their component parts-- a kind of metaphysics-- a diagram, which, in his view, belongs to the realm of science, charts forces that, like vectors, require that a given object and its sign possess an analogous structure of internal relationships. Peirce sought to make his system, which ultimately grew to embrace a 66-fold classification of signs, all-encompassing, yet is betrayed by the personification of epistomy, sketched in the margin of one of his manuscripts, his system was as much a product of compulsive obsession as of distant speculation. In laying bare the contradictions and cracks, even the breaking points of his classification, Peirce's marginalia aped the function of the often unruly, ribald, and shape-shifting images in the borders of medieval manuscripts. Oops. There they are. Whether or not one regards them as revelatory of the philosopher's unconscious concerns about the stability of his system, his drawings are symptomatic of tremendous emotional, as well as philosophical struggle. Sorry. I'm now just going to go back to where I wanted to be. Peirce's diagram of IT, part of the triad I, It, and Thou, an early attempt to reduce Kant's 12 categories of understanding to a more manageable three, once again, which he drew as he was working on his treatise, The Natural History of Words, provides a compelling example of diagrammatic contingencies. Less a table or an octagon, both words that have been used to describe it, the diagram suggests a whirling St. Catherine's wheel of words and vectors, whose centrifugal forms fly out from the center. Rather than charting the infinite qualities of quantity, indicated to the right, the spinning forms suggest a spiraling centrifuge. In fact, the drawing's forms reflect the circular blade structure of the turbine, a new technology whose forms fascinated Peirce-- this is Peirce's drawing of a turbine on the left-- just as they did his contemporary, Henry Adams, who, in perhaps the most famous chapter of his autobiography, The Education of Henry Adams, the Virgin and the Dynamo, describes the great machines at the Chicago exhibition of 1900, in Chicago, as a symbol of infinity. Adams continues, "As he grew--" he writes about himself in the third person-- "as he grew accustomed to the great gallery of machines, he began to feel the 40 foot dynamos as a moral force, much as the early Christians felt the cross." In its linking of categories of perception to vectors of motion, Peirce's drawing is uncannily reminiscent of Klee's diagram, Ich Du Erde Welt. Both embody precisely the kind of metaphorical ambiguity that Peirce's so-called Existential Graphs sought to eliminate. On the one hand, Peirce's diagrams reveal a determination to impose order on the unruly signs of the world-- I'm sorry-- on the unruly world of signs. On the other hand, they suggest a deep-seated fear of disorder. This double-faced character, rational and irrational, finds its match in medieval diagrams and marginalia. And I've deliberately put up an example of phallic marginalia, because the image at the top, by Peirce, which is a parody of his own existential graphs, also contains similar phallic imagery. In the Middle Ages, diagrams acted not only as instruments of pedagogy, they also served as a means of recording, if inadequately, prophetic and visionary experience. In other cases, in keeping with their generative as well as their representational capacities, they further provided a matrix for such experiences. Diagrams supplied an armature within which authors and artists could, to use the medieval term, figure those things that could not be captured by words and images alone. To view diagrams as instruments of mystification might seem like a contradiction in terms. Like their modern counterparts, medieval diagrams demonstrate. As soon, however, as one speaks of demonstration, a contradiction arises. The purpose of a demonstration is to make something clear. Yet an image requiring decipherment can also be used to veil the truth. Diagram's potential to clarify often, if not always, comes at the expense of metaphorical misrepresentation. Does history, or do concepts, really unfold like the ramifications of a tree? Is the Trinity really like a triangle? Even if one could draw such a figure, and that is precisely the point, how is God like a circle, whose center is nowhere and whose circumference is everywhere? In this double character of the diagram lay a large part of its appeal to medieval theologians and exegetes. Metaphor, as Umberto Eco points out, referring to Aristotle's theory of the trope, has the capacity to establish new ontologies. Whether overtly figurative or merely suggestive of figuration, the medieval diagram participated in a dynamic dialectic of revelation and concealment, one in which the viewer was invited to participate. To look at a theological or mystical diagram was not only to be instructed, it was also to be initiated. Diagrams not only make visible. Their abstract, etiolated forms hint at what remains invisible, beyond representation. With this turn, one could object, or observe that, at a fundamental level, a diagram is not a representation at all. One could even go so far as to say that diagrams ultimately have no object or referent in the world. Rather than reflect the world as it is, they first identify, then rearrange its parts into coherent configurations, that both reflect and enable, shape and structure, our patterns of thought. There is therefore a poetic dimension to the diagram. Like a metaphor, if less open-ended, a diagram posits a correlation among disparate elements, or figures of those elements, in order, not simply to reveal, but rather to create a tertium comparationis. It is, of course, impossible to press all diagrams, or even all medieval diagrams, into a single mold. Diagrams remain part of the culture and historical horizon which produced them. For the purposes of this lecture, I therefore propose presenting, very briefly, a set of diagrams created in the early 1290s, by the Dominican Friar, Berthold of Nuremberg, on whom I'm currently completing a book. According to a pair of authorial subscriptions, one at the end of the first and second parts, the first dedicated to the cross, the second to the Virgin Mary, the work was written between 1292 and 1294. In its complete form, it survives only in a single manuscript, the presentation copy. Despite the author's name, the style of the illumination points to the upper Rhine, in particular the region of Lake Constance. One of the most ambitious diagrammatic works from the entire Middle Ages, it ostensibly serves as no more than a supplement to the famous treatise by Robanus Maurus in honor of the Holy Cross, the celebrated collection of picture poems composed shortly before the year 810. Taking Robanus's magnum opus as his point of departure, Berthold traces the entire history of salvation, from Adam and Eve to the end of time, in an astonishing 120 sections, each of them prefaced by a small illustration. So at left you have the original work by Robanus Maurus, which in this manuscript is a 15th century copy, which can be shown to have replaced a 13th century original, that was there from the outset. And then you have part one of Berthold's work in which Robanus's cosmology is reorganized to tell the story of salvation in chronological order. And then in Berthold's part two, on your right, the third part of the whole, the commentary on the Virgin, her life and that of Christ, are spelled out in greater detail. By setting salvation history in the context of typology, his work is of a piece with the great typological handbooks of his day. The Biblia Pauperum, the Speculum Humanae Salvationis, and the Concordantiae Caritatis, not to mention various scholastic Marian compendia. Both Robanus and Berthold represent radical shifts in what we are accustomed to thinking of as an image. Whereas the one shifts image in the direction of text, the other shifts image in the direction of diagram. Despite its difficulty, Robanus's commentary on the cross was copied and admired throughout the Middle Ages. Written in the early 9th century, at a time when forms, such as the figured and-- oops. What am I doing here? When forms, such as the figured and historiated initial, which combined word and image in ways that were to become integral to the formal and expressive repertory of medieval art, remained of recent vintage, Robanus's work provided a benchmark of unrivaled complexity for its integration. Its reception thus provides a measure of difference in medieval aesthetics, theology, and piety. Berthold's religious commitments as a Dominican Friar, and his attitude towards images, differed dramatically from those of his Carolingian predecessor. These differences chart an epic-making shift from cross to crucifix, from a simple, stark, all-embracing sign, to a compelling, heart-rending, and ultimately abject depiction of the human body, shown hanging, not from a sign but, as in this case, from a green, living cross. The differences between Robanus's original and Berthold's re-interpretation chart the origins and development of figuration in medieval art, a passage from cross to crucifix, from book to body, from disembodied sign to embodied image, that in itself, and in keeping with Christian justifications of anthropomorphic representations of the Godhead, fulfills and demonstrates the doctrine of the incarnation. More than just signs, diagrams also embody technique. Not just knowledge, but ways of knowing. In light of the processual nature of the diagram, any adjustments to Robanus's complex, text-image amalgam therefore testify to shifting epistemologies, not to mention shifting attitudes towards images in particular, and visual experience in general. Whereas Robanus's work revolves around, and is in part generated by, intricate numerology, having no center other than the cross itself-- witness on the left his doubling of the seven gifts of the Holy Spirit, so as to be able to rearrange them as a symmetrical cross-- Berthold's is thoroughly historical, and hence, paratactic in structure, tracing the history of salvation in terms of typology from beginning to end. Whereas the disciplined construction of Robanus's work presents itself as a revelation of a temporal truth, whose validity is underscored by an abundance of number symbolism, Berthold's sprawling structure, more devotional and theological, accommodates a gradual unfolding of human events in historical time. For Robanus, locking his figurations within diagrammatic arrays of words disfigures the image, stripping it of its immateriality. For Berthold, fleshing out diagrams with figuration enhanced their ability to affect the truth of the incarnation. Development introduces the danger of teleology, a progress from abstraction to anthropomorphism. Medieval art, however, relied on multiple systems of representation, the figural as well as the abstract, the narrative as well as the hieratic. Similar cross-contamination persists in modern scientific diagrams. Witness Peter Galison's distinction, in scientific imaging, between the homologous-- that is, retaining logical relations within that which is represented-- and the homomorphic-- retaining the form of that which is represented. Those are his definitions. The re-emergence of the diagrammatic in the modern avant garde, in which figural modes of representation were largely abandoned in favor of various forms of abstraction, provides further insight into the forces at work in medieval diagrams. In diagnosing the diagrammatic aspects of Dada, David Joselit has spoken, and I quote, "of an historical rupture between the textual codes of the book and the visual codes exemplified by cinema, constituting the ground against which Dada's spectacular heteroglossia emerges." According to Deleuze and Guattari, whose concept of the rhizome is fundamentally diagrammatic, albeit by definition in the manner of a distributed nodal network, not a tree, the diagram does not function to represent even something real, but rather constructs a real that is yet to come, a new type of reality. In this definition, the diagram acquires a particular form of agency. One might thus ask, is abstraction an appropriate term applied to medieval diagrams? To apply the classic formula of E. H. Gombrich, making comes before matching. Medieval diagrams, again, make and model of the world, the only difference being that there ideations refer not to a world to come, except perhaps in an eschatological context, but rather to its prototypical existence in the mind of God. that Robanus's picture poems are all structured by the exemplary Christian diagram, the cross, they also exhibit a spectacular heteroglossia. They function simultaneously as poems, pictures, and diagrams. Their origins can be traced, not only to the antique traditions of picture poems, and opus geminata, prose and poetry combined, but also to the earliest Christian representation of the crucifixion, the staurogram, forged from parts of the Greek words for cross and crucify. In some cases, Robanus imposes diagrammatic elements on the poems in the form of geometric or figural shapes. In other cases, the shapes themselves take the form of letters pregnant with meaning. In translating Robanus's treatise into the diagrammatic mode, and in literally drawing it out at such length, Berthold disturbed its delicate balance between word and image, separating the two, and employing independent images in the form of colored diagrams, many accompanied by figural elements. The differences in formal and theological presentation between the two authors, one famous, the other forgotten, defining two ages-- the first operating in the aftermath of Byzantine iconoclasm, the second in a moment of unchecked enthusiasm for images. Both works are serial in structure-- Robanus's with 28 parts, Berthold's with 120. But each is organized according to different principles. In the latter, sacred history is not simply represented, it is effectively reenacted, using a medium which itself incorporates a temporal and corporeal dimension, as a result of the diagrams being placed in sequence, and because each diagram implicitly involves a series of connections being drawn in both material and ideational sense of the word. In this sense, Berthold's diagrams could be said to evolve in the manner of a mathematical proof, not only each one in and of itself, but also in combination with one another, as a series that both retraces and maps out the course of salvation history, from past to present to future. Words used to describe the diagrammatics of computer graphics applied just as well to Berthold's method. Quote, "Here, syntactic possibilities extend into time and space as they define transposition operations that facilitate the dynamic exploration of information. Information is entered, or played against time. The analyst enters the diagrammatic space, and what was formerly a snapshot now becomes a flow of information." That's from an article about computer graphics. As an historical form predicated on dyadic relationships, typology is fundamentally diagrammatic in form. Figura, one of the terms most commonly employed in medieval sources to designate diagrams, carries within it the connotations of the typological figure, which merely shadows the truth, which, in the fullness of time, will be fully fleshed out. Berthold effectively fuses the two connotations of the term-- the one, a geometrical figure, the other, a prophetic image. In keeping with the discourse that identified Old Testament prophecy as a mere foreshadowing of Christian truth, and the incarnation as the ultimate authorization of images, the diagram prefigures and points towards the visible and necessary reality of Christ's coming into the world. The first diagram of the Marian treatise offers an encapsulated lesson in typology. It is also modeled on that most familiar of logical diagrams, the square of opposition. Rooted in Aristotle's distinction between contradiction and contrariety, the square constituted a fundamental part of medieval teaching on logic. Four concepts or propositions are placed at the corners. The top pair, defining the squares upper horizontal, represents contrary terms. The bottom two, the subcontraries. Each term at the bottom stands as the subaltern in relation to the term directly above it along the vertical sides of the square. The square was applied to such standard topics as the relations of propositions, the construction of syllogisms, the mathematics of musical intervals, and the relationship of elements within natural philosophy. In other words, it could be found throughout the syllabus. The square, quite, simply became part of every student's toolkit. Diagrams such as the square are neither illustrations of the text which they accompany, nor mere adjuncts to argument. Rather, they are tools with which to think, and, in Berthold's hands, with which to create. In this example, Adam, at the upper left, is the partner, yet superior of the subaltern, Eve, who by virtue of having been formed from Adam's rib, could be thought of as literally being but a part of her male progenitor. In the case of Christ and Mary, the diagram posits the same relationship, although in this instance, just how one represents the particular, the other the universal, remains unclear. The efficacy of the diagram, however, extends quite far. Adam supplies the contrary of the new Adam, Christ. Moreover, Adam and Mary like Christ and Eve, are contradictions of one another. Transecting the simple yet elegant image, which is rooted in the chiastic structure of the square of opposition, the bipartite tree easily suggests both the tree of life and the tree of knowledge. By using logical diagrams as his model-- others are based on the Tree of Porphyry-- Berthold lends his diagrammatic history of salvation the force of a logical demonstration. Moreover, he does so in keeping with the pedagogic traditions of his order. The square can be found among the 29 illustrations in a little book for the consolation and instruction of novices, dated around 1300. Extant at a single early 14th century manuscript from the important Dominican library at Toulouse, its text nonetheless indicates that it was approved in 1283 at the general chapter of the order in Montpellier, suggesting wider distribution. The anonymous author elaborates his diagrammatic didacticism. "I therefore paint here certain images I made so that desolate novices might find consolation in them and so that a little flame of newly fervent and more devout devotion might be kindled making perceptible matter from celestial signification." The images, the diagrams included, are intended to ignite devotional desire. The novice is instructed in great detail in these necessary arguments, not simply how to read, but even how to lay out the diagrams, step by step. What follows, however, are not instructions in how to use the diagram to solve problems in logic, but rather how to apply a moral gloss. And this becomes clear in the subsequent illustration, in which process is key. The product of this procedure is this the second diagram, whose construction and meaning the text explains at length. Whereas the text asks the reader to populate the preexisting diagram with scenes from the parable of Dives and Lazarus, the miniature fleshes out the image right before his eyes, within an armature that structures the process of interpretation. And the relationship between the various parts of the parallel is mapped onto the square of opposition. And I won't read you the text at any length, because the moralizing is, as usual, rather dreary. Other works of the period, Tomasin von Zerklaere's Der welisch gast, a didactic work in the vernacular addressed to a literate, aristocratic readership, likewise assumes easy familiarity with the square. You see an image from one manuscript here. And so, to come full circle, and I am approaching my conclusion. To come full circle, we also find the square embedded within the sacred genealogy represented by Peter of Poitiers' Compendium, here in the same manuscript I showed you at the beginning of this talk. And I don't think it's been noticed before that Peter also makes use of the square of opposition. Techniques of representing logical and genealogical relationships overlap in Peter's representation of the complicated line of descent from Solomon. To clarify these tangled relationships, and to make them easier to remember, Peter provides a diagram based on the square of opposition. Nathan and Melchi, the two husbands in that order of Este, placed at the center, occupy the roundels constituting the upper right and left corners of the square. Whereas the sons of each of those unions, Jacob and Eli, respectively, occupied those at the lower right and left. Nathan and Jacob represent the direct line of descent indicated by the thick red line, which connects Nathan to his ancestor, Solomon, above, and to his descendant, Joseph, below. The horizontal bar connecting the two fathers reads, "connected because killed," a reference to Melchi having succeeded Nathan, following the latter's death as Este's wife. Mapped onto the square of opposition, their relationship is contrary, insofar as they cannot have been married to the same woman simultaneously. The bar connecting Eli and Jacob, the two sons, reads, uterine brothers, identifying them as stepbrother's by the same mother. Mapped onto the square in the same fashion, their relationship, analogous to that of their fathers', is subcontrary. The vertical lines connecting fathers and sons read, carnal sons, the equivalent in the square to subalternation. Finally, the diagonals linking the two fathers via Este to the two sons, read, conjugal son. These represent the lines of contradiction connecting the corners, in that Nathan is not the father of Eli, and Melchi, in turn, is not the father of Jacob. And if you don't remember that, the square of opposition would provide you with a handy little mnemonic device in order to do so. It is time to steer toward some conclusions. What better way to work them out than in an imagined diagram? I envisage not a line, leading either forward from the Middle Ages to modernity, or backwards from the present to the past, but rather a parallelogram, in which analogous relationships undergo similar shifts, if to different ends. Whether medieval or modern, diagrams mix modes of expression, visual and verbal. Diagrams are not only conceptual, they are also material. Process is not only part of their making, but also their meaning. Diagrams do not simply reflect or embody ideas, they shape them. And beyond ideas, they move like machines, to encourage action. Diagrams are relational. And very often, as in the case of Robanus and Berthold, they are serial. As for Berthold, his way of thinking about the history of salvation, typology, which pairs Christian event to Old Testament prophecy, and uses geometry to lend those pairings the character of a demonstrative proof, is fundamentally diagrammatic. Whether looking at modern or medieval art, one can ask the same question. Why resort to diagrammatic modes of representation in the first place? The answers, whether to demonstrate cosmic harmony, provoke visions, understand the prophetic course of salvation history, or, in the case of more modern examples, to disrupt established social codes, will differ dramatically. One thing, however, remains constant. The diagram, on its face a cold abstraction, permits one to delve into the desires that informed its makers. In this respect, diagrams themselves map what is perhaps the primary point of intersection between the technical and the cultural. Icons, once a source of mystery, have become as commonplace as a computer screen. The internet and the worldwide web purport to displace traditional centers and hierarchies in favor of rhizomatic, distributed networks-- in effect, diagrams or systems best understood in terms of diagrams. Of this desire disruption, bitcoin is a paradigmatic example. Taken together, such developments have transformed not only academia, but also society. And in the age of social networks, human actors themselves have become diagrams, defined less in terms of a stable, autonomous self-- if any such thing ever existed-- than of a system of relationships that go by the name of networks. It is no less the case that the algorithm, which even if not a diagram in itself is best understood in diagrammatic terms, has, in a manner of speaking, become the platonic form of everyday existence, both underlying and informing the reality of contemporary life. Diagrams are central to all those disciplines which deal with the visualization of knowledge, as well as, no less important, the critical discourses that consider the impact of modern media on ourselves and on our society. As more and more forms of social interaction become networked, and as all human activity comes to be defined in terms of links within a larger web, our world and ourselves increasingly come to resemble the mechanics of a diagram. What, one might reasonably ask, does any of this have to do with the Middle Ages? One way in which medieval art resonates with the present is that much, although not all of it, is fundamentally diagrammatic. In lieu of linear or continuous narrative of a kind brought to perfection in antiquity, medieval art, so much of which is predicated on typology, lends itself to demonstrative, didactic, and diagrammatic modes of image making. The complex armatures of stained glass windows provide only the most obvious example. Cathedral facades are another. Medieval authors and artists seized upon traditions of scientific illustration rooted in antiquity to create complex representations of Christian cosmology. These, in turn, often imbued ornament with cosmological significance. Medieval art is not simply symbolic, abstract, an anti-naturalistic, it is structured and systematic in its approach to representing a world which it perceived as being highly structured in itself. History, far from being open-ended, had a structure-- one that could be investigated and manifested in the language of geometry. There was, however, no single, medieval Weltanschauung. Multiple theologies, cosmologies, and epistemologies competed, and continued to compete to claim the mantle of just interpretation. It is within this contested field, modern as well as medieval, linking diagrams to thought in the search for truth, that the subject of diagrams and the diagrammatic in the Middle Ages and-- sorry. It is within this contested field, modern as well as medieval, linking diagrams to thought and the search for truth, that the subject of diagrams and the diagrammatic in the Middle Ages and beyond becomes so compelling. Thank you. [APPLAUSE] I realize time is getting on, but I would love to entertain some questions, if there are any. Don't be shy. And I'll just say, there are microphones to use. I started with this image of a medieval classroom with two Greek students discussing the proverbial problem of squaring the circle-- it's from a 13th century English manuscript-- to picture the dialogue that I hoped we would have. So you can't disappointment me. In your last slide, if you could go back to you last slide? By all means. Just a moment. Actually, this was my last slide. But we can go. This one. No, the square. With a cross. This? Yeah. On the left. Is there is a story embedded in the square? It's not prose. It's a poem. In one of the many Latin meters that Robanus Maurus employed. And the way these picture poems work, it's quite mind boggling-- I think of them as a combination between the world's most complex crossword puzzle and sudoko-- is that the text can be read continuously from left to right, top to bottom, the way you would read a normal text. And it reads as Latin verse. But you can also read the so-called in-text. That is, the text within the figures-- in this case, a simple cross-- as independent verses in their own right. This is one of the simpler examples in the work. In other words, some letters serve a double purpose. They make sense within two different semantic units, which are superimposed one upon the other. And very often, the arrangement of words and letters is governed by very complex numerological considerations. It's quite mind boggling, how anyone could produce such a thing. But as we know from concrete poetry, and poets like Mallarme, or Apollinaire, there's a long, long tradition, not only in the west but also in the Greek tradition, and in other cultures as well, of picture poems. Poems that double as text and as image. A famous example within English literature would be George Herbert, in the 17th century. If you look at some of the poems in his collection, The Temple, they take the form of picture poems. Does that answer your question? Yeah. It reminded me of a word crossword, where you have to find the word with-- Yes. Well. These works are not designed to be understood at first glance. They're designed to be meditated on at great length, in true monastic tradition. Irene, did you have a question? I'm pondering it still. When you discussed Peirce-- Sorry. When you discussed Peirce, you used the word, model. And how a model relates to a diagram. Might be, perhaps, unpacked a bit further by you. But also, it occurred to me that there is a difference between a model and a metaphor. And to the extent that a diagram can be read as metaphor or as model, it would make an interesting distinction. I wonder what your thoughts are. Yes, well, I am not going to be categorical. Because in some ways I'm more interested in the slippage between those two types, or categories. I think the case of Peirce-- and I don't claim to have read everything that Peirce wrote that is directly relevant to the topic, let alone have understood it-- but I think in Peirce's case, he very much wants it to be model or form, but very often it slips into metaphor, in ways that he finds deeply unsettling. One of the things I find so fascinating about the drawings in his manuscripts, which have only recently begun to receive any kind of attention, as in so many other cases, editions of texts, the drawings, the diagrams, the marginalia, simply get edited out. But one could make the case that they are, as it were, the unconscious of his text. Although I think he was very much aware of the problems. And so the drawings, as I said, I think produce metaphoric nightmares for him. Even though he insists that he dreams only in algebra, which I find one of the most remarkable statements I've ever read. Obviously, I could go further. But I think that debate as to whether diagrams are models and metaphors, is in part, for example, the debate that has played out in the history of mathematics, from the mid-19th century. And in both mathematics and philosophy, there was a desire exemplified by logicians such as Frege, and mathematicians such as Hilbert. And I would never claim that I understand their mathematics. But they rejected geometry and diagrams 100%. They wanted to convert all of mathematical representation into algebra. And since-- I would say-- well, it's really co-equal with the advent of computers. Diagrams have, of course, made a tremendous comeback, not simply as practical tools, but also as legitimate vehicles for philosophical and mathematical analysis. And so there is this debate amongst mathematicians, historians of mathematics, but also cognitive scientists, as to the validity of diagrams. To what extent they represent thought processes, model thought processes, or are merely metaphors for thought processes. I don't claim to have the answer. Yeah, Robin. Thank you, Jeffrey, for that wonderful talk. It may be fanciful, but I've long wondered whether part of Peirce's addiction to the triadic notion of semiosis was related to the fact that he spent decades working for the US Coastal Survey, and engaged in triangulation as a matter of daily practice. So I'm wondering if you have any speculations about the relationship between the diagrammatic impulses in the Middle Ages and to other practices outside these beautiful works that you're showing us, whether liturgical practices or what have you, that might shed some light on the diagrams we're seeing? Thank you for that wonderful question, Robin. I'm trying to think on my feet. In Peirce's case, he was certainly obsessed with triads, which is why I relish that detail of his conversion from Unitarianism to Trinitarianism. I'm not sure what is cause and what is effect. Although the phenomenon you point out-- surely those years of working in the Survey ingrained certain habits of mind, that could have had a powerful impact, or formative impact, on his patterns of thinking. Your question makes me think of a concept rather like Baxandall's "period eye." For those of you who've read his book on art and experience in Renaissance Italy, there's a famous passage in that book in which he talks about barrel making, and barrel gauging in Renaissance Florence. And he argues that the way in which 15th century mercantile patrons would have sized up, to use the term advisedly, would have sized up the volumetric representation of objects in three dimensional space and in perspective was conditioned by the tools of their trades. I mean, one of the problems with Baxandall is that you never quite know where to draw the line in defining the period eye. In terms of the-- guess the case that I'm making here, and more extensively in the book I've written, is that that practice is logic. And if one reads the history of diagrams in the Middle Ages, there is some attention given to logical diagrams. For example, if you read John Murdock's amazing book, The Album of Science-- he was an historian of science here at Harvard-- still, by far, the best book on medieval diagrams. He has a chapter devoted to logical diagrams, but there is no attempt to think of logic diagrams and the patterns of thought that they represent as exemplary, or as models for other systems of representation. Whereas part of what I want to argue, and I tried to demonstrate it briefly, is that you have an historian and theologian, like Peter of Poitiers, who's using logical diagrams as a way of structuring his works. You have Berthold of Nuremberg who's making extensive use of porphyrion logic diagrams, and the square of opposition, to structure his diagrams as an exegetical tool. So I don't I don't know if the realm of logic and logic diagrams is far enough outside, to answer your question. But that's the answer with which I'm comfortable at this point. I think part of what I want to do is argue that logic, and the modes of thought and representation associated with it, are paradigmatic when it comes to diagrams, to a greater extent than has been argued. So for example, if you think of the famous diagram of the triangle representing the Trinity. That is a logic diagram, because the various corners are connected by the words, est, non est-- is and is not. So it's dealing-- it's a kind of triangle of oppositions. So that's about the best answer, for now, I can get to your question. Yes. I believe, in the case of logical diagrams, it would be appropriate to distinguish the mnemonic tools, if you like, like logical square, and on the other hand, in your presentation, you had also Venn diagrams. Venn diagrams, semantic tableaux, kernel maps, are more than visual tools. They are demonstrative, and deductive, it you like, artifacts that can be used. So this is a distinction. And mind maps more for, if you like, brainstorming and creative usage. Which are also-- Yes. Yes, I think that's an important distinction. But of course, one of the reasons why-- if I remember my history of logic correctly-- one of the reasons why Peirce worked so obsessively on his existential graphs, is because he felt that there were problems with Venn diagrams, and he wanted to improve on that system. And recently, there's an historian-- not an historian-- a logician at Yale who has written on Peirce's existential graphs, with a view to demonstrating, or trying to demonstrate to her satisfaction-- I've forgotten her name for the moment-- that Peirce's system actually is an improvement. And that it works. Again, I am not sufficiently sophisticated in order to come to any final judgment myself. But the fact that such a debate exists demonstrates that logicians themselves continue to grapple with the question, to what degree a diagram has probative value. For example, in the history of mathematics, there are debates about Euclid. And there is a lively literature arguing against those who have taken the position that Euclid's diagrams are problematic. That they do not actually prove the propositions they supposedly demonstrate. And yet recently, there are historians of mathematics who have made that case. Now as for the square of opposition-- and this is perhaps the last thing I'll say in response to your very important distinction-- the square of opposition was long written off as a kind of simplistic tool for novices in logic, despite the fact that, well into the modern period, it was of such importance in the syllabus and curriculum. But even the square of opposition is making a comeback. And there have been conferences with hundreds of contributions devoted to the square of opposition and its various applications. So without wanting to adjudicate beyond my ken, it's interesting that, as in the case of Euclid, so too in the case of the square. There are now some logicians who argue that a case can be made for it, in certain contexts. I hope that goes some way to answering your question. Yes, Dan. Well, let me thank you again for that wonderful and stimulating talk. I would like to pick up on a detail from Robin's question and push it in a different direction, because you began with a T-O map. And early on, you had a map of, I think it was from Matthew Paris, with a flap extending up. Yes. And it raised a question for me about, what is the relationship between the map and the diagram? Is one a subset of the other? Is there an easy overlap between those two? What other plans do you have this evening? Dinner. Yes. Me too. That's an exceptionally fraught question. Obviously, if you think of a kind of sliding scale, there's a point at which maps become diagrammatic, and at which some diagrams become cartographic. And they both deal with spatiality, in some ways that are related. Some people have said that, unlike diagrams, maps have real reference in the world. And again, I don't want to seem like a cop out, but I think that I would have to take it on a case by case basis. I mean, maps are fascinating in that, as we all know, there's something arbitrary about any form of cartographic representation. And that's why I used the comparison to talk about a kind of Weltanschauung. Medieval maps happen to be, for the most part, exceptionally diagrammatic. I'm not sure I would call Google Maps diagrammatic, although, if you look at the under-- if you could strip away the imagery, and look at the underlying software-- that again, without knowing what that is-- but I imagine that if you stripped the images away, and could just see how it works, whether it's based on algorithms or whether it's based on certain kinds of projective geometry, that it is fundamentally diagrammatic as well. Yes. Thank you very much. Actually, you just answered the first half of my question about the difference between diagram and map. I guess my other question was, what happened? What was the afterlife of the circling of the square? Of course, it is a very-- I haven't solved the problem. It's a very, very important figure for writers, such as Dante, of course, and so on and so forth. And it persists as a figure of thought into the Renaissance period. And I was just wondering whether it was absorbed, since it stopped being represented diagrammatically. Was it absorbed by some other media? Well, again, I seem to be answering every question by saying, I don't know. But I imagine that, if one looked through the very extensive corpus of school books and treatises about pedagogy and the seven liberal arts, that one might find and prints some representations of it. But that I don't-- that I don't know. Obviously, the notion of squaring the circle has become proverbial. To that extent, it's survived as the problem that is impossible to solve. Although, of course, calculus makes it a little bit easier. I should let all of you know that I virtually failed mathematics in high school. So I feel I'm really skating on thin ice. But that's also part of what has made this line of investigation so fascinating for me. You know, better late than never. So the image I showed you of the two figures engaged in debate, brandishing their squared circles, that's an unusual image. There is a whole iconography of the classroom in the Middle Ages. And Laura Cleaver has recently written an interesting book precisely on that subject. But it's not as if there was a pictorial tradition of illustrating that problem. Of course, there are many medieval images that consist of a circle or circles in a square. I could show you those ad infinitum. In multiple media. And that geometry is more than just a decorative device, or a framing device. Although frames are of critical importance in medieval art, it is a way of imposing a kind-- or implying a kind of cosmological order. I'm afraid beyond that, I'm not sure I can answer your question. Yeah. Gina? Thank you, Jeffrey. That was wonderful. And I am sure I can ask you a question that you will know the answer to. That is, sort of following up on the questions about, I guess, imagery, this diagrammatic impulse-- and you put up how figuration and diagrammatic impulse go together in Berthold's case. And I wonder if there is ever a sort of impact on artistic practice itself. That the images of figuration actually get, like diagrammatic impulse, translated into it, in a way? I'm just coming from a tradition-- thinking about a tradition that develops am iconometric system that sort of transposes that diagrammatic thinking onto the body? So human figures and figurative representation. Well, it's interesting. If we were to page through Villard de Honnecourt's so-called sketchbook-- it's not a sketch book. It's not even really a model book. You would find many stick figures based on geometric figures. And there, clearly, it's being used as a mnemonic device for craftsmen to generate figures, or to remember how to produce a certain figure. Architecture becomes very problematic, but I think it's fair to say-- setting aside symbolic geometry, which is often very difficult to establish, much as scholars like to project it onto buildings-- geometry plays a generative role in medieval architecture. And certainly, in medieval images. Just the tools they used-- ruler, compass. It lent itself to geometric armatures. But those armatures also served as mnemonic devices. But more important, I think, as I received way of generating variations on a theme. In the case of manuscripts, of course, you have the ruling on the page. And even the pages that have illuminations on them, as received by the painter, they're already ruled, because these craftsmen or scribe who did the ruling didn't know, in most cases, where the pictures would fall. And artists do use that ruling as a matrix for their pictures. Not always, but very, very often one can show, even in the later Middle Ages, where illuminators are using more naturalistic modes of representation, that they are employing the underlying ruling as a kind of pre-established graph paper, if you will. Yeah. Yeah, Rosha? Thanks, Jeffrey, for this inspiring talk. I'm so looking forward to the book. There's a solid Jeffrey bookshelf in my office. So, especially in the case of Berthold, as you have beautifully showed and argued, there's this shift towards anthropomorphization, from the cross to the crucifix. And it strikes me that Berthold's diagrams are so figurative, and that's part of his time. That's part of the development. I understand that. But my question is the underlying metric is still the cross, isn't it? So it still goes back to that. But then, how does script come into it? So is it also script that makes, especially when we look at this image, makes it a mnemonic diagram? So my question, I guess, is what's the relationship between figurative representation and script? Yeah. Well-- That's huge, I know. No, no. But a lot of these images do involve changing constellations of word and image. And as you say, that's huge. To indulge in a really gross generalization, you could say that the history of medieval art, certainly from the Carolingian period onward, in which text and image are fused in a variety of very formative ways, historiated initials first appear in the Carolingian period. These picture poems are another salient example. That the whole history of medieval art from that point forward involves a gradual teasing out, and separation of the two elements. So that by the Gothic period, for example, text and image have largely, if not entirely, been extricated from one another. Of course, you find historiated initials in the Gothic period, but not in the way that one would, say, in Romanesque manuscripts. And then, of course, by the 15th century, you can think of book illustration, in a way rather similar to the way in which we still think of book illustration, with text and image divorced from one another, but still related in some ways. To take this particular example, the inscriptions here are selected verses from the carmina figurata, from the picture poems of Robanus. Sometimes abbreviated, but they're actually incorporated in some cases in rather awkward ways. They're simply placed around the images, wherever he can fit them in. And that awkwardness actually attests to this disintegration of the two media. Yes, at the back? Yeah, hi. You said a thing earlier that I thought was interesting. You said you were interested in the slippage between metaphor and model. And it reminded me of-- you also mentioned Deleuze a lot, which I commend you for. Deleuze has this concept, you know, of abstract machines, where you sort of take a concept, or a logic that normally wouldn't apply in another area, and you map it onto another area, just to see what you sort of get out of it. And so I'm wondering if there are any historical examples of people doing that sort of a thing. Not like concepts, like Deleuze does. Like, you know, he'll use concrete. But people forget the concrete was in liquid form before it's formed, and so people will be like, this isn't concrete. That sounds like fluid. Do people do anything like that, that you know of, with pictorial diagram explicitly? To be productive with their diagrams, in that sort of a way? If that makes sense. Medieval thinkers could be very, very playful in speculative ways. You know, witness the discussions of what would have happened if you assume the complete free will of God, and if he had come to save mankind in the form of an ass, instead of a man. So they weren't beyond entertaining wild hypotheticals in order to make certain kinds of arguments. They occasionally got in trouble for that. But I think to that extent, Deleuze and the typical medieval thinker are really very, very different. In that, as I understand him, Deleuze is always interested in that kind of disruption and in breaking down of hierarchies. That's why he attacks the tree as opposed to the rhizome, and the tree is perhaps the archetypal metaphor diagram of the Middle Ages. There's a wonderful article by-- or essay by Umberto Eco, by the way, called, "From Tree to Labyrinth," which explores very much the same set of ideas, in, I think, very brilliant ways. I was thinking-- if I could help at all-- Sure. Because Deleuze says that the point of philosophy is to-- like, what philosophers do, and the point of philosophy is to create concepts. And he has a few parts where he also says that sometimes you need a tree and other times you need a rhizome. So I think he's doing more of like a differential, or binary sort of thing there, than just privileging one over the other. And so I guess I was just wondering if there would be a medieval example of someone taking, just, like, some crazy-- I don't know-- botanical drawing, and then saying like, well, now, let's see how this explains-- Well I think-- Makes sense of something else. I think you could make the case that a medieval thinker or artist could press just about anything into a diagrammatic mode. And again, speaking in terms of gross generalizations, I would argue that, as opposed to narrative, as opposed to even allegory, that the diagrammatic is perhaps the dominant mode in medieval art. And I'm not the first to say that. Others have pointed that out as well. They've also made the connection with typology as a mode of thought. I'm not sure I agree with you completely about Deleuze, although I expect you're more widely read than I am in his works. But if you read that introductory section of his text on the rhizome, it's just a scathing attack and criticism of all forms of, you might say, arboreal thought. I think it's meant to be deeply, deeply subversive, in a way that I think is unthinkable for most medieval authors. But certainly, in the same way that, you know, in the Dada diagram that I showed you. If you read Joselit's analysis of those diagrams, he makes the case that their agenda was, again, fundamentally subversive and disruptive of established social norms. That's just not the way that medieval art works, in most cases. I nonetheless think that it's very instructive to apply the thinking of some of these modern philosophers to the medieval works. And vice versa. Yeah. Thanks. I don't want to keep anybody from their dinner. Yes. One more question at the back? A very short question. You mentioned shortly-- or you spoke mostly on two-dimensional diagrams, and mentioned shortly the three-dimensional diagrams. And I was wondering what about the fourth dimension? What about the movement or the temporality? How does that come into play in reading diagrams, or in creating diagrams, or could even a movement be part of a diagram? That would be my question. Absolutely. In terms of the thought process itself as being processual. In terms of the making of the diagram, obviously, there's a temporal aspect. And in the Dominican treatise that I referred to, at the end, it tells you step by step, how to draw the diagram, and it could have been done in a different way. But he allegorizes the process of drawing it. Having told you how to do it, because he wants you to think of its construction in a particular way. The element of time is what fundamentally distinguishes Berthold's rewriting of the work. You could say, again speaking crudely, that whereas Robanus's work is centered on the cross, and everything means the cross, and in a sense it's atemporal, it's eternal, Berthold's is put into the model of salvation history. And that's why he has to completely reorder Robanus's structure. Because, as a 13th century Dominican, he's interested in salvation history, and in time, and in history in ways that Robanus simply wasn't. And he's interested in images in a way that Robanus couldn't be, because of the context of iconoclastic debates. And then, as I tried to show in the Peter of Poitiers, where you have a timeline, the temporal element is inscribed within the work. But I don't even think you need to stop with four dimensions. I mean, since I mentioned David Hilbert, you know, he's most famous for so-called Hilbert spaces, which can involve an infinite number of dimensions. Which I gather is why Hilbert spaces are of such interest to physicists and string theorists. Don't ask me any questions. But there you're dealing with, as it were, diagrams in an infinite number of dimensions, of vector spaces, and an infinite number of dimensions. And now I'm really talking about something I know nothing about. So thank you very much to you all. [APPLAUSE]



The route has two alternative midsections, reuniting in north-east England and northern Belgium: the first devised route includes Oxford, Reading, Brugge and Antwerp and does not include the southern part of the Dutch coast. The corollary midsection, listed as the alternative route, includes Hull, Lincoln, Cambridge and Bergen op Zoom.

In Belgium the route starts from Ostend on the GR5A (Grote Route 5A) via Brugge and Antwerp. After a few kilometres in the mid-northern municipality (district) of Zoersel in Belgium it joins the GR 5 (Grande Randonnée 5), which forms the remainder of the route, passing through Luxembourg, Eastern France, then Switzerland at Lake Geneva before returning to France to finish at Nice.

The highest point of the route is at Col de l'Iseran at 2,770 metres (9,090 ft).

Sign on the path of the E2 near Vianden, Luxembourg
Sign on the path of the E2 near Vianden, Luxembourg

Great Britain

The Irish section of the route is not yet open.[1]

In Scotland the route starts from Stranraer and follows the Southern Upland Way, to Melrose, and St. Cuthbert's Way, from Melrose to Kirk Yetholm.

In England the route follows the Pennine Way, from Kirk Yetholm to Standedge, the Oldham Way, from Standedge to Mossley, the Tameside Trail, from Mossley to Broadbottom, the Etherow/Goyt Valley Way, from Broadbottom to Compstall, the Goyt Way from Compstall to Disley, the Peak Forest Canal from Disley to Marple, the Gritstone Trail, from Disley to Rushton Spencer, the Staffordshire Way, from Rushton Spencer to Canock Chase, the Heart of England Way, from Cannock Chase to Bourton on the Water, the Oxfordshire Way, from Bourton on the Water to Kirtlington, the Oxford Canal, from Kirtlington to Oxford, the Thames Path, from Oxford to Weybridge, the Wey Navigation Path, from Weybridge to Guildford, and the North Downs Way, from Guildford to Dover.

In England, the alternative midsection begins on the Pennine Way at Middleton-in-Teesdale. It then follows the Teesdale Way, the Cleveland Way, the Yorkshire Wolds Way, the Viking Way, the Hereward Way, the Fen Rivers Way, the Icknield Way Path, the Stour Valley Path and the Essex Way to Harwich International Port. Urban highlights are Lincoln and Cambridge and long coastal stretches are in the Netherlands.


In the Netherlands the eastern route begins at the ferry port of Hoek van Holland (where ferries from Harwich International Port arrive) and passes over islands and dams and through Bergen op Zoom. This corollary section picks up the main route at Sint Antonius, Zoersel in northern Belgium.


  1. ^, Irish section not yet open.

External links

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