9783796554117_LP

Je rey F. Hamburger

Spaces of Knowledge in Medieval Diagrams

Randgänge der Mediävistik

Edited by Michael Stolz

Volume 11

Jeffrey F. Hamburger

Spaces of Knowledge in Medieval Diagrams

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7 Defining the Diagram

11 From Plato to Peirce

17 The Scene of the Diagram

21 The Divided Line

25 Dimensions of Diairesis in the Pseudo-Dionysius

43 Figures of Thought

53 The Athletic Scholar

63 About the Author

Defining the Diagram

Diagrams present a special instance of the figure-ground relationship, which is as central to cognition as it is to the visual arts.1 In architectural and urban studies, a figure-ground diagram charts the relationship between built and unbuilt spaces. Built spaces are normally shown in black (or some other color); streets are left blank. Neither is more important than the other; each connotes a different category of information. Other images exploit the figure-ground relationship to create what are called bistable images, which in turn are linked to perceptual multistability, the coexistence of two mutually exclusive perceptual states. Of this type of image, perhaps the most familiar are Edgar Rubin’s vase, in which one perceives either a black vase against a white background or a pair of white faces on a black background, and Joseph Jastrow’s rabbit / duck made famous by Wittgenstein’s ‘Philosophical Investigations’, in which he used it to distinguish “seeing that” from “seeing as”, ordinary perception as opposed to seeing under an aspect.2

In ancient and medieval Greek, the word διάγραμμα (diágramma) derives from the verb διαγράφειν, “to mark out by lines”. Diagrams conventionally consist of lines drawn on a blank background, whether of parchment or paper, although they can, of course, subsist in other media, whether dust on the ground (as in Valerius Maximus’s apocryphal account of the death of Archimedes) or the virtual space

1 This essay draws to a significant extent on portions of my book, Jeffrey F. Hamburger, The Areopagite through The Ages. A Millennium of Diagramming the Pseudo-Dionysius, c. 600–c. 1600. Toronto forthcoming. I am grateful to Michael Stolz for providing me with an opportunity to expand on some its ideas here, to Eva Schlotheuber for bringing my text to his attention, to Herbert Kessler for a close reading, and to John Kee for his assistance in transcribing and translating Greek inscriptions. Any errors remain my own.

2 See Jörgen L. Pind, Edgar Rubin and Psychology in Denmark. Figure and Ground (History and Philosophy of Psychology). Cham 2014. From the enormous literature on Wittgenstein, I cite only William James Earle, Ducks and Rabbits. Visuality in Wittgenstein. In: David M. Kleinberg-Levin (ed.), Sites of Vision: The Discursive Construction of Sight in the History of Philosophy. Cambridge MA 1999, pp. 293–314; and Emmanuel Alloa, Seeing-as, Seeing-in, Seeing-with: Looking Through Images. In: Richard Heinrich et al. (eds.), Image and Imaging in Philosophy, Science and the Arts. Proceedings of the 33rd International Ludwig Wittgenstein Symposium in Kirchberg, 2010, 2 vols. (Image and Imagining in Philosophy, Science and the Arts 1; Publications of the Austrian Ludwig Wittengenstein Society, NS 16). Berlin 2011, vol. 1, pp. 179–190.

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of a computer screen.3 They consist, however, of more than lines alone. The lines serve as outlines, that is, they surround or subtend an area of a particular shape.4 Even if they are colored, they still have contours.5 This duality of line and plane poses a series of questions: when looking at a diagram, do the lines command our attention? Or rather the spaces those lines enclose? Are the lines contours demarcating spaces and denoting limits? Or are they the connective tissue within a network of nodes, as in graph theory? In either case, the answer is both, albeit to variable degrees depending on content and context.

Another way of posing these questions is to ask whether diagrams are inherently spatial. The Euclidean proof of the Pythagorean theorem offers an instructive example: the sum of the areas of the two squares on the legs of a right triangle equals the area of the square on the hypotenuse. The theorem can be expressed in either algebraic (i.e., linear or sentential) or geometric (i.e., spatial) terms. In like fashion, logical relations can be defined by statements (as in first-order logic

3 See Valerius Maximus, Memorable Doings and Sayings, vol. II: Books 6–9. Ed. and trans. by David R. Shackleton Bailey (Loeb Classical Library 493). Cambridge MA, London 2000, VIII.7.ext. 7, p. 235: “I should say that Archimedes’ diligence also bore fruit if it had not both given him life and taken it away. At the capture of Syracuse Marcellus had been aware that his victory had been held up much and long by Archimedes’ machines. However, pleased with the man’s exceptional skill, he gave out that his life was to be spared, putting almost as much glory in saving Archimedes as in crushing Syracuse. But as Archimedes was drawing diagrams with mind and eyes fixed on the ground, a soldier who had broken into the house in quest of loot with sword drawn over his head asked him who he was. Too much absorbed in tracking down his objective, Archimedes could not give his name but said, protecting the dust with his hands, ‘I beg you, don’t disturb this’, and was slaughtered as neglectful of the victor’s command; with his blood he confused the lines of his art”. 4 For a systematic treatment of the spatial dimensions of diagrams, see Jan Wöpking, Raum und Wissen. Elemente einer Theorie epistemischen Diagrammgebrauchs (Berlin Studies in Knowledge Research 8). Berlin 2016, with references to earlier literature. Wöpking’s definition of the diagram, p. 188, in part dependent on Bruno Latour’s concept of “immutable mobiles” (see Bruno Latour, Visualisation and Cognition: Drawing Things Together. Knowledge and Society: Studies in the Sociology of Culture Past and Present 6 [1986], pp. 1–40) as “eine externe Inskription” which “(i) auf komplexe, intelligente und wesentliche Weise Raumrelationen nutzt” and “(ii) insofern das Arbeiten mit ihr regelgeleitet, normative ist, (iii) und der Verfolgung epistemischer Zwecke dient”, strikes me as too narrow in that it excludes a great deal of diagrammatic material that is neither mathematical nor scientific but still epistemic. One function of diagrams is to lend material that is not necessarily systematic the appearance of normativity. Moreover, as this study seeks to show, while most medieval diagrams externalized and materialized knowledge, they hardly objectified it or succeeded in lending it stability. In applying such criteria, Wöpking, like Latour, overestimates the “optical consistency” of printed images, let alone those found in manuscripts.

5 The use of color in medieval diagrams awaits comprehensive discussion. In the meantime, see Jeffrey F. Hamburger, Color in Cusanus. Stuttgart 2021.

and Boolean algebra) or by sets (e.g., Euler circles and Venn diagrams). Although cognitive scientists sometimes question whether the two modes of demonstration can be distinguished from one another, they, in like fashion, propose alternating views of mental representation, one based on “Euclidean cognitive maps”, the other, on “graph-like representations” (with “graph” understood as it is in graph theory, as a way of representing a network).6 The diagram presents a way, not simply of representing, but also of thinking about, the world.

6 See Michael Peer et al., Structuring Knowledge with Cognitive Maps and Cognitive Graphs. Trends in Cognitive Science 25 (2021), pp. 37–54.

From Plato to Peirce

On the nature of the diagram, Charles Sanders Peirce is precise:

Hypoicons [instantiations of icons] may be roughly divided according to the mode of Firstness [that is, their pre-reflexive immediacy] of which they partake. Those which partake of simple qualities, or First Firstnesses, 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.7

The diagram is neither a symbol nor an index; rather, it is an icon – but of a particular type. Whereas indices are characterized by a factual correspondence with their object, and symbols by imputed characteristics, icons share qualities with it (as in a red painting of a red rose). Rather than first-order likenesses, diagrams root resemblance in a set of structural relationships shared with that which they represent. These relationships, “mainly dyadic” (as in a one-to-one correspondence), are what render a diagram spatial. In the words of Johanna Drucker , “Diagrammatic images spatialize relations in a meaningful way. They make spatial relations [e.g. simultaneity, hierarchy, juxtaposition, proximity, alignment, connectedness and disconnectedness] meaningful”.8

A simple line suffices to connect two or more points. Attach identifiers to these points (Peirce’s ‘analogous relations’), thereby creating a compound object consisting of text and image, and one has a diagram. A diagram of this kind is not as simple as it seems. Plato’s Divided Line (‘Republic’ 509d–511e), philosophy’s Ur-diagram, provides an exemplary instance. Speaking to Glaucon, Socrates says:

‘Imagine a line cut in two. Take two unequal segments and again cut each one in the same ratio, one for the visible class, the other for the intelligible; and you will have in the first segment of the visible section images in relation to each

7 Charles Sanders Peirce, Sundry Logical Conceptions. In: The Essential Peirce. Selected Philosophical Writings, vol. 2. Ed. by the Peirce Edition Project. Bloomington, Indianapolis 1998, pp. 267–288, at p. 274.

8 Johanna Drucker, Graphesis: Visual Forms of Knowledge Production (MetaLABprojects). Cambridge MA 2014, p. 66.

Jeffrey F. Hamburger

other by their clarity or obscurity – and by images I mean firstly shadows, the reflections in water and in those surfaces which are solid, smooth and shiny, and everything like this, if you get my meaning’. [Glaucon] ‘Well yes, I do’. [Socrates] ‘Now take the second section which this one resembles to be the living creatures around us, all natural things and the whole class of artificial things’.9

Socrates’s first division is between the visible world (the realm of the senses) and the intelligible (the realm of illumination, imagery rooted in the immediately prior passage, the analogy of the sun, according to which the idea of goodness illuminates the intelligible with truth). Each of the two realms is divided in turn, the former between eikasia (opinion rooted in empirical cognition or the imagination, often bordering on delusion)10 and pistis (belief rooted in experience), the latter, between διάνοια (thought, exemplified by mathematical knowledge) and noēsis (intuitive understanding, i.e., dialectic entirely free of likenesses).

For Plato, diagrams, including the Divided Line, whether imagined by the reader or realized in material form, themselves belong to the realm of the senses. Although the diagram assists in leading the mind toward an endpoint (literal as well as argumentative), only the object of the diagram, the idea or set of concepts it represents in terms of the relationship among its parts, can justly claim to consist of knowledge in its pure, intuitive form. In short, the knowledge achieved by the mathematician falls short of that attained by the philosopher.11 Plato would disapprove, but the purpose of this essay is to bring diagrams down to earth by underscoring some of the contingencies that affected their production, transmission, and interpretation. Diagrams do not exist in the abstract; they make speculation concrete in ways that inflect their meanings, intentionally or not. Greek mathematicians, who were essentially geometers, may have thought in diagrams so that, in Reviel Netz ’s formulation, “the diagram is the metonym of the proposi-

9 Plato, The Republic. Vol. 1: Books 1–5. Ed. and trans. by Christopher Emlyn-Jones and William Preddy. Cambridge MA 2013, p. 97.

10 Damien Story, What is eikasia? Oxford Studies in Ancient Philosophy 58 (2020), pp. 19–57.

11 For Plato’s use of mathematical diagrams, see Richard Patterson, Diagram, Dialectic, and Mathematical Foundations in Plato. Aperion 40 (2020), pp. 1–34; Duncan F. Kennedy, Metaphysics and the Mathematical Diagram. Geometry between History and Philosophy. In: Pantelis Michelakis (ed.), Classics and Media Theory (Classical Presences). Oxford 2020, pp. 77–114; and Tamsin de Waal, The Mathematicians’ Use of Diagrams in Plato. In: Nils Kürbis, Bahram Assadian and Jonathan Nassim (eds.), Knowledge, Number and Reality. Encounters with the Work of Keith Hossack. London 2022, pp. 143–160, with additional bibliography. For Greek mathematicians’ use of diagrams outside of philosophy, see Reviel Netz, The Shaping of Deduction in Greek Mathematics. A Study in Cognitive History (Ideas in Context 41). Cambridge 1999.

tion”.12 Plato, however, adopts the diagrammatic language of Greek mathematics for his own metaphysical purposes, a circumstance that constitutes a contingency of its own. Within the hierarchy of cognition charted by the Divided Line, in which intuitive cognition (noēsis) surpasses discursive thinking (dianoia), the form of technē represented by the diagram remains inherently contingent, practical rather than theoretical. This contingency cuts to the heart of what for Plato represents true understanding or wisdom as opposed to technical know-how.13 Whereas for him, ἐπιστήμη (epistēmē: knowledge, understanding) stands in opposition to δόξα (doxa: opinion), in modern discourse, at least following Foucault, the epistemic has come to stand for contingency tout court, in his words, “the conditions of possibility of all knowledge” within a given historical period. Rather than referring to timeless truth, the seeming pleonasm of “epistemic knowledge” refers less to what is known than to the time-bound rules or assumptions by which anything can be known.14 The manuscript transmission of diagrams reveals such contingency in very material ways.15 The earliest extant representation of Plato’s Divided Line occurs

12 Reviel Netz, Greek Mathematical Diagrams. Their Use and Their Meaning. For the Learning of Mathematics 18 (1998), pp. 33–39, at pp. 37–38.

13 These brief comments hardly do justice to the complexity of the relationship between episteme and techne in Plato, for which see Richard Parry, Episteme and Techne. In: The Stanford Encyclopedia of Philosophy (Spring 2024 Edition), available at: https://plato.stanford.edu/archives/ spr2024/entries/episteme-techne/ (28/7/2024).

14 Michel Foucault, The Order of Things. An Archaeology of the Human Sciences. New York 1994, p. 168: “In any given culture, there is always only one episteme that defines the conditions of possibility of all knowledge, whether expressed in a theory or silently invested in a practice”. – See, e.g., Gary Gutting, Michel Foucault’s Archaeology of Scientific Reason (Modern European Philosophy). Cambridge 1989, p. 285: “Foucault proposes, in the end, a twofold transformation of the traditional concept of philosophy. First, he turns it away from the effort at an a priori determination of the essential limits of human thought and action and instead makes it a historical demonstration of the contingency of what present themselves as necessary restrictions. Second, he no longer asks it to provide the justification for the values that guide our lives but instead employs it to clear the path of intellectual obstacles to the achievement of those values”.

15 Although this essay focuses on examples from the Neoplatonic tradition, a similar approach, adumbrated by Michael Krewet, Bilder des Unräumlichen. Zum Erkenntnispotential von Diagrammen in Aristoteleshandschriften. Wiener Studien 127 (2014), pp. 71–100, can also be applied to Aristotelian diagrams. Krewet, p. 86, summarizes Aristotle’s position as laid out in ‘De Memoria et reminiscentia’ I 449b30–450a14 as follows: “Um das Wesen einer Sache, das für sich selbst keine festgelegte Ausdehnung besitzt und nicht an eine bestimmte Materie gebunden ist, erkennen zu können, stellen wir uns das Wesen der Sache vor Augen, indem wir es in seiner Realisierung in einer bestimmten Ausdehnung – und damit bildlich – vorstellen”. For diagrams in Aristotelian manuscripts (including commentaries), see Alfred Stückelberg, Aristotle illustratus. Anschauungshilfsmittel in der Schule des Peripatos. Museum Helveticum 50 (1992), pp. 131–143.

F. Hamburger in a Byzantine manuscript of the late ninth century (Paris, Bibliothèque nationale de France, ms. grec. 1807, fol. 72r), part of a set known as the ‘Philosophical Collection’ (Fig. 1). 16

Sybille Krämer has suggested that “We cannot dismiss the possibility that the indeterminacy in Plato’s descriptions of diagrams is due to the fact that there were practices in place at the Academy which made it clear precisely how Plato’s descriptions were to be carried out graphically”.17 Elsewhere, she argues that

16 As far as I have been able to determine, the sole mention of this diagram in the vast literature that touches on the Divided Line is Sybille Krämer, ‘The Mind’s Eye’. Visualizing the Non-Visual and the ‘Epistemology of the Line’. In: Richard Heinrich et al. (eds.), Image and Imaging in Philosophy, Science and the Arts, vol. 2. Proceedings of the 33rd International Ludwig Wittgenstein Symposium in Kirchberg, 2010, 2011 (Publications of the Austrian Ludwig Wittgenstein Society. New Series 17). Frankfurt et al., pp. 275–293, at pp. 279–281, where, however, its existence is noted but its form and content, let alone the apparent contradictions in its construction, are not discussed. As part of Book VI, it is not included among the diagrammatic scholia to the Republic redrawn in Domenico Cufalo, Scholia Graeca in Platonem. Scholia ad Clitophontem et Reipublicae libros I–V continens. Tesi di Dottorato di Ricerca, Università degli Studi di Pisa 2010–2011, pp. 101–106. It is, however, reproduced schematically in Scholia Platonica. Ed. by William Chase Green. Haverford PA 1938, p. 246 (510d). For the ‘Philosophical Collection’, see Bernard Flusin, La production byzantine des livres aux siècles VII–VIII. In: Jean-Claude Cheynet (ed.), Le monde byzantin 2: L’Empire byzantin 641–1204. Paris 2006, pp. 346–347; Lidia Perria, Scrittura e ornamentazione nei codici della ‘collezione filosofica’. Rivista di studi bizantini e neoellenici 28 (1991), pp. 45–111; Annaclara Palau Cataldi, Un nuovo codice della ‘collezione filosofica’. Il palinsesto Parisinus graecus 2575. Scriptorium 55 (2001), pp. 249–274; Filippo Ronconi, La collection brisée. La face cachée de la ‘collection philosophique’. Les milieux socioculturels. In: Paolo Oderico (ed.), La face cachée de la littérature byzantine. Le texte en tant que message immédiat. Actes du colloque international, Paris, 5–7 juin 2008. Paris 2012, pp. 137–166; Didier Marcotte, La ‘collection philosophique’. Historiographie et histoire des textes. Scriptorium 68 (2014), pp. 145–165; Daniele Bianconi and Filippo Ronconi (eds.), La ‘Collection philosophique’ face à l’histoire. Péripéties et tradition. Spoleto 2020. For marginal annotations in these manuscripts, see Christian Brockmann, Scribal Annotations as Evidence of Learning in Manuscripts from the First Byzantine Humanism. The ‘Philosophical Collection’. In: Jörg B. Quenzer and Dimitri Bondarev (eds.), Manuscript Cultures: Mapping the Field. Berlin, Boston 2014, pp. 11–34.

17 Sybille Krämer, Is There a Diagrammatic Impulse with Plato? ‘Quasi-Diagrammatic Scenes’ in Plato’s Philosophy. In: Sybille Krämer and Christina Ljungberg (eds.), Thinking with Diagrams. The Semiotic Basis of Human Cognition. Berlin, Boston 2016, pp. 161–177, at pp. 165–166. Among Krämer’s many other essays on relevant topics, see ead., Notationen, Schemata, Diagramme. Über ‘Räumlichkeit’ als Darstellungsprinzip. Sechs kommentierte Thesen. In: Gabriele Brandstetter, Franck Hofmann and Kirsten Maar (eds.), Notationen und choreographisches Denken. Freiburg i. Br. 2010, pp. 27–45; ead., Gedanken sichtbar machen: Platon – Eine diagrammatologische Rekonstruktion. Ein Essay. In: Jan-Henrik Möller, Jörg Sternagel and Lenore Hipper (eds.), Paradoxalität des Medialen. Munich 2013, pp. 175–191; and ead., Point, Line, Surface as Plane. From Notational Iconicity to Diagrammatology. In: Zarco Paic and Kresimir Purgar (eds.), Theorizing Images. Cambridge 2016, pp. 202–227.

Fig. 1: The Divided Line (‘Republic’, 509d–511e). Plato, ‘Tetralogies’, 8–9, Constantinople, 850–875. Paris, Bibliothèque nationale de France, ms. gr. 1807, fol. 72r (detail). Photo: BnF.

“although the procedure of diairesis is transmitted to us in the Platonic dialogues only in written form, we can proceed from the assumption that it corresponded to actual visualizations”.18 But can we?

18 Sybille Krämer, Figuration, Anschauung, Erkenntnis. Grundlinien einer Diagrammatologie (Suhrkamp Taschenbuch Wissenschaft 2176). Berlin 2016, p. 171: “Obwohl also das Verfahren der Dihairesis uns in den platonischen Dialogen nur in Textform überliefert ist, können wir davon ausgehen, dass ihm reale Visualisierungen entsprachen”. My translation.

The Scene of the Diagram

Almost 1500 years separate the earliest extant manuscripts of Plato’s works (the Clarke Plato, Oxford, Bodleian Library, MS. Clarke 39, and the manuscript in Paris) from their author. This period witnessed profound changes, not only in the reception of classical texts, but also in the way in which they were written (in a new minuscule script), annotated (often with an extensive marginal apparatus), and disseminated.19 Even had Byzantine scholars of the ninth and tenth centuries possessed an ancient Greek copy of Plato (which they did not), they would have understood its diagrammatic illustrations, assuming there were any, in their own distinctive way.

This farfetched hypothetical poses the question of whether diagrams must always take the form of external representations. In some cases, mostly mathematical, almost certainly yes, but in others, primarily philosophical (such as Meno’s thought experiment with the slave boy in Plato’s dialogue of the same name), most likely not. Reading diagrams requires a set of skills, which developed and changed over time. Just as the shift from the oral to the written transmission of works transformed what it meant to speak of a “text” (making them more stable objects, if not as fixed as we are accustomed to thinking of them today), so too, the transition from an oral to a written setting altered the understanding and application of diagrams.20 The mental manipulation of a diagram need not have always required their external realization. Even had it done so, diagrams drawn

19 Brockmann (note 16), pp. 11–34. See also Pasquale Orsini, Pratiche collettive di scrittura a Bisanzio nei secoli IX e X. Segno e Testo: International Journal of Manuscripts and Text Transmission 3 (2005), pp. 265–342.

20 On this point, see Netz (note 12), p. 39: “they [the Greeks] did not develop mathematical notation. Because of this, they relied much more heavily upon oral aids, such as oral formulae. The verbal was typically used in the oral mode. In mathematics, then, on the one hand, the Greeks did not develop anything like our modern written symbols; on the other, they did develop a system of formulaic expressions, one which is essentially oral. They were therefore in a sense oral both in what they did not do and in what they did do. I am convinced that much of their mathematical reasoning took the shape of a silent soliloquy, unaided by writing, in front of a diagram”. For the fixity of medieval texts (or lack thereof), see Marilynn Desmond, The Visuality of Reading in Pre-Modern Textual Cultures. Australian Journal of French Studies 46 (2009), pp. 219–234, one of many such studies that could be cited.

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extemporaneously in the dust would have functioned differently than the textbook diagrams with which we are familiar today. In the words of Reviel Netz, referring to ancient Greek mathematical diagrams, they served a performative function: “one took the concrete, schematic diagram, and invested it, in one’s imagination, with the full information it was required to carry”. He continues:

The fundamental point [was] that reading, in general, was expected to involve such work of interpretation. The precise position of ancient writing – schematic, and yet pointing at something made vivid to the mind – made it very natural to embed, within it, this special object, so significant to Greek mathematics: the schematic diagram, made-believe to be the geometrical thing itself.21

Although writing centuries later than the Hellenistic period addressed by Netz, Augustine, in his ‘Confessions’ IV.16, speaks to a similar mise-en-scène in ways that reveal significant continuities and discontinuities:

What good did it do me that at about the age of twenty there came into my hands a work of Aristotle which they call the ‘Ten Categories’? My teacher in rhetoric at Carthage, and others too who were reputed to be learned men, used to speak of this work with their cheeks puffed out with conceit, and at the very name I gasped with suspense as if about to read something great and divine. Yet I read it without any expositor and understood it. I had discussions with people who said that they had understood the ‘Categories’ only with much difficulty after the most erudite teachers had not only given oral explanation but had drawn numerous diagrams in the dust [sed multa in pulvere depingentibus]. They could tell me nothing they had learnt from these teachers which I did not already know from reading the book on my own without having anyone explain it.22

As noted by James O’Donnell in reference to this passage, “[w]hat Augustine reads as a sign of intuition on his part is at the same time a fragment of the history of literacy. Where other students could approach the text only through the medium of oral discussion [of which, it should be noted, the recourse to diagrams drawn in the dust constituted a critical part], Augustine has mastered the skills

21 Reviel Netz, Why Were Greek Mathematical Diagrams Schematic? Nuncius 35 (2020), pp. 506–535, at p. 535.

22 St. Augustine, Confessions. Trans. by Henry Chadwick. Oxford 2008, p. 69.

of purely textual manipulation”.23 Augustine situates himself in a context in which diagrams move from the ground on which a given group was gathered to the plane of the page.24 Solitary, perhaps even silent, reading already existed in antiquity, but Augustine testifies to a shift in the scene of instruction: from an oral setting, in which participants in a dialogue share ideas in a quasi-public forum to that of private reading and study. From his account, however, which stresses that others required the aid of diagrams yet implies that he did not, it cannot be inferred that the text from which he taught himself without the aid of oral instruction contained the diagrams to which he refers.

Over time, however, the spaces of knowledge shift, as do its objects.25 By the time of the rhetorician C. Marius Victorinus (c. 300–after 362) and the polymath Cassiodorus (c. 485–c. 585), diagrams already had an established place in the codex.26 They had become external aids to cognition. In their landmark essay on the theory of the extended mind, Andy Clark and David Chalmers argue that, “[i]f, as we confront some task, a part of the world functions as a process which, were it done in the head, we would have no hesitation in recognizing as part of the

23 James J. O’Donnell, Augustine. ‘Confessions’, 3 vols. Oxford 1992, vol. 2, p. 265.

24 The transition in question is by no means completed with Augustine; see John D. Schaeffer, The Dialectic of Orality and Literacy. The Case of Book 4 of Augustine’s ‘De Doctrina Christiana’. In: Richard Leo Enos et al. (eds.), The Rhetoric of Saint Augustine of Hippo: ‘De doctrina Christiana’ and the Search for a Distinctly Christian Rhetoric (Studies in Rhetoric and Religion 7). Waco TX 2008, pp. 289–310, at p. 307: “The ‘De doctrina’ presents concepts of reading, praying, and style that are informed by Augustine’s attempt to bring classical rhetoric, a discipline associated with extemporaneous oral performance, to bear on Christian preaching, which was grounded in the interpretation of a written text. The profound and glacial changes that occurred while European culture was adopting Christianity, a textually based religion, are documented, at least partially, in book 4 of the ‘De doctrina’, where Augustine attempts to transform rhetoric into a tool for oral expression of interior states formed by reading and prayer. But even reading and prayer are still orally controlled in book 4, for Augustine has not interiorized silent reading or private meditation to the extent that modern Western subjects have”.

25 Cf. Netz (note 12), p. 39: “but the one fixed, solid object in Greek mathematics was not the word, but the picture. This was actually drawn, not just spoken, and therefore it was actually out there, it was the physical reality, the inter-subjective definite object. […] The verbal could not be the fixed object in Greek mathematics because of the more oral approach taken in Greek mathematics”.

26 For Victorinus, see Thomas Riesenweber, C. Marius Victorinus. ‘Commenta in Ciceronis Rhetorica’, 2 vols. Boston, Berlin 2015; for Cassiodorus, Michael Gorman, The Diagrams in the Oldest Manuscripts of Cassiodorus’ ‘Institutiones’. Revue Bénédictine 110 (2000), pp. 27–41, at p. 30: “Diagrams were an integral part of the ‘Institutiones’ as planned and executed by Cassiodorus”.

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cognitive process, then that part of the world is […] part of the cognitive process.”27 Diagrams, albeit manmade, are just such a ‘part of the world’. Yet what constitutes a diagram very much depends on how such an object – an object of thought or an actual object – works in a particular context. In effect, the diagrams invoked in Plato’s dialogues demand an inversion of Clark and Chalmers’s hypothetical: if, as we confront some task for which we now rely on an external aid (such as a diagram or, for that matter, a computer), we cannot conclude that for people living the better part of two and a half millennia before our time, that same task could not in some cases have been done in the head. Most of us also cannot recite poems the length of the ‘Iliad’ and the ‘Odyssey’ by heart.

27 Andy Clark and David J. Chalmers, The Extended Mind. Analysis 58 (1998), pp. 7–19, at p. 8. For further discussion (and demurral), see Richard Menary (ed.), The Extended Mind. Cambridge MA 2010.