Thinking with Diagrams

Sybille Krämer, Christina Ljungberg, Sybille Krämer, Christina Ljungberg

• Publisher: De Gruyter Mouton
• Year: 2016
• ISBN:
  9781501503689
• Edition: 1
• Topic: Languages & Linguistics
• Subtopic: Linguistics
• Index: Languages & Linguistics

Diagrammatic reasoning is crucial for human cognition. It is hard to think of any forms of science or knowledge without the "intermediary world" of diagrams and diagrammatic representation in thought experiments and/or processes, manifested in forms as divers as notes, tables, schemata, graphs, drawings and maps. Despite their phenomenological and structural-functional differences, these forms of representation share a number of important attributes and epistemic functions. Combining aspects of linguistic and pictorial symbolism, diagrams go beyond the traditional distinction between language and image. They do not only represent, yet intervene in what is represented. Their spatiality, materiality and operativity establish a dynamic tool to exteriorize thinking, thus contributing to the idea of the extended mind. They foster imagination and problem solving, facilitate orientation in knowledge spaces and the discovery of unsuspected relationships.

How can the diagrammatic nature of cognitive and knowledge practices be theorized historically as well as systematically? This is what this volume explores by investigating the semiotic dimension of diagrams as to knowledge, information and reasoning, e.g., the 'thing-ness' of diagrams in the history of art, the range of diagrammatic reasoning in logic, mathematics, philosophy and the sciences in general, including the knowledge function of maps.

IWhat is a diagram and how does it function?

Michael Marrinan

1On the “thing-ness” of diagrams

Michael Marrinan, Stanford

1Introduction: Peirce looks at paintings

Icons are so completely substituted for their objects as hardly to be distinguished from them. A diagram, indeed, so far as it has a general signification, is not a pure icon; but in the middle part of our reasonings we forget that abstractness in great measure, and the diagram is for us the very thing. So in contemplating a painting, there is a moment when we lose the consciousness that it is not the thing, the distinction of the real and the copy disappears, and it is for the moment a pure dream – not any particular existence, and yet not general. At that moment we are contemplating an icon.

– Charles Sanders Peirce (W 5: 163)

Any art historian interested in looking at pictures must attend to this passage in which a founding figure of the theory of signs – most commonly recognized in the triad Icon/Index/Symbol – reflects upon the act of looking at a painting. An art historian also interested in diagrams cannot afford to ignore the way Peirce couples this act of contemplation to his account of diagrams (Bender and Marrinan 2010), especially his claim that for a moment “the diagram is for us the very thing.” In this moment, when the distinction between “the real and the copy disappears,” we confront the icon. What does it mean for a diagram to become “the very thing”? How does Peirce explain this transformation? Does his account square with the way art historians look at pictures? Are there other ways to think of diagrams as “things” in their own right? These are some of the questions I will address in this essay.

2Some Vagaries of Peirce’s thinking about Diagrams

Frederik Stjernfelt’s meticulous close-reading of Peirce’s writings about diagrams makes it possible for non-specialists (like myself) to understand more fully the internal mechanics of Peirce’s thinking (Stjernfelt 2007, chapters 3 and 4; Stjernfelt 2000; see also Stjernfelt and Østergaard in this volume). There is no doubt that diagrams are necessary to rational thought, as when Peirce writes in 1885 “deduction consists in constructing an icon or diagram the relation of whose parts shall present a complete analogy with those of the parts of the objects of reasoning, of experimenting upon this image in the imagination, and of observing the results so as to discover unnoticed and hidden relations among the parts” (W 5: 164). At about the same time, he articulates four steps of deductive reasoning in which diagram construction plays the essential role, noting that “even though the diagram exists only in the imagination . . . after it has once been created, though the reasoner has power to change it, he has no power to make the creation already past and done different from what it is. It is, therefore, just as real an object as if drawn on paper” (Peirce 1976a, 4: 275–276). A few years later, he insists that “it is by icons only that we really reason, and abstract statements are valueless in reasoning except so far as they aid us to construct diagrams” (CP 4: 127). Peirce begins “Prolegomena for an Apology to Pragmatism” of 1906, his most detailed discussion of diagrams, by stating: “All necessary reasoning is diagrammatic; and the assurance furnished by all other reasoning must be based on necessary reasoning. In this sense, all reasoning depends directly or indirectly upon diagrams” (Peirce 1976b, 4: 314). Stjernfelt’s study of this essential text helps us to isolate the principal ideas underpinning Peirce’s account of diagrammatic reasoning.

Most important, Peirce insists the diagram “is an Icon of a set of rationally related objects” so that it is amenable to rational experiments and compatible with reasoning (Peirce 1976b, 4: 316). Stjernfelt points out that “as soon as an icon is contemplated as a whole consisting of interrelated parts whose relations are subject to experimental change, we are operating on a diagram” (Stjernfelt 2007: 92). Peirce writes in an alternate version of his text that “the Diagram represents a definite Form of Relation. This Relation is usually one that actually exists,” although that condition is not “essential to the Diagram as such” (Peirce 1976b, 4: 316). He suggests that “the pure Diagram is designed to represent and to render intelligible, the Form of Relation merely” (Peirce 1976b, 4: 316). Peirce wrestles with the problem of how to align mental apprehension with the fact that “Diagrams remain in the field of perception and imagination”; he settles on a two-pronged formula: the drawn diagram and its universal signification “taken together constitute what we shall not too much wrench Kant’s term in calling a Schema, which is on one side an object capable of being observed while on the other side it is General” (Peirce 1976b, 4: 318; see also Stjernfelt 2007: 82–83 and 94–95). On Peirce’s account, pure diagrams – comprised only of rational relations – depend upon what Stjernfelt calls “applied” or “token diagrams” to communicate their ideal entity (Stjernfelt 2007: 96).1 They do so because “reading rules” guide our “prescission” of “accidental characters that have no significance”: in the alternate version of his text Peirce describes these as “Conventions embodied in Habits”; in the final version he writes that “certain modes of transformation of Diagrams of the system of diagrammization used have become recognized as permissible” (Peirce 1976b, 4: compare 317 and 318). None of these modes is spelled out with great precision; Peirce brushes aside the difficulty by saying only that “sometimes in one way, sometimes in another, we need not pause to enumerate the ways” they are recognized. I want to underscore exactly this imprecision.

It is also something of an inconvenience that Peirce neglects to say much about a taxonomy of diagrams. In a circa 1895 text that sketches four steps of deductive reasoning, Peirce does invoke “a diagram, or visual image, whether composed of lines, like a geometric figure, or an array of signs, like an algebraic formula, or of a mixed nature, like a graph” that is constructed “so as to embody in iconic form, the state of things asserted in the premise,” but he assumes it “exists only in the imagination” (Peirce 1976a, 4: 275). Stjernfelt rightly remarks that pure diagrams must be coextensive with mathematics, which means that considering them also entails thinking about the nature of mathematics (Stjernfelt 2007: 111–112). On the other hand, applied or token diagrams come in many stripes and – as demonstrated by the often-cited studies of Alan Blackwell and Yuri Engelhardt – there is little agreement amongst thinkers across disciplines concerning their salient qualities (Blackwell and Englehardt 1998).2 Stjernfelt remarks somewhat laconically that “the construction of a rational taxonomy of diagrams will be a major future challenge for (not only) Peircean semiotics” (Stjernfelt 2007: 111). I will eventually suggest that one way to avoid interminable debates about the nature of diagrams is to adopt a cognitive systemic view that obviates many of the problems and vagaries of taxonomy.

Definitional dawdling is not the only soft spot in Peirce’s analysis. Crucial to his account of diagrammatic reasoning in “Prolegomena for an Apology to Pragmatism” – both the principal version and its variant – is a moment that Stjernfelt properly calls “imaginary” (Stjernfelt 2007: 83–87 and Stjernfelt 2000: 376–378). Already in Peirce’s 1895 account of the four steps of deduction (cited above) he writes about the second step: “Upon scrutiny of this diagram, the mind is led to suspect that the sort of information sought may be discovered, by modifying the diagram in a certain way. This experiment is tried” (Peirce 1976a, 4: 275). Peirce allows that something like suspicion or hunch suggests possible avenues of experiment to the interpretant. Peirce reformulates this comprehension in his 1906 description of how diagrammatic reasoning works: “The Diagram sufficiently partakes of the percussivity of a Percept to determine, as its Dynamic, or Middle, Interpretant, a state [of] activity in the Interpreter, mingled with curiosity. As usual, this mixture leads to Experimentation” (Peirce 1976b, 4: 318).3 Stjernfelt signals “a certain tension” in the “seductive welding together of object and representation in this phase which constitutes the major source of error in diagrammatic reasoning” (Stjernfelt 2007: 112). Moreover, he admits Peirce’s formulation introduces “a strange psychological tone alien to him” with its recourse to an excitement (see variant) of activity and curiosity (Stjernfelt 2007: 102).

Peirce works around this danger by embedding the incitement for experiment within a process of trial-and-error that asks: has the experiment I just performed yielded an iconic result that expands upon the initial symbol under consideration? Whatever the answer, the motivating moment of forgetfulness, lost consciousness, and dream remains just that – an instant of seduction held in check by immediate constraints of the on-going experiment and the ultimate object of our reasoning. Stjernfelt reminds us that “the whole formalist endeavor in the philosophy of mathematics and the emphasis upon symbolic calculi and mistrust of geometry since the late nineteenth century is based on attempts at getting rid of the dangers of seduction by intuition in this very moment” (Stjernfelt 2007: 112). He attempts to rescue Peirce from this danger by suggesting “the decisive thing is that this moment is made possible by structural iconicity between diagram and object – not by the psychology of he or she who contemplates that iconicity.” I am not convinced Stjernfelt’s formulation neutralizes Peirce’s remark that “the action of the Diagram . . . has the same percussive action on the Interpreter that any other Experience has” (Peirce 1976b, 4: 317, variant). Nonetheless, I agree with Stjernfelt’s conclusion that “the pragmatist trial-and-error feedback between initial and final symbols in the diagrammatic reasoning process must be the Peircean means of avoiding being caught up in the ‘imaginary moment’” (Stjernfelt 2007: 114). We can ask if Peirce is truly successful in avoiding this pitfall, but I want to take away two points from my admittedly partisan reading of Peirce’s account of diagrammatic reasoning: the crucial place in the process of an interpretant’s intutive response to the diagram; the coterminous fiction that the diagram has become a thing.

3Peirce’s Prescission: Diagrams of no Substance

Peirce put little stock in the materiality of diagrams, which makes sense because, on his account, pure diagrams of the most general application are virtual constructs of the mind. “One contemplates the Diagram,” he writes in the variant version of 1906:

. . . and one at once prescinds from the accidental characters that have no significance. They disappear altogether from one’s understanding of the Diagram . . . one can contemplate the Diagram and perceive that it has certain features which would always belong to it however its insignificant features might be changed. What is true of the geometrical diagram drawn on paper would be equally true of the same Diagram put on the blackboard (Peirce 1976b, 4: 317, note).

He remarks a bit further in the main body of the same essay:

Logic requires great subtlety of thought, throughout; and especially in distinguishing those characters which belong to the diagram with which one works, but which are not significant features of it considered as the Diagram it is taken for, from those that testify to the Form represented. For not only may a Diagram have features that are not significant at all, such as being drawn upon “laid” or upon “wove” paper; not only may it have features that are significant but not diagrammatically so; but one and the same construction may be, when regarded in two different ways, two altogether different diagrams (Peirce 1976b, 4: 324).

Nonetheless, a diagram’s field of support is significant even if its materiality is not because, on the most abstract level, the support stands for at least three distinct types of discourse. First, geometrical figures drawn upon a flat plane where “the parts of the diagram are seen in the visual image to have the relations supposed.” Second, algebraic formulas on a support where “the parts have shapes to which conventions or ‘rules’ are attached, by means of which the supposed relations are attributed, or imputed, to the parts of the diagrams” (Peirce 2010: 46–47). Finally, underpinning what Peirce calls Existential Graphs, the support is a “Sign of a logical Universe” upon which graphs are scribed with a different kind of attachment, “since the Entire Graph of the Area is after a fashion predicated of that Universe” (Peirce 1976b, 4: 322).4 Stjernfelt correctly summarizes by noting that the first two types of support are “continuous, requiring the imaginary translation of geometric objects on the surface; the latter discontinuous, involving the adding or erasing of whole discrete structures on the sheet. Thus, diagrams comprise both continuous and discontinous systems” (Stjernfelt 2007: 101). Among the latter group, Peirce singles out two instances that “are remarkable for being truly continuous both in their matter and in their corresponding Signification . . . the Graph of Identity represented by the Line of Identity, and the Graph of coëxistence, represented by the Blank” (Peirce 1976b, 4: 324). Obviously, in the last instance, the “sheet of assertion” is itself a graph (Peirce 1933b, 4: 396). I do not intend to dive into the murky waters of Existential Graphs, but I underscore Peirce’s attention to the field of diagrammatic inscription because it raises a question that interests me: what does Peirce see when looking at pictures? What part of that experience provokes in him a “moment when we lose the consciousness that it is not the thing”?

Figure 9: Pietro Perugino, Christ giving the Keys to Saint Peter, ca 1481–82. Fresco. Rome (Vatican), Sistine Chapel (330 x 550 cm). Photo credit: Public Domain via Wiki Commons.

To simplify matters, let us imagine Peirce before Perugino’s famous fresco, Christ giving the Keys to Saint Peter (Figu...

Table of contents

• Cover
• Title Page
• Copyright
• Table of contents
• Thinking and diagrams – An introduction
• I What is a diagram and how does it function?
• II Diagrammatic scenes
• References
• Index of names
• Index of subjects
• Footnotes