When looked at cumulatively, it can be said that American pragmatist philosopher Charles Sanders Peirce strove to understand cognition via his sign theory and especially his notion of existential graphs. Peirce put forth ideas for a discipline that would incorporate notions of psychology and semiotics into a unified ontological and epistemological theory of mind. The connecting link was his system of diagrammatic logic, called “existential graphs.” For Peirce a graph was more powerful than language as a means of understanding because it showed how its parts resembled relations among the parts of cognitive acts. Existential graphs show that cognition cannot be extracted from a linear or hierarchical succession of structures, but the very process of thinking itself in actu. In fact, Peirce called his graphs “moving pictures of thought” because they allow us to see how are thoughts are unfolding. In short, as Kiryuschenko (2012) puts it, “Graphic language allows us to experience a meaning visually as a set of transitional states, where the meaning is accessible in its entirety at any given here and now during its transformation” (p. 122).
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Diagrams are keys to understanding real-world phenomena, as scientists and mathematicians certainly know, allowing them to both represent their theoretical hunches and then to use these very representations to derive further ideas and to conduct experiments. Peirce certainly understood this, inventing Existential Graphs (EGs) to explain not only the nature of discovery in science but also how the visual imagery of the mind generates ideas. The focus in this paper will be on the relevance of EGs in mathematics (Danesi 2013), but the concept of EGs as “discovery devices” applies to all domains of knowledge. EGs mirror how the mind synthesizes into abstract images of real-world objects, rearranging parts in various ways to see what the blending yields. EGs offered Peirce the possibility of linking semiotics and psychology into a model of how thought and discovery unfold. As he points out in the following excerpt, diagrams are maps of thought, which may be used “to stick pins into” in order to mark anticipated changes.
But why do that [use maps] when the thought itself is present to us? Such, substantially, has been the interrogative objection raised by an eminent and glorious General. Recluse that I am, I was not ready with the counter-question, which should have run, “General, you make use of maps during a campaign, I believe. But why should you do so, when the country they represent is right there?” Thereupon, had he replied that he found details in the maps that were so far from being “right there,” that they were within the enemy’s lines, I ought to have pressed the question, “Am I right, then, in understanding that, if you were thoroughly and perfectly familiar with the country, no map of it would then be of the smallest use to you in laying out your detailed plans?” No, I do not say that, since I might probably desire the maps to stick pins into, so as to mark each anticipated day’s change in the situations of the two armies.” “Well, General, that precisely corresponds to the advantages of a diagram of the course of a discussion. Namely, if I may try to state the matter after you, one can make exact experiments upon uniform diagrams; and when one does so, one must keep a bright lookout for unintended and unexpected changes thereby brought about in the relations of different significant parts of the diagram to one another. Such operations upon diagrams, whether external or imaginary, take the place of the experiments upon real things that one performs in chemical and physical research. (CP4: 530)
The study of diagrammatic representations is a productive area of investigation in various domains of semiotics and cognitive science (Shin 1994, Chandrasekaran, Glasgow, & Narayanan 1995, Hammer 1995, Hammer & Shin 1996, 1998, Allwein & Barwise 1996, Barker-Plummer & Bailin 1997, 2001, Kulpa 2004, Stjernfelt 2007, Roberts 2009, Kiryushchenko 2012). EGs are consistent with two prominent trends in these field—namely, phenomenology and blending theory in cognitive science (for example, Lakoff & Núñez 2000)—trends that were prefigured by Peirce’s notion of “phaneroscopy,” which he described as the formal analysis of appearances apart from how they appear to interpreters and of their actual material content.
TopExistential Graphs
Peirce argued that discoveries in chemistry were phaneroscopic, because chemical compounds could be studied not as mixtures of actual substances but as diagrammatic structures. Chemists discovered that the structure of a molecule and transformations of chemical compounds themselves gave birth to the scientific language that explained them though the diagrams used to represent them. Diagrams contain within them “virtual objects,” which are like real objects and can thus be used to experiment cognitively with the latter. Peirce wrote an entry on the concept of “virtual” for Baldwin’s (1902, p. 763) Dictionary of Philosophy and Psychology, which is of relevance here: