The Image shows many different iconic forms of representation, including parentheses, blocks, containers, trees, networks, maps, rooms and paths. Each can be transformed into the others, allowing the particular structure of an iconic representation to emphasize particular interactional and instructional concepts. The organizing principle of symbolic representation is that what a symbol looks like is disconnected from what it means. This allows us to agree, for example, that X might mean all the digits from 0 to 9. We can just as easily say that digits or that bloops mean all the digits from 0 to 9. The point is that none of these names tells us anything about the actual things being named. Therefore to change a symbolic notation we simply exchange one set of memorized names for a different set of memorized names.

Icons resemble what they mean. There may still be many different ways to illustrate a concept. To indicate a direction, for example, we might draw a simple arrow, , or we might draw one of the many other icons of direction: , , , , . Iconic math uses the concept of containment. There are many different icons for containers, ranging over typographic delimiters such as ( ) , [ ] , { } , < > , 〖 〗; geometric figures such as O , , , , ; and three dimensional objects such as boxes, cups, bottles and bags. However, the concept of containment can also be indicated by arrows, relational signs, stacks, maps, and other iconic devices. While maintaining the meaning and the physical image of containment, iconic notation can exhibit a wide variety of form.


Iconic Representations

The sequence of images shows the conversion of planar containment to physical blocks to abstract graphs. To fully express the stack as a graph, each of the blocks becomes a node and each planar touching surface (formerly the boundaries between territories) becomes a link.

The stack of blocks is further extruded and then the entire structure is turned upside-down since putting the ground nodes at the bottom is the usual convention for graphs. Finally the coloring of each of the graph nodes is converted into labels that identify the node’s functionality.


This page is currently under development, March 2018. In the interim, I’ve attached some relevant papers.


A Simple Space

A Simple Space, written in 1986, describes a way of thinking about math that allows its simplicity to emerge. Here’s the abstract:

The lines in which we write mathematical symbols impose constraints upon mathematical thinking. A simple space of representation is proposed that does not enforce the linear concepts of associativity, commutativity, duplicity of representation, and binary scope. The properties of this simple space are discussed in the foundational case when the space is empty and in the self-referential case when the space contains only representations of itself. Concepts that evolve from this discussion include representational incompleteness, functional spaces, boundary objects, representational unity of object and process, and two kinds of void. The implications of a representational space without linear properties are explored for propositional calculus. A graph notation is proposed as a simplification of the traditional linear notation for logic.


Syntactic Variety

Syntactic Variety in Boundary Logic was written for the Diagrams 2006 conference. The idea, which I apparently failed to convey at the conference, is that logic itself has many different iconic representations, and many different methods of proof, all of which are anchored in concrete manipulations and in physical activity. Here’s the abstract:

Boundary logic is a formal diagrammatic system that combines Peirce’s Entitative Graphs with Spencer Brown’s Laws of Form. Its conceptual basis includes boundary forms composed of non-intersecting closed curves, void-substitution (deletion of irrelevant structure) as the primary mechanism of reduction, and spatial pattern-equations that define valid transformations. Pure boundary algebra, free of interpretation, is first briefly described, followed by a description of boundary logic. Then several new diagrammatic notations for logic derived from geometrical and topological transformation of boundary forms are presented. The algebra and an example proof of modus ponens is provided for textual, enclosure, graph, map, path and block based forms. These new diagrammatic languages for logic convert connectives into configurations of containment, connectivity, contact, conveyance, and concreteness.


Nonsymbolic Logic

Finally, Nonsymbolic Logic is an exploration of the visual and iconic forms used in logic. Jaron Lanier introduced the idea of postsymbolic communication when he was formulating the ideas of virtual reality in the late 1980s. He believes that symbols, in language and in mathematics, are middlemen. They stand between people sharing experiences. Virtual reality was intended to surplant symbolic communication. One of Jaron’s bumper-stickers, from his book You Are Not a Gadget, is that information is alienated experience. Similarly, symbolic math is alienated abstraction.

This article consists of notes and images. It shows that diagrammatic logic was used throughout history, and it shows the variety of iconic form that is available to enrich our understanding of logic and of thought.