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.
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BRICKEN, William, 1986. A Simple Space. Advanced Decision Systems. 1986. pdf 
[Accessed 15 December 2023].
I wish to discuss the space in which we record symbols. The central idea is that mathematical operations are more clearly expressed in a space which imposes fewer constraints upon the tokens it contains. Tokens placed in lines encourage sequential metaphors, while tokens spread over a plane encourage parallel metaphors. A simple space of representation frees objects within it from irrelevent syntactic constraints.
My primary motivation for writing this paper is to attempt to elucidate a different way of thinking about mathematical symbolism. This approach has been personally helpful, both in my understanding of mathematical systems and in my implementation of parallel Inference Engines for Artificial Intelligence applications.
After some context setting in Section 1, I will construct, from scratch, a simple space of representation for mathematical expressions (Section 2). I will then show how this simple space can be used to simplify propositional logic (Section 3). My main concern is with redundancy. The elegance of the representation of logic, for instance, is critical to the efficiency of automated theorem proving and of inference engines in expert systems. Unnecessary and redundant rules slow computation. The space I will describe is particularly powerful computationally because it supports parallelism in deduction. It is particularly powerful psychologically because it provides a clear way of thinking about representation.