Iconic Arithmetic

Well, the end of a seven year project. William Bricken recently met a correspondence friend for the first time. He immediately asked: “Why can’t you write a small book?”

> My honest reply was that I had tried to write for an audience, and found that it was a skill not within my reach. So I’m writing to meet my personal goals and standards, and for consistency with a couple of (um, perhaps risky) decisions made early.

The goal is to show the simplicity of iconic arithmetic with an abundance of examples in order to convey a single message: **the entire content of school mathematics can be described by a few definitions that anchor how iconic constants might behave, supported by three visually simple pattern-recognition axioms that animate the behavior and transformation of James Forms**.

The half-dozen useful theorems are simple combinations of the axioms, usually standing in place of no more than four or five substitution steps.

Volume I seeks to describe and to illustrate the visual dynamics of James algebra while emphasizing the conceptual beauty of void-based thinking and thus to motivate a completely different way of thinking about and understanding elementary arithmetic. The meta-goal is also simple: to demonstrate that symbolic mathematics is a choice rather than a necessity by providing a viable iconic alternative.

The first volume went smoothly and relatively quickly given the hundreds of accompanying illustrations.

The goal of Volume II is to relate iconic thinking to the evolution of metamathematics and to the revered contributions of the founders of modern axiomatic arithmetic. That volume developed significantly slower since it involved substantial and substantive research. The seminal work from over a century ago had not yet clearly formulated fundamental ideas about meaning and structural transformation. The formal concepts being explored were also (obviously) pre-computational and pre-electronic.

William Bricken struggled with antiquated motivations that were simultaneously prescient and anachronistic. Volume II is a journey into feeling inadequate yet opinionated. The primary challenge was learning how to see through the nearly universal acceptance of the axiomatic structure of sets and logic and whole numbers to find underneath an elegant, postsymbolic alternative.

This volume though has been the most challenging of the three, for mainly human reasons. Many technical details needed to be refined in order to expand James algebra into topics that were not originally intended but do serve as useful application examples. The goal is to present each application domain from an innovative, postsymbolic perspective.

While trying to finish the first volume William Bricken elected to put aside nearly completed explorations until the “next volume”, a decision directly caused by, yes, too many words.

> While updating the “nearly finished” content from 2015 and on, I was fully expecting an editing and compilation task. What I found was a whole lot of redundancy. Over the years the same ideas were repeated many times in many different contexts. My understanding had evolved, my errors had moved from one dimension to another, complicated dilemmas turned simple while new tangles were exposed, and worst of all, integrating a two-foot stack of new notes into already written words was a nightmare. I had thought that what was could be converted into what is, while holding a totally inappropriate belief that time is in short supply.

Truth is, time doesn’t care. And then 2020 pushed back and yet another year passed without completion. pdf

Preface Volume II

When this project began, it was envisioned as an application of boundary techniques to numeric arithmetic, to be followed by a report on William Bricken's twenty years of work with boundary logic.

> I thought that arithmetic would be a friendlier and more familiar introduction to iconic thinking than would logic.

As the chapters of Iconic Arithmetic accumulated, it became clear that there were three relatively separable areas: basic iconic arithmetic, historical grounding and the exotics of imaginary and infinite forms. Each of these areas has taken a separate volume to explore, primarily to make the technical case for iconic math both structurally and computationally, and to provide a thorough description of historical modes of thought. Several major themes have emerged:

> Perhaps needless to say, I also learn by writing.


I awoke one morning in 2015 to realize that my training in formal systems circa 1980 was, to say the least, antiquated. In the twenty-first century I was using a style of mathematical thinking from the early twentieth century, ignoring the fundamental evolution of mathematical perspective due to Grothendieck (algebraic geometry), Baez (n-categories), Chaitin (undecidability), Wolfram (universal computation) and others. I began to see that elementary logic and arithmetic are not determined, secure or natural. The iconic tools I had been working with for years have just as valid a claim to mathematical foundations as do the early explorations that led us today into set theory, Boolean logic and functional thinking. The bulk of the mathematical community still bases their formal thinking about numbers on Peano’s axioms, while the modern evolution in mathematical thought appears to have taken place in the more rarefied atmosphere of the unification of advanced abstraction. Short of returning to school, I tried to leverage my antediluvian education to build what appear to be quite different systems of elementary mathematical thinking.


Volume I of this series describes two alternative ways to conceptualize numbers.

Ensemble arithmetic mirrors the evolution of numeric math for thousands of years prior to the symbolic dominance that took hold less than two hundred years ago.

James algebra embodies iconic structure to open new perspectives on elementary arithmetic by reformulating the “laws” of algebra.

And there are plenty of surprises, especially the unexpected appearance of imaginary and infinite forms, both of which constitute the content of Volume III.

This Volume II is focused on comparative axiomatics, comparing James algebra to our current formal foundations for the arithmetic of numbers. Three simple structural James axioms ground iconic transformations throughout the three volumes. From a computational perspective nothing is remote, complicated or indirect, given the narrow focus on elementary math.

Here we explore the potential of a postsymbolic math that injects our current formal foundations with multiple heresies while still respecting the formality of mathematics itself. To justify the imposition of new iconic forms and transforms, I’ve described and compared the approaches of several technically different fields including numeric arithmetic and algebra, predicate logic, set theory, computational pattern-matching, educational methods and iconic boundary mathematics. Not only is there no agreement across fields about the basic structures of arithmetic, there is also no communality across tools and objectives. Iconic math adds yet more enrichment through diversity.

Volume II feels quite different than Volume I, with deeper, more historical questions at the foundations of the current philosophies of mathematics. This is a necessary volume to address the many technical details about the structures, assumptions and thought processes that we now expect grade school teachers and students to grasp intuitively.


Volume III is an exploration of the interplay between formal symbolic knowledge and void-based iconic innovation. The sections on history, the imaginary J, applications of James algebra and non-numeric forms are relatively independent.

The heart of arithmetic consists of three concepts: Zero, One and Infinity.

The heart of James algebra embraces three grounding concepts: the nonexistent void, the accumulator ( ) and the unifier [ ].

This volume begins with a resurrection of Euler’s work on the logarithms of negative numbers in the form of the numeric constant [<( )>], abbreviated as J.

The first three chapters explore the history, behavior and issues associated with log –1 and its apparent neglect in modern arithmetic.

The next three chapters examine the intimate relationship between π, i and J established by Euler. Then follows a collection of applications of postsymbolic formal thinking to selected subfields of elementary mathematics. If iconic arithmetic is more than an isomorphism, then its application should suggest entirely different perspectives on established mathematical systems. These four chapters are unabashedly exploratory, recounting experiments, dead ends and forced structural conclusions. The next three chapters wrestle with the role of [ ], a fundamental “unit” within the James arithmetic that does not accumulate and in that sense is non-numeric. Some call it infinity.

The final content chapter in this volume revisits the spatial dialects of Volume I.