Iconic Arithmetic Volume III: The Structure of Imaginary and Infinite Forms

In Volume III of Iconic Arithmetic the momentum generated by the first two volumes of this series is applied to several branches of elementary mathematics to yield new techniques and new insights that are not accessible to symbolic arithmetic. The rules of algebra are reduced to three simple patterns expressed as spatial containment relations. These patterns resurrect a discarded imaginary number J, the logarithm of –1, that provides an additive foundation for the multiplicative imaginary i, that replaces inverse operations by a single constant, and that eliminates signed numbers by rendering polarity as an exponent. There is no zero, absence of numeric value is replaced by complete absence. All numeric operations are replaced by the single iconic relation of containment. This volume applies postsymbolic arithmetic to reduce trigonometry to reflection along a line, to condense calculus derivatives into a single generic pattern, and to identify and organize infinite and indeterminate expressions. Other innovations include base-free logarithms, fractional polarity, bipolar numbers, and direct deletion of illusory structure introduced by typographic notation.Volume I of this series (ISBN 9781732485136) explores the visual and interactive forms of iconic arithmetic from the perspectives of historical evolution, formal mathematics, computer science and mathematics education. Volume II (ISBN 9781732485143) connects James algebra to the symbolic foundation of numerics developed over a century ago, to show that iconic form does not incorporate set theory, logic or functional thinking. The overall goal is to demonstrate a comprehensive formal system that can be interpreted as arithmetic but bears little resemblance to our current universally adopted symbolic arithmetic. Iconic Arithmetic provides a proof of concept that our understanding of arithmetic has been limited by the absence of formal techniques for sensory interaction with abstraction.Iconic Arithmetic helps us to transition into a postsymbolic world of interactive information. The innovative perspectives will be fully rewarding to scholars interested in unique mathematical ideas, to students who question conventional mathematics, to educators and designers hoping to enhance creativity, to computer scientists seeking powerful new techniques, to folks who trust their senses more than their memory, to those whose intellectual approach to the world is visual, tactile and experiential, and to explorers of the evolution of philosophy and cognition. Read more