The Image is the cover of Iconic Arithmetic Volume I: The Design of Mathematics for Human Understanding, the book version of some of the content on this website.
This book is the first of three volumes and is available from Amazon (ISBN 978-1-7324851-3-6).
The first half of Iconic Arithmetic describes ensemble arithmetic, an iconic system based on tallies, the unary arithmetic in use since the dawn of civilization. Hilbert considered this arithmetic to be so simple that it does not need any axioms. Add and multiply integers without cognitive effort! The second half of Volume I describes James algebra, an iconic system that expresses the rules and operations of arithmetic, including rational, real and imaginary numbers, using one concept (containment) and three simple axioms. Here is the text on the back cover of Volume I:
Arithmetic evolves.
The Laws of Algebra are design choices made before the Internet, before computers, before electricity.
The arbitrary symbols we memorize and try to recall are abstract, one-dimensional, sequential, detached.
Iconic arithmetic is physical, sensual, natural.
Tangible mathematics connects us to our cultural and biological heritage, to the authentic world of experience.
Icons look and feel like what they mean: embodied, multidimensional, concurrent, inclusive.
Iconic arithmetic is a form of experience.
The second volume of this series, Iconic Arithmetic Volume II: Symbolic and Postsymbolic Formal Foundations, is also available from Amazon (ISBN 978-1-7324851-4-3). Volume II compares the conventional symbolic forms of equality, composition and concurrency to the iconic perspective of James algebra. Then the structure of James algebra is compared to the foundations of arithmetic developed by Hilbert, Frege and Peano; to weaker arithmetics (Presburger, Robinson and Primitive Recursive Arithmetic); and to symbolic formalism and digital computation in general. These contrasts support a postsymbolic perspective that sets, logic and functional thinking are not necessary for formal arithmetic. Only the single concept of distinction, expressed by the single iconic relation of containment, is necessary. Here is the text on the back cover of Volume II:
Symbolic arithmetic is a belief system.
The arithmetic we are taught was designed before film, before TV, before music videos, before smart phones.
Our media are coupled to how we think and what we do.
Symbolic abstraction is too detached for this century.
Postsymbolic math reintegrates our bodies and our minds.
Sets and logic and functional thinking need to evolve to incorporate global unity and ecological diversity.
Concept is embodied, concrete, incorporated, lived.
Meaning comes from experience.
Arithmetic is multisensory pattern.
Iconic Arithmetic Volume III: The Structure of Imaginary and Infinite Forms is also available frpm Amazon (ISBN 978-1-7324851-5-0).
Volume III applies the tools developed in Volumes I and II to several areas of elementary mathematics. James arithmetic and void-based thinking are applied to imaginary and complex numbers, to the analysis of critical breakdowns in algebraic rules such as distribution of polarity across square roots, to calculus derivatives, to trigonometry on the real and complex planes, and to the organization and integration of infinite and indeterminate forms within arithmetic. Here is the text on the back cover:
Arithmetic invites invention and exploration.
The way we record arithmetic obscures its simplicity.
Postsymbolism eliminates the artifacts of symbolic notation.
Iconic patterns provide space for more elegant concepts.
Void-based structure renders numeric illusions transparent.
Meaning is in existence rather than symbolic truth or numeric value.
Numbers are bipolar. Zero is self-contradictory.
Functions are opaque. Inverses reduce to one constant.
The multiplicative √–1 decomposes into the additive log –1.
Base-free logarithms, fractional polarity, angle-free trigonometry, one generic derivative, coalition of the Many with the One.
Iconic arithmetic is embodied cognition.
SAMPLES
PDF samples of all three volumes of Iconic Arithmetic are available below. The samples have a distinct right and left page format. Please set your PDF viewer (probably Acrobat) for View>PageDisplay>TwoPageView.
Volume I Samples: FIRST and LAST
The Iconic Arithmetic Volume I Sample includes the contents, preface, bibliography and index of Volume I, as well as Chapters 1, 5, 6 and 15. Chapter 5 includes the axiomatic basis of all systems described in the three volumes. Chapter 6 outlines the perspective of iconic and void-based thinking about arithmetic.
Volume I: The MIDDLE
Chapters 2-4 include the structure of ensemble arithmetic, depth-value notation and many multidimensional dialects.
Chapters 7-14 introduce the structure of James arithmetic as it applies to thinking, counting, computing and the deconstruction of the arithmetic and algebra of numbers. Several multidimensional and interactive dialects are described.
Volume II Samples: FIRST and LAST
The Iconic Arithmetic Volume II Sample includes the contents, preface, bibliography and index of Volume II, as well as Chapters 16, 26 and 30. The first Chapter 16 introduces the objectives of iconic mathematics. Chapter 26 describes the postsymbolic perspective that symbols are not necessary for the expression of mathematical concepts. The final Chapter 30 summarizes the volume and introduces the content of Volume III.
Volume II: The MIDDLE
Chapters 17-20 explore some of the structure and tools of computation.
Chapters 21-25 examine the historical origins of formal arithmetic.
Chapters 27-30 make the case for postsymbolic formal structure.
Volume III Samples: FIRST and LAST
The Iconic Arithmetic Volume III Sample includes the contents, preface, bibliography and index of Volume III, as well as Chapters 31, 38 and 45. The first Chapter 31 introduces the objectives of the volume, particularly the introduction and use of an abandoned imaginary number J, the logarithm of –1. Chapter 38 describes how both the imaginary i and the imaginary J evolve from the structure of our number system. The final Chapter 45 summarizes the volume and briefly describes and critiques the innovations.
Volume III: The MIDDLE
Chapters 32-36 describe the history and features of the imaginary constant log –1.
Chapter 37 applies James algebra to elementary calculus derivatives.
Chapters 39-40 apply James algebra to the complex plane and to real and imaginary trigonometry.
Chapters 41-43 integrate infinite and indeterminate forms into James algebra.
Chapter 44 returns to the experiential dialects of Volume I, applying them to the innovations in this volume.
For PURCHASE
The softcover physical books can be purchased directly from Amazon. The price is intended only to cover some of the development overhead.
Iconic Arithmetic Volume I: ($30), ISBN 978-1-7324851-3-6.
Iconic Arithmetic Volume II: ($30), ISBN 978-1-7324851-4-3.
Iconic Arithmetic Volume III: ($30), ISBN 978-1-7324851-5-0.
Thanks!
Your editorial comments and error corrections are most welcomed at william@iconicmath.com