Iconic Arithmetic Volume I: The Design of Mathematics for Human Understanding - Softcover

Bricken Ph.D., William

Softcover

ISBN 10: 1732485135 ISBN 13: 9781732485136

Publisher: Unary press, 2019

Synopsis

Over the last two centuries, arithmetic has devolved from physical manipulation of objects to memorization of strings of abstract symbols. Iconic Arithmetic Volume I introduces two innovative types of postsymbolic arithmetic that reunite sensual experience with the formal structure of numbers. Iconic numbers are icons that look and feel like what they mean. This book demonstrates that the universally accepted numbers we currently use are a design choice and not the way that arithmetic must necessarily work.

In ensemble arithmetic adding is putting icons together, multiplying is putting icons inside each other. Neither requires cognitive effort. In James algebra numeric concepts and operations are expressed as different ways of arranging containers. Both systems support a multitude of physical, interactive and multidimensional dialects that are described in the book.

James algebra is based on the iconic logic developed by Charles Sanders Peirce, George Spencer Brown and Louis Kauffman. It reduces the “laws” of arithmetic and algebra to three simple patterns of containment. Two of these patterns delete unnecessary symbolic complexity found in integer, rational, irrational and complex numeric expressions.

The postsymbolic innovations of iconic arithmetic modify the formal structure of mathematics in fundamental and surprising ways:

• Arithmetic needs only one concept, distinction, and one structure, the container.
• Difference is more important than equality.
• Meaning is based on existence rather than on truth or numeric value.
• Zero is self-contradictory; instead the inside of an empty container is void.
• Objects and operations are two sides of the same boundary.
• Numbers, even exponents, do not need a base (e.g. base 10).
• There is only one inverse operation seen from different perspectives.
• Parallel deletion of structure is a more efficient way to compute.

In Volume I of this series Dr. Bricken explores 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 by Frege and Peano, and demonstrates that iconic form does not require set theory, logic or functional thinking. Volume III (ISBN 9781732485150) shows how James algebra can organize and integrate non-numeric concepts such as infinity and indeterminism, and introduces a new additive imaginary number J that underlies √–1.

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.

Edition details

• Publisher: Unary press
• Publication date: 2019
• Language: English
• ISBN 10: 1732485135
• ISBN 13: 9781732485136
• Binding: Paperback
• Number of pages: 442