A Calculus of Number Based on Spatial Forms - University of ...

University <strong>of</strong>Washington
Abstract
A <strong>Calculus</strong> <strong>of</strong> <strong>Number</strong> <strong>Based</strong> on Spatial Forms
by Jerey M. James
Chairperson <strong>of</strong> Supervisory Committee:
Pr<strong>of</strong>essor Judith Ramey
Technical Communication
A calculus for writing and transforming numbers is dened. The calculus is based
on a representational and computational paradigm, called boundary mathematics, in
which representation consists <strong>of</strong> making distinctions out <strong>of</strong> the void. The calculus
uses three boundary objects to create numbers and covers complex numbers and basic
transcendentals. These same objects compose into operations on these numbers.
Expressions transform using three spatial match and substitute rules that work in
parallel across expressions. From the calculus emerge generalized forms <strong>of</strong> cardinality
and inverse that apply identically to addition and multiplication. An imaginary
form in the calculus expresses numbers in phase space, creating complex numbers.
The calculus attempts to represent computational constraints explictly, thereby improving
our ability to design computational machinery and mathematical interfaces.
Applications <strong>of</strong> the calculus to computational and educational domains are discussed.