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

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out <strong>of</strong> very basic objects, built into structures that reveal information about their
own transformational constraints. The three axiomatic transformations given in the
next section are all that is required to demonstrate the boundary equalities.
The mechanics <strong>of</strong> number manipulation are simple in the boundary calculus. More
complicated concepts are not \built-in" but instead appear as directed use <strong>of</strong> it.
Techniques for managing numbers, such as separating magnitude or naming small
integers, exist independent <strong>of</strong> the axioms <strong>of</strong> this system. The calculus provides a
crisp mathematical substrate within which to construct and illustrate mathematical
concepts that were previously interwoven with the symbolic rules themselves.
1.2 The <strong>Calculus</strong>
This thesis denes a few mathematical forms and how they transform, i.e. it denes
a calculus. These forms represent numbers, as opposed to logic or sets. And these
forms are based on a representational paradigm that is spatial. In short, this thesis
denes a calculus <strong>of</strong> number based on spatial forms.
This material assumes a minimalist approach to representation called boundary
mathematics [3, 17]. The ideas were initiated by G. Spencer-Brown [11] and have
been extended in the domain <strong>of</strong> numbers by Spencer-Brown himself, as well as Louis
Kauman [22, 24] and William Bricken [5, 9]. This work extends theirs byintroducing
general forms for cardinality and inverse, and by dening phase. This work also
gives a functional interpretation <strong>of</strong> boundaries not found elsewhere, treating them as
logarithmic functions.
In boundary mathematics, representation begins with empty space, a complete
lack <strong>of</strong> structure, void. Upon this void, distinctions are made that impose structure
and intention. Distinction may be thought <strong>of</strong>asaboundary that delineates space.
Boundaries may be nested and collected to form dierent congurations. A conguration
<strong>of</strong> boundaries represents a mathematical expression that can be transformed
according to match and substitute rules, as dened for the mathematical system [11, 3].
The boundary numbers described here use more than one distinction, a practical
move which asyet has not been philosophically justied in the paradigm. Multiple
boundaries force a distinction that is not a cleaving <strong>of</strong> the space; instead, it is a differentiation
<strong>of</strong> the boundaries. Here, this distinction is accomplished with geometric
characteristics: the boundaries are drawn round ; square 2; or pointy 4: To hold