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

17
Figure 3.1: The Void.
Figure 3.2: A Distinction.
no structure or intention on forms which are made. In contrast, linear, token-based
representations impose a great deal <strong>of</strong> structure.
The void plays a vital role in boundary mathematics. It is anywhere and everywhere;
it is where all distinctions are made. The void does not go away: it permeates
all boundary representations [3].
Equivalences with the void can be easily mistaken for typographical errors when
written linearly. The equivalence form A = B puts no explicit boundary around the
expressions being equated so a void equivalence such asA=may at rst appear
erroneous. Since void expressions are legitimate in boundary mathematics, these
expressions should be recognized as valid.
3.2.2 Distinction
Distinction is the primary form <strong>of</strong> boundary mathematics. A distinction cleaves
space, imposing structure and intention upon it. Distinctions serve both a syntactic
and semantic role. The meaning lies in making the separation.
Figure 3.2 shows a distinction in the two-dimensional space <strong>of</strong> this page. The
shape and scale <strong>of</strong> this distinction are irrelevant. The distinction serves only to
separate the content <strong>of</strong> the distinction from the context in which itwas made. The
content and context <strong>of</strong> a distinction are labeled in Figure 3.3.
Additional distinctions are made with respect to the rst distinction. Distinction
is an action; the result <strong>of</strong> distinction is an object for further action. There are two
ways to make a second distinction: in the content or context <strong>of</strong> the rst distinction.