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

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contents, they are also drawn as delimiters: (A); [A]; :
These three boundaries are the fundamental forms <strong>of</strong> the calculus <strong>of</strong> number.
Alphabetic characters are also used to denote unknown values and constants. Equality
is expressed as usual, denoting congurations <strong>of</strong> equal numerical value.
Natural numbers can be represented by collecting a single boundary, as in a tally
system. Call this boundary instance and draw it round. Counting proceeds by
accumulating empty instance boundaries, as ; ; ;:::and so on. These numbers
add by collecting them in space, no computation is necessary. A second boundary|
call it abstract and draw it square|enables formation <strong>of</strong> multiplication and power
functions. For example, the multiplication 2 3 appears in boundary notation as
([][]) and the cube <strong>of</strong> two, 2 3 , appears as (([[]][])):
A third boundary|call it inverse|can be used to construct additive and multiplicative
inverses, and other inverse structures such as roots. Inverse extends the
calculus to the integers by serving as the additive inverse, to the rationals by serving
as the multiplicative inverse, and to algebraic irrationals in other combinations. A
full sampling <strong>of</strong> boundary numbers is shown in Table 1.2 and the boundary operations
are shown in Table 1.3.
Calculation on these forms is completely dened by three axioms: involution, distribution,
and inversion, shown in Table 1.4. Involution denes instance and abstract
to be functionally opposite. Distribution denes them to additionally have a distributive
relationship. Inversion denes the inverse boundary to behave asaninverse
operator. The three axioms govern all transformations <strong>of</strong> boundary expressions.
The calculus constructs general forms that are not available in the standard notation.
Using involution and distribution, a repeated form such asAAA can be transformed
into an equivalent expression that requires only one reference to that form,
as ([A][]). This transformation is called cardinality and is shown in Table 1.5.
This means <strong>of</strong> counting applies to repeated addition and to repeated multiplication;
it is therefore considered to be a generalized form <strong>of</strong> cardinality.
The inverse boundary is also unparalleled in standard notation. It is used to
form the additive inverse as well as the multiplicative inverse (see Table 1.3). All
<strong>of</strong> the properties <strong>of</strong> this generalized inverse apply to both <strong>of</strong> these contexts. These
properties include collection <strong>of</strong> inverses, cancellation with itself, and relocation, all
shown in Table 1.5.