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

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Resolve paradoxes. This denition is not thorough because the paradoxes <strong>of</strong>
multiple-value and zero singularity are not completely resolved. The traditional notation
addresses these paradoxes by considering the domain and range <strong>of</strong> each function
and then limiting rule application accordingly. This approach does not clearly translate
to the boundary forms because the approach relies heavily on a fairly wide context
<strong>of</strong> forms (i.e. numerical class) to determine when a rule applies. In the boundary
calculus, the paradoxes must be considered at the level <strong>of</strong> boundaries, with as little
dependence on number type as possible. A preliminary attempt at this was made in
Chapter 6.
Additional functions. The boundary calculus can express only the basic arithmetic
functions. It leaves out many useful functions found in elementary algebra. While
it includes addition, subtraction, multiplication, division, exponents, and logarithms,
those functions which utilize set theoretics, such as summations, or those which have
discontinuities, such as absolute value, are not covered.
Structures around expressions. It lacks many structures for doing ordinary algebra,
such as equality and inequality (standard linear structures were used here). It has
no larger system <strong>of</strong> truth maintenance, nor a form for abstracting functions. Such basic
tenets <strong>of</strong> mathematics are sorely necessary for any notation to be remotely useful.
Since most <strong>of</strong> predicate calculus is well understood, this extension is reasonable.
In this discourse, these issues were avoided by embedding the boundary forms
within traditional linear constructions. Because boundary calculus is spatial, many
<strong>of</strong> its advantages are lost when it is restricted to linear constructs. To maintain the
spatial advantages, the calculus most cover this completely, so that the entire notation
is spatial.
7.3 Practical
Currently, the boundary calculus is impractical for manipulating numbers, because
traditional support tools and techniques do not apply directly to its forms. The
common knowledge taught in grade school mathematics is directed towards standard
notation and must be reconsidered to use with boundary numbers (see [29]).
Problem phrasing. Traditional problem representation techniques were designed
around the conceptual objects <strong>of</strong> standard notation: numbers and functions. In
contrast, the boundary forms use distinction and collection. It is unlikely that math-