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

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Technology
interface
s<strong>of</strong>tware
Solution Loop
application
6
?
conceptual objects
6
?
computational objects
hardware
6
Activity
problem representation
translation and control
calculation
Figure 8.1: Technology and Representation.
With this new set <strong>of</strong> primitives, the activities in the solution loop become more
clearly delineated. In particular, control <strong>of</strong> the calculation steps, direction, becomes
clearly separated from the constraints <strong>of</strong> the calculation, terrain. These constraints
can be maintained throughout an algorithm so that all intermediate states are mathematically
equivalent. These primitive dynamics allow more precise control within the
representation than standard forms allow. Any given standard form is not optimally
structured for certain calculations.
Pushing the notation down to computational objects signicantly aects the tools
and mechanismsby which we do mathematics. We can streamline the entire solution
loop and the technology that assists us: the hardware that performs calculations,
the s<strong>of</strong>tware that builds and manipulates conceptual objects, our interface to these
mathematical objects, and the pedagogy <strong>of</strong> this entire loop. Each <strong>of</strong> these areas are
protable applications <strong>of</strong> this material and are briey considered in the following
sections.
8.2 Hardware
The way we conceptualize mathematics aects how we implement computational
hardware. Most machine computations are built upon a system <strong>of</strong> mathematics based
in the higher conceptual objects <strong>of</strong> the standard notation. The boundary calculus
provides lower-level forms from which to base machine computations.
The boundary calculus has computational advantages because it is inherently