ONT Re: Differential Logic
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DLOG. Note D35
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Alias and Alibi Transformations
There are customarily two modes of understanding a transformation,
at least, when we try to interpret its relevance to something in
reality. A transformation always refers to a changing prospect,
to say it in a unified but equivocal way, but this can be taken
to mean either a subjective change in the interpreting observer's
point of view or an objective change in the systematic subject of
discussion. In practice these variant uses of the transformation
concept are distinguished in the following terms:
1. A "perspectival" or "alias" transformation refers to
a shift in perspective or a change in language that
takes place in the observer's frame of reference.
2. A "transitional" or "alibi" transformation refers to
a change of position or an alteration of state that
occurs in the object system as it falls under study.
(For a recent discussion of the "alias vs. alibi" issue, as
it relates to linear transformations in vector spaces and to
other issues of an algebraic nature, see [MaB, 256, 582-4].)
Naturally, when we are concerned with the dynamical properties of a system,
the transitional aspect of transformation is the factor that comes to the
fore, and this involves us in contemplating all of the ways of changing
a universe into itself while remaining under the rule of established
dynamical laws. In the prospective application to dynamic systems,
and to neural networks viewed in this light, our interest lies
chiefly with the transformations of a state space into itself
that constitute the state transitions of a discrete dynamic
process. Nevertheless, many important properties of these
transformations, and some constructions that we need to
see most clearly, are independent of the transitional
interpretation and are likely to be confounded with
irrelevant features if presented first and only
in that association.
In addition, and in partial contrast, intelligent systems are exactly that
species of dynamic agents that have the capacity to have a point of view,
and we cannot do justice to their peculiar properties without examining
their ability to form and transform their own frames of reference in
exposure to the elements of their own experience. In this setting,
the perspectival aspect of transformation is the facet that shines
most brightly, perhaps too often leaving us fascinated with mere
glimmerings of its actual potential. It needs to be emphasized
that nothing of the ordinary sort needs be moved in carrying out
a transformation under the alias interpretation, that it may only
involve a change in the forms of address, an amendment of the terms
which are customed to approach and fashioned to describe the very same
things in the very same world. But again, working within a discipline of
realistic computation, we know how formidably complex and resource-consuming
such transformations of perspective can be to implement in practice, much less
to endow in the self-governed form of a nascently intelligent dynamical system.
Jon Awbrey
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