ONT Re: Differential Logic
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DLOG. Note D64
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Terminological Interlude
| Lastly, my attention was especially attracted, not so much to the scene,
| as to the mirrors that produced it. These mirrors were broken in parts.
| Yes, they were marked and scratched; they had been 'starred', in spite
| of their solidity ...
|
| Gaston Leroux, 'The Phantom of the Opera', [Ler, 230]
At this point several issues of terminology have accrued enough substance
to intrude on our discussion. The remarks of this Section are intended to
accomplish two goals. First, I call attention to important aspects of the
previous series of Figures, translating into literal terms what they depict
in iconic forms, and I restress the most important structural elements that
they indicate. Next, I prepare the way for taking on more complex examples
of transformations, whose target universes have more than a single dimension.
In talking about the actions of operators it is important to keep in mind the
distinctions between the operators per se, their operands, and their results.
Furthermore, in working with composite forms of operators W = <W_1, ..., W_n>,
transformations F = <F_1, ..., F_n>, and target domains X% = [x_1, ..., x_n],
we need to preserve a clear distinction between the compound entity of each
given type and any one of its separate components. It is curious, given the
usefulness of the concepts "operator" and "operand", that we seem to lack
a generic term, formed on the same root, for the corresponding result of
an operation. Following the obvious paradigm would lead on to words like
"opus", "opera", and "operant", but these words are too affected with clang
associations to work well at present, though they might be adapted in time.
One current usage gets around this problem by using the substantive "map"
as a systematic epithet to express the result of each operator's action.
I am following this practice as far as possible, for example, using the
phrase "tangent map" to denote the end product of the tangent functor
acting on its operand map.
Scholium. See [JGH, 6-9] for a good account of tangent functors
and tangent maps in ordinary analysis, and for examples of their
use in mechanics. This work as a whole is a model of clarity in
applying functorial principles to problems in physical dynamics.
Whenever we focus on isolated propositions, on single components of composite
operators, or on the portions of transformations that have 1-dimensional ranges,
we are free to shift between the native form of a proposition J : U -> B and the
thematized form of a mapping J : U% -> [x] without much trouble. In these cases
we are able to tolerate a higher degree of ambiguity about the precise nature of
the input and output domains of an operator than we otherwise might. For example,
in the preceding treatment of the example J, and for each operator W in the set
{!e!, !h!, E, D, d, r}, both the operand J and the result WJ could be viewed in
either one of two ways. On the one hand, we could regard them as propositions
J : U -> B and WJ : EU -> B, ignoring the qualitative distinction between the
range [x] ~=~ B of !e!J and the range [dx] ~=~ D of the other WJ's. This is
what we usually do when we content ourselves simply with coloring in regions
of venn diagrams. On the other hand, we could view these entities as maps
J : U% -> [x] = X% and !e!J : EU% -> [x] c EX% or WJ : EU% -> [dx] c EX%,
in which case the qualitative characters of the output features are not
allowed to go without saying, nor thus at the risk of being forgotten.
Jon Awbrey
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