User:JonAwbrey/Sandbox

Terminological Interlude
At this point several issues of terminology have accrued enough substance to intrude on our discussion. The remarks of this Subsection are intended to accomplish two goals. First, we call attention to significant aspects of the previous series of Figures, translating into literal terms what they depict in iconic forms, and we re-stress the most important structural elements they indicate. Next, we prepare the way for taking on more complex examples of transformations, those whose target universes have more than one 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 $$\mathrm{W} = (\mathrm{W}_1, \ldots, \mathrm{W}_n),\!$$ transformations $$\mathrm{F} = (\mathrm{F}_1, \ldots, \mathrm{F}_n),\!$$ and target domains $$X^\bullet = [x_1, \ldots, x_n],\!$$ 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 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. We will follow 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 $$J : U \to \mathbb{B}\!$$ and the thematized form of a mapping $$J : U^\bullet \to [x]\!$$ without much trouble. In these cases we are able to tolerate a higher degree of ambiguity about the precise nature of an operator's input and output domains than we otherwise might. For example, in the preceding treatment of the example $$J,\!$$ and for each operator $$\mathrm{W}\!$$ in the set $$\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \},\!$$ both the operand $$J\!$$ and the result $$\mathrm{W}J\!$$ could be viewed in either one of two ways. On one hand we may treat them as propositions $$J : U \to \mathbb{B}\!$$ and $$\mathrm{W}J : \mathrm{E}U \to \mathbb{B},\!$$ ignoring the distinction between the range $$[x] \cong \mathbb{B}\!$$ of $$\boldsymbol\varepsilon J\!$$ and the range $$[\mathrm{d}x] \cong \mathbb{D}\!$$ of the other types of $$\mathrm{W}J.\!$$ This is what we usually do when we content ourselves with simply coloring in regions of venn diagrams. On the other hand we may view these entities as maps $$J : U^\bullet \to [x] = X^\bullet\!$$ and $$\boldsymbol\varepsilon J : \mathrm{E}U^\bullet \to [x] \subseteq \mathrm{E}X^\bullet\!$$ or $$\mathrm{W}J : \mathrm{E}U^\bullet \to [\mathrm{d}x] \subseteq \mathrm{E}X^\bullet,\!$$ in which case the qualitative characters of the output features are not ignored.

At the beginning of this Section we recast the natural form of a proposition $$J : U \to \mathbb{B}\!$$ into the thematic role of a transformation $$J : U^\bullet \to [x],\!$$ where $$x\!$$ was a variable recruited to express the newly independent $$\check{J}.\!$$ However, in our computations and representations of operator actions we immediately lapsed back to viewing the results as native elements of the extended universe $$\mathrm{E}U^\bullet,\!$$ in other words, as propositions $$\mathrm{W}J : \mathrm{E}U \to \mathbb{B},\!$$ where $$\mathrm{W}\!$$ ranged over the set $$\{ \boldsymbol\varepsilon, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}.\!$$  That is as it should be. We have worked hard to devise a language that gives us these advantages &mdash; the flexibility to exchange terms and types of equal information value and the capacity to reflect as quickly and as wittingly as a controlled reflex on the fibers of our propositions, independently of whether they express amusements, beliefs, or conjectures.

As we take on target spaces of increasing dimension, however, these types of confusions (and confusions of types) become less and less permissible. For this reason, Tables 54 and 55 present a rather detailed summary of the notation and the terminology we are using, as applied to the case $$J = uv.\!$$ The rationale of these Tables is not so much to train more elephant guns on this poor drosophila of a concrete example but to invest our paradigm with enough solidity to bear the weight of abstraction to come.

Table 54 provides basic notation and descriptive information for the objects and operators used in this Example, giving the generic type (or broadest defined type) for each entity. Here, the sans serif operators $$\mathsf{W} \in \{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{d}, \mathsf{r} \}\!$$ and their components $$\mathrm{W} \in \{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}\!$$ both have the same broad type $$\mathsf{W}, \mathrm{W} : (U^\bullet \to X^\bullet) \to (\mathrm{E}U^\bullet \to \mathrm{E}X^\bullet),\!$$ as appropriate to operators that map transformations $$J : U^\bullet \to X^\bullet\!$$ to extended transformations $$\mathsf{W}J, \mathrm{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet.\!$$

Table 55 supplies a more detailed outline of terminology for operators and their results. Here, we list the restrictive subtype (or narrowest defined subtype) that applies to each entity and we indicate across the span of the Table the whole spectrum of alternative types that color the interpretation of each symbol. For example, all the component operator maps $$\mathrm{W}J\!$$ have $$1\!$$-dimensional ranges, either $$\mathbb{B}^1\!$$ or $$\mathbb{D}^1,\!$$ and so they can be viewed either as propositions $$\mathrm{W}J : \mathrm{E}U \to \mathbb{B}\!$$ or as logical transformations $$\mathrm{W}J : \mathrm{E}U^\bullet \to X^\bullet.\!$$ As a rule, the plan of the Table allows us to name each entry by detaching the underlined adjective at the left of its row and prefixing it to the generic noun at the top of its column. In one case, however, it is customary to depart from this scheme. Because the phrase differential proposition, applied to the result $$\mathrm{d}J : \mathrm{E}U \to \mathbb{D},\!$$ does not distinguish it from the general run of differential propositions $$\mathrm{G}: \mathrm{E}U \to \mathbb{B},\!$$ it is usual to single out $$\mathrm{d}J\!$$ as the tangent proposition of $$J.\!$$