Quickly add a free MyWikiBiz directory listing!

# Inquiry Driven Systems : Fields Of Inquiry

Author: Jon Awbrey

 The field denotes this body, and wise men call one who knows it the field-knower. — Bhagavad Gita, 13.1

## Introduction : Review and Transition

 Know me as the field-knower in all fields — what I deem to be knowledge is knowledge of the field and its knower. — Bhagavad Gita, 13.2

In this exposition the character of a logical expansion or analytic form is seen to correspond to a particular perspective on a universe of discourse, a specialized way of viewing its propositions and organizing their interpretations into comprehensible forms.

In this essay we study the use of analytic forms both in the ordinary expansions and the differential analysis of propositions. The process of logical expansion is formalized in greater detail and, to compensate for the extra level of abstraction, the resulting analytic forms are motivated on intuitive lines as variant perspectives or alternate ways of viewing the situations represented in propositions. By the end of this ascent we hope the reader finds it pleasing to contemplate the whole panorama of differential operations and group actions on a proposition as just so many facets of its differential enlargement which arise from its projection onto complementary perspectives.

### Proposition Fields

 Hear from me in summary what the field is in its character and changes, and of the field-knower's power. — Bhagavad Gita, 13.3

This section introduces the class of mathematical objects known as proposition fields. These are structures which arise in a natural way from the analysis of information systems, especially when it comes to systems with “reflective” intelligence, those which maintain and process components of information about their own states. A system of this kind has its dynamics best understood only if we interpret some of its dimensions of variation as reflecting information about other components of state, and further, only if we assume that the system itself “interprets” these variables as being informative in just this way.

A notion of intelligence has just been made to depend on a notion of interpretation. Both issues are to be side-stepped here, but we leave this note as a pointer to future work. To make these ideas explicit, we ought to say what it means for a system to be an interpreter of some qualities of its state as signs for other qualities of its state. But understanding how this is possible is tantamount to a major objective of this whole study, and can only be developed over the course of our investigation. It may not seem like good mathematical practice to let the clearing up of definitions fall to the end, but that is how it sometimes has to be.

A proposition field is a mapping from points to propositions, that is, it is a function having the general form $$U \to (V \to \mathbb{B}).\!$$ We may visualize the proposition field as “tagging” each point $$x\!$$ in one universe of discourse $$U\!$$ with a determinate proposition $$f_x : V \to \mathbb{B}~\!$$ in another universe of discourse $$V.\!$$

Under the general type of a proposition field there are numerous special cases which frequently arise. Common species of proposition fields are listed in Table 1, which displays next to the informal name of each an indication how its type is specialized.

 $$\text{Name}\!$$ $$\text{Type}\!$$ $$\text{Field}\!$$ $$U \to (V \to \mathbb{B})\!$$ $$\text{Fold}\!$$ $$U \to (U \to \mathbb{B})\!$$ $$\text{File}\!$$ $$U \to (U^\prime \to \mathbb{B})\!$$ $$\text{Flow}\!$$ $$U \to (\mathrm{d}U \to \mathbb{B})\!$$ $$\text{Fray}\!$$ $$\mathrm{d}U \to (U \to \mathbb{B})\!$$

In the midst of a particular application we often find ourselves taking a definite proposition $$F\!$$ of a certain complexity and deriving from it a proposition field $$\boldsymbol\iota F.\!$$ The proposition field $$\boldsymbol\iota F\!$$ may be viewed as setting forth one of the many ways of comprehending $$F\!$$ as an organized collection of simpler propositions, namely, by associating a unique proposition on a simpler space with each point of another simpler space.

The proposition $$F,\!$$ in turn, often arises as a tacit extension $$\boldsymbol\varepsilon f\!$$ or a differential enlargement $$\mathrm{E}f\!$$ of a prior proposition $$f.\!$$ Table 2 exhibits typical settings in which proposition fields are developed and applied, taking into consideration the kinds of fields noted above. In each case, composing the whole sequence of operations found in a given row of the Table, we define a transformation $$T,\!$$ parameterized by the subscripted argument of the last evaluation step, which takes us from propositions in one universe of discourse to propositions in that same or other related universe.

 $$\text{Origin}~ f\!$$ $$\to~\!$$ $$\text{Extension}~ F~\!$$ $$\to~\!$$ $$\text{Factorization}~ \Gamma\!$$ $$\to~\!$$ $$\text{Evaluation}~ g\!$$ $$f : U \to \mathbb{B}\!$$ $$\boldsymbol\varepsilon\!$$ $$\boldsymbol\varepsilon f : U \times V \to \mathbb{B}\!$$ $$\boldsymbol\iota_1\!$$ $$\Phi f : U \to (V \to \mathbb{B})\!$$ $$\boldsymbol\varphi_u\!$$ $$T_u f : V \to \mathbb{B}\!$$ $$f : U \to \mathbb{B}\!$$ $$\boldsymbol\varepsilon\!$$ $$\boldsymbol\varepsilon f : U \times U \to \mathbb{B}\!$$ $$\boldsymbol\iota_1\!$$ $$\Phi f : U \to (U \to \mathbb{B})\!$$ $$\boldsymbol\varphi_u\!$$ $$T_u f : U \to \mathbb{B}\!$$ $$f : U \to \mathbb{B}\!$$ $$\boldsymbol\varepsilon\!$$ $$\boldsymbol\varepsilon f : U \times U^\prime \to \mathbb{B}\!$$ $$\boldsymbol\iota_1\!$$ $$\Phi f : U \to (U^\prime \to \mathbb{B})\!$$ $$\boldsymbol\varphi_u\!$$ $$T_u f : U^\prime \to \mathbb{B}\!$$ $$f : U \to \mathbb{B}\!$$ $$\mathrm{E}\!$$ $$\mathrm{E}f : U \times \mathrm{d}U \to \mathbb{B}\!$$ $$\boldsymbol\iota_1\!$$ $$\Phi f : U \to (\mathrm{d}U \to \mathbb{B})\!$$ $$\boldsymbol\varphi_u\!$$ $$T_u f : \mathrm{d}U \to \mathbb{B}\!$$ $$f : U \to \mathbb{B}\!$$ $$\mathrm{E}\!$$ $$\mathrm{E}f : U \times \mathrm{d}U \to \mathbb{B}\!$$ $$\boldsymbol\iota_2\!$$ $$\Psi f : \mathrm{d}U \to (U \to \mathbb{B})\!$$ $$\boldsymbol\varphi_v\!$$ $$T_v f : U \to \mathbb{B}\!$$

### Analytic Expansions

 Ancient seers have sung of this in many ways, with varied meters and with aphorisms on the infinite spirit laced with logical arguments. — Bhagavad Gita, 13.4

In “Tools and Views” we considered the analytic expansions of propositions with respect to various degrees of interpretation of their variables. The significance of these expansions for logical understanding is that they factor the process of interpretation between two stages, in application giving two moments to the clarification of a problematic proposition. In type, an expansion corresponds to an isomorphism,

 $$\begin{matrix} (\mathbb{B}^n \to \mathbb{B}) & \cong & (\mathbb{B}^j \times \mathbb{B}^{n-j} \to \mathbb{B})) & \cong & (\mathbb{B}^j \to (\mathbb{B}^{n-j} \to \mathbb{B})), \end{matrix}$$

in which propositions $$F : \mathbb{B}^n \to \mathbb{B}\!$$ are reconsidered as “proposition fields” $$\boldsymbol\iota F : \mathbb{B}^j \to (\mathbb{B}^k \to \mathbb{B}),\!$$ where $$n = j + k.\!$$ This gives rise to “n choose j” renditions of the isomorphism $$\boldsymbol\iota\!$$ for each rendering of the original alphabet into a $$j\!$$-set and a $$k\!$$-set.

In the conception of an individual analytic form the set of variables is divided into two parts and the sense of each proposition is apportioned accordingly. The initial set of variables is used to sweep out a range of partial interpretations and thus to set the context of analysis. The remaining set is left uninterpreted and kept embodied in propositional coefficients which are associated with the “loci” (the points or topics) of this conceptual framework.

In appearance, the expression of the analytic form is made up of coefficient propositions on a subset of the variables taking their places next to partial interpretations over the complementary set. This may seem like a purely superficial change, since the analytic expansion preserves logical equivalence with the original proposition, but the good of a particular form for a particular situation lies precisely in the facility of its appearance being unaffected by its logical substance. Its choice and use are matters of pragmatic relevance and suitability to a purpose. This means that the worth of a particular form for a problematic situation can only be judged after the fact of its being applied, by the contribution it makes toward clarifying an opportune constellation of propositions, those which comprehend the situation and define the problem.

If we need to formalize the process, each isomorphism $$\boldsymbol\iota\!$$ is implicitly indexed by a subset $$U\!$$ of the original alphabet $$X,\!$$ but most often we consider only a single expansion at a time and can safely let context determine the sense.

Let $$X = \{ x_1, \ldots, x_m \}\!$$ be our principal set of logical variables, and let $$X\!$$ be divided arbitrarily into a pair of subsets, which we may assume without loss of generality to form an initial segment $$U = \{ x_1, \ldots, x_j \}\!$$ and a final segment $$V = \{ x_{j+1}, \ldots, x_m \},\!$$ of cardinalities $$j\!$$ and $$k,\!$$ respectively, where $$m = j + k.\!$$ Corresponding to the choice of $$U\!$$ (which determines the complementary set $$V\!$$) there are a pair of isomorphisms,

 $$\begin{matrix} \boldsymbol\iota_U : (X \to \mathbb{B}) & = & (U \times V \to \mathbb{B}) & \to & (U \to (V \to \mathbb{B})), \\ \boldsymbol\iota_V : (X \to \mathbb{B}) & = & (U \times V \to \mathbb{B}) & \to & (V \to (U \to \mathbb{B})), \end{matrix}$$

whose employments we describe by saying that the proposition $$F\!$$ can be factored into the proposition fields $$\boldsymbol\iota_U F\!$$ and $$\boldsymbol\iota_V F,\!$$ respectively. These factorizations are commonly expressed by means of the associated analytic forms, which determine a pair of logical expansions for each proposition $$F\!$$ in the universe $$X^\bullet,\!$$

 $$\begin{matrix} F(x) & = & F(u, v) & = & \sum_u \boldsymbol\varphi_u F(v) \cdot u, \\ F(x) & = & F(u, v) & = & \sum_v \boldsymbol\varphi_v F(u) \cdot v. \end{matrix}$$

### Elementary Examples

 The field contains the great elements, individuality, understanding, unmanifest nature, the eleven senses, and the five sense realms. — Bhagavad Gita, 13.5

A few examples of analytic expansions are appropriate here. The following selection exhibits the concepts and notation we need from previous discussions, indicates how we intend to adapt these materials to the present purpose, and provides a set of building blocks for future constructions.

To begin with a minimal example, the conjunction $$J(x, y) = x \cdot y\!$$ yields the partial expansion with respect to $$x,\!$$

 $$\begin{array}{lrrrr} J(x, y) & = & J(1, y) \cdot x & + & J(0, y) \cdot (x), \\ & = & \boldsymbol\varphi_x J(y) \cdot x & + & \boldsymbol\varphi_{(x)} J(y) \cdot (x), \\ & = & y \cdot x & + & 0 \cdot (x), \end{array}$$

the partial expansion with respect to $$y,\!$$

 $$\begin{array}{lrrrr} J(x, y) & = & J(x, 1) \cdot y & + & J(x, 0) \cdot (y), \\ & = & \boldsymbol\varphi_y J(x) \cdot y & + & \boldsymbol\varphi_{(y)} J(x) \cdot (y), \\ & = & x \cdot y & + & 0 \cdot (y), \end{array}$$

and the complete expansion with respect to $$x\!$$ and $$y,\!$$

 $$\begin{array}{l*{8}{r}} J(x, y) & = & J(1, 1) \cdot xy & + & J(1, 0) \cdot x(y) & + & J(0, 1) \cdot (x)y & + & J(0, 0) \cdot (x)(y), \\ & = & \boldsymbol\varphi_{xy} J \cdot xy & + & \boldsymbol\varphi_{x(y)} J \cdot x(y) & + & \boldsymbol\varphi_{(x)y} J \cdot (x)y & + & \boldsymbol\varphi_{(x)(y)} J \cdot (x)(y), \\ & = & 1 \cdot xy & + & 0 \cdot x(y) & + & 0 \cdot (x)y & + & 0 \cdot (x)(y). \end{array}$$

Notice how the results of the coefficient extraction (partial interpretation or partial evaluation) operators $$\boldsymbol\varphi_\alpha,\!$$ where the index $$\alpha\!$$ refers to a singular proposition of the relevant category of discourse, could almost be defined in terms of the analytic form, namely, as the propositions occupying the designated places of the expansion. This becomes a real possibility if we have an independent way of developing the analytic form in a constructive or computational setting.

Finally, the reason we are taking such great pains to define all entities in terms of operators on names of expressions is because we wish to anticipate those circumstances of computation when we fail to have the full expressions available all the time, for instance, to evaluate by performing substitutions on.

Figure 3 uses the “bundle of boxes” style of venn diagram to illustrate the partial expansions of the conjunction $$J(x, y).\!$$

 $$\text{Figure 3. Factorizations of Conjunction}\!$$

As a slightly more complex example, consider the boolean function determined by the equality or biconditional $$I(x, y) = ((x, y)).\!$$ This affords the partial expansion with respect to $$x,\!$$

 $$\begin{array}{lrrrr} I(x, y) & = & y \cdot x & + & (y) \cdot (x), \end{array}$$

the partial expansion with respect to $$y,\!$$

 $$\begin{array}{lrrrr} I(x, y) & = & x \cdot y & + & (x) \cdot (y), \end{array}$$

and the complete expansion with respect to $$x\!$$ and $$y,\!$$

 $$\begin{array}{l*{8}{r}} I(x, y) & = & 1 \cdot xy & + & 0 \cdot x(y) & + & 0 \cdot (x)y & + & 1 \cdot (x)(y). \end{array}$$

Figure 4 illustrates the partial expansions of the equality $$I(x, y).\!$$

 $$\text{Figure 4. Factorizations of Equality}\!$$

For our last example, the implication or conditional $$K(x, y) = (x(y))~\!$$ results in the partial expansion with respect to $$x,\!$$

 $$\begin{array}{lrrrr} K(x, y) & = & y \cdot x & + & 1 \cdot (x), \end{array}$$

the partial expansion with respect to $$y,\!$$

 $$\begin{array}{lrrrr} K(x, y) & = & 1 \cdot y & + & (x) \cdot (y), \end{array}$$

and the complete expansion with respect to $$x\!$$ and $$y,\!$$

 $$\begin{array}{l*{8}{r}} K(x, y) & = & 1 \cdot xy & + & 0 \cdot x(y) & + & 1 \cdot (x)y & + & 1 \cdot (x)(y). \end{array}$$

Figure 5 illustrates the partial expansions of the implication $$K(x, y).\!$$

 $$\text{Figure 5. Factorizations of Implication}\!$$

### Differential Enlargements

 Longing, hatred, happiness, suffering, bodily form, consciousness, resolve, thus is this field with its changes defined in summary. — Bhagavad Gita, 13.6

Another important operation treated in the Tools and Views paper was the differential enlargement of propositions. This a mapping of the type,

 $$\mathrm{E} : (U \to \mathbb{B}) \to (U \times \mathrm{d}U \to \mathbb{B}),\!$$

whose action on any proposition $$f : U \to \mathbb{B}\!$$ is defined by an equation of the form,

 $$\begin{matrix} \mathrm{E}f(x_1, \ldots, x_n, \mathrm{d}x_1, \ldots, \mathrm{d}x_n) & = & f(x_1 + \mathrm{d}x_1, \ldots, x_n + \mathrm{d}x_n). \end{matrix}$$

In the effort to comprehend what the differential enlargement of a proposition is telling us about the original proposition we usually parse it as a proposition field in either one of two obvious ways. If we factor the ordinary component out to the front we obtain the flow,

 $$\begin{matrix} \Phi f & = & \boldsymbol\iota_U \mathrm{E}f : U \to (\mathrm{d}U \to \mathbb{B}), \end{matrix}$$

If we factor the differential component out to the front we obtain fray,

 $$\begin{matrix} \Psi f & = & \boldsymbol\iota_{\mathrm{d}U} \mathrm{E}f : \mathrm{d}U \to (U \to \mathbb{B}). \end{matrix}$$

Evaluating the flow at points of the original universe, we discover the intentional attraction or influence of $$f\!$$ at each point,

 $$\begin{matrix} T_u f & = & \boldsymbol\varphi_u \Phi f & = & \boldsymbol\varphi_u \boldsymbol\iota_U \mathrm{E}f : \mathrm{d}U \to \mathbb{B}. \end{matrix}$$

Evaluating the fray along directions of the differential universe, we find the differential action or diffraction of $$f\!$$ in each direction,

 $$\begin{matrix} T_v f & = & \boldsymbol\varphi_v \Psi f & = & \boldsymbol\varphi_v \boldsymbol\iota_{\mathrm{d}U} \mathrm{E}f : U \to \mathbb{B}. \end{matrix}$$

Diffraction = differential factorization (or differential fractionation):

 $$\begin{matrix} (\mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B}) & \cong & (\mathbb{D}^n \to (\mathbb{B}^n \to \mathbb{B}). \end{matrix}$$

Between the twin perspectives afforded by these two $$\boldsymbol\iota\!$$'s we have what we need to synthesize a stereotactic grasp of the situation represented in a proposition.

## Differential Propositions and Transformation Groups

 Its hands and feet reach everywhere; its head and face see in every direction; hearing everything, it remains in the world, enveloping all. — Bhagavad Gita, 13.13

In this section we examine a number of relationships between differential operators and higher order propositions, together with the actions and characters of related transformation groups on the space of propositions.

### Differential Expansions

 Lacking all the sense organs, it shines in their qualities; unattached, it supports everything; without qualities, it enjoys them. — Bhagavad Gita, 13.14

On analogy with usage in ordinary calculus, we introduce the following terminology. Given two sets of logical features $$X \subseteq Y,\!$$ say

 $$X = \{ x_1, \ldots, x_n \}\!$$ and $$Y = \{ x_1, \ldots, x_m \},\!$$

a proposition $$F : Y \to \mathbb{B}\!$$ is called an infinitesimal with respect to the universe $$X,\!$$ written $$F \in \Upsilon(X),\!$$ if $$F\!$$ is false outside the region of the logical disjunction $$((x_1)( \ldots )(x_n)),\!$$ in other words, if $$F = 0\!$$ at the origin $$(x_1)( \ldots )(x_n)\!$$ of $$X.\!$$

 $$\text{Figure 6. Enlargement of Conjunction}\!$$

 $$\text{Figure 7. Diffraction of Conjunction}\!$$

 $$\text{Figure 8. Enlargement of Equality}\!$$

 $$\text{Figure 9. Diffraction of Equality}\!$$

 $$\text{Figure 10. Enlargement of Implication}\!$$

 $$\text{Figure 11. Diffraction of Implication}\!$$

### Partial Evaluations

 Outside and within all creatures, inanimate but still animate, too subtle to be known, it is far distant, yet near. — Bhagavad Gita, 13.15

 $$f\!$$ $$\mathrm{E}f\!$$ $$\mathrm{E}f\!$$ @ $$x \cdot y\!$$ $$\mathrm{E}f\!$$ @ $$x \cdot (y)\!$$ $$\mathrm{E}f\!$$ @ $$(x) \cdot y\!$$ $$\mathrm{E}f\!$$ @ $$(x)(y)\!$$ $$f_0\!$$ $$()\!$$ $$()\!$$ $$0\!$$ $$0\!$$ $$0\!$$ $$0\!$$ $$f_1\!$$ $$(x)(y)\!$$ $$((x,\mathrm{d}x))((y,\mathrm{d}y))\!$$ $$\mathrm{d}x ~ \mathrm{d}y\!$$ $$\mathrm{d}x (\mathrm{d}y)\!$$ $$(\mathrm{d}x) \mathrm{d}y\!$$ $$(\mathrm{d}x)(\mathrm{d}y)\!$$ $$f_2\!$$ $$(x) y\!$$ $$((x,\mathrm{d}x)) (y,\mathrm{d}y)\!$$ $$\mathrm{d}x (\mathrm{d}y)\!$$ $$\mathrm{d}x ~ \mathrm{d}y\!$$ $$(\mathrm{d}x)(\mathrm{d}y)\!$$ $$(\mathrm{d}x) \mathrm{d}y\!$$ $$f_4\!$$ $$x (y)\!$$ $$(x,\mathrm{d}x) ((y,\mathrm{d}y))\!$$ $$(\mathrm{d}x) \mathrm{d}y\!$$ $$(\mathrm{d}x)(\mathrm{d}y)\!$$ $$\mathrm{d}x ~ \mathrm{d}y\!$$ $$\mathrm{d}x (\mathrm{d}y)\!$$ $$f_8\!$$ $$x y\!$$ $$(x,\mathrm{d}x) (y,\mathrm{d}y)\!$$ $$(\mathrm{d}x)(\mathrm{d}y)\!$$ $$(\mathrm{d}x) \mathrm{d}y\!$$ $$\mathrm{d}x (\mathrm{d}y)\!$$ $$\mathrm{d}x ~ \mathrm{d}y\!$$ $$f_3\!$$ $$(x)\!$$ $$((x,\mathrm{d}x))\!$$ $$\mathrm{d}x\!$$ $$\mathrm{d}x\!$$ $$(\mathrm{d}x)\!$$ $$(\mathrm{d}x)\!$$ $$f_{12}\!$$ $$x\!$$ $$(x,\mathrm{d}x)\!$$ $$(\mathrm{d}x)\!$$ $$(\mathrm{d}x)\!$$ $$\mathrm{d}x\!$$ $$\mathrm{d}x\!$$ $$f_6\!$$ $$(x, y)\!$$ $$((x,\mathrm{d}x), (y,\mathrm{d}y))\!$$ $$(\mathrm{d}x, \mathrm{d}y)\!$$ $$((\mathrm{d}x, \mathrm{d}y))\!$$ $$((\mathrm{d}x, \mathrm{d}y))\!$$ $$(\mathrm{d}x, \mathrm{d}y)\!$$ $$f_9\!$$ $$((x, y))\!$$ $$(((x,\mathrm{d}x), (y,\mathrm{d}y)))~\!$$ $$((\mathrm{d}x, \mathrm{d}y))\!$$ $$(\mathrm{d}x, \mathrm{d}y)\!$$ $$(\mathrm{d}x, \mathrm{d}y)\!$$ $$((\mathrm{d}x, \mathrm{d}y))\!$$ $$f_5\!$$ $$(y)\!$$ $$((y,\mathrm{d}y))\!$$ $$\mathrm{d}y\!$$ $$(\mathrm{d}y)\!$$ $$\mathrm{d}y\!$$ $$(\mathrm{d}y)\!$$ $$f_{10}\!$$ $$y\!$$ $$(y,\mathrm{d}y)\!$$ $$(\mathrm{d}y)\!$$ $$\mathrm{d}y\!$$ $$(\mathrm{d}y)\!$$ $$\mathrm{d}y\!$$ $$f_7\!$$ $$(x y)\!$$ $$((x,\mathrm{d}x) (y,\mathrm{d}y))\!$$ $$((\mathrm{d}x)(\mathrm{d}y))\!$$ $$((\mathrm{d}x) \mathrm{d}y)\!$$ $$(\mathrm{d}x (\mathrm{d}y))\!$$ $$(\mathrm{d}x ~ \mathrm{d}y)\!$$ $$f_{11}\!$$ $$(x (y))\!$$ $$((x,\mathrm{d}x) ((y,\mathrm{d}y)))\!$$ $$((\mathrm{d}x) \mathrm{d}y)\!$$ $$((\mathrm{d}x)(\mathrm{d}y))\!$$ $$(\mathrm{d}x ~ \mathrm{d}y)\!$$ $$(\mathrm{d}x (\mathrm{d}y))\!$$ $$f_{13}\!$$ $$((x) y)\!$$ $$(((x,\mathrm{d}x)) (y,\mathrm{d}y))~\!$$ $$(\mathrm{d}x (\mathrm{d}y))\!$$ $$(\mathrm{d}x ~ \mathrm{d}y)\!$$ $$((\mathrm{d}x)(\mathrm{d}y))\!$$ $$((\mathrm{d}x) \mathrm{d}y)\!$$ $$f_{14}\!$$ $$((x)(y))\!$$ $$(((x,\mathrm{d}x))((y,\mathrm{d}y)))\!$$ $$(\mathrm{d}x ~ \mathrm{d}y)\!$$ $$(\mathrm{d}x (\mathrm{d}y))\!$$ $$((\mathrm{d}x) \mathrm{d}y)\!$$ $$((\mathrm{d}x)(\mathrm{d}y))\!$$ $$f_{15}\!$$ $$(())\!$$ $$(())\!$$ $$1\!$$ $$1\!$$ $$1\!$$ $$1\!$$

### Group Actions

 Undivided, it seems divided among creatures; understood as their sustainer, it devours and creates them. — Bhagavad Gita, 13.16

 $$f\!$$ $$\mathrm{E}f\!$$ $$T_{11}f\!$$ $$T_{10}f\!$$ $$T_{01}f\!$$ $$T_{00}f\!$$ $$x = x_1\!$$ $$y = x_2\!$$ $$\mathrm{sub}_i [(x_i + \mathrm{d}x_i)/x_i] F\!$$ $$\mathrm{E}f\!$$ @ $$\mathrm{d}x \cdot \mathrm{d}y\!$$ $$\mathrm{E}f\!$$ @ $$\mathrm{d}x \cdot (\mathrm{d}y)\!$$ $$\mathrm{E}f\!$$ @ $$(\mathrm{d}x) \cdot \mathrm{d}y\!$$ $$\mathrm{E}f\!$$ @ $$(\mathrm{d}x)(\mathrm{d}y)\!$$ $$f_0\!$$ $$()\!$$ $$()\!$$ $$()\!$$ $$()\!$$ $$()\!$$ $$()\!$$ $$f_1\!$$ $$(x)(y)\!$$ $$((x,\mathrm{d}x))((y,\mathrm{d}y))\!$$ $$x y\!$$ $$x (y)\!$$ $$(x) y\!$$ $$(x)(y)\!$$ $$f_2\!$$ $$(x) y\!$$ $$((x,\mathrm{d}x))(y,\mathrm{d}y)\!$$ $$x (y)\!$$ $$x y\!$$ $$(x)(y)\!$$ $$(x) y\!$$ $$f_4\!$$ $$x (y)\!$$ $$(x,\mathrm{d}x)((y,\mathrm{d}y))\!$$ $$(x) y\!$$ $$(x)(y)\!$$ $$x y\!$$ $$x (y)\!$$ $$f_8\!$$ $$x y\!$$ $$(x,\mathrm{d}x)(y,\mathrm{d}y)\!$$ $$(x)(y)\!$$ $$(x) y\!$$ $$x (y)\!$$ $$x y\!$$ $$f_3\!$$ $$(x)\!$$ $$((x,\mathrm{d}x))\!$$ $$x\!$$ $$x\!$$ $$(x)\!$$ $$(x)\!$$ $$f_{12}\!$$ $$x\!$$ $$(x,\mathrm{d}x)\!$$ $$(x)\!$$ $$(x)\!$$ $$x\!$$ $$x\!$$ $$f_6\!$$ $$(x, y)\!$$ $$((x,\mathrm{d}x), (y,\mathrm{d}y))\!$$ $$(x, y)\!$$ $$((x, y))\!$$ $$((x, y))\!$$ $$(x, y)\!$$ $$f_9\!$$ $$((x, y))\!$$ $$(((x,\mathrm{d}x), (y,\mathrm{d}y)))~\!$$ $$((x, y))\!$$ $$(x, y)\!$$ $$(x, y)\!$$ $$((x, y))\!$$ $$f_5\!$$ $$(y)\!$$ $$((y,\mathrm{d}y))\!$$ $$y\!$$ $$(y)\!$$ $$y\!$$ $$(y)\!$$ $$f_{10}\!$$ $$y\!$$ $$(y,\mathrm{d}y)\!$$ $$(y)\!$$ $$y\!$$ $$(y)\!$$ $$y\!$$ $$f_7\!$$ $$(x y)\!$$ $$((x,\mathrm{d}x)(y,\mathrm{d}y))\!$$ $$((x)(y))\!$$ $$((x) y)\!$$ $$(x (y))\!$$ $$(x y)\!$$ $$f_{11}\!$$ $$(x (y))\!$$ $$((x,\mathrm{d}x)((y,\mathrm{d}y)))\!$$ $$((x) y)\!$$ $$((x)(y))\!$$ $$(x y)\!$$ $$(x (y))\!$$ $$f_{13}\!$$ $$((x) y)\!$$ $$(((x,\mathrm{d}x))(y,\mathrm{d}y))~\!$$ $$(x (y))\!$$ $$(x y)\!$$ $$((x)(y))\!$$ $$((x) y)\!$$ $$f_{14}\!$$ $$((x)(y))\!$$ $$(((x,\mathrm{d}x))((y,\mathrm{d}y)))\!$$ $$(x y)\!$$ $$(x (y))\!$$ $$((x) y)\!$$ $$((x)(y))\!$$ $$f_{15}\!$$ $$(())\!$$ $$(())\!$$ $$(())\!$$ $$(())\!$$ $$(())\!$$ $$(())\!$$ $$\text{Total Number of Fixed Points:}\!$$ $$4\!$$ $$4\!$$ $$4\!$$ $$16\!$$

### Derivations

 The light of lights beyond darkness it is called; knowledge attained by knowledge, fixed in the heart of everyone. — Bhagavad Gita, 13.17

 $$\mathrm{D}f = f + \mathrm{E}f\!$$ $$\mathrm{D}f~\!$$ @ $$x \cdot y\!$$ $$\mathrm{D}f~\!$$ @ $$x \cdot (y)\!$$ $$\mathrm{D}f~\!$$ @ $$(x) \cdot y\!$$ $$\mathrm{D}f~\!$$ @ $$(x)(y)\!$$ $$f_0\!$$ $$() + ()\!$$ $$0\!$$ $$0\!$$ $$0\!$$ $$0\!$$ $$f_1\!$$ $$(x)(y) + ((x,\mathrm{d}x))((y,\mathrm{d}y))\!$$ $$\mathrm{d}x ~ \mathrm{d}y\!$$ $$\mathrm{d}x (\mathrm{d}y)\!$$ $$(\mathrm{d}x) \mathrm{d}y\!$$ $$((\mathrm{d}x)(\mathrm{d}y))\!$$ $$f_2\!$$ $$(x) y + ((x,\mathrm{d}x)) (y,\mathrm{d}y)\!$$ $$\mathrm{d}x (\mathrm{d}y)\!$$ $$\mathrm{d}x ~ \mathrm{d}y\!$$ $$((\mathrm{d}x)(\mathrm{d}y))\!$$ $$(\mathrm{d}x) \mathrm{d}y\!$$ $$f_4\!$$ $$x (y) + (x,\mathrm{d}x) ((y,\mathrm{d}y))\!$$ $$(\mathrm{d}x) \mathrm{d}y\!$$ $$((\mathrm{d}x)(\mathrm{d}y))\!$$ $$\mathrm{d}x ~ \mathrm{d}y\!$$ $$\mathrm{d}x (\mathrm{d}y)\!$$ $$f_8\!$$ $$x y + (x,\mathrm{d}x) (y,\mathrm{d}y)\!$$ $$((\mathrm{d}x)(\mathrm{d}y))\!$$ $$(\mathrm{d}x) \mathrm{d}y\!$$ $$\mathrm{d}x (\mathrm{d}y)\!$$ $$\mathrm{d}x ~ \mathrm{d}y\!$$ $$f_3\!$$ $$(x) + ((x,\mathrm{d}x))\!$$ $$\mathrm{d}x\!$$ $$\mathrm{d}x\!$$ $$\mathrm{d}x\!$$ $$\mathrm{d}x\!$$ $$f_{12}\!$$ $$x + (x,\mathrm{d}x)\!$$ $$\mathrm{d}x\!$$ $$\mathrm{d}x\!$$ $$\mathrm{d}x\!$$ $$\mathrm{d}x\!$$ $$f_6\!$$ $$(x, y) + ((x,\mathrm{d}x), (y,\mathrm{d}y))\!$$ $$(\mathrm{d}x, \mathrm{d}y)\!$$ $$(\mathrm{d}x, \mathrm{d}y)\!$$ $$(\mathrm{d}x, \mathrm{d}y)\!$$ $$(\mathrm{d}x, \mathrm{d}y)\!$$ $$f_9\!$$ $$((x, y)) + (((x,\mathrm{d}x), (y,\mathrm{d}y)))\!$$ $$(\mathrm{d}x, \mathrm{d}y)\!$$ $$(\mathrm{d}x, \mathrm{d}y)\!$$ $$(\mathrm{d}x, \mathrm{d}y)\!$$ $$(\mathrm{d}x, \mathrm{d}y)\!$$ $$f_5\!$$ $$(y) + ((y,\mathrm{d}y))\!$$ $$\mathrm{d}y\!$$ $$\mathrm{d}y\!$$ $$\mathrm{d}y\!$$ $$\mathrm{d}y\!$$ $$f_{10}\!$$ $$y + (y,\mathrm{d}y)\!$$ $$\mathrm{d}y\!$$ $$\mathrm{d}y\!$$ $$\mathrm{d}y\!$$ $$\mathrm{d}y\!$$ $$f_7\!$$ $$(x y) + ((x,\mathrm{d}x) (y,\mathrm{d}y))\!$$ $$((\mathrm{d}x)(\mathrm{d}y))\!$$ $$(\mathrm{d}x) \mathrm{d}y\!$$ $$\mathrm{d}x (\mathrm{d}y)\!$$ $$\mathrm{d}x ~ \mathrm{d}y\!$$ $$f_{11}\!$$ $$(x (y)) + ((x,\mathrm{d}x) ((y,\mathrm{d}y)))\!$$ $$(\mathrm{d}x) \mathrm{d}y\!$$ $$((\mathrm{d}x)(\mathrm{d}y))\!$$ $$\mathrm{d}x ~ \mathrm{d}y\!$$ $$\mathrm{d}x (\mathrm{d}y)\!$$ $$f_{13}\!$$ $$((x) y) + (((x,\mathrm{d}x)) (y,\mathrm{d}y))\!$$ $$\mathrm{d}x (\mathrm{d}y)\!$$ $$\mathrm{d}x ~ \mathrm{d}y\!$$ $$((\mathrm{d}x)(\mathrm{d}y))\!$$ $$(\mathrm{d}x) \mathrm{d}y\!$$ $$f_{14}\!$$ $$((x)(y)) + (((x,\mathrm{d}x))((y,\mathrm{d}y)))\!$$ $$\mathrm{d}x ~ \mathrm{d}y\!$$ $$\mathrm{d}x (\mathrm{d}y)\!$$ $$(\mathrm{d}x) \mathrm{d}y\!$$ $$((\mathrm{d}x)(\mathrm{d}y))\!$$ $$f_{15}\!$$ $$(()) + (())\!$$ $$0\!$$ $$0\!$$ $$0\!$$ $$0\!$$

## Generalized Projections : Analytic Operators and Fields of Inquiry

 Arjuna, know that anything inanimate or alive with motion is born from the union of the field and its knower. — Bhagavad Gita, 13.26

Projections, Dissections, Distortions, Perspectives (Outlooks, Opinions, Local Views),

Partial Derivatives, Fields of Inquiry, Surveys, Test Fields, Textures, Cultures, Biases, Warps.

Relation to questions of value-free inquiry and inquiry into values, to what extent these projects are possible or approachable from the stance of a concrete interpreter.

Analytic operators or derivations, all of type $$q : (\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^n \to \mathbb{B}).\!$$ Includes projections and partial derivatives. Equivalence to proposition fields, question fields, fields of inquiry, as follows:

 $$Q : \mathbb{B}^n \to (F : (\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}).\!$$

Note relation between operator symmetry and dimension reduction. When the question field $$Q\!$$ is based on a symmetric operation, $$f(x, y) = f(y, x),\!$$ then the corresponding analytic operator $$q\!$$ may be viewed under a dimension-reducing type as $$q : (\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^{n-1} \to \mathbb{B}).\!$$

 $$f\!$$ $$\theta f_{xy1}\!$$ $$\theta f_{xy0}\!$$ $$\theta f_{1yz}\!$$ $$\theta f_{0yz}\!$$ $$\theta f_{x1z}\!$$ $$\theta f_{x0z}\!$$ $$f_0\!$$ $$0\!$$ $$1\!$$ $$(z)\!$$ $$(z)\!$$ $$(z)\!$$ $$(z)\!$$ $$f_1\!$$ $$(x)(y)\!$$ $$((x)(y))\!$$ $$(z)\!$$ $$(y, z)\!$$ $$(z)\!$$ $$(x, z)\!$$ $$f_2\!$$ $$(x) y\!$$ $$((x) y)\!$$ $$(z)\!$$ $$((y, z))\!$$ $$(x, z)\!$$ $$(z)\!$$ $$f_4\!$$ $$x (y)\!$$ $$(x (y))\!$$ $$(y, z)\!$$ $$(z)\!$$ $$(z)\!$$ $$((x, z))\!$$ $$f_8\!$$ $$x y\!$$ $$(x y)\!$$ $$((y, z))\!$$ $$(z)\!$$ $$((x, z))\!$$ $$(z)\!$$ $$f_3\!$$ $$(x)\!$$ $$x\!$$ $$(z)\!$$ $$z\!$$ $$(x, z)\!$$ $$(x, z)\!$$ $$f_{12}\!$$ $$x\!$$ $$(x)\!$$ $$z\!$$ $$(z)\!$$ $$((x, z))\!$$ $$((x, z))\!$$ $$f_6\!$$ $$(x, y)\!$$ $$((x, y))\!$$ $$(y, z)\!$$ $$((y, z))\!$$ $$(x, z)\!$$ $$((x, z))\!$$ $$f_9\!$$ $$((x, y))\!$$ $$(x, y)\!$$ $$((y, z))\!$$ $$(y, z)\!$$ $$((x, z))\!$$ $$(x, z)\!$$ $$f_5\!$$ $$(y)\!$$ $$y\!$$ $$(y, z)\!$$ $$(y, z)\!$$ $$(z)\!$$ $$z\!$$ $$f_{10}\!$$ $$y\!$$ $$(y)\!$$ $$((y, z))\!$$ $$((y, z))\!$$ $$z\!$$ $$(z)\!$$ $$f_7\!$$ $$(x y)\!$$ $$x y\!$$ $$(y, z)\!$$ $$z\!$$ $$(x, z)\!$$ $$z\!$$ $$f_{11}\!$$ $$(x (y))\!$$ $$x (y)\!$$ $$((y, z))\!$$ $$z\!$$ $$z\!$$ $$(x, z)\!$$ $$f_{13}\!$$ $$((x) y)\!$$ $$(x) y\!$$ $$z\!$$ $$(y, z)\!$$ $$((x, z))\!$$ $$z\!$$ $$f_{14}\!$$ $$((x)(y))\!$$ $$(x)(y)\!$$ $$z\!$$ $$((y, z))\!$$ $$z\!$$ $$((x, z))\!$$ $$f_{15}\!$$ $$1\!$$ $$0\!$$ $$z\!$$ $$z\!$$ $$z\!$$ $$z\!$$