Inquiry Into Inquiry

Riffs and Rotes • Happy New Year 2025

Posted on January 1, 2025 by Jon Awbrey

$\text{Let} ~ p_n = \text{the} ~ n^\text{th} ~ \text{prime}.$

$\text{Then} ~ 2025 = 81 \cdot 25 = 3^4 5^2 = {p_2}^4 {p_3}^2 = {p_2}^{{p_1}^{p_1}} {p_3}^{p_1} = {p_{p_1}}^{{p_1}^{p_1}} {p_{p_2}}^{p_1} = {p_{p_1}}^{{p_1}^{p_1}} {p_{p_{p_1}}}^{p_1}$

No information is lost by dropping the terminal 1s. Thus we may write the following form.

$2025 = {p_p}^{p^p} {p_{p_p}}^p$

The article linked below tells how forms of that sort correspond to a family of digraphs called riffs and a family of graphs called rotes. The riff and rote for $2025$ are shown in the next two Figures.

Riff 2025

Rote 2025

Reference

Riffs and Rotes

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Posted in Algebra, Arithmetic, Combinatorics, Computation, Graph Theory, Group Theory, Logic, Mathematics, Number Theory, Recursion, Representation, Riffs and Rotes, Semiotics, Visualization | Tagged Algebra, Arithmetic, Combinatorics, Computation, Graph Theory, Group Theory, Logic, Mathematics, Number Theory, Recursion, Representation, Riffs and Rotes, Semiotics, Visualization | Leave a comment

Differential Propositional Calculus • 37

Posted on December 31, 2024 by Jon Awbrey

Foreshadowing Transformations • Extensions and Projections of Discourse

And, despite the care which she took to look behind her at every moment, she failed to see a shadow which followed her like her own shadow, which stopped when she stopped, which started again when she did, and which made no more noise than a well‑conducted shadow should.

— Gaston Leroux • The Phantom of the Opera

Many times in our discussion we have occasion to place one universe of discourse in the context of a larger universe of discourse. An embedding of the type $[\mathcal{X}] \to [\mathcal{Y}]$ is implied any time we make use of one basis $\mathcal{X}$ which happens to be included in another basis $\mathcal{Y}.$ When discussing differential relations we usually have in mind the extended alphabet $\mathfrak{Y}$ has a special construction or a specific lexical relation with respect to the initial alphabet $\mathfrak{X},$ one which is marked by characteristic types of accents, indices, or inflected forms.

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Posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Category Theory, Change, Cybernetics, Differential Analytic Turing Automata, Differential Calculus, Differential Logic, Discrete Dynamics, Equational Inference, Functional Logic, Graph Theory, Hologrammautomaton, Indicator Functions, Inquiry Driven Systems, Leibniz, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Propositional Calculus, Time, Topology, Visualization | Tagged Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Category Theory, Change, Cybernetics, Differential Analytic Turing Automata, Differential Calculus, Differential Logic, Discrete Dynamics, Equational Inference, Functional Logic, Graph Theory, Hologrammautomaton, Indicator Functions, Inquiry Driven Systems, Leibniz, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Propositional Calculus, Time, Topology, Visualization | 3 Comments

Differential Propositional Calculus • 36

Posted on December 30, 2024 by Jon Awbrey

Transformations of Discourse

It is understandable that an engineer should be completely absorbed in his speciality, instead of pouring himself out into the freedom and vastness of the world of thought, even though his machines are being sent off to the ends of the earth; for he no more needs to be capable of applying to his own personal soul what is daring and new in the soul of his subject than a machine is in fact capable of applying to itself the differential calculus on which it is based. The same thing cannot, however, be said about mathematics; for here we have the new method of thought, pure intellect, the very well‑spring of the times, the fons et origo of an unfathomable transformation.

— Robert Musil • The Man Without Qualities

Here we take up the general study of logical transformations, or maps relating one universe of discourse to another. In many ways, and especially as applied to the subject of intelligent dynamic systems, the argument will develop the antithesis of the statement just quoted. Along the way, if incidental to my ends, I hope the present essay can pose a fittingly irenic epitaph to the frankly ironic epigraph inscribed at its head.

The goal is to answer a single question: What is a propositional tangent functor? In other words, the aim is to develop a clear conception of what manner of thing would pass in the logical realm for a genuine analogue of the tangent functor, an object conceived to generalize as far as possible in the abstract terms of category theory the ordinary notions of functional differentiation and the all too familiar operations of taking derivatives.

As a first step we examine the types of transformations we already know as extensions and projections and we use their special cases to illustrate several styles of logical and visual representation which figure in the sequel.

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Differential Propositional Calculus • 35

Posted on December 29, 2024 by Jon Awbrey

Example 2. Drives and Their Vicissitudes (concl.)

Applied to the example of $4^\text{th}$ ‑gear curves, the indexing scheme results in the data of the next two Tables, showing one period for each orbit.

The states in each orbit are listed as ordered pairs $(p_i, q_j),$ where $p_i$ may be read as a temporal parameter indicating the present time of the state and where $j$ is the decimal equivalent of the binary numeral $s.$

Grasped more intuitively, the Tables show each state $q_s$ with a subscript $s$ equal to the numerator of its rational index, taking for granted the constant denominator $2^4 = 16.$ In that way the temporal succession of states can be reckoned by a parallel round‑up rule. Namely, if $(d_k, d_{k+1})$ is any pair of adjacent digits in the state index $r$ then the value of $d_k$ in the next state is ${d_k}^\prime = d_k + d_{k+1}.$

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Differential Propositional Calculus • 34

Posted on December 29, 2024 by Jon Awbrey

Example 2. Drives and Their Vicissitudes (cont.)

With a little thought it is possible to devise a canonical indexing scheme for the states in differential logical systems. A scheme of that order allows for comparing changes of state in universes of discourse that weigh in on different scales of observation.

To that purpose, let us index the states $q \in \mathrm{E}^m X$ with the dyadic rationals (or the binary fractions) in the half-open interval $[0, 2).$ Formally and canonically, a state $q_r$ is indexed by a fraction $r = \tfrac{s}{t}$ whose denominator is the power of two $t = 2^m$ and whose numerator is a binary numeral formed from the coefficients of state in a manner to be described next.

The differential coefficients of the state $q$ are the values $\mathrm{d}^k\!A(q)$ for $k = 0 ~\text{to}~ m,$ where $\mathrm{d}^0\!A$ is defined as identical to $A.$ To form the binary index $d_0.d_1 \ldots d_m$ of the state $q$ the coefficient $\mathrm{d}^k\!A(q)$ is read off as the binary digit $d_k$ associated with the place value $2^{-k}.$ Expressed in algebraic terms, the rational index $r$ of the state $q$ is given by the following equivalent formulations.

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Differential Propositional Calculus • 33

Posted on December 28, 2024 by Jon Awbrey

Example 2. Drives and Their Vicissitudes (cont.)

Expressed in the language of drives and gears our next Example may be described as the family of fourth‑gear curves through the fourth extension $\mathrm{E}^4 X = \langle A, ~\mathrm{d}A, ~\mathrm{d}^2\!A, ~\mathrm{d}^3\!A, ~\mathrm{d}^4\!A \rangle.$ Those are the trajectories generated subject to the dynamic law $\mathrm{d}^4 A = 1,$ where it’s understood all higher order differences are equal to $0.$

Because $\mathrm{d}^4 A$ and all higher differences $\mathrm{d}^k A$ are fixed, the state vectors vary only with respect to their projections as points of $\mathrm{E}^3 X = \langle A, ~\mathrm{d}A, ~\mathrm{d}^2\!A, ~\mathrm{d}^3\!A \rangle.$ Thus there is just enough space in a planar venn diagram to plot the orbits and show how they partition the points of $\mathrm{E}^3 X.$ It turns out there are just two possible orbits, of eight points each, as shown in the following Figure.

$\text{Example 2. Fourth Gear Orbits}$

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Differential Propositional Calculus • 32

Posted on December 28, 2024 by Jon Awbrey

I open my scuttle at night and see the far‑sprinkled systems,
And all I see, multiplied as high as I can cipher, edge but
the rim of the farther systems.

— Walt Whitman • Leaves of Grass

Example 2. Drives and Their Vicissitudes

Before we leave the one‑feature case let’s look at a more substantial example, one which illustrates a general class of curves through the extended feature spaces and affords an opportunity to discuss important themes concerning their structure and dynamics.

As before let $\mathcal{X} = \{ x_1 \} = \{ A \}.$ The discussion to follow considers a class of trajectories having the property that $\mathrm{d}^k A = 0$ for all $k$ greater than a fixed value $m$ and indulges in the use of a picturesque vocabulary to describe salient classes of those curves.

Given the above finite order condition, there is a highest order non‑zero difference $\mathrm{d}^m A$ exhibited at each point of any trajectory one may consider. With respect to any point of the corresponding curve let us call that highest order differential feature $\mathrm{d}^m A$ the drive at that point. Curves of constant drive $\mathrm{d}^m A$ are then referred to as $m^\text{th}$ ‑gear curves.

Note. The fact that a difference calculus can be developed for boolean functions is well known and was probably familiar to Boole, who was an expert in difference equations before he turned to logic. And of course there is the strange but true story of how the Turin machines of the 1840s prefigured the Turing machines of the 1940s. At the very outset of general purpose mechanized computing we find the motive power driving the Analytical Engine of Babbage, the kernel of an idea behind all of his wheels, was exactly his notion that difference operations, suitably trained, can serve as universal joints for any conceivable computation.

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Differential Propositional Calculus • 31

Posted on December 27, 2024 by Jon Awbrey

Tacit Extensions

Returning to the Table of Differential Propositions, let’s examine how the general concept of a tacit extension applies to the differential extension of a one‑dimensional universe of discourse, where $\mathcal{X} = \{ A \}$ and $\mathcal{Y} = \mathrm{E}\mathcal{X} = \{ A, \mathrm{d}A \}.$

Each proposition $f_i : X \to \mathbb{B}$ has a canonical expression $e_i$ in the set $\{ 0, \texttt{(} A \texttt{)}, A, 1 \}.$ The tacit extension $\boldsymbol\varepsilon f_i : \mathrm{E}X \to \mathbb{B}$ may then be expressed as a logical conjunction $f_i = e_i \cdot \tau,$ where $\tau$ is a logical tautology using all the variables in $\mathcal{Y} - \mathcal{X}.$ The following Table shows how the tacit extensions $\boldsymbol\varepsilon f_i$ of the propositions $f_i$ may be expressed in terms of the extended basis $\{ A, \mathrm{d}A \}.$

$\text{Tacit Extension of}~ [A] ~\text{to}~ [A, \mathrm{d}A]$

In its bearing on the singular propositions over a universe of discourse $X$ the above analysis has an interesting interpretation. The tacit extension takes us from thinking about a particular state, like $A$ or $\texttt{(} A \texttt{)},$ to considering the collection of outcomes, the outgoing changes or singular dispositions springing or stemming from that state.

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Differential Propositional Calculus • 30

Posted on December 26, 2024 by Jon Awbrey

I would really like to have slipped imperceptibly into this lecture, as into all the others I shall be delivering, perhaps over the years ahead.

— Michel Foucault • The Discourse on Language

Tacit Extensions

In viewing the previous Table of Differential Propositions it is important to notice the subtle distinction in type between a function $f_i : X \to \mathbb{B}$ and its inclusion as a function $g_j : \mathrm{E}X \to \mathbb{B},$ even though they share the same logical expression. Naturally, we want to maintain the logical equivalence of expressions representing the same proposition while appreciating the full diversity of a proposition’s functional and typical representatives. Both perspectives, and all the levels of abstraction extending through them, have their reasons, as will develop in time.

Because this special circumstance points to a broader theme, it’s a good idea to discuss it more generally. Whenever there arises a situation like that above, where one basis $\mathcal{X}$ is a subset of another basis $\mathcal{Y},$ we say any proposition $f : \langle \mathcal{X} \rangle \to \mathbb{B}$ has a tacit extension to a proposition $\boldsymbol\varepsilon f : \langle \mathcal{Y} \rangle \to \mathbb{B}$ and we say the space $(\langle \mathcal{X} \rangle \to \mathbb{B})$ has an automatic embedding within the space $(\langle \mathcal{Y} \rangle \to \mathbb{B}).$

The tacit extension operator $\boldsymbol\varepsilon$ is defined in such a way that $\boldsymbol\varepsilon f$ puts the same constraint on the variables of $\mathcal{X}$ within $\mathcal{Y}$ as the proposition $f$ initially put on $\mathcal{X},$ while it puts no constraint on the variables of $\mathcal{Y}$ beyond $\mathcal{X},$ in effect, conjoining the two constraints.

Indexing the variables as $\mathcal{X} = \{ x_1, \ldots, x_n \}$ and $\mathcal{Y} = \{ x_1, \ldots, x_n, \ldots, x_{n+k} \}$ the tacit extension from $\mathcal{X}$ to $\mathcal{Y}$ may be expressed by the following equation.

$\boldsymbol\varepsilon f(x_1, \ldots, x_n, \ldots, x_{n+k}) ~=~ f(x_1, \ldots, x_n).$

On formal occasions, such as the present context of definition, the tacit extension from $\mathcal{X}$ to $\mathcal{Y}$ is explicitly symbolized by the operator $\boldsymbol\varepsilon : (\langle \mathcal{X} \rangle \to \mathbb{B}) \to (\langle \mathcal{Y} \rangle \to \mathbb{B}),$ where the bases $\mathcal{X}$ and $\mathcal{Y}$ are set in context, but it’s normally understood the $``\boldsymbol\varepsilon"$ may be silent.

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Differential Propositional Calculus • 29

Posted on December 25, 2024 by Jon Awbrey

I guess it must be the flag of my disposition, out of hopeful
green stuff woven.

— Walt Whitman • Leaves of Grass

Back to the Feature

Let’s assume the sense intended for differential features is well enough established in the intuition for now to continue outlining the structure of the differential extension $[\mathrm{E}\mathcal{X}] = [A, \mathrm{d}A].$

The extended alphabet $\mathrm{E}\mathcal{X} = \{ x_1, \mathrm{d}x_1 \} = \{ A, \mathrm{d}A \}$ of cardinality $2^n = 2$ generates the terms of description for the extended space $\mathrm{E}X$ of cardinality $2^{2n} = 4$ according to the following series of equations.

$\begin{array}{lll} \mathrm{E}X & = & \langle A, \mathrm{d}A \rangle \\[4pt] & = & \{ \texttt{(} A \texttt{)}, A \} ~\times~ \{ \texttt{(} \mathrm{d}A \texttt{)}, \mathrm{d}A \} \\[4pt] & = & \{ \texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)},~ \texttt{(} A \texttt{)} \mathrm{d}A,~ A \texttt{(} \mathrm{d}A \texttt{)},~ A ~ \mathrm{d}A \}. \end{array}$

The space $\mathrm{E}X$ may be given the nominal type $\mathbb{B} \times \mathbb{D},$ at root isomorphic to $\mathbb{B} \times \mathbb{B} = \mathbb{B}^2.$ An element of $\mathrm{E}X$ may be regarded as a disposition at a point or a situated direction, in effect, a singular mode of change occurring at a single point in the universe of discourse. In practice the modality of those changes may be interpreted in various ways, for example, as expectations, intentions, or observations with respect to the behavior of a system.

To construct the extended universe of discourse $\mathrm{E}X^\bullet = [A, \mathrm{d}A]$ the basic dispositions in $\mathrm{E}X$ need to be extended to the full set of differential propositions $\mathrm{E}X^\uparrow = \{ g : \mathrm{E}X \to \mathbb{B} \},$ each of type $\mathbb{B} \times \mathbb{D} \to \mathbb{B}.$ There are $2^{2^{2n}} = 16$ propositions in $\mathrm{E}X^\uparrow,$ as detailed in the following Table.

$\text{Differential Propositions}$

Aside from changing the names of variables and shuffling the order of rows, the Table follows the format previously used for boolean functions of two variables. The rows are grouped to reflect natural similarity classes holding among the propositions. In a future discussion the classes will be given additional explanation and motivation as the orbits of a certain transformation group acting on the set of 16 propositions. Notice that four of the propositions, in their logical expressions, resemble those given in the table for $X^\uparrow.$ Thus the first set of propositions $\{ f_i \}$ is automatically embedded in the present set $\{ g_j \}$ and the corresponding inclusions are indicated at the far left margin of the Table.

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	Differential Logic •… on Survey of Animated Logical Gra…
	Differential Logic •… on Survey of Differential Logic •…
	Differential Logic •… on Logic Syllabus
	Survey of Differenti… on Differential Logic • 2
	Differential Logic •… on Survey of Animated Logical Gra…
	Differential Logic •… on Survey of Differential Logic •…
	Differential Logic •… on Logic Syllabus
	Differential Logic •… on Survey of Animated Logical Gra…
	Differential Logic •… on Survey of Differential Logic •…
	Differential Logic •… on Logic Syllabus

M	T	W	T	F	S	S
						1
2	3	4	5	6	7	8
9	10	11	12	13	14	15
16	17	18	19	20	21	22
23	24	25	26	27	28

Riff 2025

Rote 2025

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Foreshadowing Transformations • Extensions and Projections of Discourse

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Transformations of Discourse

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Example 2. Drives and Their Vicissitudes (concl.)

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Example 2. Drives and Their Vicissitudes (cont.)

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Example 2. Drives and Their Vicissitudes (cont.)

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Example 2. Drives and Their Vicissitudes

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Tacit Extensions

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Tacit Extensions

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Back to the Feature

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