Frankl, My Dear : 12

It is one of the rules of my system of general harmony, that the present is big with the future, and that he who sees all sees in that which is that which shall be.

Leibniz • Theodicy

We continue with the differential analysis of the proposition in Example 1.

Example 1

 (1)

Like any moderately complex proposition, the difference map of a proposition has many equivalent logical expressions and can be read in many different ways.

 (5)

The expansion of $\mathrm{D}f$ computed in Post 9 and further discussed in Post 10 is shown again below with the terms arranged by number of positive differential features, from lowest to highest.

$\begin{array}{*{4}{l}} \multicolumn{4}{l}{\mathrm{D}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =} \\[10pt] & \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & \texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} \cdot \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} \\[10pt] + & \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} & + & \texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} \cdot \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} \\[10pt] + & \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} & + & \texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} \cdot \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} \end{array}$

$\begin{array}{*{4}{l}} + & \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} & + & \texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} \cdot \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} \\[10pt] + & \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} & + & \texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} \cdot \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} \\[10pt] + & \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} & + & \texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} \cdot \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} \\[10pt] + & \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} & + & \texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \cdot \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} \end{array}$

The terms of the difference map $\mathrm{D}f$ may be obtained from the table below by multiplying the base factor at the head of each column by the differential factor that appears beneath it in the body of the table.

The full boolean expansion of $\mathrm{D}f$ may be condensed to a degree by collecting terms that share the same base factors, as shown in the following display:

$\begin{array}{*{4}{c}} \multicolumn{4}{l}{\mathrm{D}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =} \\[10pt] & \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} & \cdot & \texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{))} \\[4pt] + & \texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} & \cdot & \texttt{~~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)~} \\[4pt] + & \texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} & \cdot & \texttt{~(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)~} \\[4pt] + & \texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} & \cdot & \texttt{~(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~~} \\[4pt] + & \texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} & \cdot & \texttt{~~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)~} \\[4pt] + & \texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} & \cdot & \texttt{~~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~~} \\[4pt] + & \texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} & \cdot & \texttt{~(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~~} \\[4pt] + & \texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} & \cdot & \texttt{~~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~~} \end{array}$

This amounts to summing terms along columns of the previous table, as shown at the bottom margin of the next table:

Collecting terms with the same differential factors produces the following expression:

$\begin{array}{*{4}{c}} \multicolumn{4}{l}{\mathrm{D}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =} \\[10pt] & \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & \cdot & q \texttt{~} r \\[4pt] + & \texttt{~} \mathrm{d}q \texttt{~(} \mathrm{d}p \texttt{)(} \mathrm{d}r \texttt{)} & \cdot & p \texttt{~} r \\[4pt] + & \texttt{~} \mathrm{d}r \texttt{~(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} & \cdot & p \texttt{~} q \\[4pt] + & \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} & \cdot & \texttt{((} p \texttt{,} q \texttt{))} ~ r \\[4pt] + & \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}r \texttt{~(} \mathrm{d}q \texttt{)} & \cdot & \texttt{((} p \texttt{,} r \texttt{))} ~ q \\[4pt] + & \texttt{~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~(} \mathrm{d}p \texttt{)} & \cdot & \texttt{((} q \texttt{,} r \texttt{))} ~ p \\[4pt] + & \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} & \cdot & p \texttt{~} q \texttt{~} r \\[4pt] + & \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} & \cdot & \texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \end{array}$

This is roughly what one would get by summing along rows of the previous tables.

To be continued …

❦ Pyramus & Thisbe ❦

It’s all about love
And the knots thereof
I have known beauty
I’ll bring it to you

Jon Awbrey • 12 Nov 2014

Posted in Anthem, Mantra, Verse | Tagged , , | Leave a comment

Frankl, My Dear : 11

Let’s take a moment from the differential analysis of the proposition in Example 1 to form a handy compendium of the results obtained so far.

 (1)

 (3)

 (4)

Difference Map $\mathrm{D}(pqr)$ of the Conjunction $pqr$

 (5)

To be continued …

Continuity, Generality, Infinity, Law, Synechism : 1

Peircers,

The concept of continuity that Peirce highlights in his synechism is a logical principle that is somewhat more general than the concepts of either mathematical or physical continua.

Peirce’s concept of continuity is better understood as a concept of lawful regularity or parametric variation. As such, it is basic to the coherence and utility of science, whether classical, relativistic, quantum mechanical, or any conceivable future science that deserves the name. (As Aristotle already knew.)

Perhaps the most pervasive examples of this brand of continuity in physics are the “correspondence principles” that describe the connections between classical and contemporary paradigms.

The importance of lawful regularities and parametric variations is not diminished one bit in passing from continuous mathematics to discrete mathematics, nor from theory to application.

Here are some further points of information, the missing of which seems to lie at the root of many recent disputes on the Peirce List:

It is necessary to distinguish the mathematical concepts of continuity and infinity from the question of their physical realization. The mathematical concepts retain their practical utility for modeling empirical phenomena quite independently of the (meta-)physical question of whether these continua and cardinalities are literally realized in the physical universe. This is equally true of any other domain or level of phenomena — chemical, biological, mental, social, or whatever.

As far as the mathematical concept goes, continuity is relative to topology. That is, what counts as a continuous function or transformation between spaces is relative to the topology under which those spaces are considered and the same spaces may be considered under many different topologies. What topology makes the most sense in a given application is another one of those abductive matters.

Regards,

Jon

Frankl, My Dear : 10

 (5)

Figure 5 shows the 14 terms of the difference map $\mathrm{D}f$ as arcs, arrows, or directed edges in the venn diagram of the original proposition $f(p, q, r) = pqr.$ The arcs of $\mathrm{E}f$ are directed into the cell where $f$ is true from each of the other cells. The arcs of $\boldsymbol\varepsilon f$ are directed from the cell where $f$ is true into each of the other cells.

The expansion of $\mathrm{D}f$ computed in the previous post is shown again below with the terms arranged by number of positive differential features, from lowest to highest.

$\begin{array}{*{4}{l}} \multicolumn{4}{l}{\mathrm{D}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =} \\[10pt] & \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & \texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} \cdot \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} \\[10pt] + & \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} & + & \texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} \cdot \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} \\[10pt] + & \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} & + & \texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} \cdot \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} \end{array}$

$\begin{array}{*{4}{l}} + & \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} & + & \texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} \cdot \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} \\[10pt] + & \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} & + & \texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} \cdot \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} \\[10pt] + & \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} & + & \texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} \cdot \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} \\[10pt] + & \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} & + & \texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \cdot \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} \end{array}$

To be continued …

Frankl, My Dear : 9

“It doesn’t matter what one does,” the Man Without Qualities said to himself, shrugging his shoulders. “In a tangle of forces like this it doesn’t make a scrap of difference.” He turned away like a man who has learned renunciation, almost indeed like a sick man who shrinks from any intensity of contact. And then, striding through his adjacent dressing-room, he passed a punching-ball that hung there; he gave it a blow far swifter and harder than is usual in moods of resignation or states of weakness.

Robert Musil • The Man Without Qualities

We continue with the differential analysis of the proposition in Example 1.

Example 1

 (1)

The difference operator $\mathrm{D}$ is defined as the difference $\mathrm{E} - \boldsymbol\varepsilon$ between the enlargement operator $\mathrm{E}$ and the tacit extension operator $\boldsymbol\varepsilon.$

The difference map $\mathrm{D}f$ is the result of applying the difference operator $\mathrm{D}$ to the function $f.$ When the sense is clear, we may refer to $\mathrm{D}f$ simply as the difference of $f.$

In boolean spaces there is no difference between the sum $(+)$ and the difference $(-)$ so the difference operator $\mathrm{D}$ is equally well expressed as the exclusive disjunction or symmetric difference $\mathrm{E} + \boldsymbol\varepsilon.$ In this case the difference map $\mathrm{D}f$ can be computed according to the formula $\mathrm{D}f = (\mathrm{E} + \boldsymbol\varepsilon)f = \mathrm{E}f + \boldsymbol\varepsilon f.$

The action of $\mathrm{D}$ on our present example, $f(p, q, r) = pqr,$ can be computed from the data on hand according to the following prescription.

The enlargement map $\mathrm{E}f,$ computed in Post 5 and graphed in Post 6, is shown again here:

$\mathrm{E}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =$

$\begin{smallmatrix} & p q r \,\cdot\, \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & p q \texttt{(} r \texttt{)} \,\cdot\, \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \mathrm{d}r & + & p \texttt{(} q \texttt{)} r \,\cdot\, \texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} & + & p \texttt{(} q \texttt{)(} r \texttt{)} \,\cdot\, \texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \, \mathrm{d}r \\[4pt] + & \texttt{(} p \texttt{)} q r \,\cdot\, \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & \texttt{(} p \texttt{)} q \texttt{(} r \texttt{)} \,\cdot\, \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)} \mathrm{d}r & + & \texttt{(} p \texttt{)(} q \texttt{)} r \,\cdot\, \mathrm{d}p \, \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} & + & \texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \,\cdot\, \mathrm{d}p \, \mathrm{d}q \, \mathrm{d}r \end{smallmatrix}$

The tacit extension $\boldsymbol\varepsilon f,$ computed in Post 7 and graphed in Post 8, is shown again here:

$\boldsymbol\varepsilon f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =$

$\begin{array}{*{8}{l}} & p q r ~ \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} & + & p q r ~ \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} \\[4pt] + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} & + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} \end{array}$

The difference map $\mathrm{D}f$ is the sum of the enlargement map $\mathrm{E}f$ and the tacit extension $\boldsymbol\varepsilon f.$

Here we adopt a paradigm of computation for $\mathrm{D}f$ that aids not only in organizing the stages of the work but also in highlighting the diverse facets of logical meaning that may be read off the result.

The terms of the enlargement map $\mathrm{E}f$ are obtained from the table below by multiplying the base factor at the head of each column by the differential factor that appears beneath it in the body of the table.

The terms of the tacit extension $\boldsymbol\varepsilon f$ are obtained from the next table below by multiplying the base factor at the head of the first column by each of the differential factors that appear beneath it in the body of the table.

Finally, the terms of the difference map $\mathrm{D}f$ are obtained by overlaying the displays for $\mathrm{E}f$ and $\boldsymbol\varepsilon f$ and taking their boolean sum entry by entry.

Notice that the “loop” or “no change” term $p q r \cdot \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}$ cancels out, leaving 14 terms in the end.

To be continued …

Frankl, My Dear : 8

 (4)

Figure 4 shows the eight terms of the tacit extension $\boldsymbol\varepsilon f$ as arcs, arrows, or directed edges in the venn diagram of the original proposition $f(p, q, r) = pqr.$ Each term of the tacit extension $\boldsymbol\varepsilon f$ corresponds to an arc that starts from the cell where $f$ is true and ends in one of the eight cells of the venn diagram.

For ease of reference, here is the expansion of $\boldsymbol\varepsilon f$ from the previous post:

$\boldsymbol\varepsilon f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =$

$\begin{array}{*{8}{l}} & p q r ~ \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} & + & p q r ~ \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} \\[4pt] + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} & + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} \end{array}$

Two examples suffice to convey the general idea of the extended venn diagram:

• The term $pqr \cdot \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}$ is shown as a looped arc starting in the cell where $pqr$ is true and returning back to it. The differential factor $\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}$ corresponds to the fact that the arc crosses no logical feature boundaries from its source to its target.
• The term $pqr \cdot \mathrm{d}p \; \mathrm{d}q \, \mathrm{d}r$ is shown as an arc going from the cell where $pqr$ is true to the cell where $\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)}$ is true. The differential factor $\mathrm{d}p \; \mathrm{d}q \, \mathrm{d}r$ corresponds to the fact that the arc crosses all three logical feature boundaries from its source to its target.

To be continued …