From 38f418e0b206b372b9dafeecc6536f574eb502ea Mon Sep 17 00:00:00 2001
From: grr <grr@lo2.org>
Date: Sat, 25 May 2024 01:56:58 -0400
Subject: pile of fixes for studies in constructed memories

---
 essays/studies_in_constructed_memories.tex | 261 ++++++++++++++++-------------
 1 file changed, 144 insertions(+), 117 deletions(-)

(limited to 'essays/studies_in_constructed_memories.tex')

diff --git a/essays/studies_in_constructed_memories.tex b/essays/studies_in_constructed_memories.tex
index 2612ead..03690ce 100644
--- a/essays/studies_in_constructed_memories.tex
+++ b/essays/studies_in_constructed_memories.tex
@@ -8,47 +8,31 @@ In these studies, I will define some conscious memory-systems, and
 investigate them. The investigation will be mathematical. In fact, the nearest 
 precedent for it is perhaps the geometry of Nicholas Lobachevski. 
 Non-Euclidian geometry had many founders, but Lobachevski in particular 
-spoke of his system as an "imaginary geometry." Lobachevski's system was, 
-so to speak, the physical geometry of an "imaginary," or constructed, space. 
+spoke of his system as an \enquote{imaginary geometry.} Lobachevski's system was, 
+so to speak, the physical geometry of an \enquote{imaginary,} or constructed, space. 
 By analogy, my investigation could be called a psychological algebra of 
 constructed minds. It is too early to characterize the investigation more 
-exactly. Let us just remember Rudoiph Carnap's Principle of Tolerance in 
+exactly. Let us just remember Rudoiph Carnap's \uline{Principle of Tolerance} in 
 mathematics: the mathematician is free to construct his system in any way 
 he chooses. 
 
-I will begin by introducing a repertory of concepts informally, 
-becoming more formal as I go along. Consider ongoing actions, which by 
-definition extend through past, present, and future. For example, "I am 
-making the trip from New York to Chicago." Consider also past actions 
-which have probable consequences in the present. "I have been heating this 
-water" (entailing that it isn't frozen now). I will be concerned with such 
-actions as these. 
-
-Our language provides for the following assertion: "I am off to the 
-country today; I could have been off to the beach; I could not possibly have 
-been going to the center of the sun". We distinguish an actual action from a 
-possible action; and distinguish both from an action which is materially 
-impossible. People insist that there are things they could do, even though 
-they don't choose to do them (as opposed to things they couldn't do). What 
-distinguishes these possible actions from impossible ones? Rather than 
-trying to analyze such everyday notions in terms of the logic of 
-counterfactual conditionals, or of modalities, or of probability, I choose to 
-take the notions at their face value. My concern is not to philosophize, but 
-to assemble concepts with which to define an interesting memory system. 
+I will begin by introducing a repertory of concepts informally, becoming more formal as I go along. Consider ongoing actions, which by definition extend through past, present, and future. For example, \enquote{I am making the trip from New York to Chicago.} Consider also past actions which have probable consequences in the present. \enquote{I have been heating this water} (entailing that it isn't frozen now). I will be concerned with such actions as these. 
+
+Our language provides for the following assertion: \enquote{I am off to the country today; I could have been off to the beach; I could not possibly have been going to the center of the sun}. We distinguish an actual action from a possible action; and distinguish both from an action which is materially impossible. People insist that there are things they could do, even though they don't choose to do them (as opposed to things they couldn't do). What distinguishes these possible actions from impossible ones? Rather than trying to analyze such everyday notions in terms of the logic of counterfactual conditionals, or of modalities, or of probability, I choose to take the notions at their face value. My concern is not to philosophize, but to assemble concepts with which to define an interesting memory system. 
 
 What is the introspective psychological difference between a thought 
 that has the force of a memory, and a thought that has the force of a 
 fantasied past, a merely possible past? I am not asking how I know that a 
 verbalized memory is true; I am asking what quality a naive thought has that 
-marks it as a memory. Let Alternative E be that I went to an East Side 
-restaurant yesterday, and Alternative W be that I went to a West Side one. 
-By the "thought of E" I mean mainly the visualization of going into the East 
-Side restaurant. My thought of E has the force of memory. It actually 
-happened. W is something I could have done. I can imagine I did do W. There 
-is nothing present which indicates whether I did E or W. Yet W merely has 
+marks it as a memory. Let Alternative $E$ be that I went to an East Side 
+restaurant yesterday, and Alternative $W$ be that I went to a West Side one. 
+By the \enquote{thought of $E$} I mean mainly the visualization of going into the East 
+Side restaurant. My thought of $E$ has the force of memory. It actually 
+happened. $W$ is something I could have done. I can imagine I did do W. There 
+is nothing present which indicates whether I did $E$ or $W$. Yet $W$ merely has 
 the force of possibility, of fantasy. How do the two thoughts differ? Is the 
-thought of E involuntarily more vivid? Is there perhaps an "attitude of 
-assertion" involuntarily present in the thought of E? 
+thought of $E$ involuntarily more vivid? Is there perhaps an \enquote{attitude of 
+assertion} involuntarily present in the thought of $E$? 
 
 Consider the memory that I was almost run down by a truck yesterday: 
 I could have been run down, but wasn't. In such a case, the possibility that I 
@@ -57,7 +41,7 @@ wasn't. (Is it not insanity, when a person is overwhelmed by the fear of a
 merely possible past event?) My hold on sanity here would be the awareness 
 that I am alive and well today. 
 
-In dreams, do we not wholeheartedly "remember" that a misfortune 
+In dreams, do we not wholeheartedly \enquote{remember} that a misfortune 
 has befallen us, and begin to adjust emotionally to it? Then we awake, and 
 wholeheartedly remember that the misfortune has not befallen us. The 
 thought that had the force of memory in the dream ceases to have that force 
@@ -74,24 +58,28 @@ moment?
 I will now specify a situation involving possible actions and 
 remembering. 
 
-Situation 1. "I could have been accomplishing G by doing $A_{a_1}$, or by 
-doing $A_{a_2}$, \ldots, or by doing $A_{a_n}$; but I have actually been accomplishing G by 
-doing $A_{a_1}$." Here the ongoing actions $A_{a_i}$, $i=1,...,n$,$a_i\neq a_h if i\neq h$, are 
-the possible methods of accomplishing G. (The subscripts are supposed to 
+\newenvironment{hangers}{\vskip 0.5em\begin{hangparas}{3em}{1}}{\end{hangparas}\vskip 0.5em}
+
+\begin{hangers}
+\textbf{Situation 1.} \enquote{I could have been accomplishing $G$ by doing $A_{a_1}$, or by 
+doing $A_{a_2}$, \ldots, or by doing $A_{a_n}$; but I have actually been accomplishing $G$ by 
+doing $A_{a_1}$.} Here the ongoing actions $A_{a_i}$, $i=1,\ldots,n$,$a_i\neq a_h$ if $i\neq h$, are 
+the possible methods of accomplishing $G$. (The subscripts are supposed to 
 indicate that the methods are distinct and countable, but not ordered.) The 
 possible methods cannot be combined, let us assume. 
+\end{hangers}
 
 In such a situation, perhaps the thought that I have been doing $A_{a_1}$
-would be distinguished from similar thoughts about $A_{a_2}, ..., A_{a_n}$ by the
-presence of the "attitude of assertion". Since the possible methods are 
+would be distinguished from similar thoughts about $A_{a_2}, \ldots, A_{a_n}$ by the
+presence of the \enquote{attitude of assertion}. Since the possible methods are 
 ongoing actions, the thought that I have been doing $A_{a_i}$ has logical or 
 probabie consequences I can check against the present. 
 
 Now $A_{a_1}$, is actual and $A_{a_2}$ is not, so that $A_{a_1}$, simply cannot have 
-possible jar in $A_{a_3}$ to contain it. The only "connection" $A_{a_1}$ could have
+possible jar in $A_{a_3}$ to contain it. The only \enquote{connection} $A_{a_1}$ could have
 material contact with $A_{a_2}$. An actual liquid in $A_{a_1}$ could not require a 
 with $A_{a_2}$, would be verbal and gratuitous. Therefore, in order to be possible 
-methods, $A_{a_2}$, ..., $A_{a_n}$ must be materially separable. A liquid in $A_{a_2}$ must
+methods, $A_{a_2}$, \ldots, $A_{a_n}$ must be materially separable. A liquid in $A_{a_2}$ must
 not require a jar in $A_{a_3}$ to contain it. If it did, $A_{a_2}$ couldn't be actualized 
 while $A_{a_3}$, remained only a possibility. 
 
@@ -103,11 +91,13 @@ Enough concepts are now at hand for the studies to begin in earnest.
 \newcommand{\assumption}[1]{\textit{Assumption #1.}}
 \newcommand{\conclusion}[1]{\textbf{Conclusion #1.}}
 
-\definition Given the sentences "I have actually been doing $A_{a_i}$", where 
+\begin{hangers}
+\definition\ Given the sentences \enquote{I have actually been doing $A_{a_i}$}, where 
 the $A_{a_i}$ are non-combinable possible methods as in Situation 1, an 
-"M-Memory" is a memory of a conscious organism such that the organism 
+\enquote{M-Memory} is a memory of a conscious organism such that the organism 
 can think precisely one of the sentences at a time, and any of the sentences 
 has the force of memory. 
+\end{hangers}
 
 This definition refers to language, mind, and the rest of reality in their 
 interrelations, but the crucial reference is to a property of certain sentences. 
@@ -118,35 +108,36 @@ by a linguistic property). I can say at once that the conditions must enable
 the organism to think the sentences at will, and they must provide that the 
 memory is consistent with the organism's present awareness. 
 
-\definition The "P-Memory" of a conscious organism is its conscious 
+\begin{hangers}
+\definition\ The \term{P-Memory} of a conscious organism is its conscious 
 memory of what it did and what happened to it, the past events of its life. I 
-want to distinguish here the "personal" memory from the preconscious. 
+want to distinguish here the \enquote{personal} memory from the preconscious. 
 
-\definition An "L-Memory" is a linguistic P-Memory having no 
+\definition\ An \term{L-Memory} is a linguistic P-Memory having no 
 extra-linguistic component. Of course, the linguistic component has 
-extra-linguistic mental associations which give it "meaning"--otherwise the 
+extra-linguistic mental associations which give it \enquote{meaning}--otherwise the 
 memory wouldn't be conscious. But these associations lack the force of a 
 mental reliving of the past independent of language. An L-Memory amounts 
 to extra-linguistic amnesia. 
 
 \assumption{1.1} With respect to normal human memory, when I forget 
-whether I did x, I can't voluntarily give either the thought that I did x, or 
-the thought that I didn't do x, the force of memory. I know that I either did 
-or didn't do x, but I can create no conviction for either alternative. (An 
+whether I did $x$, I can't voluntarily give either the thought that I did $x$, or 
+the thought that I didn't do $x$, the force of memory. I know that I either did 
+or didn't do $x$, but I can create no conviction for either alternative. (An 
 introspective observation.) 
 
 \conclusion{1.2} An L-Memory is not sufficient for an M-Memory, even 
 in the trivial case that the $A_{a_i}$ are beyond perception (as internal bodily 
 processes are). True, there would be no present perceptions to check the 
-sentences "I have actually been doing $A_{a_i}$" against. True, the L-Memory 
-precludes any extra-linguistic memory-"feelings" which would conflict with 
+sentences \enquote{I have actually been doing $A_{a_i}$} against. True, the L-Memory 
+precludes any extra-linguistic memory-\enquote{feelings} which would conflict with 
 the sentences. But the L-Memory is otherwise normal. And \textit{Assumption 1.1}
 indicates that normally, either precisely one of a number of mutually 
 exclusive possibilities has the force of memory; or else the organism can give 
 none of them the force of memory. 
 
 \assumption{1.3} I cannot, from within a natural dream, choose to swith 
-to another dream. (An introspective observation. A "natural" dream is a 
+to another dream. (An introspective observation. A \enquote{natural} dream is a 
 dream involuntarily produced internally during sleep.) 
 
 \conclusion{1.4} An M-Memory could not be produced by natural 
@@ -156,10 +147,10 @@ is such that the organism can choose one sentence-memory one moment and
 another the next. See Assumption 1.3. 
 
 \assumption{1.5} Returning to the example of the restaurants, I find 
-that months after the event, my thought of E no longer has the force of 
-memory. All I remember now is that I used to remember that I did E. I 
-remember that I did E indirectly, by remembering that I remembered that I 
-did E. (My memory that I did E is becoming an L-Memory.) The assumption 
+that months after the event, my thought of $E$ no longer has the force of 
+memory. All I remember now is that I used to remember that I did $E$. I 
+remember that I did $E$ indirectly, by remembering that I remembered that I 
+did $E$. (My memory that I did $E$ is becoming an L-Memory.) The assumption 
 is that a memory of one's remembering can indicate, if not imply, that the 
 event originally remembered occurred. 
 
@@ -169,16 +160,17 @@ of an M-Memory.
 \item The sentences are the organism's only memory of which 
 method he has been using. 
 
-\item When the organism thinks "I have actually been doing $A_{a_i}$".
+\item When the organism thinks \enquote{I have actually been doing $A_{a_i}$}.
 then (he artificially dreams that) he has been doing $A_{a_i}$ --- and is 
 now doing it. 
 
 \item When the dream ends, he does not remember that he 
-remembered that "he has been doing $A_{a_i}$," That is, he does not remember 
+remembered that \enquote{he has been doing $A_{a_i}$,} That is, he does not remember 
 the dream; and he does not remember that he thought the sentence. These 
 conditions would permit the existence of an M-Memory or else a memory 
 indistinguishable to all intents and purposes from an M-Memory. 
 \end{enumerate}
+\end{hangers}
 
 What I have in mind in \conclusion{1.6} is dreams which are produced 
 artificially but otherwise have all the remarkable qualities of natural dreams. 
@@ -190,13 +182,14 @@ Consider now what it is like as a mental experience to have an M-Memory.
 What present or ongoing awareness accompanies an M-Memory? 
 \conclusion{1.6.2} already told what the remembering is like. For the rest, I will 
 informally sketch some conclusions. The organism can extra-linguistically 
-image the $A_{a_i}$. The organism can think "I could have been doing $A_{a_i}$." When 
+image the $A_{a_i}$. The organism can think \enquote{I could have been doing $A_{a_i}$.} When 
 not remembering, the organism doesn't have to do any $A_{a_i}$, or he can do any 
 one of them. The organism must not do anything which would liquidate a 
 possble method, render the action no longer possible for him. 
 
+\begin{hangers}
 \assumption{2.1} A normal dream can combine two totally different 
-past episodes in my life into a fused episode, or amalgam; so that I "relive" it 
+past episodes in my life into a fused episode, or amalgam; so that I \enquote{relive} it 
 without doubts as.a single episode, and yet remain vaguely aware that 
 different episodes are present in it. Dreams have the capacity not only to 
 falsify my world, but to make the impossible believable. (An introspective 
@@ -208,11 +201,14 @@ contact which is out of the question in a normal Situation 1. The dream is so
 flexible that the organism can dream that an (actual) liquid is\slash was contained 
 by a jar in a possible method. See \assumption{2.1} Thus, the $A_{a_i}$ do not have 
 to be separable to be possible methods. 
+\end{hangers}
 
 I will now introduce further concepts pertaining to the mind. 
 
-\definition\ A "mental state" is a mental "stage" or "space" or "mood" 
+\begin{hangers}
+\definition\ A \term{mental state} is a mental \enquote{stage} or \enquote{space} or \enquote{mood} 
 in which visualizing, remembering, and all imaging can be carried on. 
+\end{hangers}
 
 Some human mental states are stupor, general anxiety, empathy with 
 another person, dizziness, general euphoria, clearheadedness (the normal 
@@ -220,15 +216,16 @@ state in which work is performed), and dreaming. In all but the last state,
 some simple visualization routine could be carried out voluntarily. Even ina 
 dream, I can have visualizations, although here I can't have them at will. The 
 states are not defined by the imaging or activities carried on while in them, 
-but are "spaces" in which such imaging or activities are carried on. 
+but are \enquote{spaces} in which such imaging or activities are carried on. 
 
 By definition. 
 
+\begin{hangers}
 \conclusion{3.2} An M-Memory has to occur within the time which the 
-possible methods require, the time required to accomplich G. By definition. 
+possible methods require, the time required to accomplish G. By definition. 
 
-\definition An "$M^*$-Memory" is an M-Memory satisfying these 
-conditions. 
+\definition\ An \term{M*-Memory} is an M-Memory satisfying these 
+conditions. \label{mstardef}
 \begin{enumerate}
 \item $A_{a_i}$, for the entire time it requires, involves the voluntary 
 assuming of mental states. $i=1,...,n$.
@@ -237,13 +234,14 @@ possible methods, the cross-method contact, is specifically some sort of
 contact between states. 
 \end{enumerate}
 
-\conclusion{3.3} For an $M^*$-Memory, to remember is to choose the 
+\conclusion{3.3} For an M*-Memory, to remember is to choose the 
 mental state in which the remembering is required to occur (by the 
 memory). After all, for any M-Memory, to remember is to choose all the 
 $A_{a_i}$-required things you are doing while you remember. 
+\end{hangers}
 
 By now, the character of this investigation should be clearer. I seek to 
-stretch our concepts, rather that to find the "true" ones. The investigation 
+stretch our concepts, rather that to find the \enquote{true} ones. The investigation 
 may appear similar to the old discipline of philosophical psychology, but its 
 thrust is rather toward the modern axiomatic systems. The reasoning is 
 loose, but not arbitrary. And the investigation will become increasingly 
@@ -252,81 +250,95 @@ mathematical.
 
 \section{D-Memories}
 
-\definition\ A "D-Memory" is a memory such that measured past time 
-appears in it only in the following sentences: "$Event_j$ occurred in the interval 
+\begin{hangers}
+\definition\ A \term{D-Memory} is a memory such that measured past time 
+	appears in it only in the following sentences: \enquote{$Event_j$ occurred in the interval 
 % TODO\<F11><F12> ? whats up with AF
-of time which is $x_j-x_{j-1}$ long and ended at $x_j$ AF, and is Yj long and ended $z_j$
-\ ago," where $x_j$, $y_j$ and $z_j$ are positive numbers of time units (such as hours) 
-and '$AF$' means "after a fixed beginning time." $x_O=O;$ $x_j> x_{j-1}$; and at any 
+of time which is $x_j-x_{j-1}$ long and ended at $x_j$ $AF$, and is $y_j$ long and ended $z_j$
+\ ago,} where $x_j$, $y_j$ and $z_j$ are positive numbers of time units (such as hours) 
+and \enquote{$AF$} means \enquote{after a fixed beginning time.} $x_O=O;$ $x_j> x_{j-1}$; and at any 
 one fixed time, the intervals $|z_j, z_j+y_j|$ nowhere overlap. $y_j+z_j\leq x_j$ For an
 integer $m$, the $m$th sentence acquires the force of memory, is added to the 
-memory, at the fixed time $x_m$. $j=1, ..., f(t)$, where the number of sentences 
+memory, at the fixed time $x_m$. $j=1, \ldots, f(t)$, where the number of sentences 
 $f(t)$ is written as a function of time $AF$. Then $f(t)=m$ when $x_m \leq t \less x_{m+1}$. 
 The sentences have the force of memory involuntarily. The organism does 
 not make them up at will. 
+\end{hangers}
 
 Let me explain what the D-Memory involves. $Event_j$ is assigned to an 
-abnormal "interval," a dual interval defined in two unrelated ways. The 
+abnormal \enquote{interval,} a dual interval defined in two unrelated ways. The 
 intervals defined by the $y_j$ and $z_j$ are tied to the present instant rather than to 
 a fixed time, and could be written $|N-z_j-y_j, N-z_j|$, where '$N$' means "the time 
 of the present instant relative to the fixed beginning time." 
 
 \newcommand{\proof}{\textit{Proof}}
 
+\begin{hangers}
 \conclusion{4} The intervals $|N-z_j-y_j, N-z_j|$ nowhere overlap. 
 
 \proof: By definition, the intervals $|z_j, z_j+y_j|$ nowhere overlap. If $j\neq k$,
 $|z_j, z_j+y_j|\cap|z_k, z_k+y_k|=\emptyset$ 
-This fact implies that \eg $z_j\less z_j+y_j\less z_k\less z_k+y_k$.
+This fact implies that e.g. $z_j\less z_j+y_j\less z_k\less z_k+y_k$.
 Then $N-z_k-y_k\less N-z_k\less N-z_j-y_j\less N-z_j$.
 Then $|N-z_k-y_k, N-z_k|\cap|N-z_j-y_j, N-z_j|=\emptyset$
 At any one time, the organism can think of all the sliding intervals, and they 
 partly cover the time up to now without overlapping. 
+\end{hangers}
 
-Suppose you find the deck of n cards 
+Suppose you find the deck of $n$ cards 
 
-{ \centering
-\framebox[1.1\width]{
-	\centering
-	$event_j$ \linebreak
-	$z_j$ ago}}
+{ \centering {\vskip 1em}
+	\framebox[1.1\width]{
+		\parbox{1in}{
+			\centering
+			\large $event_{j}$ 
+			{\vskip 0.25em}
+			$z_{j}$ ago
+		}
+	}
+	\par {\vskip 0.5em}
+}
 
 
-($j=1,...,n$ and $z_j$ is a positive number of days), and you have no 
+
+($j=1,\ldots,n$ and $z_j$ is a positive number of days), and you have no 
 information to date them other than what they themselves say. If you 
 believe the cards, your mental experience will be a little like having a 
 D-Memory. Then, the definition does not require that $y_j=x_j-x_{j-1}$. Again, it is 
-not that two concepts of "length" are involved, but that the "interval" is 
+not that two concepts of \enquote{length} are involved, but that the \enquote{interval} is 
 abnormal. Of course this is all inconsistent, but I want to study the 
 conditions under which a mind will accept inconsistency. 
 
+\begin{hangers}
 \assumption{5.1} With respect to normal human memory, it is possible 
 to forget what day it is, even though one remembers a past date. (An 
 empirical observation.) 
 
 \assumption{5.2} This assumption is based on the fact that the sign 
-'CLOSED FOR VACATION. BACK IN TWO WEEKS' was in the window of 
+\enquote{\textsc{Closed for Vacation. Back in two weeks.}} was in the window of 
 a nearby store for at least a month this summer; and the fact that a 
-filmmaker wrote in a newspaper, "When an actor asks me when the film will 
-be finished, I say 'In two months," and two months later I give the same 
-answer, and I'm always right.' Even in normal circumstances, humans can 
-maintain a dual and outright inconsistent awareness of measured time. [n 
+filmmaker wrote in a newspaper, \enquote{When an actor asks me when the film will 
+be finished, I say \enquote{In two months,} and two months later I give the same 
+answer, and I'm always right.} Even in normal circumstances, humans can 
+maintain a dual and outright inconsistent awareness of measured time. In 
 general, inconsistency is a normal aspect of human thinking and even has 
 practical value. 
+\end{hangers}
 
 Imagine a child who has been told to date events by saying, for 
-example, x happened two days ago, and a day later saying again, x happened 
+example, $x$ happened two days ago, and a day later saying again, $x$ happened 
 two days ago---and who has not been told that this is inconsistent. What 
 conditions are required for the acceptance of this dating system? It is 
 precisely because of Assumptions 5.1 and 5.2 that a certain answer cannot 
 be given to this question. The human mind is so flexible and malleable that 
 there is no telling how much inconsistency it can absorb. I can only study 
-what flaws might lead the child to reject the system. The child might "feel" 
+what flaws might lead the child to reject the system. The child might \enquote{feel} 
 that an event recedes into the past, something the memory doesn't express. 
 An event might be placed by the memory no later than another, and yet 
-"feel" more recent than the other. I speculate that if anything will discredit 
-the system, it will be its conflict with naive, "felt," extra-linguistic memory. 
+\enquote{feel} more recent than the other. I speculate that if anything will discredit 
+the system, it will be its conflict with naive, \enquote{felt,} extra-linguistic memory. 
 
+\begin{hangers}
 \conclusion{5.3} The above dating system would be acceptable to an 
 organism with an L-Memory. 
 
@@ -337,6 +349,7 @@ past would have no form independent of language. Anyway, time is gone for
 good, leaving nothing that can be checked directly. Without an 
 extra-linguistic memory to fall back on, and considering Assumptions 5.1 
 and 5.2, the dual temporal memory shouldn't be too much to absorb. 
+\end{hangers}
 
 As I said, the real difficulty with this line of investigation is putting 
 limits on anything so flexible as the mind's capacity to absorb inconsistency. 
@@ -346,18 +359,19 @@ $\delta(S^D_j)$ be the minimum number of time units it takes to think the jth
 D-sentence. This function, abbreviated '$\delta_j$', is the duration function of the 
 D-sentences. 
 
+\begin{hangers}
 \conclusion{6.1} If $\delta_j\greater z_j$, the memory of the interval defined by $y_j$ and 
 $z_j$ places the end of the interval after the beginning of the memory of it, or 
 does something else equally unclear. If $\delta_j\greater y_j+z_j$, the entire interval is placed 
 after the beginning of the memory of it. When $\delta_j\greater z_j$, let us say that the end 
 of the remembered interval falis within the interval for the memory of it, or 
-that the situation is an "\textsc{infall}." (Compare \said{The light went out a half-second 
+that the situation is an \enquote{infall.} (Compare \enquote{The light went out a half-second 
 ago}.)
 
 \conclusion{6.2} If $\delta_j\greater x_{j+k}-x_j$, then $S^D_{j+k}$ is added to the preconscious 
-before $S^D_j$ can be thought once. The earliest interval during which the jth 
-sentence can be thought "passes over" the (j+k)th interval. Let us say that 
-the situation is a "\textsc{passover}." (Something of the sort is true of humans, 
+	before $S^D_j$ can be thought once. The earliest interval during which the $j^{th}$ 
+	sentence can be thought \enquote{passes over} the $(j+k)^{th}$ interval. Let us say that 
+the situation is a \enquote{passover.} (Something of the sort is true of humans, 
 whose brains contain permanent impressions of far more sensations than can 
 be thought, remembered in consciousness.) 
 
@@ -366,44 +380,52 @@ cannot both think the sentences during the earliest intervals possible and be
 aware of the passovers. 
 
 \proof: The only way the organism can be aware of $\delta(S_j)$
-is for $event_{j+h}$ (h a positive integer) to be the thinking of $S_j$. 
+is for $event_{j+h}$ ($h$ a positive integer) to be the thinking of $S_j$. 
 If the thinking of $S_j$ takes piace as the $(j+1)^{th}$ event, then the organism gets two 
 values for $\delta(S_j)$, namely $x_{j+1}-x_j$ and $y_{j+1}$. Assume that only $x_{j+1}-x_j$
 is allowed as a measure of $\delta(S_j)$. Since $\delta(S_j)=x_{j+1}-x_j$, there is no passover. If 
 the thinking of $S_j$ takes place as the $(j+2)^{th}$ event, then $x_{j+2}-x{j+1}=\delta(S_j)$
 could be greater than $x_{j+1}-x_j$. But since $S_j$ goes into the preconscious at $x_j$, 
 $S_j$ is not actually thought in the earliest interval during which it could be 
-thought. See the diagram. 
+thought. See diagram \ref{dmemdiag}.
 
-\imgw{4in}{img/dmemdiag}
+\begin{figure}
+	\centering
+	\includegraphics[width=4in]{img/dmemdiag}
+	\caption{tktk}
+	\label{dmemdiag}
+\end{figure}
 
-\conclusion{6.4} Let there be an \textsc{infall} in the case where $event_{j+1}$ is the 
+\conclusion{6.4} Let there be an \term{infall} in the case where $event_{j+1}$ is the 
 thinking of $S_j$. $\delta(S_j)=x_{j+1}-x_j$ and $\delta(S_j)\greater z_j$. $S_{j+1}$ gives $\delta(S_j)$, 
 so that the organism can be aware of it. 
 It is greater than $z_j$. Thus, the organism can be 
-aware of the \textsc{infall}. However, the \textsc{infall} will certainly be no more difficult to 
+aware of the infall. However, the infall will certainly be no more difficult to 
 accept than the other features of the D-Memory. And the thinking of $S_j$ has 
 to be one of the events for the organism to be aware of the infall. 
+\end{hangers}
 
 \section{$\Phi$-Memories}
 I will conclude these studies with two complex constructions. 
 
-\definition A "$\Phi$-Memory" is a memory which includes an $M^*$-Memory 
+\begin{hangers}
+\definition\ A \enquote{$\Phi$-Memory} is a memory which includes an M*-Memory 
 and a D-Memory, with the following conditions. 
 \begin{enumerate}
-\item The goal G, for the $M^*$-Memory, is to move from one point to another. 
+\item The goal $G$, for the M*-Memory, is to move from one point to another. 
 
-\item For the D-Memory, "$event_j$" becomes a numerical term, the decrease in the organism's distance 
-from the destination point during the temporal interval. \said{A 3-inch move 
-toward the destination} is the sort of thing that "$event_j$' here refers to. 
+\item For the D-Memory, \enquote{$event_j$} becomes a numerical term, the decrease in the organism's distance 
+from the destination point during the temporal interval. \enquote{A 3-inch move 
+toward the destination} is the sort of thing that \enquote{$event_j$} here refers to. 
 
 \item The number of $A_{a_i}$ equals the number of D-sentences factorial. The number 
 of D-sentences, of course, increases. 
 \end{enumerate}
+\end{hangers}
 
 Consider the consecutive thinking of each D-sentence precisely once, in 
 minimum time, while the number of sentences remains constant. Such a 
-"D-paragraph" is a permutation of the D-sentences. Let $\mathparagraph^m$ be a 
+\enquote{D-paragraph} is a permutation of the D-sentences. Let $\mathparagraph^m$ be a 
 D-paragraph when the number of sentences equals the integer m. There are 
 $m!$ $\mathparagraph^m$s. When $f(t)=m=3$, one of the six $\mathparagraph^3$s is $S^D_3 S^D_1 S^D_2$, 
 thought in 
@@ -415,36 +437,40 @@ $$ \triangle(\mathparagraph^m)=\sum_{j=1}^{m} \delta_j $$
 The permutations of the D-sentences, as well as the D-paragraphs, can be 
 indexed with the $a_i$, just as the possible methods are. 
 
-Definition. A "$\Phi^*$-Memory" is a $\Phi$-Memory in which the order of the 
-sentences in the $a_i$th $\mathparagraph^m$ has the meaning of \said{I have actually been doing $A_{a_i}$}
+\begin{hangers}
+\definition\ A \enquote{$\Phi^*$-Memory} is a $\Phi$-Memory in which the order of the 
+sentences in the $a_i$th $\mathparagraph^m$ has the meaning of \enquote{I have actually been doing $A_{a_i}$}
 assigned to it. The order is the indication that $A_{a_i}$ has actually been used; it 
-is the $a_j$th $M^*$-assertion. \said{I have actually been doing $A_{a_i}$} is merely an English 
+is the $a_j$th $M^*$-assertion. \enquote{I have actually been doing $A_{a_i}$} is merely an English 
 translation, and does not appear in the $\Phi^*$-Memory. 
 
 \conclusion{7} Given a $\Phi^*$-Memory, if one D-sentence is forgotten, not 
 only will there be a gap in the awareness of when what events occurred; it 
 will be forgotten which method has actually been used. 
+\end{hangers}
 
 This conclusion points toward a study in which deformations of the 
 memory language are related to deformations of general consciousness. 
 
-\definition A "$\Phi^*$-Reflection," or reflection in the present of a 
+\begin{hangers}
+\definition\ A \enquote{$\Phi^*$-Reflection,} or reflection in the present of a 
 $\Phi^*$-Memory, is a collection of assertions about the future, derived from a 
 $\Phi^*$-Memory, as follows. 
 \begin{enumerate}
-	\item There are the sentences "$Event_j$ will occur in the 
+	\item There are the sentences \enquote{$Event_j$ will occur in the 
 interval of time which is $x_j-x_{j-1}$ long, and begins at twice the present time 
-$AF$, minus $x_j AF$; and which is $y_j$ long and begins $z_j$ from now." If $event_j$ was 
-a 3-inch move toward the destination in the "$\Phi^*$-Memory, the sentence in the 
+$AF$, minus $x_j AF$; and which is $y_j$ long and begins $z_j$ from now.} If $event_j$ was 
+		a 3-inch move toward the destination in the \enquote{$\Phi^*$-Memory,} the sentence in the 
 $\Phi^*$-Reflection says that a 3-inch move will be made in the future temporal 
 interval. 
 	\item The $a_i$th permutation of the sentences defined in (1) is an 
-assertion which has the meaning of \said{I will do $A_{a_i}$}; and the organism can 
+assertion which has the meaning of \enquote{I will do $A_{a_i}$}; and the organism can 
 think precisely one permutation at a time. The $A_{a_i}$, $x_j$, $y_j$, $z_j$, and the rest are 
 defined as before (so that in particular the permutations can be indexed with 
 the $a_i$). 
 \end{enumerate}
-
+\end{hangers}
+\begin{hangers}
 \conclusion{8} Given that the $\Phi^*$-Memory's temporal intervals $|x_{j-1}, x_j|$
 are reflected as $|2N-x_j, 2N-x_{j-1}|$, the reflection preserves the intervals' 
 absolute distances from the present. 
@@ -452,12 +478,13 @@ absolute distances from the present.
 \proof: The least distance of $|x_{j-1}, x_j|$
 from $N$ is $N-x_j$; the greatest distance is $N-x_{j-1}$. Adding the least distance, and 
 then the greatest distance, to $N$, gives $|2N-x_j, 2N-x_{j-1}|$.
+\end{hangers}
 
 I will end with two problems. If a $\Phi^*$-Memory exists, under what 
 conditions will a $\Phi^*$-Reflection be a precognition? Under what conditions 
-will every assertion be prescience or foreknowledge? By a "precognition" I 
+will every assertion be prescience or foreknowledge? By a \enquote{precognition} I 
 don't mean a prediction about the future implied by deterministic laws; I 
-mean a direct "memory" of the future unconnected with general principles. 
+mean a direct \enquote{memory} of the future unconnected with general principles. 
 
 Finally, what would a precognitive $\Phi^*$-Reflection be like as a mental 
 experience? What present or ongoing awareness would accompany a 
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