chugging through syntax soup

1 files changed, 167 insertions, 283 deletions

diff --git a/blueprint.tex b/blueprint.tex
index 3499bd2..63af6d4 100644
--- a/blueprint.tex
+++ b/blueprint.tex
@@ -3499,69 +3499,50 @@ not a real infall, but it should give the reader some intuitive notion of an
 infall. 
 
 
-"Infalls" 
+\newcommand{\midheading}[1]{
+	{ \centering \large \textsc{#1} \par}}
 
+\midheading{"Infalls"}
 
-A "sentence" is a D-sentence, in a D-Memory such that event) + 4 is the first 
+\begin{sysrules}
+	A "sentence" is a D-sentence, in a D-Memory such that $event_{j+1}$ is the first 
 thinking of the jth D-sentence, for all j. 
 
 Two sentences "imply" another if and only if all three are the same; or else 
-the three are adjacent {and can be written Sit: S;, Si-1 ), and are such 
-that 6 5 = xj44-Xj raat Sy is the implicand. (The function of Sj+4 is to 
-give the duration 6,= +1 -%; of Sj. Sj states that event;, the first 
-dae' of s? "4, ended ata aitence: Zj inte the past, where zj is smaller 
-than s \$s own vduretian The diagram indicates the relations.) 
-
-
-G2: evenby obi: event 3 
-occurred in [X5-40° x5 I occurred in ia, Xa 
-
-
-shia and in IN-25-Y5) N-z.; and inI N- "Ared ya Needl oP 2 
-
-
-event itd 
-
-
-events 42 
-xs 544] t 
-Bi ese *y+4 A542 
-
-
-"evenby ended 25 ago" "evenly 44 inI 
+the three are adjacent (and can be written $S_{j+1},S_j,S_{j-1}$), and are such 
+that $\delta_j=x_{j+1}-x_j\greater z_j,$ $S^D_{j-1}$ is the implicand. (The function of $S_{j+1}$ is to 
+give the duration $\delta_j=x_{j+1}-x_j$ of $S_j$. $S_j$ states that $event_j$, the first 
+thinking of $S^{D}_{j-1}$, ended at a distance $z_j$ into the past, where $z_j$ is smaller 
+than $S^D_j$'s own duration. The diagram indicates the relations.) 
 
+\img{infallsdiag}
 
 In this variety of D-Memory, the organism continuously thinks successive 
 D-sentences, which are all different, just as the reader of the above exercise 
 continuously reads successive and different sentences. Thus, the possibility 
 of repeating a sentence depends on the possibility of thinking it while one is 
-thinking another sentence--a possibility which may be far-fetched, but which 
-
-
-96 
-
-
+thinking another sentence---a possibility which may be far-fetched, but which 
 is not explicitly excluded by the definition of a "D-Memory." If the 
-possibility is granted, then "Infalls" has Properties 1-5. Direct implication is 
+possibility is granted, then "\textsc{Infalls}" has Properties 1--5. Direct implication is 
 completely mechanical; it is subjective only in that the involuntary 
-determination of the z; and other aspects of the memory is a 'subjective' 
+determination of the $z_j$ and other aspects of the memory is a 'subjective' 
 process of the organism. Compound implication is also mechanical to an 
 outside observer of the memory, but if the organism itself is to be aware of 
 it, it has to perform fantastic feats of multiple thinking. 
 
-"Dream Amaigams" and "Infalls" are systems constructed with 
+"\textsc{Dream Amalgams}" and "\textsc{Infalls}" are systems constructed with 
 imaginary elements, systems whose "notation" is drawn from an imaginary 
 object or system. Such systems have no descriptive syntax. Imaginary objects 
 were introduced into mathematics, or at least into geometry, by Nicholas 
 Lobachevski, and now I am using them as a notation. For these systems to 
 be nonisomorphic to any ink-on-paper systems, the mathematician must be 
-the organism with the M*-Memory or the D-*Memory. But this means that 
+the organism with the M*-Memory or the D*-Memory. But this means that 
 in this case, the mathematics which is nonisomorphic to any ink-on-paper 
 system can be performed only in an imaginary mind. 
 
 Now for a different approach. Carnap said that we are free to choose 
 the rules of a system arbitrarily. Let us take Carnap literally. I want to 
-construct more systems in constructed memories--so why not construct the 
+construct more systems in constructed memories---so why not construct the 
 system by a procedure which ensures that constructed memories are 
 involved, but which is otherwise arbitrary? Why not suspend the striving 
 after "interesting" systems, that last vestige of the striving after 
@@ -3571,16 +3552,20 @@ system by a chance procedure?
 To construct a system, we have to fill in the blanks in the following rule 
 schema in such a way that grammatically correct sentences result. 
 
+\newcommand{\blankspace}{\_\_\_\_\_\_\_\_\_\_}
+
+\midheading{Rule Schema}
 
-Rule Schema 
+\begin{sysrules}
+A "sentence" is a(n) \blankspace.
 
-A"sentence" isa(n)_ 
+Two sentences "imply" a third if and only if the two sentences \blankspace\ the third. 
 
-Two sentences "imply" a third if and only if the two sentences 
-the third. 
+An "axiom" is a sentence that \blankspace.
+\end{sysrules}
 
 
-I now spread the pages of 'Studies in Constructed Memories" on the floor. 
+I now spread the pages of \essaytitle{Studies in Constructed Memories} on the floor. 
 With eyes closed, I hold a penny over them and drop it. I open my eyes and 
 copy down the expressions the penny covers. By repeating this routine, I 
 obtain a haphazard series of expressions concerning constructed memories. It 
@@ -3588,34 +3573,31 @@ is with this series that I will fill in the blanks in the rule schema. In the ne
 stage, I fill the first (second, third) blank with the ceries of expressions 
 preceding the-first (second, third) period in the entire series. 
 
+\midheading{"Haphazard System"}
 
-"Haphazard System" 
-
-A "sentence" is a the duration D-sentences A (@") conclude these 
-"*-Reflection," or the future Assumption voluntarily force of 
+\begin{sysrules}
+A "sentence" is a the duration D-sentences \triangle\ ($\mathparagraph^m$) conclude these 
+"\Phi*-Reflection," or the future Assumption voluntarily force of 
 conviction for conclusion the Situation or by ongoing that this 
 system? be given telling between the Situation 1. 
 
-Two sentences "imply" a third if and only if the two sentences is/ was 
+Two sentences "imply" a third if and only if the two sentences is\slash was 
 contained not have to the acceptance that a certain and malleable 
 study what an event involves material specifically mathematics: 
 construct accompanies the rest, extra-linguistically image organism 
-can fantasy not remembering ® *-Memory, the future interval defined 
+can fantasy not remembering \Phi*-Memory, the future interval defined 
 in dream the third. 
 
 An "axiom" is a sentence that internally D-sentences, just as the 
-
-
-"}*-Memory" sentences Ay is A,.. 
-1 2 
-
+"\Phi*-Memory" sentences $A_{a_1}$ is $A_{a_2}$. 
 
 In the final stage, I cancel the smallest number of words I have to in 
 order to make the rules grammatical. 
+\end{sysrules}
 
+\midheading{"Fantasied Amnesia"}
 
-"Fantasied Amnesia" 
-
+\begin{sysrules}
 A "sentence" is a duration or the future force of conviction for the Situation 
 or this system given Situation 1. 
 
@@ -3623,9 +3605,10 @@ Two sentences "imply" a third if and only if the two sentences have the
 acceptance that a certain and malleable study extra-linguistically can 
 fantasy not remembering the future interval defined in the third. 
 
-An "axiom" is a sentence that internally just sentences ay: 
+An "axiom" is a sentence that internally just sentences $A_{a_2}$.
+\end{sysrules}
 
-It becomes clear in thinking about "Fantasied Amnesia' that its 
+It becomes clear in thinking about "Fantasied Amnesia" that its 
 metametamathematics is dual. Describing the construction of the rules, the 
 metamathematics, by a systematic performance, is one thing. Taking the 
 finished metamathematics at face value, independently of its origin, and 
@@ -3633,38 +3616,26 @@ studying it in the usual manner, is another. Let us take "Fantasied Amnesia"
 at face value. As one becomes used to its rules, they become somewhat more 
 meaningful. I will say that an "interpretation" of a haphazard system is an 
 explanation of its rules that makes some sense out of what may seem 
-senseless. 'Interpreting' is somewhat like finding the conditions for the 
+senseless. "Interpreting" is somewhat like finding the conditions for the 
 existence of a constructed memory which seemingly cannot exist. The first 
 rule of "Fantasied Amnesia" is a disjunction of three substantives. The 
-
 "Situation" referred to in the second substantive expression is either 
-
 Situation 1 or else an unspecified situation. The third substantive expression 
-
-apparently means 'this system, assuming Situation 1,' and refers to 
-
-"Eantasied Amnesia" itself. The definition of 'sentence' is thus meaningful, 
-
+apparently means "this system, assuming Situation 1," and refers to 
+"Fantasied Amnesia" itself. The definition of "sentence" is thus meaningful, 
 but very bizarre. The second rule speaks of "the acceptance" as if it were a 
-
 written assent. The rule then speaks of a "malleable study" as "fantasying" 
-
-
-98 
-
-
 something. This construction is quite weird, but let us try to accept it. The 
 third rule speaks of a sentence that "sentences" (in the legal sense) a possible 
 method. So much for the meaning of the rules. 
 
-
 Turning to the nine properties of formalist systems, the reference to 
-"the future interval' in the implication rule of "Fantasied Amnesia" 
+"the future interval" in the implication rule of "Fantasied Amnesia" 
 indicates that the system has Property 2; and the system can perfectly well 
 have Property 8. It does not have Property 6 in any known sense. Certainly 
 it does have Property 9. it just might have Property. 1. But as for the other 
 four properties, it seems out of the question to decide whether "Fantasied 
-Amnesia' has them. For whatever it is worth, "Fantasied Amnesia' is on 
+Amnesia" has them. For whatever it is worth, "Fantasied Amnesia" is on 
 balance incomparable to formalist systems. 
 
 My transformation rule schema has the form of a biconditional, in 
@@ -3675,12 +3646,10 @@ then the latter would vary "trivially." 1 will say that a system whose
 transformation rule can vary non-trivially is a "heterodeterminate" system. 
 Since 1 have constructed a haphazard metamathematics, why not a 
 heterodeterminate metamathematics? Consider a mathematician with an 
-M-Memory, such that each Ag. is the consistent use of a different 
-
-
+M-Memory, such that each $A_{a_i}$. is the consistent use of a different 
 transformation rule, a different definition of "imply," for the mathematics 
 in which the mathematician is discovering theorems. The consistent use of a 
-transformation rule is after all a method--a method for finding the 
+transformation rule is after all a method---a method for finding the 
 commitments premisses make, and for basing conclusions in premisses. When 
 the mathematician goes to remember which rule of inference he has actually 
 been using, he "chooses" which of the possible methods is remembered to 
@@ -3691,23 +3660,19 @@ memory.
 
 We are now in the realm of mathematical systems which cannot be 
 written out, but can only be described metametamathematically. I will 
-present a final system of this sort. It is entitled "System Such That No One 
-Knows What's Going On." One just has to guess whether this system exists, 
+present a final system of this sort. It is entitled \textsc{"System Such That No One 
+Knows What's Going On."} One just has to guess whether this system exists, 
 and if it does what it is like. The preceding remark is the 
 metametamathematical description, or definition, of the system. 
 
-
-99 
-
-
-1.3 Epilogue 
+\subsection{Epilogue}
 
 Ever since Carnap's Principle of Tolerance opened the floodgates to 
 arbitrariness in mathematics, we have been faced with the prospect of a 
 mathematics which is  indistinguishable from  art-for-art's-sake, or 
 amusement-for-amusement's-sake. But there is one characteristic which saves 
 mathematics from this fate. Mathematics originated by abstraction from 
-primitive technology, and is indispensable to science and technology--in 
+primitive technology, and is indispensable to science and technology---in 
 short, mathematics has scientific applications. The experience of group 
 theory has proved, I hope once and for all, the bankruptcy of that narrow 
 practicality which would limit mathematics to what can currently be applied 
@@ -3729,22 +3694,16 @@ lesson to be learned from the history of formalism is that an idea does not
 have to be "true" to be productive. 
 
 
-Note 
-Early versions of "tllusions" and "Innperseqs" appeared in my essay 
+\section{Note}
+Early versions of \textsc{"Illusions"} and \textsc{"Innperseqs"} appeared in my essay 
 "Concept Art," published in An Anthology, ed. La Monte Young, New 
-York, 1963. An early, July 1961 version of "System Such That No One 
-Knows What's Going On" appeared in dimension 14, Ann Arbor, 1963, 
+York, 1963. An early, July 1961 version of \textsc{"System Such That No One 
+Knows What's Going On"} appeared in dimension 14, Ann Arbor, 1963, 
 published by the University of Michigan College of Architecture and Design. 
 
+\section{Studies in Constructed Memories}
 
-100 
-
-
-2. Studies in Constructed Memories 
-
-
-2.1 Introduction 
-
+\subsection{Introduction}
 
 The memory of a conscious organism is a phenomenon in which 
 interrelations of mind, language, and the rest of reality are especially evident. 
@@ -3752,7 +3711,7 @@ 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, 
+spoke of his system as an "imaginary geometry." Lobachevski's system was, 
 so to speak, the physical geometry of an "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 
@@ -3762,10 +3721,10 @@ 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, "1! am 
-making the trip from New York to chicago." Consider also past actions 
+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 
+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 
@@ -3783,15 +3742,10 @@ 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 
+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 
-
-
-101 
-
-
 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 
@@ -3800,7 +3754,7 @@ thought of E involuntarily more vivid? Is there perhaps an "attitude of
 assertion" involuntarily present in the thought of E? 
 
 Consider the memory that I was almost run down by a truck yesterday: 
-! could have been run down, but wasn't. In such a case, the possibility that I 
+I could have been run down, but wasn't. In such a case, the possibility that I 
 could have been run down would be more vivid than the actuality that I 
 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 
@@ -3820,72 +3774,40 @@ control their content? Can the hypnotist give his subject one false memory
 one moment, and replace it with a contradictory memory the next 
 moment? 
 
-I will now = specify a_ situation involving possible actions and 
+I will now specify a situation involving possible actions and 
 remembering. 
 
-Situation 7. "! could have been accomplishing G by doing Aa, or by 
-
-
-doing Aayy ..., or by doing A, ; but I have actually been accomplishing G by 
-n 
-doing Aas" Here the ongoing actions Age i= 1, ..., 9, a; * a, if ixh, are 
-
-
+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 
 indicate that the methods are distinct and countable, but not ordered.) The 
 possible methods cannot be combined, let us assume. 
 
-In such a situation, perhaps the thought that I have been doing Aa, 
-
-
-3 by the 
-
-. . us - > n 
-
-presence of the "attitude of assertion'. Since the possible methods are 
-
-ongoing actions, the thought that I have been doing A,. has logical or 
-i 
-
-
-would be distinguished from similar thoughts about Aan! wy A 
-
-
+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 
+ongoing actions, the thought that I have been doing $A_{a_i}$ has logical or 
 probabie consequences I can check against the present. 
-Now Aa, is actual and Aao is not, so that Aa, simply cannot have 
-
-
-102 
-
-
-material contact with ay' An actual liquid in Aay could not require a 
-a, could have 
-1 
-
-with A, would be verbal and gratuitous. Therefore, in order to be possible 
-methods, Aan' . 
-
-
-not require a jar in Aas to contain it. If it did, Aan couldn't be actualized 
-
-
-possible jar in Pao to contain it. The only "connection" A 
-.., A, must be materially separable. A liquid in Aan must 
-n 
-
-
-while Aj, remained only a possibility. 
 
+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
+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
+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. 
 
 Enough concepts are now at hand for the studies to begin in earnest. 
 
+\subsection{M-Memories}
 
-2.2 M- Memories 
-Definition. Given the sentences 'I have actually been doing A,.', where 
-i 
-
+\newcommand{\definition}{\textbf{Definition.}}
+\newcommand{\assumption}[1]{\textit{Assumption #1.}}
+\newcommand{\conclusion}[1]{\textbf{Conclusion #1.}}
 
-the A,. are non-combinable possible methods as in Situation 1, an 
+\definition Given the sentences "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 
 can think precisely one of the sentences at a time, and any of the sentences 
 has the force of memory. 
@@ -3899,52 +3821,44 @@ 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 
+\definition The "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. 
 
-Definition. An "L-Memory" is a linguistic P-Memory having no 
+\definition An "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 
 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 
+\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 
 introspective observation.) 
 
-Conclusion 1.2. An L-Memory is not sufficient for an M-Memory, even 
-in the trivial case that the Aa. are beyond perception (as internal bodily 
-
-
-103 
-
-
+\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 '! have actually been doing A,." against. True, the L-Memory 
-i 
-
-
+sentences "I have actually been doing $A_{a_i}$" against. True, the L-Memory 
 precludes any extra-linguistic memory-"feelings" which would conflict with 
-the sentences. But the L-Memory is otherwise normal. And Assumption 1.1 
+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 1.3.1 cannot, from within a natural dream, choose to swith 
+\assumption{1.3} I cannot, from within a natural dream, choose to swith 
 to another dream. {An introspective observation. A "natural" dream is a 
 dream involuntarily produced internally during sleep.) 
 
-Conclusion 1.4. An M-Memory could not be produced by natural 
+\conclusion{1.4} An M-Memory could not be produced by natural 
 dreaming. It is true that in one dream one sentence could have the force of 
 memory, and in another dream a different sentence could. But an M-Memory 
 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 
+\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 ! 
@@ -3952,58 +3866,56 @@ 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. 
 
-Conclusion 1.6. The following are adequate conditions for the existence 
-of an M-Memory. 1. The sentences are the organism's only memory of which 
-method he has been using. 2. When the organism thinks 'I have actually been 
-doing A,.'. then (he artificially dreams that) he has been doing Ag,-and is 
-now doing it. 3. When the dream ends, he does not remember that he 
-remembered that "he has been doing A,.," That is, he does not remember 
+\conclusion{1.6} The following are adequate conditions for the existence 
+of an M-Memory. 
+\begin{enumerate}
+\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 $A_{a_i}$".
+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 
 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. 
 
-What I have in mind in Conclusion 1.6 is dreams which are produced 
+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. 
 There would have to be a state of affairs such that the sentence would 
 instantly start the dream going. 
 
 So much for the conditions for the existence of an M-Memory. 
 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 
+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 Aa: The organism can think 'l could have been doing Aa; When 
-
-
-104 
-
-
-not remembering, the organism doesn't have to do any Ag., or he can do any 
-
-
+image the $A_{a_i}$. The organism can think "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. 
 
-Assumption 2.1. A normal dream can combine two totally different 
+\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 
 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 
 observation.) 
 
-Conclusion 2.2. The conditions for the existence of an M-Memory 
+\conclusion{2.2} The conditions for the existence of an M-Memory 
 further permit material contact between the possible methods, the very 
 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) fiquid is/was contained 
-by a jar in a possible method. See Assumption 2.1. Thus, the A,, do not have 
+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. 
 
 I will now introduce further concepts pertaining to the mind. 
 
-Definition. A "mental state" is a mental "stage" or "space" or "mood" 
+\definition\ A "mental state" is a mental "stage" or "space" or "mood" 
 in which visualizing, remembering, and all imaging can be carried on. 
 
-
 Some human mental states are stupor, general anxiety, empathy with 
 another person, dizziness, general euphoria, clearheadedness (the normal 
 state in which work is performed), and dreaming. In all but the last state, 
@@ -4014,108 +3926,87 @@ but are "spaces" in which such imaging or activities are carried on.
 
 By definition. 
 
-Conclusion 3.2. An M-Memory has to occur within the time which the 
+\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. 
 
-Definition. An "M*-Memory" is an M-Memory satisfying these 
-conditions. 1. Agi: for the entire time it requires, involves the voluntary 
-
-
-assuming of mental states. i = 1, ..., n. 2. The material contact between the 
+\definition An "M*-Memory" is an M-Memory satisfying these 
+conditions. 
+\begin{enumerate}
+\item $A_{a_i}$, for the entire time it requires, involves the voluntary 
+assuming of mental states. $i=1,...,n$.
+\item The material contact between the 
 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. 
 
-
-A,.-required things you are doing while you remember. 
-i 
 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 
 may appear similar to the old discipline of philosophical psychology, but its 
-
-
-105 
-
-
 thrust is rather toward the modern axiomatic systems. The reasoning is 
 loose, but not arbitrary. And the investigation will become increasingly 
 mathematical. 
 
 
-2.3 D-Memories 
-
-
-Definition. A "D-Memory" is a memory such that measured past time 
-
-
-appears in it only in the following sentences: "Event; occurred in the interval 
-
-
-of time which is xX] long and ended at Xj AF, and is Yj long and ended 2; 
-
-
-ago," where Xj, 
-and 'AF' means "after a fixed beginning time." XQ = 'O; XjPX5A and at any 
-
-
-Yje and zj are positive numbers of time units (such as hours) 
-
+\subsection{D-Memories}
 
-one fixed time, the intervals IZj. zjtyjl nowhere overlap. Vit ZS%- For an 
-
-
-integer m, the mth sentence acquires the force of memory, is added to the 
-memory, at the fixed time x,,.j =1, ..., f(t), where the number of sentences 
-f(t) is written as a function of time AF. Then f(t) = m when x,,<t<x,, 44. 
+\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 
+% 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." $x_O=O;$ $x_j\greater x_{j-1}$; 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 
+$f(t)$ 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. ; : : 
+not make them up at will. 
 
-Let me explain what the D-Memory involves. Event; is assigned to an 
+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 
-intervals defined by the Yj and z; are tied to the present instant rather than to 
-a fixed time, and could be written IN-2;-Yj, N-zjl, where 'N' means "the time 
+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." 
 
-Conclusion 4. The intervals IN-2)-Yj, N-Z;I nowhere overlap. Proof: By 
-definition, the intervals IZj, zi+y;I nowhere overlap. If j #k, IZj, Ztyillz,, 
-Zz. +y¥pl = 0. This fact implies that e.g. ZjZjtVjZKS ZK +YK- Then 
-N-2-¥_SN-2<N-2)-9; <N-Zj. Then IN-2p-y,, N-z, 1 N-2jy 7, N-z)I = ¢. At 
-any one time, the organism can think of all the sliding intervals, and they 
+\newcommand{\proof}{\textit{Proof}}
+
+\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 $z_j\less z_j+y_j\less z_k\less z_k+y_k$.
+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. 
 
 Suppose you find the deck of n cards 
 
-
-event j 
-Zz i oa" 
-(jj = 1, .., n and z, is a positive number of days), and you have no 
+{\centering
+\framebox[1.1\width]{
+	$event_j$ \\
+	$z_j$ ago}
+\par}
 
 
-J 
+($j=1,...,n$ 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 Yj = Xt Again, it is 
-
-
-106 
-
-
+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 
-abnormal. Of course this is ali inconsistent, but I want to study the 
+abnormal. Of course this is all inconsistent, but I want to study the 
 conditions under which a mind will accept inconsistency. 
 
-Assumption 5.1. With respect to normal human memory, it is possible 
+\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 
+\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 
 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 
@@ -4127,21 +4018,21 @@ practical value.
 
 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 
-two days ago--and who has not been told that this is inconsistent. What 
+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 "fee!" 
+what flaws might lead the child to reject the system. The child might "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. 
 
-Conclusion 5.3. The above dating system would be acceptable to an 
-organism with an L—Memory. 
+\conclusion{5.3} The above dating system would be acceptable to an 
+organism with an L-Memory. 
 
-Conclusion 5.4. The existence of an L-Memory is an adequate condition 
+\conclusion{5.4} The existence of an L-Memory is an adequate condition 
 for the existence of a D-Memory. With extra-linguistic amnesia, the 
 structure of the language would be the structure of the past in any case. The 
 past would have no form independent of language. Anyway, time is gone for 
@@ -4153,15 +4044,8 @@ 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. 
 
 Now the thinking of a sentence in a D-Memory itself takes time. Let 
-'tS; be the minimum number of time units it takes to think the jth 
-
-
-D-sentence. This function, abbreviated '8y), is the duration function of the 
-
-
-107 
-
-
+$\delta(S^D_j)$ 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. 
 
 Conclusion 6.1. If 5j>Z), the memory of the interval defined by Yj and